CCSNe are also of interest for GW astronomy as targets in their own right. As the sensitivity of GW detectors increases, they will begin to detect not only binary mergers but also other lower amplitude sources of GWs such as CCSNe. Accurate knowledge of the GW emission from CCSNe will be essential for detection and parameter estimation. The GW signal from rotational core bounce has already been well covered in the literature (e.g. Dimmelmeier et al. 2008; Abdikamalov et al. 2014; Fuller et al. 2015; Richers et al. 2017). In the non-rotating case, the GW emission from the post-bounce phase has been studied using self-consistent 3D simulations by many groups (Kuroda, Kotake & Takiwaki 2016; Andresen et al. 2017, 2019; Kuroda et al. 2017, 2018; Powell & Müller 2019, 2020; Radice et al. 2019; Andresen, Glas & Janka 2020; Mezzacappa et al. 2020; Pan et al. 2020). The structure of the GW emission has shown common features in different simulations from recent years. The dominant emission feature in the GW emission is due to the quadrupolar surface f/g mode 1 of the proto-neutron star (PNS), which produces GW frequencies rising in time from a few hundred Hz up to a few kHz (Müller, Janka & Wongwathanarat 2012; Sotani et al. 2017; Kuroda et al. 2018; Morozova et al. 2018; Torres-Forné et al. 2018, 2019). In addition, some models (Kuroda et al. 2016, 2017; Andresen et al. 2017; Mezzacappa et al. 2020; Powell & Müller 2020) exhibit low-frequency GW emission due to the standing accretion shock instability (SASI; Blondin, Mezzacappa & DeMarino 2003; Blondin & Mezzacappa 2006; Foglizzo et al. 2007). In rapidly rotating models, very strong GW emission can also occur during the post-bounce phase due to a corotation instability (Takiwaki & Kotake 2018). The emerging understanding of the GW emission features has led to the formulation of universal relations for the GW emission (Torres-Forné et al. 2019) and paved the way for phenomenological modelling for CCSN signals (Astone et al. 2018). Further work is still needed, however, to extend these models to fully explore CCSN GW signals from across the progenitor parameter space. The majority of 3D simulations that include GW emission are for progenitor stars below $30\, \mathrm{M}_{\odot }$. In this paper, we perform simulations of high-mass Population III (Pop-III) stars in the pulsational pair instability regime to expand the parameter space coverage of 3D simulations and to provide further insights into the massive and very massive star remnant BH population.


Several faint and sparse knots are detected over all the field of observations, especially around the systemic velocity of IC443 vLSR = −4.5 km s−1 (Hewitt et al. 2006). These structures, noticeable between v = −7.5 km s−1 and v = −1.5 km s−1, might either correspond to a slice of turbulent medium driven by the SN shockwave and/or belong to the ambient gas associated with the NW-SE molecular cloud in which IC443G is embedded (Lee et al. 2012). Other than that, the description of the region probed by our observations can be divided into the following six distinct structures:

1.Cloudlet: in the upper part of the field we observe a large (~ 5′ × 2′, i.e., ~ 2.8 × 1.1 pc) elongated cloudlet detected between v = −7.0 km s−1 and v = −5.5 km s−1 (indicated by the letter “A” in Fig. 2), which is also detected in 12CO(3–2). This structure was labeled G1 by Zhang et al. (2010), as part of the double-peaked morphology of the extended G region. The 13CO J = 1–0 counterpart of this structure is much brighter than the other main structures in the field, and it is also detected in the transitions J = 2–1 and J = 3–2, and in C18O J = 1–0 and J = 2–1. This structure was also presented and characterized by Lee et al. (2012) who proposed the label SC 03, among a total of 12 SCs (of size ~1′) found in IC443.
2.Ring-like structure: a ring-like structure seemingly lying in the center of the field (indicated by the letter “B” in Fig. 2), appearing between v = −5.5 km s−1 and v = −4.5 km s−1 and also detected in 12CO(3–2). It has a semimajor axis of 1.5′, or 0.8 pc. This structure might be spurious and is likely to be physically connected to the elongated cloudlet as both are spatially contiguous and their emission lines are spectrally close. It is partially detected as well in our observations of 13CO J = 1–0, J = 2–1 and J = 3–2, and also has a faint, partial counterpart in C18O J = 1–0 and J = 2–1. To understand the nature of this region we searched for counterparts in Spitzer Multiband Imaging Photometer (MIPS), WISE, DSS and XMM-Newton data; and in near-infrared and optical point source catalogs (Sect. 5), without success. Owing to projection effects, this apparent circular shape could also be explained by an unresolved and clumpy distribution of gas.
3.Shocked clump: in the lower part of the field we identify a very bright clump emitting between v = −31.0 km s−1 and v = 16 km s−1. This structure of size ~2′ × 0.75′ (~ 1.1 × 0.4 pc), which is detected in the 12CO(3–2) transition as well, belongs to the southwestern ridge of the molecular shell of the SNR and has been described as a shocked molecular structure by several studies (van Dishoeck et al. 1993; Cesarsky et al. 1999; Snell et al. 2005; Shinn et al. 2011; Zhang et al. 2010). The core of the shocked clump (indicated by the letter “C” in Fig. 2) is also detected in 13CO in the transitions J = 1–0, J = 2–1, and J = 3–2, and in C18O J = 1–0 and J = 2–1.
4.Shocked knot: an additional shocked knot (indicated by the letter “D” in Fig. 2) is also detected to the west of the previously described structure. This fainter and smaller structure is spatially separated from the main shocked clump.
5.At the same position as the shocked clump and extending southward and westward, we find a faint, elongated clump emitting between v = 5.0 km s−1 and v = 7.5 km s−1. This structure (indicated by the symbol “*” in Fig. 2) spatially coincides with the shocked clump, yet the peak velocity is not exactly the same (see developments on kinematics of the region in Sect. 3.3). It has a faint counterpart in 13CO(1–0). Observations of the ambient molecular cloud by Lee et al. (2012) indicate this structure as part of a faint NE-SW complex of molecular gas in the velocity range +3 km s−1 vLSR  +10 km s−1.
6.Finally, the 13CO(1–0) map (Fig. 4, right panel and Fig. D.2) indicates a large clump of gas extending from the bottom center to the right end of the field, with a bright knot in the bottom right corner of the field. However, this structure has no bright, well-defined counterpart in any of the 12CO transitions maps. It is spatially and kinematically correlated with the faint and diffuse 12CO J = 1–0 and J = 2–1 emission seen in the velocity range −5.5 km s−1 vLSR  −2 km s−1. From the comparison with the 12CO observations of Lee et al. (2012) and 13CO observations of Su et al. (2014) toward the SNR, we conclude that this structure is part of the western molecular complex observed in the velocity range −10 km s−1 vLSR  0 km s−1.


The total energy at each point is defined as a sum of four separate components; the CCSD(T) energy extrapolated to the complete basis set (CBS) limit and three energy increments. Each increment represents a source of electron correlation omitted from the above CBS energy: core-correlation, scalar-relativistic correlation, and higher order dynamic correlation. A nomenclature has been used in the past to describe this composite theory by its components, where C represents CBS-extrapolation, cC represents core-correlation, R represents scalar-relativity, and E represents higher order correlation (Fortenberry et al. 2011). For example, the CcCRE potential surface (equation 2) includes all of the corrections, but the CcCR omits the higher order correlation correction.
(2)
\begin{equation*}
E_\text{CcCRE} = E_\text{CBS} + \Delta E_\text{core} + \Delta E_\text{rel} + \Delta E_\text{hlc}
\end{equation*}
Two different procedures are used to extrapolate the CCSD(T) energy to the CBS limit. The first scheme used is the three-point total energy extrapolation shown in equation (3, Martin & Lee 1996).
(3)
\begin{equation*}
E(X) = E_\text{CBS}+ A(X+1/2)^{-4} + B(X+1/2)^{-6}
\end{equation*}
The second scheme extrapolates the Hartree–Fock energy and the correlation energy, represented here as ε, separately using equations (4) and (5), respectively (Feller 1993; Helgaker et al. 1997).
(4)
\begin{equation*}
E^\text{HF}(X) = E^\text{HF}_\text{CBS} + A e^{BX}
\end{equation*}
(5)
\begin{equation*}
\varepsilon ^\text{(T)}(X) = \varepsilon ^\text{(T)}_\text{CBS} + A X^{-3}
\end{equation*}
The resulting CCSD(T)/CBS energy in this latter scheme is $E^{\prime }_\text{CBS} = E^\text{HF}_\text{CBS} + \varepsilon ^\text{(T)}_\text{CBS}$, where the prime is simply used to distinguish the two CBS extrapolations. Each energy correction (increment) is defined as the difference between the energy with the correlation effect considered and the energy without it when using the same basis set. The core-correlation correction, ΔEcore, is EMTc − EMT as defined with the geometry correction. Scalar relativity is incorporated through ΔErel, which is taken as the difference in the all-electron CCSD(T)/cc-pCV(T+d)Z-DK energy with and without the Douglas–Kroll–Hess (DKH) corrections to second-order (Reiher & Wolf 2004). The energy contribution from non-iterative quadruple excitations in the coupled cluster wavefunction, ΔEhlc, is defined as the difference between CCSDT(Q)/cc-pV(T+d)Z and CCSD(T)/cc-pV(T+d)Z at each point.


However, in our model, the interactions between strangeons are dominated by strong force. The detailed calculations of the shear modulus is still not performed, but it should be much larger than that of the NS’s crust. If the burst oscillation frequencies observed in low-mass X-ray binaries correspond to the first few torsional modes of SSs, the shear modulus should be about one thousand times of the NS’s crust, say $\mu \simeq 4\times 10^{32}\, \rm erg\, cm^{-3}$ (Xu 2003; Owen 2005). We do not know how large the tidal deformability will deviate from the fluid case for SSs, but we can obtain some key insights of this problem from the calculations of tidal deformabilities for crystalline colour superconducting phase (Lau, Leung & Lin 2017, 2019). QSs composed of crystalline colour superconducting phase are rigid with extremely high shear modulus (Alford, Bowers & Rajagopal 2001; Mannarelli, Rajagopal & Sharma 2007). The shear modulus is approximately given by (Mannarelli et al. 2007)
(77)$$\begin{eqnarray*}
\mu =2.47 \, \mathrm{MeV} / \mathrm{fm}^{3}\left(\frac{\Delta }{10 \, \mathrm{MeV}}\right)^{2}\left(\frac{\mu _{\rm q}}{400 \, \mathrm{MeV}}\right)^{2}\, ,
\end{eqnarray*}$$where μq is the average quark chemical potential and Δ is the gap parameter which is in the range of 5–$25\, \rm MeV$. If the shear modulus is taken to be $\mu =4\times 10^{32}\, \rm erg\, cm^{-3}$ ($\Delta \simeq 5\, \rm MeV$), according to the results in Lau et al. (2017, 2019), the deviations of the tidal deformabilities from the fluid case are in the order of $1{{\ \rm per\ cent}}$. For larger gap parameters, the deviations can be very large and more interestingly, they found that the universal relations between the moment of inertia and tidal deformability will deviate from the ones for fluid stars. If we take the shear modulus to be about $\mu \simeq 4\times 10^{32}\, \rm erg\, cm^{-3}$, the corrections may not be very large and the results discussed in our work is a good approximation. But if the shear modulus is actually larger, then we need to consider the corrections due to the elasticity.


We derive leftover planetesimal populations directly from our simulations sets that incorporate planetesimals (see Table 1 and the left panel of Figure 1), and perform an additional batch of ∼105 simulations of the final few giant impacts in our systems to derive collisional fragment distributions. This additional suite of computations utilizes a modified version of the Mercury integration package described in Chambers (2013) that incorporates collisional parameter space mapped in Stewart & Leinhardt (2012) to track hit-and-run and fragmenting impacts. Aside from the fragmentation algorithm, the integration parameters (i.e., time step, ejection radius, etc.) are identical to those utilized in Section 2.1. To prevent the calculation from becoming intractable, the user must establish a minimum fragment mass (MFM; 0.005 M
⊕ in our simulations based on previous experience with the code; Quintana et al. 2016; Clement et al. 2019b). When the algorithm detects a fragmenting collision, the total remnant mass is divided into a number of equal-mass fragments with M > M
MFM that are ejected in uniformly spaced directions in the collisional plane at v ≃ 1.05v
esc. If the total eroded mass is less than the MFM, the collision is considered totally accretionary. While this treatment of fragmentation is obviously somewhat contrived (see Section 4 for a discussion of some of these caveats), a first-order dissection of the distinctive populations of collisionally generated debris and unaltered planetesimals is important for our study as the fragment material might posses enhanced volatile and light element inventories if it is ejected from near the surface of partially or fully differentiated proto-planets. Moreover, since we utilize these simulations to infer the orbital distribution of debris given a supposed size frequency distribution (SFD), our particular selection of MFM is not carried forward into our study of late bombardment, and is therefore only important for keeping our experiments computationally feasible.


Recent MIR studies have provided us with a much more realistic view of the central part of the AGNs. Spitzer studies of nearby Compton-thick AGNs have shown that even Compton-thick AGNs, especially low-luminosity ones, often show only modest to moderate silicate absorption at

 μm (e.g., Hao et al. 2007; Goulding et al. 2012). A classical smooth torus model, such as that of Pier & Krolik (1992), predicts deeper absorption in proportion to the X-ray absorption column density. On the other hand, if the torus is made of a collection of clouds, each cloud is heated to ∼300 K to emit MIR emission while absorbing the background light when the foreground cloud is cooler than the one behind. The radiation transfer effect among the clouds significantly reduces the net silicate absorption even when the torus is seen edge-on (Nenkova et al. 2002, 2008a, 2008b; Hönig et al. 2006; Hönig & Kishimoto 2010; Stalevski et al. 2012, 2016). Meanwhile, recent MIR interferometric studies of nearby AGNs have started to directly reveal the dust distribution in the vicinity of the AGNs at parsec scales. In some best-studied AGNs, extended optically thin dust emission elongated toward the system’s polar direction (e.g., direction of the extended narrow-line region or outflow) is typically found in addition to the compact disk-like component (e.g., Raban et al. 2009; Hönig et al. 2012, 2013; Tristram et al. 2012, 2014; López-Gonzaga et al. 2016; see also Asmus et al. 2016 for the single-dish study; see Netzer 2015 for a review). Such extended polar emission is clearly inconsistent with the classical idea of the dusty torus in the unification theory, and its nature is under debate. Some proposed ideas are that it originates from the inner funnel of an extended dust distribution above and below the torus and/or the dusty outflow within the ionizing cone that is radiatively driven from the inner wall of the compact dusty torus (e.g., Hönig et al. 2012, 2013; Keating et al. 2012; Roth et al. 2012; Tristram et al. 2014).


The EPS model developed by Bond et al. (1991) is based on the excursion set formalism. For each collapsed region one constructs random ‘trajectories’ of the linear density contrast δ(M) as a function of the variance σ2(M). Defining

(3)

\begin{equation}
\omega \equiv \delta _{\rm c}(z)\quad {\rm {and}}\quad S\equiv \sigma ^{2}(M),
\end{equation}

we use ω and S to label redshift and mass, respectively. If the initial density field is a Gaussian random field smoothed under a sharp k-space filter, increasing S (corresponding to a decreased in the filter mass M) results in δ(M) starting to wander away from zero, executing a random walk. The fraction of matter in collapsed objects in the mass interval M, M + dM at redshift z is associated with the fraction of trajectories that have their first upcrossing through the barrier ω in the interval S, S + dS, which is given by (Bond et al. 1991; Bower 1991; Lacey & Cole 1993)

(4)

\begin{equation}
f(S,\omega ){\rm d}S=\frac{1}{\sqrt{2\pi }}\frac{\omega }{S^{3/2}}\rm {exp}\left[-\frac{\omega ^{2}}{2S}\right]dS,
\end{equation}

where S is defined as

(5)

\begin{equation}
S(M) = \frac{1}{2\pi ^{2}}\int _{0}^{\infty }P(k)\hat{W}^{2}(k;R)k^{2}{\rm d}k.
\end{equation}

Here P(k) is the linear power spectrum and $\hat{W}(k;R)$ is the Fourier transform of a top hat window function. The probability function in equation (4) yields the PS mass function and gives the probability for a change ΔS in a time-step Δω, since for random walks the upcrossing probabilities are a Markov process (i.e. are independent of the path taken). The analytic function given by equation (4) provides the basis for the construction of merger trees. Neistein et al. (2006) derived a differential equation for the average halo mass history over an ensemble of merger trees from the EPS formalism. Defining MEPS(z) to be the mass of the most massive halo (main progenitor), along the main branch of the merger tree, as a function of redshift, they obtained the differential equation (see derivation in Appendix A)

(6)

\begin{eqnarray}
\frac{{\rm d}M_{\rm {EPS}}}{{\rm d}z}&=& \sqrt{\frac{2}{\pi }}\frac{M_{\rm {EPS}}}{\sqrt{S_{q}-S}}\frac{1.686}{D(z)^{2}}\frac{{\rm d}D(z)}{{\rm d}z},
\end{eqnarray}

where Sq = S(MEPS(z)/q) and S = S(MEPS(z)). The value of q needs to be obtained empirically so that MEPS reproduces halo mass histories from cosmological simulations. Neistein et al. (2006) showed that the uncertainty of q is an intrinsic property of EPS theory, where different algorithms for constructing merger trees may correspond to different values of q.


Finally, putting everything together, and using dΩu = dϕvdθusin θu, the cross-correlation $\rho _{\ell \ell ^{\prime }ij}^{uv}(\beta , b, \bar{n}_{g})$ can be written as
(28)$$\begin{eqnarray*}
\rho _{\ell \ell ^{\prime }ij}^{uv}(\beta , b, \bar{n}_{g}) = \delta _{ij}^{K} \frac{\kappa _{\ell \ell ^{\prime }}^{uv}(k_{i},\beta ,b,\bar{n}_{g})}{\kappa _{\ell \ell ^{\prime }}(k_{i},\beta ,b,\bar{n}_{g})},
\end{eqnarray*}$$with, at leading order in $(\bar{n}_{g}P_m^\mathrm{lin}(k))^{-1}$ (e.g. Meiksin & White 1999; Howlett & Percival 2017),
(29)$$\begin{eqnarray*}
\kappa _{\ell \ell ^{\prime }}^{uv}(k,\beta ,b,\bar{n}_{g}) &=& \int _{0}^{2\pi } \mathrm{ d}\phi _{v} \int _{0}^{\pi } \mathrm{ d}\theta _{u}\sin {\theta _{u}} \nonumber \\
&&\times \, \bigg[ b^{2}\left(1 + \beta \cos ^{2}{\theta _{u}}\right) \left(1 + \beta \cos ^{2}{\theta _{v}}\right) P_m^\mathrm{lin}(k) \nonumber \\
&& +\, \frac{1}{\bar{n}_{g}} e^{-k^{2}\left(\cos ^{2}{\theta _{u}} + \cos ^{2}{\theta _{v}} - 2 \cos {\theta _{u}}\cos {\theta _{v}} \cos \theta _{uv} \right) f^{2} \sigma _{d}^{2}/2} \bigg]^{2} \nonumber \\
&&\times \, \mathcal {L}_{\ell }(\cos {\theta _{u}})\mathcal {L}_{\ell ^{\prime }}(\cos \theta _{v}),
\end{eqnarray*}$$
 (30)$$\begin{eqnarray*}
&&\kappa _{\ell \ell ^{\prime }}(k,\beta ,b,\bar{n}_{g}) \nonumber \\
&&\quad = 2\pi \int _{0}^{\pi } \mathrm{ d}\theta _{u}\sin {\theta _{u}} \left[ b^{2} \left(1 + \beta \cos ^{2}{\theta _{u}}\right)^{2} P_m^\mathrm{lin}(k) + \frac{1}{\bar{n}_{g}} \right]^{2} \nonumber \\
&&\qquad\times \, \mathcal {L}_{\ell }(\cos {\theta _{u}})\mathcal {L}_{\ell ^{\prime }}(\cos {\theta _{u}}).
\end{eqnarray*}$$In the limit that there is no shot noise, $1/\bar{n}_{g} \rightarrow 0$, and the $P_m^\mathrm{lin}(k)$ in equation (29) cancels with the $P_m^\mathrm{lin}(k)$ in equation (30), removing any k-dependence on the cross-correlation. If the LOS u and v are orthogonal, then we arrive at the same expression as equations (18) and (19).


In addition, star formation in the halo is modeled by assuming a conversion efficiency from gas into stars ϵ
⋆. In our fiducial case, we adopt ϵ
⋆ = 0.05 (e.g., Visbal et al. 2015), which is motivated by abundance matching and the observed UV luminosity function of galaxies at z ≃ 6 (Bouwens et al. 2015). Note that the efficiency is calculated as the average value over time and scales in galaxies. Alternatively, we suppose that a star formation episode with a high value of ϵ
⋆ lasts within a few megayears in the nuclear region before SN explosions of massive stars begin to occur and regulate the star formation efficiency (SFE) in the bulge. According to numerical simulations that study star cluster formation from a giant molecular cloud with a size of 10–100 pc, the SFE becomes as high as ϵ
⋆ ≳ 0.2–0.3 when the initial gas surface density is higher than ∼103
M
⊙ pc−2 (Fukushima et al. 2020; Fukushima & Yajima 2021). Referring to those observational and theoretical studies, we assume f

n
 = f
0(1 − ϵ
⋆), where f
0 = 1 and ϵ
⋆ = 0.05 (fiducial case) and f
0 = 4 and ϵ
⋆ = 0.5 (high SFE case). Following the definition of the SFE, ϵ
⋆, the SFR is approximated as
32

SFR=4πϵ⋆1−ϵ⋆ρ(r)r2Vc,≃0.11ϵ⋆f0Tvir,43/2M⊙yr−1,

where V
c is the halo circular velocity. This SFR is used to estimate the emissivity of stellar irradiation (LW and X-rays) and to calculate the bulge growth in mass. Note that in most cases, the gas mass within r ≃ r
c for the given initial condition is lower than the total mass of newly forming stars.
10

10
Although we do not consider the self-gravity of gas in our simulations, it would dominate the bulge gravity at the intermediate scale of ∼O(10 pc) only in the early stage of the bulge formation where M
⋆ ≲ 106
M
⊙. In this case, however, radiative feedback associated with BH accretion blows the gas outward, and then the gas self-gravity eventually becomes less important (see Section 3.2.2). Therefore, our star formation model implicitly assumes that the bulge growth is not led by in situ star formation but by efficient migration of stars formed at larger radii with morphological evolution owing to stellar relaxation (see also Appendix A).


To obtain reaction rates in a wide temperature range

T

9

=
0.001

–10 (where T9 is in units of 109 K) for BBN calculation inputs, continuous excitation functions over

10

−
8

 to a few MeV are needed for numerical integration. Therefore we performed an R-matrix analysis to compile both the present and the previous data of these three reaction channels by using AZURE2 (Azuma et al. 2010) including all the relevant partial widths

Γ

n

,

Γ

p
0

,

Γ

p
1

, and

Γ

α

. The 7Be

(
n
,

p

0

)

 data set used for the R-matrix analysis is based on the same selection as used in Damone et al. (2018), Damone (2018; part of Dam18 + Sek76 as displayed in Figure 2), and the present THM data, Bor63, and Pop76 are added. For the 7Be

(
n
,

p

1

)

 and 7Be

(
n
,
α
)

 channels, all the labeled data shown in Figure 2 (except for p1 Tom19 and

γ
α

 Bar16) were used without any data cutoff. Appendix B shows the adopted data sets more explicitly. We included nine known levels below the 8Be excitation energy of

E

x

=
24

MeV

 (Tilley et al. 2004) and an additional nonresonant background pole, with a common channel radius of 5 fm for each channel in the same manner as Adahchour & Descouvemont (2003). The most significant channel 7Be

(
n
,

p

0

)

7

Li is dominated by four of these levels as described in detail by a former R-matrix study (Adahchour & Descouvemont 2003; Descouvemont et al. 2004); the 2− resonance near the neutron threshold (labeled as level I) principally dominates the cross section up to the BBN energies, the two 3+ states at

E

x

=
19.24

MeV

 (level II) and

E

x

=
21.5

MeV

 well characterize the corresponding single peaks, and a 2+ background pole accounts for the enhancement at high energies. The most important resonances to expand analysis to the 7Be

(
n
,

p

1

)

7

Li* and 7Be

(
n
,
α
)

4

He channels are the 1− state at

E

x

=
19.4

MeV

 (level III) and the 2+ state at

E

x

=
20.1

MeV

 (level IV), respectively; the former expresses the 7Be

(
n
,

p

1

)

 resonance behavior peaked at

E

c
.
m
.

∼
0.5

MeV

 tailing down to the thermal neutron energy by the

1

/

v

 law, and the latter forms the first peak in the 7Be

(
n
,
α
)

 spectrum around

E

c
.
m
.

∼
1

MeV

. Despite its importance in the

(
n
,
α
)

 channel, level IV is much less significant in the total cross section especially at lower energies due to its p-wave nature. Therefore we imposed some restrictions on level IV, which made the analysis much simpler; fixing

Γ

p
0

,

Γ

p
1

, and

Γ

α

 at a known ratio

Γ

α

/

Γ

p

∼
4.5

 (Tilley et al. 2004), and freeing

Γ

n

 and resonance energy (refitted to be 19.87 MeV) not to significantly exceed the known total width

Γ
=
880

keV

 (Tilley et al. 2004). The other four higher-lying levels (0+ at

E

x

=
20.2

MeV

, 2+ at

E

x

=
22.24

MeV

, 1− at

E

x

=
22

 MeV, and 2− at

E

x

=
24

 MeV) play rather supplementary roles mainly for the higher-energy behavior. The 4+ and 4− higher-spin states (at

E

x

=
19.86

MeV

 and 20.9 MeV, respectively) were not included due to their limited influences. We do not introduce the γ-emission channels to fit the Bar16 plots (representing the 7Be

n
,
γ
α

4

He reaction channel Barbagallo et al. 2016), which appears significant only below the BBN energies. See Appendix C for more details on the present R-matrix analysis.


The bolometric luminosity of 3C 84 is about 0.4% of the Eddington luminosity. Thus, the accretion flow of 3C 84 is likely to be a radiatively inefficient accretion flow (RIAF: Narayan & Yi 1995) rather than a standard disk (Shakura & Sunyaev 1973). However, we note that 3C 84 has a cold (

T

e

∼

10

4

 K) disk-like accretion flow, as identified by FFA of the emission from the counter jet in the parsec scale (Walker et al. 2000) and inhomogeneous gas distribution around the black hole (Fujita et al. 2016). A number of theoretical studies predicted that the accretion flow components of hot geometrically thick (RIAF-like) and cold geometrically thin can coexist in either horizontal or vertical stratification (e.g., Miller & Stone 2000; Merloni & Fabian 2002; Liu et al. 2007; Ho 2008; Liu & Taam 2013). The measured Faraday rotation can be caused by such an RIAF-like component. We thus estimate the accretion rate of the RIAF-like component using the measured RM. For a simplicity, we assume that the RIAF-like component is a quasi-spherical Bondi accretion flow with a power-law density profile. We can calculate the accretion rate, following the formulation as follows (Quataert & Gruzinov 2000; Marrone et al. 2006; Kuo et al. 2014).

M

˙

=

1.3
×

10

−
10

[

1
−

(

r

out

/

r

in

)

−
(
3
β
−
1
)

/

2

]

−
2

/

3

×

M

BH

8.0
×

10

8

M

⊙

4

/

3

2

3
β
−
1

−
2

/

3

r

in

7

/

6

RM

rad

m

−
2

2

/

3

.

For an inner effective radius

r

in

 of 1 pc (

1.3
×

10

4

R

s

), where the hotspot is located, the observed RM implies an accretion rate of

∼
4.3
×

10

−
2

M

⊙

 yr−1 and

∼
8.6
×

10

−
2

M

⊙

 yr−1 for

β
=
0.5

 and

β
=
1.5

, which are corresponding to convection-dominated accretion flow (CDAF: Narayan et al. 2000; Quataert & Gruzinov 2000) and advection-dominated accretion flow (ADAF: Ichimaru 1977; Narayan & Yi 1995), respectively. Here we assumed the outer effective radius

r

out

 of

10

5

R

s

 (∼8 pc), which is approximately the same with the Bondi radius of 8.6 pc (Fujita et al. 2016). The derived accretion rate is roughly consistent with that estimated from the bolometric luminosity with a black hole mass of

8
×

10

8

M

⊙

 and a radiative efficiency of 10% (

M

˙

∼

L

bol

/

(
0.1

c

2

)
≃
7.1

×

10

−
2

M

⊙

 yr−1).


We tested the optimization of the physical model for each of these candidates. For each candidate, we constructed a grid of parameters (λL, βL, ϕ0, ψ0, Ia, Ib). The grid for the orientation of the angular momentum vector was constructed by distributing ten different positions evenly on the celestial sphere. Standard Euler angles at t0 were arranged at 60° intervals. In addition, the moments of inertia were sequenced at 0.01 intervals from 0.01 to 0.99 in the case of SAM, and intervals of 0.1 from 1.1 to 10 in the case of LAM. Optimization was performed using the grid as the initial parameter set. The photometric model of the surface properties of airless body was considered as the Hapke model (Hapke 1993). The Hapke model parameters were optimized using initial values of a typical S-type asteroid: ϖ = 0.23, g = −0.27, h = 0.08, B0 = 1.6, and

$\bar{\theta} = 20^{\circ}$

θ
¯

=

20
°

 (Li et al. 2015), where ϖ denotes the single scattering albedo, g denotes the particle phase function parameter, h and B0 respectively denote the width and amplitude of the opposition surge, and

$\bar{\theta}$

θ
¯

 denotes the macroscopic roughness angle. Additionally, because the light curves used in this study covered only phases angles from 14.0° to 28.5°, the parameters for opposition surge (h and B0) were fixed during the optimization process. The

$\bar{\theta}$

θ
¯

 was also fixed because it cannot be reliably obtained from our data. Thus, only two parameters, ϖ and g, were optimized. We found that the physical model is not sensitive to the Hapke model parameters. Solutions corresponding to the 2016 and 2017 data converged at one or two global minima for each frequency combination. However, we found that the values between the moment of inertia obtained from the shape model (assuming constant density) and the dynamical moments of inertia were different in all solutions except for SAM1. Because we preferred the physically self-consistent model, we excluded the combination of LAM and SAM2. In addition, solutions corresponding to the 2016 and 2017 data were similar to SAM1. Therefore, we accepted values for Pψ and Pϕ based on SAM1.


We conclude by displaying in Figure 9 the NS moment of inertia calculated from the EoSs without Δ's and with Δ's (with couplings xσΔ = xωΔ = 1.15 and xρΔ = 0). The results have been obtained by solving Hartle’s slow-rotation differential equation for the moment of inertia in general relativity coupled to the TOV equations (Hartle 1967). Astronomical observations of binary pulsars may provide information on the moment of inertia of NSs, ultimately offering possible limits on the underlying EoS (Lattimer & Schutz 2005). The only double-pulsar system known to date is PSR J0737-3039. The mass of its primary component PSR J0737-3039A, or pulsar A, is of 1.338 M⊙. It is expected that a precise measurement regarding the moment of inertia of this slowly rotating pulsar will be obtained in the near future from radio observations of PSR J0737-3039 (Burgay et al. 2003; Lyne et al. 2004). In a recent work, Landry & Kumar (2018) (also see Kumar & Landry 2019) use approximately universal relations among NS observables (Yagi & Yunes 2013a, 2013b), known as the binary-Love and I-Love relations (recall that the tidal deformability is related to the Love number), to determine a range of

 g cm2 for I of pulsar A from the 90% credible limits on the tidal deformability of a 1.4 M⊙ NS reported by LIGO-Virgo from the GW170817 event (Abbott et al. 2019). Landry & Kumar (2018) also facilitate a wider range I ≤ 1.67 × 1045 g cm2 for I of pulsar A from the less restrictive upper limit on the tidal deformability obtained by the LIGO-Virgo collaboration in their initial data analysis of GW170817 in Abbott et al. (2017). We have plotted in Figure 9 these two constraints reported in Landry & Kumar (2018) for I of pulsar A (we show both of the constraints because there may be a certain dependence on the analysis and on the assumed boundaries for Λ of a 1.4 M⊙ star). We can see in Figure 9 that the moment of inertia is an observable highly sensitive to the existence of Δ isobars in the composition of the core of the star. The presence of these particles, in particular for smaller values of the coupling of the Δ to the ρ-meson, improves the agreement with the constraints on the moment of inertia of pulsar A, whose measurement is anticipated within the next few years.


The radioactive contribution to the light curve follows the analytic estimation of Dado & Dar (2015) exactly, taking as inputs from our models the Ni mass (Table 1), the ejecta asymptotic velocity, Vej, and the ejecta mass, $\mathcal {M}_{\rm ej}$. All these quantities are approximate and not the result of a sufficiently long and detailed calculation. We estimate an upper bound to the ejecta mass as the difference between the stellar mass at the brink of collapse, M0, and the PNS mass at tpb = tf (i.e. $\mathcal {M}_{\rm ej}:=M_0-M_{{\small PNS}}(t_{\rm f})$). We note that $\mathcal {M}_{\rm ej}$ is significantly greater than the mass unbound at tpb = tf (Mej; see Fig. 3) or the mass enclosed by the shock at that time (Msh, e; Table 1). At tpb = tf the explosion is ongoing and both Mej and Msh, e are still increasing until they reach their final values. The asymptotic ejecta velocity is estimated as $V_{\rm ej}=\sqrt{2E_{\rm k}/\mathcal {M}_{\rm ej}} = \sqrt{2 (\mathcal {T}_{{\small PNS}}f_{\rm er} + E_{\rm exp})/ \mathcal {M}_{\rm ej}}\approx \sqrt{2 \mathcal {T}_{{\small PNS}}f_{\rm er} / \mathcal {M}_{\rm ej}}$, where we allow for the possibility that only a fraction fer of the PNS rotational energy can be converted into kinetic energy of the ejecta, Ek. The last approximation results from the fact that the $\mathcal {T}_{{\small PNS}}$ ($\gt 10^{52}\,$erg) is one order of magnitude (or more) greater than Eexp ($\lt 2\times 10^{51}\,$ erg; see Table 1). Typical ejecta velocities in all our models are $V_{\rm ej} \approx (0.4-1.6)\times 10^{9}\, \rm{cm\,s}^{-1}$. The corresponding luminosity reads (Dado & Dar 2015)
(8)$$\begin{eqnarray}
L_{\mathrm{nuc}}(t)=\frac{{\rm e}^{-t^{2} / 2 t_{\mathrm{r}}^{2}}}{t_{\mathrm{r}}^{2}} \int _{0}^{t} x {\rm e}^{x^{2} / 2 t_{\mathrm{r}}^{2}} \skew{4}\dot{E} \, {\rm d} x ,
\end{eqnarray}$$where
(9)$$\begin{eqnarray}
t_{\rm r}=\sqrt{ \frac{3\mathcal {M}_{\rm ej}f_{\rm e} \sigma _{\rm T}}{8\pi m_{\rm p} c V_{\rm ej}} },
\end{eqnarray}$$σT is the Thompson cross-section, mp is the proton mass, c is the speed of light, fe ≈ 0.275 is the fraction of free electrons, and $\skew{4}\dot{E}=\skew{4}\dot{E}_\gamma + \skew{4}\dot{E}_{e^+}$, where
(10)$$\begin{eqnarray}
\skew{4}\dot{E}_{\gamma } &=& \frac{M_\mathrm{Ni}}{{\rm M}_{\odot }}\left[7.78 A_{\gamma }^\mathrm{Ni} {\rm e}^{-t / 8.76 \mathrm{~d}}\right. \\
&&\left.+\,1.50 A_{\gamma }^\mathrm{Co}\left[{\rm e}^{-t / 111.27 \mathrm{~d}}-{\rm e}^{-t / 8.76 \mathrm{~d}}\right]\right] 10^{43} \mathrm{erg}\, \mathrm{s}^{-1}
\end{eqnarray}$$is the power supplied by gamma rays, and
(11)$$\begin{eqnarray}
\skew{4}\dot{E}_{\mathrm{e}^{+}} = \frac{M_\mathrm{Ni}}{{\rm M}_{\odot }} A_{\mathrm{e}} \left[{\rm e}^{-t / 111.27 \mathrm{~d}}-{\rm e}^{-t / 8.76 \mathrm{~d}}\right] 10^{43} \mathrm{erg}\, \mathrm{s}^{-1}
\end{eqnarray}$$is the power supply in the form of the kinetic energy of positrons. Ae ≈ 0.05 is the ratio of the energy released as positron kinetic energy and as gamma-ray energy in the decay of 56Co. $A_\gamma ^\mathrm{Ni}$ and $A_\gamma ^\mathrm{Co}$ are the absorbed fractions of the gamma-ray energy in the SN ejecta from the decay of 56Ni (decay time of 8.76 days) and 56Co (111.27 days), respectively. They are computed as $A_\gamma \approx 1-{\rm e}^{-\tau _\gamma }$, with $\tau _\gamma =3\mathcal {M}_{\rm ej}\sigma _t/(8\pi m_p V^2_{\rm ej}t^2)$, and $\sigma _t=9.5\times 10^{-26}\,$ cm2 ($\sigma _t=8.7\times 10^{-26}\,$ cm2) for 56Ni (56Co).


Instabilities, driven by departures from local thermodynamic equilibrium (LTE), are frequently credited with affecting the behavior of rapidly evolving plasma systems, e.g., the expanding solar wind (Matthaeus et al. 2012). To quantify these departures, the underlying charged particle velocity distribution functions (VDFs) are typically modeled as bi-Maxwellians, having anisotropic temperatures T
⊥,j
 and T
∥,j
 with respect to the local magnetic field
B
; relative field-aligned drifts between each pair of constituent VDF components i and j being Δv

i,j
=(
V

i
 −
V

j
) ·
B
/∣
B
∣; and temperature disequilibrium between species T

i
 ≠ T

j
 (Marsch et al. 1982). These anisotropies, drifts, and disequilibrium serve as distinct sources of free energy capable of driving the growth of a number of distinct unstable solutions (see, e.g., Section 5 of Verscharen et al. 2019). The presence of multiple free-energy sources makes it difficult to determine which subset of sources drives a given instability; parametric models accounting for a single source of free energy—e.g., the temperature anisotropy of a single population (Gary et al. 1997; Yoon 2017)—do not account for the diminishment or enhancement of predicted linear growth rates associated with the introduction of other departures from LTE, e.g., relative drifts of proton beams (Daughton & Gary 1998; Woodham et al. 2019; Liu et al. 2021), helium (Podesta & Gary 2011; Bourouaine et al. 2013; Verscharen et al. 2013), or their combined effects (Chen et al. 2016). To account for these effects, previous studies implemented Nyquist’s instability criterion (Nyquist 1932; Klein et al. 2017) on limited sets of in situ measurements from the Wind (Klein et al. 2018), Parker Solar Probe (Verniero et al. 2020; Klein et al. 2021), and Helios (Klein et al. 2019) missions, finding that a majority of intervals were unstable, that the kinds of waves driven unstable were very sensitive to the model used to describe the VDF, and that inclusion of multiple ion populations could both enhance or diminish the predicted growth rates.


The Virgo Cluster radio galaxy M87 (NGC 4486), located at a distance of

 Mpc (Mei et al. 2007) and believed to harbour a BH of mass

, was the first extragalactic source detected at VHE energies (Aharonian et al. 2003). Given its proximity, M87 has been a prime target to probe scenarios for the formation of relativistic jets with high-resolution radio observations exploring scales down to some tens of rg, and much effort has recently been dedicated in this direction (e.g., Acciari et al. 2009; Doeleman et al. 2012; Hada et al. 2014, 2016; Akiyama et al. 2015, 2017; Kino et al. 2015). At VHE energies, M87 has revealed at least three active γ-ray episodes, during which day-scale flux variability (i.e.,

) has been observed (Aharonian et al. 2006; Albert et al. 2008; Acciari et al. 2009; Abramowski et al. 2012; Aliu et al. 2012). The VHE spectrum is compatible with a relatively hard power law (photon index ∼2.2) extending from 300 GeV to beyond 10 TeV, while the corresponding TeV output is relatively moderate, with an isotropic equivalent luminosity of

 erg s−1. The inner, parsec-scale jet in M87 is considered to be misaligned by

–25°, resulting in modest Doppler boosting of its jet emission and creating challenges for conventional jet models to account for the observed VHE characteristics (see, e.g., Rieger & Aharonian 2012 for review and references). Gap-type emission models offer a promising alternative and different realizations have been proposed in the literature (e.g., Neronov & Aharonian 2007; Levinson & Rieger 2011; Broderick & Tchekhovskoy 2015; Vincent 2015; Ptitsyna & Neronov 2016). M87 is overall highly underluminous with characteristic estimates for its total nuclear (disk and jet) bolometric luminosity not exceeding

 erg s−1 by much (e.g., Owen et al. 2000; Whysong & Antonucci 2004; Prieto et al. 2016), suggesting that accretion onto its BH indeed occurs in a non-standard, advective-dominated (ADAF) mode characterized by an intrinsically low radiative efficiency (e.g., Di Matteo et al. 2003; Nemmen et al. 2014), with inferred accretion rates possibly ranging up to

 (e.g., Levinson & Rieger 2011) and a BH spin parameter close to its maximum one (e.g., Feng & Wu 2017). For these values of the accretion rate, the soft photon field (see Equations (2) and (3)) is sufficiently sparse that the maximum Lorentz factor

 of the magnetospheric particles is essentially determined by the curvature mechanism. The observed VHE variability is in principle compatible with

, so that the different dependence of the gap power on β, Equation (20), does not necessarily (in the absence of other, intrinsic considerations of gap closure) imply a strong difference in the extractable gap powers. Figure 2 shows a representative point for M87 (taking

). The observed VHE luminosity of M87 is some orders of magnitudes lower than the maximum possible gap power (given by the dotted line) and within the bound imposed by ADAF considerations (vertical line). The observed VHE flaring events thus appear consistent with a magnetospheric origin. VLBI observations of (delayed) radio core flux enhancements indeed provide support for the proposal that the variable VHE emission in M87 originates at the jet base very near to the BH (e.g., Acciari et al. 2009; Beilicke 2012; Hada et al. 2012, 2014).


In addition to the photons from the UVB, accounted for in Haardt Madau (2001) (HM01), ionizing photons ($\rm h\nu \ge 1\, Ryd$) emitted by nearby young stars, in particular these belonging to the halo in consideration, can contribute substantially to the total Ly emission of star-forming galaxies. The strength of this contribution depends strongly on the interstellar dust and gas geometry and kinematics (Kunth etal. 2003; Verhamme etal. 2012), as those determine how many Ly photons escape from the star-forming regions. In a first, conservative approximation, we assume that all the ionizing photons from stars are absorbed by dust or photoionizing neutral gas in the ISM. Since the dust attenuation is poorly constrained at low redshift, we will consider a simplistic model for the emitted Ly$\rm \alpha$ photons. We start with the prescription from Furlanetto etal. (2005) to estimate the intrinsic Ly luminosity from ionizing photons in the absence of dust: $L\rm ^{stars}_{Ly\alpha } \ [erg\, s^{-1}] = 10^{42}\ SFR\ [M_{\odot }\ yr^{-1}]$. We compute the SFR of each halo from the mass of young stars, using the continuous star formation approximation (Kennicutt 1998):

(1)
$$\begin{eqnarray*}
\rm SFR [M_{\odot }\, yr^{-1}] = \frac{\mathit{ M}_{stars \lt 10^8\ yr}\ [M_{\odot }]}{10^8\ [yr]}.
\end{eqnarray*}
$$
Using COS data of low-redshift ($z \sim 0.03$) star-forming galaxies, Wofford, Leitherer Salzer (2013) measured a Ly escape fraction ranging from 1percent to $\rm 10{{\ \rm per\ cent}}$. They estimate that this fraction is sensitive to the presence of dust and to the Hi column density, the Ly photons escaping more easily from holes of low Hi and dust column densities, resulting in a large scatter. Winds can also have a strong effect and can help Ly photons to escape (Dijkstra Jeeson-Daniel 2013) but we do not consider winds in our model. As we do not have any model for either the dust or radiative transfer, we will stay conservative in our assumptions. Hayes etal. (2011) find Ly escape fractions between $\rm 0.1{{\ \rm per\ cent}}$ and $\rm 1{{\ \rm per\ cent}}$ for low-z galaxies. Therefore, we will consider two extreme cases for the Ly luminosity from the stellar contribution at low redshift: one with a Ly escape fraction of $\rm 1{{\ \rm per\ cent}}$, and another with a Ly escape fraction of $\rm 0.1{{\ \rm per\ cent}}$. At higher redshift (z  1), we adopt a Ly escape fraction of $\rm 10{{\ \rm per\ cent}}$ that is within the prediction of Hayes etal. (2011). We are aware that different works predict very different escape fractions for Ly (e.g. Wofford etal. 2013; Naidu etal. 2017, but we choose to follow the trend observed in Hayes etal. (2011). Predicting the spatial and spectral profiles of such emission would require the full calculation from radiative transfer techniques (Verhamme etal. 2006, 2012; Rosdahl etal. 2013; Lake etal. 2015), which is beyond the scope of this work. However, as our goal is to study the detectability of such emission with upcoming instruments, we chose to make the simple assumption that all the Ly photons only go through one absorption/re-emission process before leaving the cloud. Also, we assume that all of the Ly photons are emitted from the centre of the galaxy. We then weigh the profile proportionally to the total hydrogen density of the gas cell and by its inverse squared distance to the centre. This gives us, for each cell j the luminosity $L_{j}^{\star }$:

(2)
$$\begin{eqnarray*}
L_{j}^{\star }\rm [erg\, s^{-1}] = \mathit{ f}_{esc} (Ly\alpha) \frac{\frac{\mathit{ n}_{\mathit{ j},H}}{\mathit{ R}_\mathit{ j}^2}}{\int _{\mathit{ R}_{vir}} \frac{\mathit{ n}_H}{\mathit{ R}^2}} 10^{42} SFR_\mathit{ j}\, [M_{\odot }\, yr^{-1}] .
\end{eqnarray*}
$$


For GRB 140629A, our analysis suggests that the optical and X-ray afterglows are from a narrow jet (

 rad) with a low B

 in a dense medium (n = 60 cm−3). In addition, the radiation efficiency of GRB 140629A is extremely low. We test whether or not it satisfies various empirical relations reported in the literature derived from observations of the prompt gamma-ray phase and the multi-wavelength afterglows. By estimating the jet opening angle with a jet-like break time tj in late multi-wavelength light curves, Ghirlanda et al. (2004a) derived a tight correlation between geometrically corrected jet energy

 and the peak energy

 of

 spectrum in the burst frame, i.e.,

. The

 value inferred from the Ghirlanda relation is 46 keV for GRB 140629A, which is definitely inconsistent with the data, i.e.,

. Liang & Zhang (2005) derived an empirical relation between

,

, and the jet break time (

) in the burst frame, i.e.,

/

/

/

Based on this relation, an isotropic energy

 erg is obtained, which is larger than that observed by more than one order of magnitude. These results suggest that GRB 140629A does not follow these two relations (Ghirlanda et al. 2004a; Liang & Zhang 2005), although both tight correlations have been used for measuring the cosmological parameters with GRBs (e.g., Dai et al. 2004; Ghirlanda et al. 2004b; Liang & Zhang 2005; Wang et al. 2015a). Note that the observed jet break time of GRB 140629A is much earlier, hence the inferred θj is much lower than those of the GRBs used to derive these relations (e.g., Frail et al. 2001; Bloom et al. 2003). It is unclear whether the violation of GRB 140629A is due to the selection effect or other physical reasons. For example, two-component jet models composed of a narrow and a wide component have been proposed to explain the data of some GRBs (e.g., Huang et al. 2004; Racusin et al. 2008). In these cases, the high-energy emission was proposed to be emitted by the narrow jet. However, one cannot exclude the possibility that the observed gamma-ray energy would be dominated by the wide jet component under certain conditions. Meanwhile, the early break time for GRB 140629A is likely due to the effect of the narrow jet component but not the wide one. If this is the case, the inconsistency between the jet energy and the opening angle would result in this violation of GRB 140629A. Liang et al. (2015) discovered a tight empirical correlation between Liso,

, and Γ0 to reveal the direct connection between the gamma-ray and afterglows,
7

Based on the equation above, we get

 for GRB 140629A, where the error is calculated from the uncertainties in

 and Γ0 only. The derived

 is well consistent with the observed one,

 erg s−1, as shown in Figure 6. Note that the initial Lorentz factor of the ejecta Γ0 is sensitive to the deceleration time (the peak time of the onset bump), but not strongly related to the jet break time. The onset of the afterglow bump is usually bright (Liang et al. 2010, 2013; Li et al. 2012; Wang et al. 2013), and it is easier to identify than the jet break time from an observed light curve.10

10
The jet break is usually detected in late optical afterglow light curves. It is dim and also contaminated by emission from the host galaxy and/or associated supernovae (e.g., Li et al. 2012). This is also an issue in identifying an observed jet break as the narrow or the wide component in the case of a two-component jet.
 The consistency of GRB 140629A with

 suggests that this relation is more robust than the Ghirlanda or Liang–Zhang relations, since it is not sensitive to the jet opening angle θj.


None of the galaxies in this paper would be classified as an AGN on the basis of WISE mid-IR colors; they do not meet the

 (Vega) threshold from Stern et al. (2012) or the reliable AGN criteria from Assef et al. (2013). There are two galaxies with weak X-ray detections (J1506+5402 and J1613+2834, four X-ray counts each) from Sell et al. (2014) that indicate

 erg s−1 and four galaxies with X-ray upper limits (J0826+4305, J0944+0930, J1558+3957, J2140+1209) that imply

 erg s−1. All of these galaxies are consistent with the relationship between X-ray luminosity and mid-IR luminosity for starburst galaxies (Asmus et al. 2011; Mineo et al. 2014; Sell et al. 2014). One of the galaxies with a weak X-ray detection (J1506+5402) also has a clear detection of [Ne v]

 and an emission-line ratio

, which is the highest ratio in the sample and consistent with a composite Baldwin, Phillips, and Terlevich (BPT) classification (Baldwin et al. 1981; Kewley et al. 2001; Kauffmann et al. 2003; Sell et al. 2014). These [Ne v] and [O iii] emission lines could be produced either by an extreme (

 Myr) starburst or an AGN that contributes

% of the mid-IR continuum (Diamond-Stanic et al. 2012; Geach et al. 2013; Sell et al. 2014). The galaxy with the slowest outflow velocity in the sample (J2140+1209) has a weak broad Mg ii line, which is a clear indication of a type 1 AGN component, and spectral modeling suggests a

% AGN continuum contribution at

 (Sell et al. 2014); the other 11/12 galaxies show no evidence for any AGN continuum contributions at near-UV or optical wavelengths on the basis of high signal-to-noise spectroscopy (as described above and shown in Figure 1, which we describe below). In summary, there is some evidence for AGN activity in several of these 12 galaxies, but in all cases, the upper limits on the AGN emission indicate that it contributes a small fraction of the galaxy’s bolometric luminosity.


Equation (25) shows that entropy advection can easily result in tens of per cent increase in the outer boundary temperature, and possibly substantially more depending on the depth to which the flow penetrates and whether the Bondi radius is larger than the Hill radius. How will this impact the cooling luminosity? We can estimate the effect on the luminosity if we assume that the radiative zone is isothermal (which applies only to low-metallicity envelopes), so that the temperature at the radiative–convective boundary (RCB) is the same as the outer boundary temperature, TRCB = Tadv. The luminosity, set at the RCB, is
(26)$$\begin{eqnarray*}
L\approx {64\pi GM_\mathrm{ p}\sigma T_{\rm RCB}^4 \nabla _{\rm ad}\over 3\kappa P_{\rm RCB}}\propto {T_{\rm RCB}^4\over \kappa P_{\rm RCB}}
\end{eqnarray*}$$(Arras & Bildsten 2006; Piso & Youdin 2014) (again we will neglect the self-gravity of the envelope). For a power-law dependence of opacity κ ∝ ρaTb ∝ PaTb − a, we find that for a fixed internal entropy so that $P_{\rm RCB}\propto T_{\rm RCB}^{\gamma /(\gamma -1)}$, the luminosity varies with RCB temperature as
(27)$$\begin{eqnarray*}
{\mathrm{ d}\ln L\over \mathrm{ d}\ln T_{\rm RCB}} = 4 - b+a - (1+a){\gamma \over \gamma -1}.
\end{eqnarray*}$$Taking the values a = 2/3 and b = 3 for molecular opacity from Bell & Lin (1994), we find that a change in RCB temperature ΔTRCB leads to a change in luminosity
(28)$$\begin{eqnarray*}
{\Delta L\over L} &\approx & -4 {\Delta T_\mathrm{RCB}\over T_\mathrm{RCB}}\approx -4{T_\mathrm{adv}-T_\mathrm{ d}\over T_\mathrm{ d}}\nonumber \\
&=& -{8\over 5} \left({R_\mathrm{ B}\over R_{\rm out}}\right) \left[{R_{\rm out}\over R_{\rm adv}}-1\right].
\end{eqnarray*}$$A hotter outer boundary leads to a lower luminosity. For Radv/Rout = 0.3 (0.1) and assuming Rout = RB, the predicted decrease in luminosity is a factor of ≈4 (≈14). For dust opacity, Bell & Lin (1994) give a = 0 and b = 1/2 (metal grains) or b = 2 (ice grains). In this case, dln L/dln TRCB = (1/2) − b, so the change in luminosity will be smaller than for molecular opacity.


Next, we try to convert the Galactic SN rate into a constraint on the Galactic SFR. According to Horiuchi et al. (2011) and Botticella et al. (2012), a cosmological SN rate is linked to the cosmological SFR as
3

RSN(z)=∫mlSNmuSNϕIMF(m)dm∫mlmumϕIMF(m)dmψSFR(z)≡kSNψSFR(z),

where

RSN(z)

 is the SN rate, ψ
SFR(z) is the SFR as a function of redshift (z), ϕ
IMF is the initial mass function (IMF), m is the mass of a star, (m

l
–m

u
) is the mass range of IMF, (

mlSN

–

muSN

) is the mass range of SN stars, and

kSN

 is the scaling factor between

RSN

 and ψ
SFR by the number fraction of stars per unit mass. We apply this relationship for the Galactic SN rate and averaged Galactic SFR; therefore,

RSNgal=kSNψSFRgal

. We show the constraint on

ψSFRgal

 and

kSN

 with 90% CL as shown in Figure 4. The yellow vertical band is an expected range of

kSN

 = (0.0068–0.0088)

M⊙−1

 assuming the (modified) Salpeter-type IMF (Salpeter 1955; Horiuchi et al. 2011; Madau & Dickinson 2014), which contains uncertainties from astronomical observations and models. The lowest value of

kSN

 is calculated from the IMF with a power index of the mass γ = −2.35 in the mass range of 0.1–100 M
⊙. On the other hand, the highest value of

kSN

 is obtained by combining the IMF with γ = −1.5 in the mass range of 0.1–0.5 M
⊙ and the one with γ = −2.35 in the mass range of 0.5–100 M
⊙. In either case, the mass range of SN stars is set in 8–40 M
⊙. The colored horizontal lines and bands correspond to an allowed

ψSFRgal

 range reproduced from the astronomical observation to be (1–2) M
⊙ yr−1 (Murray & Rahman 2009; Robitaille & Whitney 2010; Chomiuk & Povich 2011; Davies et al. 2011; Licquia & Newman 2015). Our result thus provides the upper limit on the SFR as

ψSFRgal

  (17.5–22.7) M
⊙ yr−1 with 90% CL assuming the Salpeter-type IMF within our Galaxy. This result disfavors a large SFR in our Galaxy and is consistent with the constraints from the astronomical observations.


The empirical relation between the mass of SMBHs and the properties of their host galaxies is considered to be one of the most important outcomes caused by their coevolution over the cosmic timescale (e.g., Magorrian et al. 1998; Ferrarese & Merritt 2000; Kormendy & Ho 2013). Theoretical models for explaining the tight correlations have been proposed, but the origin is still unclear. To understand the nature of these correlations, it is critically important to study them beyond the local universe, characterizing how and when the relations have been established and evolved until now. So far, a large number of observational studies have extensively investigated the redshift dependence of the BH-to-bulge mass ratio of M
•/M
⋆ and overall suggested its positive redshift dependence, i.e., the ratio increases with redshift (Bennert et al. 2011; Schramm & Silverman 2013; Ding et al. 2020). Beyond z ∼ 6, the Atacama Large Millimeter/submillimeter Array (ALMA) is a powerful tool to measure the dynamical mass of gas in quasar host galaxies and allows us to explore the early stage of the BH/galaxy correlation (e.g., Wang et al. 2010, 2013; Venemans et al. 2017). In addition, observations with the Subaru HSC provide low-luminosity and less-massive BH samples, which are unique populations to determine the M
•/M
⋆ ratio at z > 6 (Izumi et al. 2019, 2021). Figure 11 shows the distribution of z > 6 quasars compiled in Izumi et al. (2021), together with those in the local universe (Kormendy & Ho 2013). First, the brightest z > 6 quasars with M
1450  −25 mag tend to have M
•/M
⋆ ratios higher than those seen in the local universe. Namely, the mass ratio for those brightest objects is boosted by a factor of ∼10 (blue dashed line; Pensabene et al. 2020). On the other hand, the fainter quasars with M
1450 >−25 mag appear to follow the local relation, although those BHs are considered to grow at rates of ≳ 0.05 SFR and will be overmassive at lower redshifts.
15

15
In this paper, overmassive BHs are referred to as a BH population with a BH-to-galaxy mass ratio higher than that observed in the local universe;

M•/M⋆≳4.9−0.5+0.6×10−3

 (Kormendy & Ho 2013). We employ this terminology to clearly contrast the difference between the overmassive and undermassive BH population with respect to the local value (see Figure 11). Note that a previous study by Agarwal et al. (2013) used a term of “obese BH,” which refers to a BH population dominating over the stellar mass of its host galaxy at least in the initial growing stage (i.e., M
•/M
⋆ > 1). We note that for all of the z > 6 samples, the values of the x-axis are not the bulge mass of their host galaxies but the dynamical mass measured by [C ii] 158 μm lines. In general, the dynamical mass is considered to be higher than the true bulge mass. With a high-resolution ALMA observation, Izumi et al. (2021) found that the gas dynamics of the core component of a low-luminosity quasar at z = 7.07 (HSC J1243+0100) is governed by rotation associated with a compact bulge and estimated its mass as ∼50% of the [C ii]-based dynamical mass. Therefore, the correlation at z > 6 might be shifted to the left if the conversion factor from the dynamical mass to the bulge mass is taken into account.