Primordial black holes as dark matter and generators of cosmic structure
PPrimordial black holes as dark matter andgenerators of cosmic structure
Bernard Carr
School of Physics and Astronomy, Queen Mary University of London
Abstract.
Primordial black holes (PBHs) could provide the dark mat-ter but a variety of constraints restrict the possible mass windows to10 − g, 10 − g and 10 − M (cid:12) . The last possibility is ofspecial interest in view of the recent detection of black-hole mergersby LIGO. PBHs larger than 10 M (cid:12) might have important cosmologicalconsequences even if they have only a small fraction of the dark matterdensity. In particular, they could generate cosmological structures eitherindividually through the ‘seed’ effect or collectively through the ‘Poisson’effect, thereby alleviating some problems associated with the standardcold dark matter scenario. Primordial black holes (PBHs) have been a source of interest for nearly 50 years[1], despite the fact that there is still no evidence for them. One reason forthis interest is that only PBHs could be small enough for Hawking radiationto be important [2]. This has not yet been confirmed experimentally and thereremain major conceptual puzzles associated with the process. Nevertheless, thisdiscovery is generally recognised as one of the key developments in 20th centuryphysics because it beautifully unifies general relativity, quantum mechanics andthermodynamics. The fact that Hawking was only led to this discovery throughcontemplating the properties of PBHs illustrates that it can be useful to studysomething even if it does not exist! But, of course, the situation is much moreinteresting if PBHs do exist.PBHs smaller than about 10 g would have evaporated by now with many in-teresting cosmological consequences [3]. Studies of such consequences have placeduseful constraints on models of the early Universe and, more positively, evap-orating PBHs have been invoked to explain certain features: for example, theextragalactic [4,5] and Galactic [6,7] γ -ray backgrounds, antimatter in cosmicrays [8,9], the annihilation line radiation from the Galactic centre [10,11], thereionisation of the pregalactic medium [12,13] and some short-period γ -ray bursts[14,15]. However, there are usually other possible explanations for these features,so there is no definitive evidence for evaporating PBHs. Only the original papersfor each topic are cited here and a more comprehensive list of references can befound in Ref. [3]. a r X i v : . [ a s t r o - ph . C O ] J a n Carr
Attention has therefore shifted to the PBHs larger than 10 g, which areunaffected by Hawking radiation. Such PBHs might have various astrophysicalconsequences, such as providing seeds for the supermassive black holes in galacticnuclei [16,17], the generation of large-scale structure through Poisson fluctuations[18,19] and important effects on the thermal and ionisation history of the Uni-verse [20,21]. Again only the original papers are cited here. But perhaps the mostexciting possibility is that they could provide the dark matter which comprises25% of the critical density [22,23], an idea that goes back to the earliest daysof PBH research [24]. Since PBHs formed in the radiation-dominated era, theyare not subject to the well-known cosmological nucleosynthesis constraint thatbaryons can have at most 5% of the critical density [25]. They should thereforebe classed as non-baryonic and behave like any other form of cold dark matter(CDM).As with other CDM candidates. there is still no compelling evidence thatPBHs provide the dark matter. There have been claims that the microlensingof quasars could indicate dark matter in jupiter-mass PBHs [26] but these arecontroversial. There was also a flurry of excitement about PBHs in 1997, whenthe MACHO microlensing results [27] suggested that the dark matter could bein compact objects of mass 0 . M (cid:12) . Alternative dark matter candidates couldbe excluded and PBHs of this mass might naturally form at the quark-hadronphase transition at 10 − s [28]. Subsequently, however, it was shown that suchobjects could comprise only 20% of the dark matter and indeed the entire massrange 10 − M (cid:12) to 10 M (cid:12) is now excluded from providing the dark matter [29].In recent decades attention has focused on other mass ranges in which PBHscould provide the dark matter and numerous constraints allow only three pos-sibilities: the asteroid mass range (10 – 10 g), the sublunar mass range (10 – 10 g) and the intermediate mass black hole (IMBH) range (10 – 10 M (cid:12) ).There is particular interest in the last possibility because the coalescing blackholes detected by LIGO [30] could be of primordial origin, although this wouldnot necessarily require the PBHs to provide all the dark matter. Also PBHscould have important cosmological consequences even if they provide only asmall fraction of the dark matter, so we explore this possibility below. PBHs could have been produced during the early Universe due to various mech-anisms. Matching the cosmological density at a time t are the big bang withdensity required to form a PBH of mass M implies that the PBH mass is com-parable to the horizon mass at formation [31,32]: M ∼ c tG ∼ (cid:18) t − s (cid:19) g . (1)Hence PBHs could span an enormous mass range: those formed at the Plancktime (10 − s) would have the Planck mass (10 − g), whereas those formed at 1 swould be as large as 10 M (cid:12) . By contrast, black holes forming at the present rimordial black holes 3 epoch (eg. in the final stages of stellar evolution) could never be smaller thanabout 1 M (cid:12) . In some circumstances PBHs may form over an extended period,corresponding to a wide range of masses, but their spectrum could be extendedeven if they form at a single epoch.As discussed in numerous papers, starting in the 1990s [33,34,35,36], thequantum fluctuations arising in various inflationary scenarios are a possiblesource of PBHs. In some of these scenarios the fluctuations generated by inflationare “blue” (i.e. decrease with increasing scale) and this means that the PBHsform shortly after reheating. Others involve some form of “designer” inflation, inwhich the power spectrum of the fluctuations — and hence PBH production —peaks on some scale. In other scenarios, the fluctuations have a “running index”,so that the amplitude increases on smaller scales but not according to a simplepower law. PBH formation may also occur due to some sort of parametric reso-nance effect before reheating, in which case the fluctuations tend to peak on ascale associated with reheating. This is usually very small but several scenariosinvolve a secondary inflationary phase which boosts this scale into the macro-scopic domain. There are too many papers on these topics to cite here but acomprehensive list of references can be found in Ref. [3].Whatever the source of the inhomogeneities, PBH formation would be en-hanced if there was a reduction in the pressure at some epoch - for example,at the QCD era [37,38] or if the early Universe went through a dust-like phaseas a result of being dominated by non-relativistic particles for a period [39,40]or undergoing slow reheating after inflation [41,42]. Another possibility is thatPBHs might have formed spontaneously at some sort of phase transition, even ifthere were no prior inhomogeneities, for example from the collisions of bubblesof broken symmetry [43,44] or the collapse of cosmic strings [45,46] or domainwalls [47,48]. Further references for such models can be found in Ref. [3].The fraction of the mass of the Universe in PBHs is time-dependent but itsvalue at the PBH formation epoch is of particular interest. If the PBHs formedat a redshift z or time t and contribute a fraction f ( M ) of the dark matter onsome mass scale M , then the collapse fraction is [49] β ( M ) = f ( M ) (cid:18) z z eq (cid:19) ∼ − f ( M ) (cid:18) t (cid:19) / ∼ − f ( M ) (cid:18) M g (cid:19) / , (2)where we assume the PBHs form in the radiation-dominated era, z eq ≈ z ) factor arises because the radiation density scales as (1 + z ) , whereas thePBH density scales as (1 + z ) . Any limit on f ( M ) (eg. f ≤ M > g)therefore places a constraint on β ( M ), which is necessarily tiny.On the other hand, one also expects the collapse fraction to be small. Forexample, if the PBHs form from primordial inhomogeneities which are Gaussianwith rms amplitude δ H ( M ) at the horizon epoch, one predicts [49] β ( M ) ≈ erfc (cid:18) δ c δ H (M) (cid:19) , (3) Carr where ‘erfc’ is the complimentary error function and δ c ≈ . β ∼ . δ H ( M ) , corresponding to the probability of sufficient spherical sym-metry, but this is still small [52,53]. In the other scenarios, β depends upon somecosmological parameter (eg. the string tension or bubble formation rate). The constraints on f ( M ), the fraction of the halo in PBHs of mass M , are sum-marised in Fig. 1, which is taken from Ref. [23], although some of them havenow been revised. All the limits assume that the PBHs have a monochromaticmass function and cluster in the Galactic halo in the same way as other formsof CDM. The effects are extragalactic γ -rays from evaporation (EG) [3], fem-tolensing of γ -ray bursts (F) [54], white-dwarf explosions (WD) [55], neutron-starcapture (NS) [56], Kepler microlensing of stars (K) [57], MACHO/EROS/OGLEmicrolensing of stars (ML) [29] and quasar microlensing (ML) [58], survival of astar cluster in Eridanus II (E) [59], wide-binary disruption (WB) [60], dynamicalfriction on halo objects (DF) [61], millilensing of quasars (mLQ) [62], generationof large-scale structure through Poisson fluctuations (LSS) [19], and accretioneffects (WMAP, FIRAS) [21].As indicated by the arrows in Fig. 1 , the permittted mass windows for f ∼ M (cid:12) ); (B) the sublunarmass range (10 – 10 g); and (C) the asteroid mass range (10 – 10 g).However, there are further limits since Fig. 1 was produced and some peopleclaim that even these windows are now excluded. For example, scenario C maybe ruled out by observations of the Galactic γ -ray background [63] or positronflux [64]. One problem with scenario A is that such objects would disrupt widebinaries in the Galactic disc. It was originally claimed that this would excludeobjects above 400 M (cid:12) but more recent studies may reduce this mass [65], so thenarrow window between the microlensing and wide-binary bounds is shrinking.There are new microlensing constraints in the lunar-mass range from the Subarutelescope [66] and in the intermediate mass range from supernovae [67]. Also theCMB accretion constraints have been revised and are now weaker [68], althoughthere are new accretion limits from X-ray observations [69,70]. Two talks at thissymposum imply interesting new constraints associated with tidal streams [71]and lensing substructure [72].The PBHs in either scenario A and B could be generated by inflation buttheorists are split as to which window they favour. For example, Inomata etal. [73] argue that doube inflation can produce a peak at around 10 g, whileClesse and Garcia-Beillido [74] argue that hybrid inflation can produce a peakat around 10 M (cid:12) , this being favored by the LIGO results. A peak at this masscould also be produced by a reduction in the pressure at the quark-hadron phasetransition [38], even if the primordial fluctuations have no feature on that scale.There is a parallel here with the search for particle dark matter, where there isalso a split between groups searching for light and heavy candidates. rimordial black holes 5 - - - - M / g f M / M ⊙ KEG F WDNS ML E WB mLQLSS WMAPFIRASDF
C AB
Fig. 1.
Constraints on f ( M ) from Ref. [23] for a monochromatic mass function from avariety of evaporation (magenta), dynamical (red), lensing (cyan), large-scale structure(green) and accretion (orange) effects. See text for references and more recent limits.The accretion limit is shown with a broken line since this has now been improved. Thearrows indicate the three mass windows where f can be close to 1. The constraints discussed above assume that the PBH mass function ismonochromatic (i.e. with a width ∆M ∼ M ). However, there are many scenar-ios in which one would expect the mass function to be extended. For example.inflation tends to produce a lognormal mass function [75] and critical collapsegenerates an extended low mass tail [76,77]. In the context of the dark-matterproblem, this is a two-edged sword [23]. On the one hand, it means that the total PBH density may suffice to explain the dark matter, even if the density inany particular mass band is small and within the observational bounds discussedabove. On the other hand, even if PBHs can provide all the dark matter at somemass-scale, the extended mass function may still violate the constraints at someother scale. This issue been addressed in a number of recent papers [78,79,80],though with somewhat different conclusions.
PBHs of mass m provide a source of fluctuations for objects of mass M intwo ways: (1) via the seed effect, in which the Coulomb effect of a single blackhole generates an initial density fluctuation m/M ; (2) via the Poisson effect,in which the √ N fluctuation in the number of black holes generates an initialdensity fluctuation ( f m/M ) / for a PBH dark matter fraction f . Both types offluctuations then grow through gravitational instability to bind regions of mass M . Each of these proposals has a long history and detailed references can be Carr found in Ref. [81]. The relationship between the two mechanisms is subtle, sowe will consider both of them below and determine the dominant one for eachmass scale.If the PBHs have a single mass m , the initial fluctuation in the matter densityon a scale M is δ i ≈ m/M (seed) , ( f m/M ) / (Poisson) , (4)where M excludes the radiation content. If PBHs provide the dark matter, f ∼ M but we also consider scenarios with f (cid:28)
1. The Poisson effect then dominates for
M > m/f and the seed effect for
M < m/f . Indeed, the first expression in (4) only applies for f < m/M , sinceotherwise a region of mass M would be expected to contain more than one blackhole. The dependence of δ i on M is indicated in Fig. 2(a). The fluctuation growsas (1 + z ) − from the redshift of matter-radiation equality, z eq ≈ δ ≈
1. Therefore the mass binding at redshift z B is M ≈ mz − B (seed) , f mz − B (Poisson) , (5)as illustrated in Fig. 2(b). The CDM fluctuations are shown for comparison.These always dominate at sufficiently large scales but the PBHs provide anextra peak in the power spectrum on small scales. ! " M | m/f_f (a) | m_1 > M eq | _10 /5 M CDM | m/f3 | | < z B m _ 1 | M o _ M o _M CDM z CDM | _ Fig. 2. (a) Form of initial fluctuation δ i as a function of M for the seed and Poissoneffect with fixed f , the first dominating at small M if f is small but the second alwaysdominating if f ∼
1. (b) Mass M binding at redshift z B for fixed f , the Poisson effectdominating for low z if f is small but at all z if f ∼
1. Also shown by dashed lines arethe forms for δ i and M ( z B ) predicted by the CDM model, this indicating the range M > M
CDM and z B < z CDM for which CDM fluctuations dominate. From Ref. [81].
One can place interesting upper limits on f ( m ) by requiring that varioustypes of structure do not form too early [81]. One can also take a more positiveapproach, exploring the possibility that PBHs may have helped the formation ofthese objects, thereby complementing the standard CDM scenario of structureformation. If the PBHs have a monochromatic mass function and provide all the rimordial black holes 7 dark matter ( f ∼ m < M (cid:12) . This implies thatPBHs can only bind subgalactic masses but still allows them to play a role inproducing the first bound baryonic clouds or the SMBHs which power quasars.For f (cid:28)
1, the seed effect dominates on small scales and can bind a regionof up to 4000 times the PBH mass. It is known that most galaxies containcentral supermassive black holes with a mass proportional to the bulge mass[82] and this correlation is naturally explained by the seed effect if the blackholes are primordial - rather than forming after galaxies - with an extended massfunction. However, limits on the µ -distortion in the CMB due to the dissipationof fluctuations before decoupling exclude many PBHs larger than 10 M (cid:12) unlessone invokes non-Gaussian fluctuations or accretion [83]. The proposal that the dark matter could comprise PBHs in the intermediatemass range has attracted much attention recently as a result of the LIGO de-tections of merging binary black holes with mass around 30 M (cid:12) [84,85,86,87].Since the black holes are larger than initially expected, it has been suggestedthat they could represent a new population. One possibility is that they were ofPopulation III origin (i.e. forming between decoupling and galaxies). The sugges-tion that LIGO might detect gravitational waves from coalescing intermediatemass Population III black holes was first made more than 30 years ago [88] and -rather remarkably - Kinugawa et al. predicted a Population III coalescence peakat 30 M (cid:12) shortly before the first LIGO detection [89].Another possibility - more relevant to the present considerations - is thatthe LIGO black holes were primordial, as first discussed in Ref. [90]. This doesnot necesarily require the PBHs to provide all the dark matter. While severalauthors have made this connection [91,92], the predicted merger rate depends onwhen the binaries form and uncertain astrophysical factors, so others argue thatthe dark matter fraction could be small [93,94,95]. Indeed the LIGO results havebeen used to constrain the PBH dark matter fraction [96,97]. Note that the PBHdensity should peak at a lower mass than the coalescence signal for an extendedPBH mass function, since the gravitational wave amplitude scales as the blackhole mass. Indeed, Clesse & Garcia-Bellido argue that a lognormal distributioncentred at around 3 M (cid:12) would naturally explains both the dark matter and theLIGO bursts without violating any of the current PBH constraints [92].A population of massive PBHs would also be expected to generate a stochas-tic background of gravitational waves [98], whether or not they form binaries. Ifthe PBHs have an extended mass function, incorporating both dark matter atthe low end and galactic seeds at the high end, this would have important impli-cations for the predicted gravitational wave background. Theorists usually focuson the gravitational waves generated by either stellar black holes (detectableby LIGO) or supermassive black holes (detectable by LISA). However, with anextended PBH mass function, the gravitational wave background should encom- Carr pass both these limits and also every intermediate frequency. If the PBHs formfrom scalar perturbations of inflatioary origin, there is also a gravitational wavebackground due to associated second-order tensor perturbations and this givesvery strong potential limits on f ( M ) [99,100,101]. In recent years PBHs have been invoked for three purposes: (1) to explain thedark matter; (2) to provide a source of LIGO coalescences; (3) to alleviate someof the problems associated with the CDM scenario. In principle, these are distinctroles and any one of them would justify the study of PBHs. On the other hand,if PBHs have an extended mass function, they could play more than one or evenall these roles.As regards (1), there are only a few mass ranges in which PBHs could providethe dark matter. We have focused particularly on the intermediate mass range10 M (cid:12) < M < M (cid:12) , because this may be relevant to (2), but the sublunarrange 10 – 10 g also remains viable. The asteroid range 10 – 10 g isprobably the least plausible. We have not discussed the possibility that stablePlanck-mass relics of PBH evaporations provide the dark matter [102]. Thisscenario cannot be excluded but it is impossible to test since Planck-scale relicswould be undetectable except through their gravitational effects.Presumably most participants at this meeting would prefer the dark matterto be elementary particles rather than PBHs, so it may be reassuring that formost of the last 50 years the study of PBHs has been a minority interest. On theother hand, as illustrated in Fig. 3, PBHs have become increasingly popular inrecent years, at least as measured by the annual publication rate on the topic.Indeed, turning to role (3), perhaps the most important point to emphasize,is that PBHs in the intermediate to supermassive mass range could play animportant cosmological role even if the do not provide the dark matter. Perhapsthis also applies for the particle candidates. Few people would now argue thatneutrinos provide the dark matter but they are still extraordinarily important. Acknowledgments
This talk is dedicated to the memory of my friend and mentor Stephen Hawking.If PBHs turn out to exist, then his pioneering work on this topic will have beenone of his most prescient and important scientific contributions. I thank theSimons Foundation for their generous hospitality at this conference and mymany PBH coathors over 45 years for an enjoyable collaboration.
References
1. Ya.B. Zel’dovich, I. Novikov, Sov. Astron. , 602 (1967)2. S.W. Hawking, Nature , 30 (1974). DOI 10.1038/248030a0rimordial black holes 9 |1982 |1993 |1997 |1999 |2005 |2010 PBHs of M~10 -3 M form at quark-hadron era Crawford & Schramm
Microlensing of QSOs è M>10 -3 M O Hawkins
6y MACHO results è M>0.5M O Alcock et al
PBHs of M~0.5M form at quark-hadron era Jedamizk & Nemeyer,
Microlensing constraints
Hamadache et al P O P U L A R I T Y PRIMORDIAL BLACK HOLEs = PBHs |1971
PBHs form from inhomogeneities
Hawking, Carr
Dark matter in Planck relicsor sublunar or IMBHs
Dynamical/accretionlimits exclude
LIGO
Fig. 3.
History of popularity of PBHs, as indicated by publication rate (top left).3. B.J. Carr, K. Kohri, Y. Sendouda, J. Yokoyama, Phys. Rev.
D81 , 104019 (2010).DOI 10.1103/PhysRevD.81.1040194. D.N. Page, S.W. Hawking, Astrophys. J. , 1 (1976). DOI 10.1086/1543505. B.J. Carr, Astrophys. J. , 8 (1976). DOI 10.1086/1543516. E.L. Wright, Astrophys. J. , 487 (1996). DOI 10.1086/1769107. R. Lehoucq, M. Casse, J.M. Casandjian, I. Grenier, Astron. Astrophys. , 37(2009). DOI 10.1051/0004-6361/2009119618. P. Kiraly, J. Szabelski, J. Wdowczyk, A.W. Wolfendale, Nature , 120 (1981).DOI 10.1038/293120a09. J.H. MacGibbon, B.J. Carr, Astrophys. J. , 447 (1991). DOI 10.1086/16990910. P.N. Okele, M.J. Rees, Astron. Astrophys. , 263 (1980)11. C. Bambi, A.D. Dolgov, A.A. Petrov, Phys. Lett. B670 , 174 (2008). DOI10.1016/j.physletb.2009.10.053,10.1016/j.physletb.2008.10.057. [Erratum: Phys.Lett.B681,504(2009)]12. M. Gibilisco, in
General Relativity and Gravitational Physics , ed. by M. Bassan,V. Ferrari, M. Francaviglia, F. Fucito, I. Modena (1997), p. 41313. K.M. Belotsky, A.A. Kirillov, JCAP (01), 041 (2015). DOI 10.1088/1475-7516/2015/01/04114. A.A. Belyanin, V.V. Kocharovsky, V.V. Kocharovsky, Mon. Not. Roy. Astron.Soc. , 626 (1996). DOI 10.1093/mnras/283.2.62615. D.B. Cline, D.A. Sanders, W. Hong, Astrophys. J. , 169 (1997). DOI 10.1086/30448016. B.J. Carr, M.J. Rees, Mon. Not. Roy. Astron. Soc. , 801 (1984)17. R. Bean, J. Magueijo, Phys. Rev.
D66 , 063505 (2002). DOI 10.1103/PhysRevD.66.06350518. P. M´esz´aros, Astron. Astrophys. , 5 (1975)19. N. Afshordi, P. McDonald, D.N. Spergel, Astrophys. J. Lett. , L71 (2003).DOI 10.1086/3787630 Carr20. B.J. Carr, Mon. Not. Roy. Astron. Soc. , 639 (1981). DOI 10.1093/mnras/194.3.63921. M. Ricotti, J.P. Ostriker, K.J. Mack, Astrophys. J. , 829 (2008). DOI 10.1086/58783122. P.H. Frampton, Mod. Phys. Lett. A31 (16), 1650093 (2016). DOI 10.1142/S021773231650093023. B. Carr, F. Kuhnel, M. Sandstad, Phys. Rev.
D94 (8), 083504 (2016). DOI10.1103/PhysRevD.94.08350424. G.F. Chapline, Nature , 251 (1975). DOI 10.1038/253251a025. R.H. Cyburt, B.D. Fields, K.A. Olive, Phys. Lett.
B567 , 227 (2003). DOI10.1016/j.physletb.2003.06.02626. M.R.S. Hawkins, Nature , 242 (1993). DOI 10.1038/366242a027. C. Alcock, et al., Astrophys. J. , 697 (1997). DOI 10.1086/30453528. K. Jedamzik, Phys. Rept. , 155 (1998). DOI 10.1016/S0370-1573(98)00067-229. P. Tisserand, et al., Astron. Astrophys. , 387 (2007). DOI 10.1051/0004-6361:2006601730. B.P. Abbott, et al., Phys. Rev. Lett. (6), 061102 (2016). DOI 10.1103/PhysRevLett.116.06110231. S. Hawking, Mon. Not. Roy. Astron. Soc. , 75 (1971)32. B.J. Carr, S.W. Hawking, Mon. Not. Roy. Astron. Soc. , 399 (1974)33. B.J. Carr, J.H. Gilbert, J.E. Lidsey, Phys. Rev.
D50 , 4853 (1994). DOI 10.1103/PhysRevD.50.485334. P. Ivanov, P. Naselsky, I. Novikov, Phys. Rev.
D50 , 7173 (1994). DOI 10.1103/PhysRevD.50.717335. L. Randall, M. Soljacic, A.H. Guth, Nucl. Phys.
B472 , 377 (1996). DOI 10.1016/0550-3213(96)00174-536. J. Garcia-Bellido, A.D. Linde, D. Wands, Phys. Rev.
D54 , 6040 (1996). DOI10.1103/PhysRevD.54.604037. M. Crawford, D.N. Schramm, Nature , 538 (1982). DOI 10.1038/298538a038. C.T. Byrnes, M. Hindmarsh, S. Young, M.R.S. Hawkins, JCAP (08), 041(2018). DOI 10.1088/1475-7516/2018/08/04139. A.G. Polnarev, M.Y. Khlopov, Sov. Astron. , 391 (1982)40. B. Carr, T. Tenkanen, V. Vaskonen, Phys. Rev. D96 (6), 063507 (2017). DOI10.1103/PhysRevD.96.06350741. M.Y. Khlopov, B.A. Malomed, Y.B. Zeldovich, Mon. Not. Roy. Astron. Soc. ,575 (1985)42. B. Carr, K. Dimopoulos, C. Owen, T. Tenkanen, Phys. Rev.
D97 (12), 123535(2018). DOI 10.1103/PhysRevD.97.12353543. H. Kodama, M. Sasaki, K. Sato, Prog. Theor. Phys. , 1979 (1982). DOI10.1143/PTP.68.197944. S.W. Hawking, I.G. Moss, J.M. Stewart, Phys. Rev. D26 , 2681 (1982). DOI10.1103/PhysRevD.26.268145. S.W. Hawking, Phys. Lett.
B231 , 237 (1989). DOI 10.1016/0370-2693(89)90206-246. A. Polnarev, R. Zembowicz, Phys. Rev.
D43 , 1106 (1991). DOI 10.1103/PhysRevD.43.110647. S.G. Rubin, A.S. Sakharov, M.Y. Khlopov, J. Exp. Theor. Phys. , 921 (2001).DOI 10.1134/1.138563148. V. Dokuchaev, Y. Eroshenko, S. Rubin, Grav. Cosmol. , 99 (2005)49. B.J. Carr, Astrophys. J. , 1 (1975). DOI 10.1086/15385350. I. Musco, J.C. Miller, L. Rezzolla, Class. Quant. Grav. , 1405 (2005). DOI10.1088/0264-9381/22/7/013rimordial black holes 1151. T. Harada, C.M. Yoo, K. Kohri, Phys. Rev. D88 (8), 084051 (2013). DOI10.1103/PhysRevD.88.084051,10.1103/PhysRevD.89.029903. [Erratum: Phys.Rev.D89,no.2,029903(2014)]52. M.Y. Khlopov, A.G. Polnarev, Phys. Lett.
B97 , 383 (1980). DOI 10.1016/0370-2693(80)90624-353. T. Harada, C.M. Yoo, K. Kohri, K.I. Nakao, Phys. Rev.
D96 (8), 083517 (2017).DOI 10.1103/PhysRevD.96.08351754. A. Barnacka, J.F. Glicenstein, R. Moderski, Phys. Rev.
D86 , 043001 (2012). DOI10.1103/PhysRevD.86.04300155. P.W. Graham, S. Rajendran, J. Varela, Phys. Rev.
D92 (6), 063007 (2015). DOI10.1103/PhysRevD.92.06300756. F. Capela, M. Pshirkov, P. Tinyakov, Phys. Rev.
D87 (12), 123524 (2013). DOI10.1103/PhysRevD.87.12352457. K. Griest, A.M. Cieplak, M.J. Lehner, Astrophys. J. (2), 158 (2014). DOI10.1088/0004-637X/786/2/15858. E. Mediavilla, J.A. Munoz, E. Falco, V. Motta, E. Guerras, H. Canovas,C. Jean, A. Oscoz, A.M. Mosquera, Astrophys. J. , 1451 (2009). DOI10.1088/0004-637X/706/2/145159. T.D. Brandt, Astrophys. J. (2), L31 (2016). DOI 10.3847/2041-8205/824/2/L3160. D.P. Quinn, et al., Mon. Not. Roy. Astron. Soc. Lett. , L11 (2009). DOI10.1111/j.1745-3933.2009.00652.x61. B.J. Carr, M. Sakellariadou, Astrophys. J. , 195 (1999). DOI 10.1086/30707162. P.N. Wilkinson, et al., Phys. Rev. Lett. , 584 (2001). DOI 10.1103/PhysRevLett.86.58463. B.J. Carr, K. Kohri, Y. Sendouda, J. Yokoyama, Phys. Rev. D94 (4), 044029(2016). DOI 10.1103/PhysRevD.94.04402964. M. Boudaud, M. Cirelli, (2018)65. M.A. Monroy-Rodr´ıguez, C. Allen, Astrophys. J. (2), 159 (2014). DOI 10.1088/0004-637X/790/2/15966. H. Niikura, M. Takada, N. Yasuda, R.H. Lupton, T. Sumi, S. More, A. More,M. Oguri, M. Chiba, (2017)67. M. Zumalacarregui, U. Seljak, (2017)68. Y. Ali-Hamoud, M. Kamionkowski, Phys. Rev.
D95 (4), 043534 (2017). DOI10.1103/PhysRevD.95.04353469. V. Poulin, P.D. Serpico, F. Calore, S. Clesse, K. Kohri, Phys. Rev.
D96 (8), 083524(2017). DOI 10.1103/PhysRevD.96.08352470. Y. Inoue, A. Kusenko, JCAP (10), 034 (2017). DOI 10.1088/1475-7516/2017/10/03471. J. Bovy, D. Erkal, J.L. Sanders, Mon. Not. Roy. Astron. Soc. (1), 628 (2017).DOI 10.1093/mnras/stw306772. Y.D. Hezaveh, et al., Astrophys. J. (1), 37 (2016). DOI 10.3847/0004-637X/823/1/3773. K. Inomata, M. Kawasaki, K. Mukaida, Y. Tada, T.T. Yanagida, Phys. Rev.
D96 (4), 043504 (2017). DOI 10.1103/PhysRevD.96.04350474. S. Clesse, J. Garca-Bellido, Phys. Rev.
D92 (2), 023524 (2015). DOI 10.1103/PhysRevD.92.02352475. A. Dolgov, J. Silk, Phys. Rev.
D47 , 4244 (1993). DOI 10.1103/PhysRevD.47.424476. J. Yokoyama, Phys. Rev.
D58 , 107502 (1998). DOI 10.1103/PhysRevD.58.10750277. I. Musco, J.C. Miller, Class. Quant. Grav. , 145009 (2013). DOI 10.1088/0264-9381/30/14/1450092 Carr78. A.M. Green, Phys. Rev. D94 (6), 063530 (2016). DOI 10.1103/PhysRevD.94.06353079. B. Carr, M. Raidal, T. Tenkanen, V. Vaskonen, H. Veerme, Phys. Rev.
D96 (2),023514 (2017). DOI 10.1103/PhysRevD.96.02351480. F. K¨uhnel, K. Freese, Phys. Rev.
D95 (8), 083508 (2017). DOI 10.1103/PhysRevD.95.08350881. B. Carr, J. Silk, Mon. Not. Roy. Astron. Soc. (3), 3756 (2018). DOI 10.1093/mnras/sty120482. A.E. Reines, M. Volonteri, Astrophys. J. , 82 (2015). DOI 10.1088/0004-637X/813/2/8283. T. Nakama, B. Carr, J. Silk, Phys. Rev.
D97 (4), 043525 (2018). DOI 10.1103/PhysRevD.97.04352584. B.P. Abbott, et al., Phys. Rev. X6 (4), 041015 (2016). DOI 10.1103/PhysRevX.6.04101585. B.P. Abbott, et al., Phys. Rev. Lett. (24), 241102 (2016). DOI 10.1103/PhysRevLett.116.24110286. B.P. Abbott, et al., Phys. Rev. Lett. (24), 241103 (2016). DOI 10.1103/PhysRevLett.116.24110387. B.P. Abbott, et al., (2018)88. J.R. Bond, B.J. Carr, Mon. Not. Roy. Astron. Soc. , 585 (1984). DOI 10.1093/mnras/207.3.58589. T. Kinugawa, K. Inayoshi, K. Hotokezaka, D. Nakauchi, T. Nakamura, Mon. Not.Roy. Astron. Soc. (4), 2963 (2014). DOI 10.1093/mnras/stu102290. T. Nakamura, M. Sasaki, T. Tanaka, K.S. Thorne, Astrophys. J. Lett. , L139(1997). DOI 10.1086/31088691. S. Bird, I. Cholis, J.B. Mu˜noz, Y. Ali-Ha¨ımoud, M. Kamionkowski, E.D. Kovetz,A. Raccanelli, A.G. Riess, Phys. Rev. Lett. (20), 201301 (2016). DOI 10.1103/PhysRevLett.116.20130192. S. Clesse, J. Garca-Bellido, Phys. Dark Univ. , 142 (2017). DOI 10.1016/j.dark.2016.10.00293. M. Sasaki, T. Suyama, T. Tanaka, S. Yokoyama, Phys. Rev. Lett. (6), 061101(2016). DOI 10.1103/PhysRevLett.121.059901,10.1103/PhysRevLett.117.061101.[Erratum: Phys. Rev. Lett.121,no.5,059901(2018)]94. T. Nakamura, et al., PTEP (9), 093E01 (2016). DOI 10.1093/ptep/ptw12795. M. Sasaki, T. Suyama, T. Tanaka, S. Yokoyama, Class. Quant. Grav. (6),063001 (2018). DOI 10.1088/1361-6382/aaa7b496. M. Raidal, V. Vaskonen, H. Veerme, JCAP , 037 (2017). DOI 10.1088/1475-7516/2017/09/03797. Y. Ali-Hamoud, E.D. Kovetz, M. Kamionkowski, Phys. Rev. D96 (12), 123523(2017)98. B.J. Carr, Astron. Astrophys. , 6 (1980)99. R. Saito, J. Yokoyama, Prog. Theor. Phys. , 867 (2010). DOI 10.1143/PTP.126.351,10.1143/PTP.123.867. [Erratum: Prog. Theor. Phys. , 351 (2011)]100. E. Bugaev, P. Klimai, Phys. Rev. D83 , 083521 (2011). DOI 10.1103/PhysRevD.83.083521101. N. Bartolo, V. De Luca, G. Franciolini, M. Peloso, D. Racco, A. Riotto, (2018)102. J.H. MacGibbon, Nature329