Primordial Black Holes as Seeds for Cosmic Structures

# Primordial Black Holes as Seeds for Cosmic Structures

Bernard Carr School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London E1 4NS, UK    Joseph Silk Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 20218, USA
July 12, 2019
###### Abstract

Primordial black holes (PBHs) could provide the dark matter in various mass windows below and those of might explain the LIGO events. PBHs much larger than this might have important consequences even if they provide only a small fraction of the dark matter. In particular, they could generate cosmological structure through either the ‘seed’ effect or the ‘Poisson’ effect, thereby alleviating some problems associated with the standard CDM scenario. If the PBHs all have a similar mass and make a small contribution to the dark matter, then the seed effect dominates on small scales, in which case PBHs could seed the supermassive black holes in galactic nuclei or even galaxies themselves. If they have a similar mass and provide the dark matter, the Poisson effect dominates on all scales and the first bound clouds would form earlier than in the usual scenario, with interesting observational consequences. If the PBHs have an extended mass spectrum, which is more likely, they could fulfill all three roles – providing the dark matter, binding the first bound clouds and generating galaxies. In this case, the galactic mass function naturally has the observed form, with the galaxy mass being simply related to the black hole mass. The stochastic gravitational wave background from the PBHs in this scenario would extend continuously from the LIGO frequency to the LISA frequency, offering a potential goal for future surveys.

###### pacs:
04.70.Bw, 97.60.Lf, 95.35.+d

## I Introduction

The standard Cold Dark Matter (CDM) scenario is characterised by two assumptions: the dark matter comprises some form of weakly interacting massive particle (WIMP); and cosmic structures – from the first bound clouds through galaxies to clusters of galaxies – form from initial inhomogeneities through a process of hierarchical build-up. However, both these assumption may be questioned. After many decades of searching, there is still no evidence for WIMPs, either from accelerator experiments or from dark matter searches divalentino (), and simulations of structure formation in the CDM scenario reveal several well-known problems on the scale of galaxies, including missing satellites, cores versus cusps, too big-to-fail, frequency of ultra-diffuse galaxies and the baryon fraction CDM (). Another problem is that some observational anomalies may require the existence of non-linear structures early in the history of the Universe dolgov (). In particular, it is now known that most galactic nuclei contain supermassive black holes (SMBHs), with mass extending from around to and already in place by a redshift of about kormendy (). These SMBHs are usually assumed to form as a result of dynamical processes after galaxy formation but it may be hard to explain how they could have formed so early in the standard picture, especially in dwarf galaxies silk2017 ().

In this paper we point out that many of these problems may be solved by invoking a population of primordial black holes (PBHs) which formed in the early Universe ch1974 (). This view has also been advocated in a series of papers by Clesse and Garcia-Bellido clesse (); gb2017 (); cgb2017 (). First, there are general arguments that PBHs rather than WIMPs may provide the dark matter. This is because the density of such black holes is not constrained by the limits on the baryonic density implied by big bang nucleosynthesis (BBNS), so they would be natural CDM candidates. Furthermore, this has the advantage that – unlike the situation for WIMPs or other particle candidates – there is no need to invoke new physics frampton (). The PBH dark matter proposal has been emphasized from the earliest days of PBH research chapline (); carr1975 () but it has become particularly popular recently cks (); chapfram () - especially since the discovery of black hole coalesences by LIGO bird (); clesse2 (), although this may only require a small fraction of the dark matter to be in PBHs sasaki (). However, there are only a few permissible mass windows in which PBHs could contribute significantly to the dark matter cksy (). The most interesting for present considerations is the intermediate mass range () but there are also windows in the lunar-mass (g) and the asteroid-mass (g) ranges.

Second, and most relevant to the considerations of this paper, there are various ways in which sufficiently massive PBHs could affect the development of large-scales structure and thus help resolve the problems of the CDM scenario. For example, sufficiently large PBHs might grow enough through accretion to seed the SMBHs which reside in AGN bean (); clesse (); habouzit (). Or if the SMBHs are themselves primordial, they might play a role in generating galaxies, either on account of their gravitational Coulomb effect hoyle () or through the Poisson fluctuations in their number density Meszaros (). In the latter case, they would need to have an initial mass of at least but their contribution to the dark matter density need only be . Such large PBHs could have other interesting observational consequences. For example, they could allow the first baryonic clouds to bind earlier than usual, with important implications for observations in the dark ages, such as the generation of an infrared background kash (). This could also modify baryonic feedback in dwarf galaxies silk () and have other knock-on effects for the development of cosmic structure.

All these features apply within the standard models of particle physics and cosmology, so this proposal should be regarded as complementing the CDM scenario rather than rivalling it. One just needs to invoke extra non-Gaussian power on scales well below those observable in the CMB or galaxy surveys. Indeed, this illustrates an important principle: one expects the first bound objects to be much smaller than galaxies in most cosmological scenarios and - as discussed by Carr & Rees carr-rees () even before the advent of the CDM scenario - many astrophysical processes associated with these objects could generate larger scale density fluctuations. Thus structure on the scale of galaxies and clusters need not derive entirely from primordial fluctuations.

But is the existence of such huge PBHs plausible? A PBH forming at a time after the big bang would have a mass of order the particle horizon size , so this depends on how late they can form. It is sometimes argued that this should be before weak freeze-out at  s, corresponding to a maximum mass of . This is because PBH production usually requires large inhomogeneities, which might be expected to disturb the usual BBNS scenario. However, this argument is not clear-cut because the fraction of the universe in PBHs at a time after the Big Bang is only , where is the current PBH density in units of the critical density carr1975 (), so this would be at most at weak freeze-out. Therefore it is not clear that this disturbs BBNS, although it does require fine-tuning of the collapse fraction.

Even if the fornation of such large PBHs is not precluded, could they be expected to form? As reviewed in Ref. carr (), PBHs may be generated by three mechanisms: through some form of cosmological phase transition, through a temporary softening of the equation of state or through the collapse of large inhomogeneities. The first two mechanisms are unlikely to be relevant after  s but the third one could be. For example, hybrid inflation could produce a spike in the power-spectrum of density fluctuations at a mass-scale which is essentially arbitrary gb1996 (). Many people have argued for a spike or non-Gaussianity in the intermediate mass range () in order to explain the dark matter with PBHs byrnes2012 (); frametal (); morales1 (); morales2 (); motohashi ().

The proposals that the dark matter comprises PBHs and that supermassive PBHs provide seeds for galaxies are essentially independent, since the mass scales and dark matter fractions are very different. One requires and , for the dark matter but and for galactic seeds. Clearly each scenario is of interest in its own right. However, it is important to note that generic initial conditions of PBH formation suggest that the PBH masses should extend over a wide range cks (), so it is possible that they could serve both functions. This means that one could have a significant density of PBHs well above the mass of those which provide the DM and possibly as large as . This would also have important implications for the existence of a stochastic gravitational wave background clesse2 (). A similar view has been advocated by Clesse and Garcia-Bellido clesse (), whose work very much complements our own.

The plan of this paper is as follows: Sec. II summarizes constraints on the fraction of the dark matter in large PBHs. Sec. III reviews previous work on the expected PBH mass function for various scenarios. Sec. IV discusses the generation of fluctuations by the seed and Poission effect for both a monochromatic and extended PBH mass function, identifying the dominant effect in various astronomical contexts. Sec. V derives constraints on the PBH dark matter fraction in order to avoid cosmic structures forming too early. Sec. VI considers whether the SMBHs in galactic nuclei could be primordial and thus seed galaxies, pointing out that sufficiently large PBHs would swallow their host galaxy entirely. It also considers the effects of PBHs on the formation of the first bounds clouds. Sec. VII discusses the possible gravitational wave background generated by PBHs with an extended mass function. Sec. VIII draws some general conclusions.

## Ii Constraints on massive PBHs

We now briefly review various constraints which can be imposed on the density of PBHs large enough to affect the development of cosmic structures. These constraints have been discussed in many recent works but an up-to-date summary for a monochromatic PBH mass function is shown in Fig. 1. This is part of a figure in Ref. cksy2 (), which provides a comprehensive review of the constraints over a much wider range of masses. All of the constraints come with various caveats but the main message of Fig. 1 is that – while PBHs with a single mass are excluded from providing all the dark matter () over the entire mass range above about – this may not exclude them having the small density () required for cosmic structure effects. One possible exception is the constraint from the -distortion in the CMB expected if PBHs are generated by primordial inhomogeneities on scales which are later dissipated. This may exclude PBHs in the mass range entirely, so we give this special consideration.

The numerous dynamical limits have been discussed by Carr and Sakellariadou  carsak () but are usually dependent on various astrophysical assumptions. They apply providing there is at least one PBH within the Galactic halo, which corresponds to the condition

 f(m)>(m/Mhalo),Mhalo≈3×1012M⊙. (1)

The most stringent limit at low is associated with the disruption of wide binaries in the Galactic disk, these being especially vulnerable to disruption from PBH encounters. By comparing the result of simulations with observations, Yoo et al. yoo () ruled out objects with from providing most of the halo mass. Later Quinn et al. quinn () argued that one of the widest-separation binaries was spurious, leading to the weaker constraint . However, the most recent analysis comes from Monroy-Rodriguez and Allen monroy () and may reduce the limiting mass to around . As a compromise, we take the limit to be and the constraint then becomes

 f(m)<{(m/102M⊙)−1(102M⊙103,M⊙), (2)

the encounters becoming non-impulsive above . Using a similar argument, the survival of globular clusters against tidal disruption by passing PBHs gives a limit

 f(m)<⎧⎪⎨⎪⎩(m/3×104M⊙)−1(3×104M⊙1011M⊙), (3)

although this depends on the mass and the radius of the typical cluster. The last expression just corresponds to condition (1). Related but somewhat stronger constraints are associated with the survival of Segue I kous () and a star cluster in the dwarf galaxy Eridanus II brandt () but these involve less certain astrophysical assumptions. Halo objects will also overheat the stars in the Galactic disc unless one has

 f(m)<{(m/3×106M⊙)−1(m<3×109M⊙),(m/Mhalo)(m>3×109M⊙). (4)

The upper limit of agrees with the more precise calculations by Lacey and Ostriker ostriker (), although they argued that black holes with could explain some features of disc heating. Constraint (4) bottoms out at with a value . On still larger scales, the dynamical friction of the halo will drag black holes into the Galactic nucleus but we do not include this because it is very model-dependent.

Another important constraint comes from accretion effects. PBHs cannot accrete appreciably in the radiation-dominated era but they might still do so in the matter-dominated period after decoupling and a Bondi-type analysis should then apply. The associated accretion and emission of radiation could have an important effect on the thermal history of the Universe, as first analysed by Carr carr81 (). This possibility was investigated in more detail by Ricotti et al. mor (), who studied the effects of such accreting PBHs on the ionisation and temperature evolution of the Universe. The emitted X-rays would produce anisotropies and spectral distortions in the cosmic microwave background (CMB). Using WMAP data to constrain the first, they obtained the constraint:

 f(m)<{(m/30M⊙)−2(30M⊙

The limit flattens off above because the black hole acretion rate then exceeds the Eddington limit. The spectral distortion limit implied by FIRAS data has a similar form but extends down to a lower mass; it excludes above but bottoms out a larger value of . These limits are not shown in Fig. 1 because they are very model-dependent.

Recently the accretion constraints have been reconsidered by several groups, who argue that the limits are weaker than indicated in Ref. mor (). Ali-Haimoud and Kamionkowski ali () calculate the accretion on the assumption that it is suppressed by Compton drag and Compton cooling from CMB photons, allowing for the PBH velocity relative to the background gas. They find the spectral distortions are too small to be detected, while the anisotropy constraints only exclude above . Horowitz horowitz () and Chen et al. chen () perform a similar analysis and obtain upper limits of and , respectively. However, neither of these analyses includes the the flattening of the limit on above some mass due to the accretion rate exceeding the Eddington limit. The CMB anisotropy constraints are even stronger if the PBHs form accretion discs poulin (). Although all these analyses exclude PBHs comprising the dark matter above some critical mass, they do not exclude the small fraction required to seed galaxies. In this context, we note that the fraction of the mass in SMBHs in galactic nuclei today is .

Another interesting limit comes from the dissipation of density fluctuations after s by Silk damping. This results in a -distortion in the CMB spectrum chluba (), leading to an upper limit over the mass range . This limit was first given in Ref. cl (), based on a result in Ref. barrow (), but the limit on is now much stronger. When the PBH formation probability is relatively large, the dispersion of primordial fluctuations is also expected to be large. Silk damping would then produce unacceptably large distortions if dissipated during the redshift interval . In principle, observational limits on the distortions can be translated into upper limits on the PBH abundance. However, this is a limit on the density fluctuations from which the PBHs derive and can only be translated into a limit on the PBHs themselves if one assumes a model for their formation.

If the fluctuations are Gaussian and the PBHs form on the high- tail, as in the simplest scenario carr1975 (), one finds a constraint on in the range kohri (). However, the assumption that the PBHs form on the high- tail of Gaussian density fluctuations may be incorrect. For example, Nakama et al. nsy () have proposed a “patch” model, in which the relationship between the background inhomogeneities and the overdensity in the tiny fraction of the volume which collapses to PBHs is modified. The -distortion constraint could thus be much weaker. One therefore needs to consider the dependence of the distortion limits on the possible non-Gaussianity of primordial fluctuations. A phenomenological description of such non-Gaussianity was introduced in Ref. nakama () and involves a parameter , such that – for a fixed PBH formation probability – the dispersion of the primordial fluctuations becomes smaller as is reduced, thereby reducing the distortion.

Recently Nakama et al. ncs () have calculated the -distortion constraints on , using both the FIRAS limit of planck () and the projected upper limit of from PIXIE abit (). They use Eq. (7) of Ref. nsk () to convert PBH mass to wave-number and Eq. (20) to convert to . The limits are very strong and essentially rule out PBHs over the entire mass range indicated in Fig. 1 unless the primordial fluctuations are highly non-Gaussian. It would therefore be more plausible to invoke smaller PBHs with initial masses of which undergo substantial accretion between the -disortion era and the time of matter-radiation equality,

## Iii The PBH mass function

In many scenarios, one would expect PBHs to form with an extended mass function. This is interesting because it would allow them to play a variety of cosmological roles. In this section, we discuss four such scenarios, with particular regard to the question of whether PBHs could provide both the dark matter and the seeds for cosmic structure. The first assumes that the PBHs form from scale-invariant primordial fluctuations or the collapse of cosmic strings, the second that they form in an early matter-dominated era, the third that they form from initial inhomogeneities of inflationary origin, and the fourth that they form from critical collapse. In each of these cases, we will give the form of the mass function and the relative densities of the PBHs which provide the dark matter and the cosmic seeds. If the SMBHs in galactic nuclei are primordial, observations require the ratio of the densities to be of order .

### iii.1 Collapse from scale-invariant fluctuations or cosmic strimgs

If the PBHs from from scale-invariant fluctuations (i.e. with constant amplitude at the horizon epoch), their mass spectrum should have the power-law form carr1975 ()

 dndm∝m−αwithα=2(1+2γ)1+γ, (6)

where specifies the equation of state () at PBH formation. The exponent arises because the background density and PBH density have different redshift dependencies. The mass function is also proportional to the probability that an overdense region of mass has a size exceeding the Jeans length at maximum expansion, so that it can collapse against the pressure. In this case, should be scale-independent, so if the horizon-scale fluctuations have a Gaussian distribution with dispersion , one expects carr1975 ()

 β≈erfc(δc/δH). (7)

Here erfc is the complimentary error function and is the threshold for PBH formation. A simple analytic argument suggest but more precise arguments – both numerical musco () and analytical harada () – suggest a somewhat larger value. At one time it was argued that the primordial fluctuations would be expected to be scale-invariant hz () but this does not apply in the inflationary scenarios (discussed below). Nevertheless, one would still expect the above equations to apply if the PBHs to form from the collapse of cosmic loops because the collapse probability is then scale-invariant.

One usually assumes , corresponding to , in which case most of the density is in the smallest PBHs and the density of those larger than is

 ρ(m)=∫mmaxmm(dn/dm)dm∝m2−α(mmin

where and are the upper and lower cut-offs for the mass function. If we assume that the PBHs contain a fraction of the dark matter, this implies that the fraction of the DM density in PBHs of mass larger than is

 f(m)≡ρ(m)/ρdm≈fdm(mdm/m)α−2(mmin

where is the mass-scale which contains most of the dark matter. [Alternatively, one could define as the fraction in PBHs in the mass interval (), which is smaller by a factor .] In a radiation-dominated era, which is most likely, and the exponent in Eq. (9) becomes . There is then a simple relationship between the density of the primordial SMBHs, taken to have a mass , and ones which provide the dark matter:

 fsmbh/fdm∼(mdm/msmbh)1/2∼10−4(mdm/10M⊙)1/2(msmbh/109M⊙)−1/2. (10)

If one wants to identify the SMBHs with those in galactic nuclei, this ratio must be around , which requires . In a more general scenario, in which is regarded as a free parameter, unrelated to , one requires .

### iii.2 Collapse in a matter-dominated era

PBHs form more easily if the Universe becomes pressureless (i.e. matter-dominated) for some period. For example, this may arise due to some form of phase transition in which the mass is channeled into non-relativistic particles kp (); pk () or due to slow reheating after inflation malomed (); cgl (). Since the value of in the above analysis is for , one might expect to increases logarithmically with . However, the analysis breaks down in this case because the Jeans length is much smaller than the particle horizon, so pressure does not inhibit collapse. Instead, collapse is prevented by deviations from spherical symmetry and the probabiity of PBH formation can be shown to be

 β(m)=0.02δH(m)5 (11)

and this leads to a mass function

 dndm∝m−2δH(m)5. (12)

The factor was first derived in Refs. kp (); pk () and is in agreement with the more recent analysis of Ref. hyknj (). is small for but much larger than the exponentially suppressed fraction in the radiation-dominated case. If the matter-dominated phase extends from to , PBH formation is enhanced over the mass range

 mmin∼MH(t1)

The lower limit is the horizon mass at the start of matter-dominance and the upper limit is the horizon mass at the epoch when the region which binds at the end of matter-dominance enters the horizon. This scenario has recently been studied in Ref. ctv ()

Since the primordial fluctuations must be approximately scale-invariant (even in the inflationary scenario), is nearly constant, so Eq. (6) applies with . Thus the mass function is uniquely determined by the values of and . Although it could well be extended enough to incorporate both dark matter and SMBH scales, should only have a weak dependence on (logarithmic if is exactly constant). Therefore the ratio woud be too large for PBHs to fulfill both roles. However, we note that in the PBH scenario advocated by Vilenkin et al. vilenkin (), one expects a combination of nass functions of the form (6), with below some critical mass and above it.

### iii.3 Collapse from inflationary fluctuations

If the fluctuations generated by inflation have a blue spectrum (i.e. decrease with increasing scale) and the PBHs form from the high- tail of the fluctuation distribution, then the exponential factor in Eq. (7) might suggest that the PBH mass function should have an exponential upper cut-off at the horizon mass when inflation ends cgl (). This corresponds to the reheat time , which the CMB quadrupole anisotropy requires to exceed s. In this case, should fall off exponentially above the reheat horizon mass, precluding any possibility of PBHs providing both dark matter and SMBHs. However, a more careful analysis gives a different result. If the fluctuations result from a smooth symmetric peak in the inflationary power spectrum, the PBH mass fuction should have the lognormal form

 dndm∝1m2exp[−(logm−logmc)22σ2]. (14)

This was first suggested in Ref. ds () and later in Ref. clesse (). It has been demonstrated both numerically green () and analytically kmrv () for the case in which the slow-roll approximation holds. It is therefore representative of a large class of inflationary scenarios, including the axion-curvaton and running-mass infation models considered in Ref. cks ().

Equation (14) implies that the mass function is symmetric about its peak at and described by two parameters: the mass-scale itself and the width of the distribution . The integrated mass function is

 f(m)=∫mmdndmdm≈erfc(lnm/σ). (15)

As in the first two scenarios, this can explain the dark matter and galactic seeds fairly naturally and it does not require such a broad spread of masses. The ratio can be expressed analytically and it is larger than indicated by the naive argument of Ref. cgl (). However, not all inflationary scenarios produce the mass function (14). Inomata et al. inomata () propose a scenario which combines a broad mass function at low (to explain the dark matter) with a sharp one at high (to explain the LIGO events). On the other hand, one could also envisage a scenario in which the sharp peak is in the SMBH range.

### iii.4 Critical collapse

It is well known that black hole formation is associated with critical phenomena choptuik () and the application of this to PBH formation was first studied in Refs. EC (); Koike (); NJ (). The conclusion was that the mass function still has an upper cut-off at around the horizon mass but there is also a low-mass tail yoko (). If we assume for simplicity that the density fluctuations have a monochromatic power spectrum on some mass scale and identify the amplitude of the density fluctuation when that scale crosses the horizon, , as the control parameter, then the black hole mass is NJ (); EC (); Koike ()

 m=K(δ−δc)c. (16)

Here is the critical density fluctuation required for PBH formation ( in a radiation-dominated era), the exponent has a universal value and . Although the scaling relation (16) is expected to be valid only in the immediate neighborhood of , most black holes should form from fluctuations with this value because the probability distribution function declines exponentially beyond if the fluctuations are blue. Hence it is sensible to calculate the expected mass function of PBHs using Eq. (16). This allows us to estimate the mass function independently of the specific form of the PDF of primordial density fluctuations. A detailed calculation gives the mass function yoko ()

 dndm∝(mγMf)1/c−1exp[−(1−c)(mγMf)1/c], (17)

where

 γ≡(1−cs)c,s=δc/σ,Mf=K (18)

and is the dispersion of . For , this gives

 dndm∝m1.85exp[−s(m/Mf)2.85]. (19)

The function would have a similar form but with an exponent of in the first term. In this case, the PBH density is too concentrated around a single mass to produce both the dark matter and the SMBHs in galactic nuclei. However, the above analysis depends on the assumption that the power spectrum of the primordial fluctuations is monochromatic. As shown in Ref. kuhnel2016 (), for a variety of inflationary models, when a realistic model of the power spectrum underlying PBH production is used, the inclusion of critical collapse can lead to a significant shift, lowering and broadening of the PBH mass spectra – sometimes by several orders of magnitude. Nevertheless, it still seems unlikely that the PBHs can play more than one role.

## Iv Seed versus Poisson fluctuations

PBHs of mass provide a source of fluctuations for objects of mass in two ways: (1) via the seed effect, in which the Coulomb effect of a single black hole binds a larger region; and (2) via the Poisson effect, in which the fluctuation in the number of black holes in the larger region binds it. Both these proposals have a long history, although the early literature tends not to be cited in recent work. The seed mechanism was first proposed by Hoyle and Narlikar hoyle () in the context of the Steady State model and then later by Ryan ryan () and Carr and Rees  carr-rees (). The Poisson mechanism was first suggested by Meszaros Meszaros (), although Carr pointed out that he had overestimated the effect carr1977 (), and it was explored in many subsequent papers freese (); carr-silk (); chisholm (); afshordi (); kash () The relationship between the two mechanisms is subtle, so we will consider both of them in the following discussion and determine the dominant one for each mass scale. We first assume that the PBHs have a monochromatic mass function and then consider the effect of an extended mass function, which is more plausible. Note that the seed need not be a black hole; a bound cluster of smaller objects carr-lacey (); metcalfe () or Ultra Compact Mini Halos (UCMHs) ricotti () would serve equally well. Indeed, the density fluctuations required to form UCMHs would be much smaller, so they would generally be more numerous than PBHs kohri ().

### iv.1 Monochromatic PBH mass function

If the PBHs have a single mass , the initial density fluctuation on a scale is

 δi≈{m/M(seed)(fm/M)1/2(Poisson), (20)

where is the fraction of the dark matter in the PBHs. If PBHs provide the dark matter, and the Poisson effect dominates for all but we also consider scenarios with . The Poisson effect then dominates for and the seed effect for . Indeed, the first expression in (20) only applies for , since otherwise a region of mass would be expected to contain more than one black hole of mass , i.e. the mass bound by a single seed can never exceed because of competition from other seeds. The dependence of on the ratio is indicated in Fig. 3(a).

It should be stressed that the fluctuation does not initially correspond to a fluctuation in the total density because at formation each PBH is surrounded by a region which is underdense in its radiation density. (This was the source of the error in Meszaros’s initial analysis.) However, because the radiation density falls off faster than the black hole density, a fluctuation in the total density does eventually develop and this has amplitude at the horizon epoch. Thereafter one can show Meszaros2 () that the fluctuation evolves as

 δ=δH(1+3ρB(t)2ρr(t))(1+3ρB(tH)2ρr(tH))−1, (21)

where and are the mean black hole and radiation densities, respectively. Therefore the fluctuation is frozen during the radiation-dominated era but it starts growing as from the start of the matter-dominated era. Since this corresponds to a redshift and an overdense region binds when , the mass binding at redshift is

 M≈{4000mz−1B(seed)107fmz−2B(Poisson). (22)

Note that one also expects the peculiar velocity of the PBHs to induce Poisson fluctuations on the scales they can traverse in a cosmological time carr-rees (). In this context, Meszaros considers fluctuations of the form , on the assumption that this corresponds to a situation in which the black holes are distributed on a lattice, with their positions being random only on scales smaller than the lattice. However, in this situation it can be shown that the effective fluctuation is really carr1975 () and this is never important. In fact, the above analysis applies even if there are no peculiar velocities.

In applying Eq. (22), we must first determine which effect dominates and this depends on the dark matter fraction . For a given value of , Eq. (22) and the condition imply that the seed effect dominates for , whereas the Poisson effect dominates for . This condition is easily understood: since fluctuations grow as after , the fraction of the universe in gravitationally bound regions at redshift is . Thus only a small fraction of the universe is bound by the seeds for . In particular, only a small fraction is bound by the seeds at the present epoch for . On the other hand, for the bound fraction at would exceed , so competition between the seeds will reduce the mass of each bound region to at most . But this is precisely the value of above which the Poisson effect dominates.

If is is treated as a free parameter, unconstrained by observations, the dependence of on the redshift is as indicated in Fig. 3(b). However, it is interesting to obtain the constraints on the function implied by the limits on discussed in Sec. II. If the PBHs provide the dark matter (), the Poisson effect always dominates and Eq. (22) and the condition imply . More generally, the wide-binary constraint (2) and the second expession in Eq. (22) imply

 M<⎧⎪⎨⎪⎩107mz−2B(m≲102M⊙)109z−2BM⊙(102M⊙103M⊙), (23)

where the seed effect dominates for

 zB>{104(m/104M⊙)−1(102M⊙103M⊙). (24)

The last expression in Eq. (23) can be large if is but – unless one invokes highly non-Gaussian fluctuations or appreciable PBH accretion in the radiation-dominated era — the -distortion upper limit on of implies . The combined constraints on for different values are indicated in Fig. 3(c).

It is interesting to compare the seed and Poisson fluctuations with the primordial fluctuations implied by the CDM model. At the time of matter-radiation equality, y, when the PBH fluctuations start to grow, the CDM fluctuations have the form

 δeq∝{M−1/3(MMeq), (25)

where is the horizon mass at . These fluctuations and the effect on the binding mass are shown by the red lines in Fig. 3. In the mass range relevant to the present considerations, the CDM fluctuations fall off slower than both the Poisson and seed fluctuations, so they necessarily dominate on sufficiently large scales (i.e. the standard scenario is unchanged). However, there is always a mass below which the PBH fluctuations dominate, so this produces extra power at small scales. In the mass range , the fluctuations fall off slower than the seed fluctuation but faster than the Poisson fluctuation, so the latter could still dominate on very large scales. However, this ony applies on scales which are currently unbound.

These effects have been invoked to produce three types of structure: the first bound baryonic objects kash (), the Lyman- forest afshordi () and galaxies carr-silk (). If one has a monochromatic mass function, Fig. 3(c) shows that one can only bind objects as large as galaxies if one invokes either highly non-Gaussian fluctuations or appreciable accretion after the -distortion epoch. However, the PBH mass function is likely to be extended and one needs this anyway to produce an extended mass function for galaxies, so we now discuss this case.

### iv.2 Extended PBH mass function

If the PBHs have an extended mass function, both the seed and Poission effects could operate on different scales. Indeed, in principle, one could have two distinct PBH populations, both monochromatic but with a different mass. In this case, one population might provide the dark matter and generate a Poisson effect, while the other provides a low density of SMBHs which generate a seed effect. However, this seems rather contrived, so the following analysis assumes that the PBHs have a continuous mass spectrum. We first discuss the power-law case, since this is easiest to analyse and conveys the essential qualitative features. We will then consider the other possible mass functions described in Sec. III.

We first note that the competition between seeds can be neglected providing the fraction of the universe bound by them is small. In the power-law case, for seeds of mass , this requires the mass of the bound regons to satisfy . Since this is an increasing function of for , we need to satisfy this requirement at the lowest value of , leading to the condition . In this case, each bound region will contain a single seed and its mass will exceed that of the black hole by a factor of . This is the simplest scenario and has obvious appeal as a model for galaxy formation.

If the filling factor of the bound regions exceeds one, the situation is more complicated. In the monchromatic case, we saw that this corresponds to the Poisson effect becoming more important than the seed effect. In the extended case, it leads to a combination of the two effects. Since , Eq. (20) implies that the biggest effect is associated with the largest holes providing . One expects this, for example, if the PBHs form from scale-invariant fluctuations and . In this case, the dominant Poisson fluctuation on scale is associated with the largest hole expected to be contained in such a region. Providing this is less than , the effective value of is , so the Poisson scenario reduces to the seed scenario with

 mseed(M)=(fdmMmα−2dm)1/(α−1)(2<α<3). (26)

This necessarily increases with for and the mass binding at redshift is

 M(zB)∼mdmf1/(α−2)dm(zB/104)(α−1)/(2−α) (27)

from Eq. (22). (Note that this equation breaks down for , as applies if the PBHs form in an early matter-dominated era.) Thus, with an extended mass function, the seed mass is not fixed but depends on the mass of the region being bound. However, for sufficiently large that the mass given by Eq. (26) exceeds , the dominant effect is the Poisson fluctuation associated with the mass and

 Nmax∼Mm1−αmaxmα−2dmfdm. (28)

In this case, Eq. (22) implies that the mass binding at redshift is

 M(zB)∼mα−2dmm3−αmaxfdm(zB/104)−2. (29)

From comparison with Eq. (27), there is a change of slope at

 z∗∼104fdm(mdm/mmax)α−2(3>α>2), (30)

this being the redshift below which the Poisson effect dominates, and the associated mass is

 M∗≡M(z∗)∼m2−αdmmα−1maxf−1dm. (31)

Note that decreases and increases as increases and as decreases. Indeed, as indicated in Fig. 3(b), the Poisson effect dominates for all masses binding in the matter-dominated era () if

 fdm>(mdm/mmax)α−2. (32)

For a monochromatic mass function, and , so Eq. (31) just gives the mass which follows from Eq. (20).

If the PBHs form from scale-invariant fluctuations in the radiation era, and so Eq. (27) implies that the mass binding from the seed effect at redshift is

 M(zB)∼1012z−3Bf2dmmdm. (33)

For and , this is of order a galactic mass for and of order the mass of the first bound clouds for . We discuss each of these cases in more detail later but numerical calculations would be needed to elucidate our treatment.

This analysis can be extended to the other extended mass function scenarios. In the matter-domination case, we can put in the above analysis. However, many of the equations are invalid for because the PBH density increases logarithmically with , so most of the density is no longer in the smallest PBHs. One can consider either a scenario with and the Poisson effect or a scenario with and the seed effect. In the inflationary case, the mass function has a well defined peak but it is broad and falls off relatively slowly either side of the peak. In the critical collapse case, the dominant contribution to both the dark matter density and the Poisson effect comes from the mass-scale , so this is like the monochromatic situation.

### iv.3 Effect of compensating voids

An important caveat is that one might expect each PBH seed to be initially surrounded by a compensating void, so there would be no excess mass to generate a Coulomb effect. However, two processes would quickly remove this cancellation: (1) the black hole may escape the surrounding void due to the peculiar velocity generated by any asymmetry in its collapse fitch (); (2) the void may escape the black hole by expanding until it is larger than the region being bound (Musco, private communication). This expansion is to be expected and if it occurs at the speed of light, it would have reached a radius of around  kpc by the time of matter-radiation equality (y), when the growth of the fluctuations is assumed to start. Coincidentally, this is comparable to the size of a galaxy. Cai et al. cai () have argued that the compensating void effect is always unimportant.

## V Constraints on f(m) from formation of cosmic structures

Even if PBHs do not play a role in generating cosmic sructures, one can still place interesting upper limits of the fraction of dark matter in them by requiring that various types of structure do not form too early. In this section, we will consider the constraints associated with the Lyman- forest, galaxies, clusters of galaxies and the first bound clouds. We will then combine them into a single constraint on the function . Throughout this section we assume that the PBHs have a monochromatic mass function. How one can apply these limits for the case of an extended mass function is discussed in Refs. cks (); ctv ().

### v.1 Lyman-α forest

Afshordi et al. [14] used observations of the Lyman- forest – taken to be the precursors of galaxies, somewhat smaller than galaxies themselves – to obtain an upper limit of about on the mass of PBHs which provide the dark matter. This conclusion was based on numerical simulations, in which the Lyman- forest was taken to have an extended mass distribution. Carr et al. cksy () obtained a related result for the case in which the PBHs provide a fraction of the dark matter and the Lyman- forest is associated with a single mass . Since the Poisson fluctuation in the number of PBHs on this mass scale grows between the redshift of CDM domination () and the redshift at which the Lyman- forest is observed () by a factor , the forest will form earlier than observed unless

 f(m)<{(m/104M⊙)−1(104M⊙

The first expression can also be obtained by putting and in Eq. (22). The second expression corresponds to having just one PBH per Lyman- cloud, so the limit bottoms out at with a value . This limit is shown in Fig. 3.

The second condition in Eq. (34) also ensures that the seed effect dominates the Poisson effect (). Indeed, since the initial seed fluctuation is , the seed mass required for the Lyman- forest to bind at is immediately seen to be . However, there is no constraint on PBHs below this line because the fraction of the Universe going into the Lyman- clouds is only small, with most of the baryons presumably going into the intergalactic medium. Therefore the seed effect does not modify the form of the limit shown in Fig. 3 but comes into effect at the minimum. We stress that this limit is not precisely equivalent to that of Afshordi et al. because they make the (more realistic) assumption that the Lyman- clouds have a rnage of masses and form over a range of redshifts, but it is at least qualitatively similar.

### v.2 Galaxies and clusters

One can also apply the above argument for galaxies, assuming these have a typical mass of and must not bind before . Then Eq. (34) is replaced by

 f(m)<{(m/106M⊙)−1(106M⊙

this bottoming out at with a value . This constraint has a similar form to the Lyman- one but applies for larger . If we apply the same argument for clusters of galaxies, assuming these have a mass of and must not bind before , we obtain

 f(m)<{(m/107M⊙)−1(107M⊙

this bottoming out at with a value . On smaller scales one can apply the argument to the first dwarf galaxies, assuming these have a mass of and do not bind before . Then Eq. (34) is replaced by

 f(m)<{(m/103M⊙)−1(103M⊙

this bottoming out at with a value . All these constraints are shown in Fig. 3. We do not apply this argument to the formation of SMBHs in galactic nuclei because these are more likely to have arisen from accretion onto individual PBHs than the seed or Poisson effect.

### v.3 Combined limits

Although we are treating the various types of cosmic structures as distinct, it is clear that the above analysis can be applied to bound structures of any mass . If these structures are required to form after some redhsift , the constraint on bottoms out at

 mmin=3×10−4MBzB,fmin=3×10−4zB, (38)

so we can merge the different limits in Fig. 3 into a combined limit, providing we know how the mass depends on the redshift . Tthe CDM scenario has nearly scale-invariant fluctuations at the horizon epoch and - as indicated by Eq. (25) - this implies that the density fluctuations at matter-radiation equality scale as in the mass range of interest. Then Eq. (38) gives

 zB∝M−1/3B⇒MBzB∝z−2B⇒fmin∝m−1/2min. (39)

The combined limit on is therefore as indicated by the broken line in Fig. 3. It scales as at the low end, as at the high end and as for intermediate . In order to compare this constraint to the other limits on , it is also indicated by the bold line in Fig. 1. It is not as strong as the dynamical friction and X-ray background limits but the latter are more tentative.

## Vi The role of PBHs in the formation of cosmic structure

In the last section, we emphasised the constraints that can be placed on the number of large PBHs from the requirement that various types of objects do not form too early. In this section, we take a more positive approach, exploring the possibility that PBHs may have helped the formation of these objects, thereby complementing the standard CDM scenario of structure formation.

### vi.1 Intermediate mass PBHs as seeds for SMBHs in galactic nuclei

There is clear evidence that SMBHs with mass reside in the centres of most galaxies kormendy (); magorrian (); richstone (), with observations of quasars suggesting that these were already in place at very early times (). This includes the recent discovery of quaars powered by black holes of at wu () and at banados (). There is also the well-known correlation between by the mass of the SMBH and the bulge mass of the host galaxy, with the ratio being of order graham (); bogdan (); sun (). The standard view is that these SMBHs formed through dynamical processes in galactic nuclei after galaxy formation rees (). There are two possible pathways – direct collapse to black holes habouzit () or super-Eddington growth pez (). Both of these have been explored in a series of paper by Agarwal et al. agarwal1 (); agarwal2 (); agarwal3 (); agarwal4 () but they are not without difficulties. In the former case, the seeds are rare; in the latter case, they should be ultra-luminous and visible in deep X-ray surveys.

There is also evidence that the linear relation between the SMBH and bulge mass steepens at low mass grahamscott (), with several cases of central black holes in dwarf galaxies valluri (). These are smaller than and the % occupation number found in nearby dwarfs suggests that they were of either Population III or primordial origin nguyen (). There is also an ultra-compact dwarf galaxy of containing a SMBH seth (), possibly the stripped core of a previously massive galaxy ahn (). At high redshifts, SMBHs are predicted to be obese if early growth occurs agarwal2 () but current data are inconclusive shankar ().

It is therefore interesting to consider the possibility that quasars are seen at high redshift because they are powered by SMBHs which formed before galaxies, In this case, they could be primordial duech (); krs (); bean (); clesse () and this would lead to three possibilities. The first is that the PBHs were themselves supermassive, so that they can be directly identified with the SMBHs. In this case, as discussed below, the black holes could also help to generate galaxies themselves through either the seed or Poisson effect, the fluctuations growing by a factor of between the time of matter-radiation equality and today. This naturally expains the proportionality between the black hole and galaxy mass and it could provide an early mode of galaxy formation that might be important for the reionization of the universe chevallard (). This would also have implications for 21cm observations of HI absorption in the dark ages, because of the longer path length of X-ray photons fialkov ().

The second possibility is that the PBHs had a more modest (intermediate) mass and then grew through Eddington-limited accretion. This scenario was first suggested by Bean and Magueijo bean (), although they overestimated the amount of accretion in the very early phase, and it has subsequently been advocated by several other authors krs (); clesse (). Bean and Magueijo argued that it needs a very narrow PBH mass function to reproduce the observed distribution of SMBHs and Kawasaki et al. kky () suggest a specific inflationary scenario to account for this. However, most of the accretion still occurs after decoupling, so it may be difficult to distinguish this observationally from a scenario in which the black holes are non-primordial. In both cases, one would expect a lot of radiation to have been generated and this may naturally explain part of the observed X-ray background soltan ().

The third possibility is that the PBHs had a more modest mass and generated the SMBHs in galactic nuclei through the seed or Poisson effect. For example, to produce a SMBH with by , the considerations of Sec. IV show that one requires for the seed effect or for the Poisson effect. However, the largest SMBHs have a mass thomas (), so in this case we would require for the seed effect or for Poisson effect. Of course, one still has to explain how the bound region around an intermediate mass PBH or a bound cluster of intermediate mass PBHs can evolve to a single SMBH. Accretion and merging could be important and only some fraction of the bound region may end up in the central black hole in the Poisson case.

In the second and third scenarios, the SMBHs are not in place early enough for the galactic-scale fluctuations to experience the full growth factor of , so galaxies would have to form from primordial fluctuations in the usual way. Nevertheless, the presence of intermediate mass PBHs could still be advantageous in resolving various issues in dwarf galaxy formation silk17 (), especially since black hole recoils following a merged binary may suppress the presence of IMBHs in dwarfs that have undergone mergers as part of their formation history. One would not expect to end up with a single IMBH or SMBH at the centre of the galaxy in the second scenario but one might in the third. However, one needs considerable accretion in order to avoid the constraint. Since the latter applies above , a PBH of final mass must increases its mass by at least a factor

### vi.2 Supermassive PBHs as seeds for galaxies

We start with some historical remarks. Hoyle and Narlikar hoyle () first suggested a version of the seed picture of galaxy formation in the context of the steady state theory. Their model starts with a fluctuation of the form , like ours, but they are not constrained by the existence of a radiation-dominated era in selecting the time at which the fluctuation begins to grow. For reasons which are specific to a steady state model, they assume that growth begins when the second-order term in becomes comparable to the first-order term. Using a typical galactic mass , they require . Although the steady state theory is now superseded, Hoyle and Narlikar also discuss how deviations from spherical symmetry could give the range of shapes observed in elliptical galaxies, and how spirals could form from rotational effects. These features should apply in any seed theory.

Subsequently, Ryan ryan () also argued that SMBHs could seed galaxies. Using a spherically symmetric Newtonian cosmology, he showed that the hydrodynamic equations permit a solution in which the density contrast has a particular form in the radiation-dominated and matter-dominated eras. This gives an expression for the galactic mass and radius . For our Galaxy, Ryan obtained , which encompasses the now established mass of eckart (); ghez (). This analysis preceded the discovery of dark matter in galaxies and so is no longer applicable but it was still very prescient.

Gunn & Gott gunn () pointed out that one can make a very specific prediction about the structure of the galaxy resulting from the seed theory. If we assume that each shell of gas virializes after it has stopped expanding (i.e. settles down with a radius of about half its radius at maximum expansion), then one would expect the resultant galaxy to have a density profile . This is because the shell with mass binds at redshift , with an associated radius and density . This does not agree with the standard NFW profile nfw (), which goes from at small radius to at large radius, but one would not expect this to apply within the radius of gravitational influence of the central black hole anyway gondolo (). Indeed, hierarchical merging should rapidly erase memory of the profiles around the initial seeds, much as found in the highest resolution dark matter simulations angulo ().

We now turn to our own seed scenario for galaxy formation. In this context, we must first decide whether the PBHs have a monochromatic or extended mass function. In the monochromatic case, all the SMBHs in galactic nuclei would start off with the same mass and the galaxy mass just depends on the redshift at which it binds. Specifically, Eq. (20) and the linear growth law for imply that a mass binds at a time

 tB(M)∼teq(Mm)3/2∼1010(M1012M⊙)3/2(m108M⊙)−3/2y, (40)

so larger galaxies would form later, as in the standard CDM model. However, this scenario does not explain the observed correlation between the mass of the galaxy and the central black hole. This might still arise if the black holes subsequently increase their mass through accretion to a value proportional to the bulge mass. For example, the observed bulge/SMBH ratio of [or dark-matter/SMBH ratio of ] might in principle be explained by Eddington-limited accretion rees-silk (). However, in this case Eq. (40) no longer applies since the initial seed mass is much reduced. Therefore invoking a monochromatic PBH mass function does not seem very plausible. The more general effect of the galactic halo on the evolution of the central black hole has been discussed by Volonteri et al. vol1 (); vol2 ().

For an extended PBH mass function, the PBH seeds will naturally produce a range of galactic masses at a given redshift. However, there are two distinct situations. If the PBHs are sufficiently rare that there is only one per galaxy, which requires , then the galaxy mass is necessarily proportional to the seed mass. This naturally explains why the bulge mass is proportional to the SMBH mass, with the ratio just being the growth factor of fluctuations between the redshift of matter-radiation equality () and the redshift when galaxies bind (). (In the Poisson case, the ratio is , which is too high.) However, there should be variances due to obese SMBHs in massive galaxies or anorexic IMBHs in dwarf galaxies. One would also expect the galactic mass function to be the same as the PBH mass function.

While this scenario is attractive, it only permits a small fraction of the universe to go into galaxies. In the second situation, the filling factor of the bound regions approaches , so that the competition between seeds becomes important and one can no longer assume . This situation is more complicated but there should still be a simple relation between the mass spectrum of the holes and that of the resulting galaxies. If , we expect the number of galaxies with mass in the range () to be where

 dNg/dM∝M(1−γ−α)/γ. (41)

In this case, Eq. (26) suggests , so Eq. (41) gives

 dNg/dM∝M−2(M

This is independent of the value of and converges to the usual equal mass per logarithmic mass interval limit. The upper cut-off in in part reflects the value of but may also be determined by accretion effects, as discussed later. For comparison, the Schechter luminosity function is schechter ()

 Φ(L)∝L−1.07exp(−L/L∗), (43)

while the Press-Schechter mass function is press-schechter ()

 dNg/dM∝M−2exp(−M/M∗), (44)

with an exponential upper cut-off at . The integrated density is then logarithmically divergent at the low mass end. It is striking that the observed mass function matches the prediction of Eq. (42), providing one can explain the exponential cut-off in some way. The first scenario more naturally produces the proportionality between the black hole and galaxy mass but it only yields the Press-Schechter mass function if .

We now consider the other mass functions discussed in Sec. III. Since the predicted galaxy mass function is independent of the value of , the above analysis should still apply in the matter-dominated scenario (). The only difference is that the upper limit is now determined by the end of the matter-dominated epoch. One problem with a very extended PBH mass function is that the situation is dynamically complicated, with the larger PBHs tending to sink to the centre of the galaxy through dynamical friction and then merging to form a single SMBH. Numerical simulations would be requred to ascertain even the qualitative features of this scenario. We have seen that a lognormal mass-function is expected in some inflationary scenarios. In this case, the spread of masses is much less, so the complications are reduced. For example, if the PBHs at the peak of the lognormal distribution provide the dark matter, then the ones on the high-mass tail would be sufficiently rare and massive to seed galaxies clesse4 (). The PBHs would also form clusters and this would have important consequences for their gravitational lensing effects clesse2 (). In the critical collapse case, the mass function is effectively monochromatic, since the PBH density falls off very fast below the upper cut-off, so one cannot produce both the dark matter and seeds for galaxies.

### vi.3 Upper limit on mass of galaxy seeded by PBH from accretion

What determines the value of in Eq. (42)? Two factors impose an upper limit on the galactic mass which can be bound by a PBH seeed. The first is that if is larger than about , the bound region will not form a single galaxy but fragment into a cluster of galaxies silk2 (). In this case, each galaxy would initially possess a central PBH seed but when the central galaxies merge to form a CD galaxy, the black holes might also merge to form a single SMBH. The proportionality between the galaxy and black hole mass might still pertain but clearly a more complicated dynamical analysis is required in this case. The second factor is accretion bower (), which we have so far neglected. Accretion of the baryons cannot begin before decoupling because the Compton drag of the background photons prevents motion relative to the CMB. However, the baryons will quickly fall into the potential well created by the dark matter after decoupling and accretion of dark matter will effectively begin from the time of matter-radiation equality ().

During the radiation era, the sound-speed is and the accretion radius is just the Schwarschild radius, so the Bondi formula bondi () gives

 dm/dt∼R2acsρ∼(Gm2)/(c3t2). (45)

Integrating this equation gives

 1/m−1/mi∼(G/c3)(1/t−1/ti) (46)

and hence

 m∼mi/[1−mi/MH(ti)+Gmi/(c3t)]. (47)

Therefore there is very little accretion for (i.e. for PBHs initially much smaller than the horizon). Although Eq. (47) suggests for , implying that a PBH with the horizon mass at formation should continue to grow like the horizon, this neglects the cosmic expansion. A more careful analysis shows that self-similar growth is impossible, so that accretion is always negligible in the radiation era ch1974 ().

During the matter-dominated era after , is increased (since falls below ) and so the accretion rate is also increased. Providing the matter temperature T follows the usual background evolution (i.e. before reheating), the Bondi formula gives

 dm/dt∼R2acsρ∼(G2m2)/(Gc3st2)∼Gm2(kTeq/mp)−3/2t−2eq. (48)

Integrating this gives

 1/m−1/mi∼−ηtwithη≡G(mp/kTeq)3/2t−2eq. (49)

Hence

 m≈mi/(1−miηt), (50)

which diverges at a time

 τ=1/(ηmi)∼(Meq/mi)(ceq/c)3teq, (51)

where is the horizon mass at y and . Thus the mass diverges at a time which precedes the present epoch (y) for zoltan ()

 mi>Meq(teq/to)∼1010M⊙. (52)

This suggests that PBHs larger than should not be found at the centers of galaxies because they would have swallowed the entire galaxy. This argument complements a recent one of Inayoshi and Haiman ih (), who find that small-scale accretion physics and angular momentum transfer ultimately limits the SMBH mass to . It is therefore interesting that observations at both low and high redshift indicate a maximum SMBH mass of order .

The above analysis assumes that the density and temperature at the accretion radius correspond to the mean cosmological conditions. A more complicated analysis would be required if the growing bound cloud around the PBH ever became larger than the accretion radius. Note also that the accretion rate reaches the Eddington limit when

 dm/dt∼ηm2∼m/tED, (53)

where y is the Salpeter timescale salpeter (). Hence we would only have super-Eddington accretion for

 m>(ηtED)−1∼Meq(teq/tED)∼1012M⊙. (54)

But this never applies for the SMBHs of interest.

### vi.4 First clouds

Population III stars are made in the first clouds. They might also be responsible for forming IMBHs in numbers that are marginally sufficient to seed the most massive SMBH seen at , provided that super-Eddington accretion occurred (2017MNRAS.471..589P, ). However we have shown that one can also obtain a mass fraction of order in intermediate mass PBHs (as required) with conservative assumptions about the PBH mass function. Hence one would not need the first clouds to collapse monolithically in order to form a population of IMBHs. At the very least, this seems to require rather special fine-tuning (2016MNRAS.463..529H, ).

Moreover, the mass of the first clouds is sufficiently small that the Poisson effect alone can bind them PBHs contirnute sufficiently to the dark matter. Let us consider the fiducial example of PBHs contributing a fraction to the dark matter density. In the canonical LCDM scenario, Jeans mass fluctuations of mass form the first DM-dominated dwarf galaxies at . These dwarfs, forming before reionization, are the building blocks of the next generation of dwarf galaxies, some of which may correspond to the extremely metal-poor ultra-faint dwarfs detected in recent deep surveys.

The PBH imprint (i.e. Poisson fluctuation) on these scales is This means that the first structures form at , which is much earlier than in the usual scenario. For the fiducial parameters, one has to carefuly reexamine the limits from recombination due to Bondi accretion of gas onto the PBH, as discussed by Ricotti et al. mor (). We avoid this problem by considering the more conservative case in which PBHs of mass are subdominant (eg. with ).

For the LCDM scenario, one can estimate the sizes, velocity dispersions and virial temperatures of the first systems as

 R∼400z−110M1/36pc,σ∼3z1/210M1/36km/s,T∼1000z10M2/36K. (55)

Residual ionization in these clouds leads to formation, eventually forming trace amounts of that allow cooling, fragmentation and formation of massive Population III stars. These short-lived stars generate metallicity and pollute the IGM sufficiently to eventually lead to enhanced cooling and formation of dwarf galaxies. It is notoriously difficult to suppress fragmentation except in the vicinity of enhanced UV fields from neighbouring Population III star clusters. With PBHs, the first cloud parameters are dramatically changed because of their boosted amplitude and earlier formation. They become

 R∝M1/3z−1B∝f−1/20.01M5/6J,6m−1/2100pc,σ∝f1/40.01M1/12J,6m1/4100km/s,T∼f1/20.01M1/6J,6m1/2100K. (56)

In this case, it seems likely that fragmentation is largely suppressed because of the lack of coolants, and that runaway growth of the PBHs may ensue. Hence IMBH formation could precede the formation of the first dwarf galaxies.

Kashlinksy kash () has also stressed that the Poisson fluctuations in PBH dark matter should lead to more abundant early collapsed halos than in the standard scenario. He makes the interesting suggestion that the black holes might generate the cosmic infrared background fluctuations detected by the Spitzer/Akari satellites kash2 (); helgason (). These should correlate with the X-ray background fluctuations measured by Chandra and a recent paper suggests that this can be explained by accreting black holes of possible primordial origin cappelluti ().

## Vii Gravity waves

The proposal that the dark matter could comprise PBHs in the IMBH range has attracted much attention recently as a result of the LIGO detections LIGO1 (); LIGO2 (); LIGO3 () of merging binary black holes with mass around . Since the black holes are larger than initially expected, it has been suggested that they could represent a new population. One possibility is that they were of Population III origin (i.e. forming between decoupling and galaxies). The suggestion that LIGO might detect gravitational waves from coalescing intermediate mass Population III black holes was first made by Bond and Carr bc () more than 30 years ago and - rather remarkably - Kinugawa et al. kinugawa () predicted a Population III coalescence peak at shortly before the first LIGO detection.

Another possibility - more relevant to the considerations of the present paper and explored by many previous authors - is that the LIGO black holes were primordial. This does not necesarily require the PBHs to provide all the dark matter; the predicted merger rate depends on too many uncertain astrophysical factors for the PBH number density to be specified precisely. However, several authors have made this connection bird (); clesse (), with Clesse and Garcia-Bellido arguing that a lognormal distribution centred at around naturally explains both the dark matter and the LIGO bursts without violating any of the current PBH constraints cgb2017 (). On the other hand, others argue that the PBH density would need to be much less than the dark matter density to explain the LIGO results sasaki (). Indeed, several groups have now used the LIGO results to constrain the PBH dark matter fraction raidal (); kovetz (). Which alternative pertains depends on the epoch at which the binaries form, whether they are clustered and the PBH mass function. In the latter context, it should be stressed that the PBH density should peak at a lower mass than the coalescence signal for an extended PBH mass function, since the amplitude of the gravitational waves scales as the black hole mass.

Although the origin of the the black holes associated with the LIGO events is still uncertain, future LIGO results and data from other gravitational-wave detectors – such as eLISA  seto () and Pre-DECIGO kawamura () – might be able to distinguish between binary black holes of Population II, Population III or primordial origin nakamura2 (). For example, Pre-DECIGO will be able to measure the mass spectrum and -dependence of the merger rate. Another important clue may come from the spin distribution suyama () and orbital eccentricities cholis () of the coalescing black holes.

As first stressed in Ref. carr80 (), a population of massive PBHs would also be expected to generate a stochastic background of gravitational waves and this would be especially interesting if some of the PBHs were in binaries coalescing due to gravitational radiation losses at the present epoch. This was discussed in Ref. bc () in the context of Population III black holes and in Refs nakamura1 (); ioka () in the context of PBHs. Stochastic gravitational-wave backgrounds from black-hole binaries offer another way of distinguishing between the progenitors of the binaries. Indeed, LIGO data had already placed weak constraints on the PBH scenarios a decade ago abbott2007 () and an updated analysis in the light of the recent merger events can be found in Refs. LIGO4 (); dvorkin (); cgb2016 ().

If PBHs have an extended mass function, incorporating both dark matter at the low end and galactic seeds at the high end, this will have important implications for the predicted gravitational wave background. Theorists usually focus on the gravitational waves generated by either dark matter black holes (detectable by LIGO) or supermassive black holes in galactic nuclei (detectable LISA). However, with an extended PBH mass function, the gravitational wave background should encompass both these limits and also every intermediate frequency. This point has also been emphasized in Ref. clesse2 ().

## Viii Discussion

We have seen that PBHs in the intermediate to supermassive mass range could play several important cosmological roles. They could (1) explain the dark matter, (2) provide a source of LIGO coalescences, and (3) alleviate some of the problems associated with the CDM scenario - including the formation of the SMBHs in galactic nuclei or even the first galaxies themselves. Although the main focus of this paper has been (3), with particular emphasis on the seed or Poisson effect, it is important to consider all three roles together. At one extreme, PBHs may play none of these roles, although our considerations still place interesting constraints on the fraction of the dark matter in PBHs. At the other extreme, as advocated in Ref. cgb2017 (), they may play all three roles.

Roles (1) and (2) are rather easily reconciled, since the mass scales involved are quite close and both in the IMBH range. Indeed, there is already a considerable literature on this topic. Reconciling (1) and (3) is more challenging, since the mass scales are very different, and this topic is relatively unexplored. Whether it is possible depends crucially on the PBH mass distribution. In principle, one could invoke two separate PBH populations and this might conceivably arise in some inflationary scenarios. However, this seems less natural than invoking a single PBH population, in which case we need to distinguish between a monochromatic and extended mass function.

For a monochromatic mass function, if the PBHs provide all the dark matter (), then the Poisson effect dominates on all scales and various astrophysical constraints require . This implies that PBHs can only bind subgalactic masses but still allows them to play a role in producing the first bound baryonic clouds or the SMBHs which power quasars. For , the seed effect dominates on small scales and can bind a region of up to times the PBH mass. However, limits on the -distortion in the CMB due to the dissipation of fluctuations before decoupling may exclude any PBHs larger than unless one invokes substantial accretion or highly non-Gaussian fluctuations. So the seed effect may also only bind subgalactic scales.

If the PBHs have an extended mass function, they could both provide the dark matter and seed structure on the galactic scale. For a power-law mass function with up to some cut-off mass , most of the mass is in the smallest PBHs for (as expected) and the seed effect dominates below . However, this situation is dynamically complicated because of the large PBH mass range. For a lognormal mass function, the PBH mass range is much narrower, so the scenario is easier to understand and probably more plausible. For a critical mass function, most of the density is still concentrated at a single mass-scale, so the situation resembles the monochromatic one and PBHs cannot both provide the dark matter and galactic seeds.

We stress that our proposal should be regarded as complementing rather than rivalling the CDM scenario, since there is no denying the success of the latter. We also emphasize that our proposal has observational consequences that can be probed by future deep surveys in the optical, radio and X-ray frequency regimes. In particular, 21cm dark-age experiments could play an important role in evaluating the contribution of PBHs to early heating and/or ionization of the universe. Such experiments could potentially discriminate between primordial and conventional sources of ionization in the dark ages because of the differing redshift dependences and consequent implications for the hydrogen spin temperature evolution.

One aspect of our proposal which has been rather neglected in this paper is PBH accretion in the period after decoupling. It is clear that this is crucial in conventional scenarios for SMBH formation in galactic nuclei and it could be equally important in the primordial context. For this reason observations associated with accretion could be a poor discriminant of the primordial scenario. This also relates to the issue of whether the PBHs were initially in the intermediate or supermassive mass range. In order to generate cosmic structures without violating the current constraints, we either need to invoke intermediate mass PBHs plus accretion or supermassivr PBHs plus high non-Gaussianity.

Finally we comment on the implications of supermassive PBHs for primordial nucleosynthesis. It is often claimed that the success of the BBNS scenario excludes PBHs forming after weak feeze-out, corresponding to initial PBH masses above . However, this limit is overstated because at most of the mass of the universe can be in PBHs at this time, even if they provide the dark matter today. On the other hand, even a small fraction of PBHs could have interesting consequences for primordial nucleosynthesis. For example, if we consider PBHs of forming at s, there should be an overproduction of helium around each one because of the local density overenhancement, The net cosmological effect will be at most because of the rarity of collapsed regions. Nevertheless, if mixing is inefficient, one might expect rare regions on dwarf-galaxy scales, optimistically 1 in with anomalous primordial nucleosynthesis abundances.

## Acknowledgments

We thank A. Babul, S. Clesse, J. Garcia-Bellido, S. Khochfar, I. Musco, T. Nakama and M. Volonteri for useful discussions. BC thanks Institut d’Astrophysique in Paris for hospitality received during this work. JS acknowledges the support of the European Research Council via grant 267117.

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