File size: 16,792 Bytes
b13a737 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 |
arXiv:1001.0008v2 [hep-th] 6 Jan 2010Multi-Stream Inflation: Bifurcations and Recombinations i n the Multiverse Yi Wang∗ Physics Department, McGill University, Montreal, H3A2T8, Canada In this Letter, we briefly review the multi-stream inflation s cenario, and discuss its implications in the string theory landscape and the inflationary multiverse . In multi-stream inflation, the inflation trajectory encounters bifurcations. If these bifurcation s are in the observable stage of inflation, then interesting observational effects can take place, such as do main fences, non-Gaussianities, features and asymmetries in the CMB. On the other hand, if the bifurcat ion takes place in the eternal stage of inflation, it provides an alternative creation mechanism of bubbles universes in eternal inflation, as well as a mechanism to locally terminate eternal inflation , which reduces the measure of eternal inflation. I. INTRODUCTION Inflation [1] has become the leading paradigm for the very early universe. However, the detailed mechanism for inflation still remains unknown. Inspired by the pic- ture of string theory landscape [2], one could expect that the inflationary potential has very complicated structure [3]. Inflation in the string theory landscape has impor- tantimplicationsinbothobservablestageofinflationand eternal inflation. The complicated inflationary potentials in the string theory landscape open up a great number of interest- ing observational effects during observable inflation. Re- searchesinvestigatingthecomplicatedstructureofthein- flationary potential include multi-stream inflation [4, 5], quasi-single field inflation [6], meandering inflation [7], old curvaton [8], etc. Thestringtheorylandscapealsoprovidesaplayground for eternal inflation. Eternal inflation is an very early stage of inflation, during which the universe reproduces itself, so that inflation becomes eternal to the future. Eternal inflation, if indeed happened (for counter ar- guments see, for example [9]), can populate the string theory landscape, providing an explanation for the cos- mological constant problem in our bubble universe by anthropic arguments. In this Letter, we shall focus on the multi-stream infla- tion scenario. Multi-stream inflation is proposed in [4]. And in [5], it is pointed out that the bifurcations can lead to multiverse. Multi-stream inflation assumes that during inflation there exist bifurcation(s) in the inflation trajectory. For example, the bifurcations take place nat- urally in a random potential, as illustrated in Fig. 1. We briefly review multi-stream inflation in Section II. The details of some contents in Section II can be found in [4]. We discuss some new implications of multi-stream inflation for the inflationary multiverse in Section III. ∗wangyi@hep.physics.mcgill.ca FIG. 1. In this figure, we use a tilted random potential to mimic a inflationary potential in the string theory landscap e. One can expect that in such a random potential, bifurcation effects happens generically, as illustrated in the trajecto ries in the figure. FIG. 2. One sample bifurcation in multi-stream inflation. The inflation trajectory bifurcates into AandBwhen the comoving scale k1exits the horizon, and recombines when the comoving scale k2exits the horizon. II. OBSERVABLE BIFURCATIONS In this section, we discuss the possibility that the bi- furcation of multi-stream inflation happens during the observable stage of inflation. We review the production of non-Gaussianities, features and asymmetries [4] in the2 FIG. 3. In multi-stream inflation, the universe breaks up into patches with comoving scale k1. Each patch experienced inflation either along trajectories AorB. These different patches can be responsible for the asymmetries in the CMB. CMB, and investigate some other possible observational effects. To be explicit, we focus on one single bifurcation, as illustrated in Fig. 2. We denote the initial (before bifur- cation) inflationary direction by ϕ, and the initial isocur- vature direction by χ. For simplicity, we let χ= 0 before bifurcation. When comoving wave number k1exits the horizon, the inflation trajectory bifurcates into Aand B. When comoving wave number k2exits the horizon, the trajectories recombines into a single trajectory. The universe breaks into of order k1/k0patches (where k0de- notes the comoving scale of the current observable uni- verse), each patch experienced inflation either along tra- jectories AorB. The choice of the trajectories is made by the isocurvature perturbation δχat scale k1. This picture is illustrated in Fig. 3. We shall classify the bifurcation into three cases: Symmetric bifurcation . If the bifurcation is symmetric, in other words, V(ϕ,χ) =V(ϕ,−χ), then there are two potentially observable effects, namely, quasi-single field inflation, and a effect from a domain-wall-like objects, which we call domain fences. As discussed in [4], the discussion of the bifurcation effect becomes simpler when the isocurvature direction has mass of order the Hubble parameter. In this case, except for the bifurcation and recombination points, tra- jectoryAand trajectory Bexperience quasi-single field inflation respectively. As there are turnings of these tra- jectories, the analysis in [6] can be applied here. The perturbations, especially non-Gaussianities in the isocur- vature directions are projected onto the curvature direc- tion, resultingin a correctionto the powerspectrum, and potentially large non-Gaussianities. As shown in [6], the amount of non-Gaussianity is of order fNL∼P−1/2 ζ/parenleftbigg1 H∂3V ∂χ3/parenrightbigg/parenleftBigg˙θ H/parenrightBigg3 , (1) whereθdenotes the angle between the true inflation di- rection and the ϕdirection. As shown in Fig. 3, the universe is broken into patches during multi-stream inflation. There arewall-likebound- aries between these patches. During inflation, theseboundaries are initially domain walls. However, after the recombination of the trajectories, the tensions of these domain walls vanish. We call these objects domain fences. As is well known, domain wall causes disasters in cosmology because of its tension. However, without tension, domain fence does not necessarily cause such disasters. It is interesting to investigate whether there are observational sequences of these domain fences. Nearly symmetric bifurcation If the bifurcation is nearly symmetric, in other words, V(ϕ,χ)≃V(ϕ,−χ), but not equal exactly, which can be achieved by a spon- taneous breaking and restoring of an approximate sym- metry, then besides the quasi-single field effect and the domain fence effect, there will be four more potentially observable effects in multi-stream inflation, namely, the features and asymmetries in CMB, non-Gaussianity at scalek1and squeezed non-Gaussianity correlating scale k1and scale kwithk1< k < k 2. The CMB power asymmetries are produced because, as in Fig. 3, patches coming from trajectory AorBcan have different power spectra PA ζandPB ζ, which are de- termined by their local potentials. If the scale k1is near to the scale of the observational universe k0, then multi- stream inflation provides an explanation of the hemi- spherical asymmetry problem [10]. The features in the CMB (here feature denotes extra large perturbation at a single scale k1) are produced as a result of the e-folding number difference δNbetween two trajectories. From the δNformalism, the curvature perturbation in the uniform density slice at scale k1has an additional contribution δζk1∼δN≡ |NA−NB|. (2) These features in the CMB are potentially observable in the future precise CMB measurements. As the addi- tional fluctuation δζk1does not obey Gaussian distribu- tion, there will be non-Gaussianity at scale k1. Finally, there are also correlations between scale k1 and scale kwithk1< k < k 2. This is because the ad- ditional fluctuation δζk1and the asymmetry at scale k are both controlled by the isocurvature perturbation at scalek1. Thus the fluctuations at these two scales are correlated. As estimated in [4], this correlation results in a non-Gaussianity of order fNL∼δζk1 ζk1PA ζ−PB ζ PA ζP−1/2 ζ. (3) Non-symmetric bifurcation If the bifurcation is not symmetric at all, especially with large e-folding number differences (of order O(1) or greater) along different tra- jectories, the anisotropy in the CMB and the large scale structure becomes too large at scale k1. However, in this case, regions with smaller e-folding number will have exponentially small volume compared with regions with larger e-folding number. Thus the anisotropy can behave in the form of great voids. We shall address this issue in more detail in [11]. Trajectories with e-folding number3 difference from O(10−5) toO(1) in the observable stage of inflation are ruled out by the large scale isotropy of the observable universe. At the remainderof this section, we would like to make several additional comments for multi-stream inflation: The possibility that the bifurcated trajectories never re- combine. In this case, one needs to worry about the do- main walls, which do not become domain fence during inflation. These domain walls may eventually become domain fence after reheating anyway. Another prob- lem is that the e-folding numbers along different tra- jectories may differ too much, which produce too much anisotropies in the CMB and the large scale structure. However, similar to the discussion in the case of non- symmetric bifurcation, in this case, the observable effect could become great voids due to a large e-folding number difference. The case without recombination of trajectory also has applications in eternal inflation, as we shall dis- cuss in the next section. Probabilities for different trajectories . In [4], we con- sidered the simple example that during the bifurcation, the inflaton will run into trajectories AandBwith equal probabilities. Actually, this assumption does not need to be satisfied for more general cases. The probability to run into different trajectories can be of the same order of magnitude, or different exponentially. In the latter case, there is a potential barrier in front of one trajec- tory, which can be leaped over by a large fluctuation of theisocurvaturefield. Alargefluctuationoftheisocurva- ture field is exponentially rare, resulting in exponentially different probabilities for different trajectories. The bi- furcation of this kind is typically non-symmetric. Bifurcation point itself does not result in eternal infla- tion. As is well known, in single field inflation, if the inflaton releases at a local maxima on a “top of the hill”, a stage of eternal inflation is usually obtained. However, at the bifurcation point, it is not the case. Because al- though the χdirection releases at a local maxima, the ϕ direction keeps on rolling at the same time. The infla- tiondirectionisacombinationofthesetwodirections. So multi-stream inflation can coexist with eternal inflation, but itself is not necessarily eternal. III. ETERNAL BIFURCATIONS In multi-stream inflation, the bifurcation effect may ei- ther take place at an eternal stage of inflation. In this case, it provides interesting ingredients to eternal infla- tion. These ingredients include alternative mechanism to producedifferentbubble universesandlocalterminations for eternal inflation, as we shall discuss separately. Multi-stream bubble universes . The most discussed mechanisms to produce bubble universes are tunneling processes, such as Coleman de Luccia instantons [12] and Hawking Moss instantons [13]. In these processes, the tunneling events, which are usually exponentially sup- pressed, create new bubble universes, while most parts FIG. 4. Cascade creation of bubble universes. In this figure, we assume trajectory Ais the eternal inflation trajectory, and trajectory Bis the non-eternal inflation trajectory. of the spatial volume remain in the old bubble universe at the instant of tunneling. If bifurcations of multi-stream inflation happen dur- ing eternal inflation, two kinds of new bubble universes can be created with similar probabilities. In this case, at the instant of bifurcation, both kinds of bubble uni- verseshavenearlyequalspatialvolume. Withachangeof probabilities, the measures for eternal inflation should be reconsideredformulti-streamtype bubble creationmech- anism. If the inflation trajectories recombine after a period of inflation, the different bubble universes will eventually have the same physical laws and constants of nature. On the other hand, if the different inflation trajectories do not recombine, then the different bubble universes cre- ated by the bifurcation will have different vacuum ex- pectation values of the scalar fields, resulting to different physical laws or constants of nature. It is interesting to investigate whether the bifurcation effect is more ef- fective than the tunneling effect to populate the string theory landscape. Note that in multi-stream inflation, it is still possi- ble that different trajectorieshaveexponentiallydifferent probabilities, as discussed in the previous section. In this case, multi-stream inflation behaves similar to Hawking Moss instantons during eternal inflation. Local terminations for eternal inflation . It is possible that during multi-stream inflation, a inflation trajectory bifurcates in to one eternal inflation trajectory and one non-eternal inflation trajectory with similar probability. Inthiscase,theinflatonintheeternalinflationtrajectory frequently jumps back to the bifurcation point, resulting in a cascade creation of bubble universes, as illustrated in Fig. 4. This cascade creation of bubble universes, if4 realized, is more efficient in producing reheating bubbles than tunneling effects. Thus it reduces the measure for eternal inflation. There are some other interesting issues for bifurcation in the multiverse. For example, the bubble walls may be observable in the present observable universe, and the bifurcations can lead to multiverse without eternal infla- tion. These possibilities are discussed in [5]. IV. CONCLUSION AND DISCUSSION To conclude, webriefly reviewedmulti-stream inflation during observable inflation. Some new issues such as do-main fences and connection with quasi-single field infla- tion are discussed. We also discussed multi-stream infla- tion in the context of eternal inflation. The bifurcation effect in multi-stream inflation provides an alternative mechanism for creating bubble universes and populating the string theory landscape. The bifurcation effect also provides a very efficient mechanism to locally terminate eternal inflation. ACKNOWLEDGMENT We thank Yifu Cai for discussion. This work was sup- ported by NSERC and an IPP postdoctoral fellowship. [1] A. H. Guth, Phys. Rev. D 23, 347 (1981). A. D. Linde, Phys. Lett. B 108, 389 (1982). A. J. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. 48, 1220 (1982). [2] R. Bousso and J. Polchinski, JHEP 0006, 006 (2000) [arXiv:hep-th/0004134]. S. B. Giddings, S. Kachru and J. Polchinski, Phys. Rev. D 66, 106006 (2002) [arXiv:hep-th/0105097]. S. Kachru, R. Kallosh, A. Linde and S. P. Trivedi, Phys. Rev. D 68, 046005 (2003) [arXiv:hep-th/0301240]. M. R. Douglas, JHEP 0305, 046 (2003) [arXiv:hep-th/0303194]. [3] Q. G. Huang and S. H. Tye, arXiv:0803.0663 [hep-th]. [4] M. Li and Y. Wang, JCAP 0907, 033 (2009) [arXiv:0903.2123 [hep-th]]. [5] S. Li, Y. Liu and Y. S. Piao, arXiv:0906.3608 [hep-th]. [6] X. Chen and Y. Wang, arXiv:0909.0496 [astro-ph.CO]. X. Chen and Y. Wang, arXiv:0911.3380 [hep-th]. [7] S. H. Tye and J. Xu, arXiv:0910.0849 [hep-th]. [8] J. O. Gong and M. Sasaki, JCAP 0901, 001 (2009) [arXiv:0804.4488 [astro-ph]]. J. O. Gong, C. Lin andY. Wang, arXiv:0912.2796 [astro-ph.CO]. [9] V. F. Mukhanov, L. R. W. Abramo and R. H. Bran- denberger, Phys. Rev. Lett. 78, 1624 (1997) [arXiv:gr-qc/9609026]. Y. F. Cai and Y. Wang, JCAP0706, 022 (2007) [arXiv:0706.0572 [hep-th]]. Q. G. Huang, M. Li and Y. Wang, JCAP 0709, 013 (2007) [arXiv:0707.3471 [hep-th]]. Y. Wang, arXiv:0805.4520 [hep-th]. [10] H. K. Eriksen, F. K. Hansen, A. J. Banday, K. M. Gorski and P. B. Lilje, Astrophys. J. 605, 14 (2004) [Erratum- ibid.609, 1198 (2004)] [arXiv:astro-ph/0307507]. A. L. Erickcek, M. Kamionkowski and S. M. Carroll, Phys. Rev. D 78, 123520 (2008) [arXiv:0806.0377 [astro-ph]]. [11] N. Afshordi, A. Slosar and Y. Wang, in preparation. [12] S. R. Coleman and F. De Luccia, Phys. Rev. D 21, 3305 (1980). [13] S. W. Hawking and I. G. Moss, Phys. Lett. B 110, 35 (1982). |