Primordial Black Holes as Dark Matter: Recent Developments
Bernard Carr, Florian Kuhnel
I Introduction
Primordial black holes (PBHs) have been a source of interest for nearly 50 years Zel’dovich and Novikov 1967, despite the fact that there is still no evidence for them. One reason for this interest is that only PBHs could be small enough for Hawking radiation to be important Hawking 1974. This discovery has not yet been confirmed experimentally and there remain major conceptual puzzles associated with the process. Nevertheless, it is generally recognised as one of the key developments in 20th century physics because it beautifully unifies general relativity, quantum mechanics and thermodynamics. The fact that Hawking was only led to this discovery through contemplating the properties of PBHs illustrates that it can be useful to study something even if it does not exist! But, of course, the situation is much more interesting if PBHs do exist.
PBHs smaller than about g would have evaporated by now with many interesting cosmological consequences Carr et al. 2010. Studies of such consequences have placed useful constraints on models of the early Universe and, more positively, evaporating PBHs have been invoked to explain certain features: for example, the extragalactic Page and Hawking 1976; Carr 1976 and Galactic Wright 1996; Lehoucq et al. 2009 -ray backgrounds, antimatter in cosmic rays Kiraly et al. 1981; MacGibbon and Carr 1991, the annihilation line radiation from the Galactic centre Okele and Rees 1980; Bambi et al. 2008, the reionisation of the pregalactic medium Belotsky and Kirillov 2015 and some short-period -ray bursts Belyanin et al. 1996; Cline et al. 1997. However, there are usually other possible explanations for these features, so there is no definitive evidence for evaporating PBHs. Only the original papers for each topic are cited here and a more comprehensive list of references can be found in Reference Carr et al. 2010.
Attention has therefore shifted to the PBHs larger than g, which are unaffected by Hawking radiation. Such PBHs might have various astrophysical consequences, such as providing the seeding of the supermassive black holes (SMBHs) in galactic nuclei Carr and Rees 1984; Bean and Magueijo 2002, the generation of large-scale structure through Poisson fluctuations Mészáros 1975 and important effects on the thermal and ionisation history of the Universe Carr 1981. Again only the original papers are cited here. But perhaps the most exciting possibility — and the main focus of this review — is that they could provide the dark matter (DM) which comprises of the critical density Carr et al. 2016a, an idea that goes back to the earliest days of PBH research Chapline 1975. Since PBHs formed in the radiation-dominated era, they are not subject to the well-known big bang nucleosynthesis (BBN) constraint that baryons can have at most of the critical density Cyburt et al. 2003. They should therefore be classified as non-baryonic and behave like any other form of cold dark matter (CDM) Frampton 2016. It is sometimes assumed that they must form before BBN, implying an upper limit of , but the fraction of the Universe in PBHs at that time would be tiny, so the effect on BBN might only be small.
As with other CDM candidates, there is still no compelling evidence that PBHs provide the dark matter. However, there have been claims of evidence from dynamical and lensing effects. In particular, there was a flurry of excitement in 1997, when the microlensing results of massive compact halo objects (MACHOs) suggested that the dark matter could be compact objects of mass Alcock et al. 1997. Alternative microlensing candidates could be excluded and PBHs of this mass might naturally form at the quark-hadron phase transition at s Jedamzik 1998. Subsequently, however, it was shown that such objects could comprise for only of the dark matter and indeed the entire mass range to was later excluded from providing all of it Alcock et al. 2000; Tisserand et al. 2007. In recent decades attention has focused on other mass ranges in which PBHs could have a significant density and numerous constraints allow only three possibilities: the asteroid mass range ( – g), the sublunar mass range ( – g) and the intermediate mass range ( – ).
We discuss the constraints on , the fraction of the halo in PBHs of mass , in Section III; this is a much reduced version of the recent review by Carr et al. Carr et al. 2020a. The results are summarised in Figure 1, all the limits assuming that the PBHs have a monochromatic mass function and cluster in the Galactic halo in the same way as other forms of CDM. Although, for the sake of completeness, we include evaporating PBHs, we do not focus on them in this review except insomuch as they may leave stable Planck-mass relics because these could also be dark-matter candidates. However, if Hawking evaporation were avoided for some reason, it is worth stressing that PBHs could provide the dark matter all the way down to the Planck mass, with few (if any) non-gravitational constraints.
At first sight, the implication of Figure 1 is that PBHs are excluded from having an appreciable density in almost every mass range. However, our intention is not to put nails in the coffin of the PBH scenario because every constraint is a potential signature. In particular, we have mentioned that there are still some mass windows in which PBHs could provide the dark matter. PBHs could be generated by inflation in all of these windows but theorists are split as to which one they favour. For example, Inomata et al. Inomata et al. 2017a argue that double inflation can produce a peak at around g, while Clesse and García-Bellido Clesse and García-Bellido 2015 argue that hybrid inflation can produce a peak at around . A peak at the latter mass could also be produced by a reduction in the pressure at the quark-hadron phase transition Byrnes et al. 2018, 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 is also a split between groups searching for light and heavy candidates.
It should be stressed that non-evaporating PBHs are dark even if they do not provide all the dark matter, so this review does not focus exclusively on the proposal that PBHs solve the dark-matter problem. Many objects are dark, so it is not implausible that the dark matter comprises some mixture of PBHs and WIMPs. Indeed, we will see that this situation would have interesting consequences for both. Also, even if PBHs provide only a small fraction of the dark matter, they may still be of great cosmological interest. For example, they could play a rôle in generating the supermassive black holes in galactic nuclei and these have obvious astrophysical significance even though they provide only of the dark matter.
The constraints shown in Figure 1 assume that the PBH mass function is monochromatic (i.e. with a width ). However, there are many scenarios in which one would expect the mass function to be extended. For example, inflation often produces a lognormal mass function Dolgov and Silk 1993 and critical collapse generates an extended low mass tail Yokoyama 1998a. In the context of the dark-matter problem, this is a double-edged sword. On the one hand, it means that the total PBH density may suffice to explain the dark matter, even if the density in any particular mass band is small and within the observational bounds. On the other hand, even if PBHs can provide all the dark matter at some mass scale, the extended mass function may still violate the constraints at some other scale Green 2016. While there is now a well-understood procedure for analysing constraints in the extended case Carr et al. 2017a, identifying the optimal PBH mass window remains problematic Kühnel and Freese 2017.
The proposal that the dark matter could comprise PBHs in the intermediate mass range has attracted much attention recently as a result of the LIGO/Virgo detections of merging binary black holes with mass in the range – Abbott et al. 2016a; Abbott et al. 2016b; Abbott et al. 2018a. Since the black holes are larger than initially expected, it has been suggested that they could represent a new population, although the mainstream view remains that they are the remnants of ordinary stars Belczynski et al. 2016. One possibility is that they were of Population III origin (i.e. forming between decoupling and galaxy formation). Indeed, the suggestion that LIGO might detect gravitational waves from coalescing intermediate mass Population III black holes was first made more than 30 years ago Bond and Carr 1984 and, rather remarkably, Kinugawa et al. predicted a Population III coalescence peak at shortly before the first LIGO detection of black holes of that mass Kinugawa et al. 2014. Another possibility, more relevant to the present considerations, is that the LIGO/Virgo black holes are primordial, as first discussed in Reference Nakamura et al. 1997. However, this does not require the PBHs to provide all the dark matter. While this possibility has been suggested Bird et al. 2016, the predicted merger rate depends on when the binaries form and uncertain astrophysical factors, so the dark-matter fraction could still be small Sasaki et al. 2016; Nakamura et al. 2016; Sasaki et al. 2018. Indeed, the LIGO/Virgo results have already been used to constrain the PBH dark-matter fraction Raidal et al. 2017, although the limit is sensitive to the predicted merger rate, which is very model-dependent Ali-Haïmoud et al. 2017. Note that the PBH density should peak at a lower mass than the coalescence signal for an extended PBH mass function, since the gravitational-waves amplitude scales as the black-hole mass.
The plan of this review paper is as follows: In Section II we elaborate on several aspects of PBH formation, including a general discussion of their mass and density, a review of PBH formation scenarios, and a consideration of the effects of non-Gaussianity and non-sphericity. In Section III we review current constraints on the density of PBH with a monochromatic mass function, these being associated with a variety of lensing, dynamical, accretion and gravitational-wave effects. At first sight, these seem to exclude PBHs providing the dark matter in any mass range but this conclusion may be avoided for an extended mass function and most limits are subject to caveats anyway. More positively, in Section IV we overview various observational conundra which can be explained by PBHs, especially those associated with intermediate mass and supermassive black holes. In Section V we discuss how the thermal history of the Universe naturally provides peaks in the PBH mass function at the mass scales associated with these conundra, the bumpy mass function obviating some of the limits discussed in Section III. We also present a recently-developed mechanism which helps to resolve a long-standing fine-tuning problem associated with PBH formation. In Section VI we discuss scenarios which involve a mixture of PBHs and particle dark matter. In Section VII we draw some general conclusions about PBHs as dark matter.
II Primordial Black Hole Formation
PBHs could have been produced during the early Universe due to various mechanisms. For all of these, the increased cosmological energy density at early times plays a major rôle Hawking 1971; Carr and Hawking 1974, yielding a rough connection between the PBH mass and the horizon mass at formation:
The fraction of the mass of the Universe in PBHs on some mass scale is epoch-dependent but its value at the formation epoch of the PBHs is denoted by . For the standard CDM model, in which the age of the Universe is , the Hubble parameter is Ade et al. 2016 and the time of photon decoupling is Hinshaw et al. 2009. If the PBHs have a monochromatic mass function, the fraction of the Universe’s mass in PBHs at their formation time is related to their number density by Carr et al. 2010
The current density parameter for PBHs which have not yet evaporated is
where is critical density. Equation (II.3) can be expressed in terms of the ratio of the current PBH mass density to the CDM density:
where is the PBH mass fraction at matter-radiation equality and we use the most recent value indicated by Planck Aghanim et al. 2018. The ratio of the energy densities of matter and radiation (all relativistic species) at any time is
where is the ratio of the dark matter and baryonic densities. At PBH formation, the fraction of domains that collapse is
where is the total dark-matter fraction and the square-bracketed term has a value of order .
II.2 Formation Scenarios
We now review the large number of scenarios which have been proposed for PBH formation and the associated PBH mass functions. We have seen that PBHs generally have a mass of order the horizon mass at formation, so one might expect a monochromatic mass function (i.e. with a width ). However, in some scenarios PBHs form over a prolonged period and therefore have an extended mass function (e.g. with its form of the mass function depending on the power spectrum of the primordial fluctuations). As discussed below, even PBHs formed at a single epoch may have an extended mass function.
The most natural possibility is that PBHs form from primordial density fluctuations. Overdense regions will then stop expanding some time after they enter the particle horizon and collapse against the pressure if they are larger than the Jeans mass. If the horizon-scale fluctuations have a Gaussian distribution with dispersion , one expects for the fraction of horizon patches collapsing to a black hole to be Carr 1975
II.2.2 Collapse from Scale-Invariant Fluctuations
If the PBHs form from scale-invariant fluctuations (i.e. with constant amplitude at the horizon epoch), their mass spectrum should have the power-law form Carr 1975
where specifies the equation of state () at PBH formation. The exponent arises because the background density and PBH density have different redshift dependencies. At one time it was argued that the primordial fluctuations would be expected to be scale-invariant Harrison, Phys. Rev. D1, 2726-2730 (1970) 1970; Zeldovich 1972 but this does not apply in the inflationary scenario. Nevertheless, one would still expect the above equations to apply if the PBHs form from cosmic loops because the collapse probability is then scale-invariant. If the PBHs contain a fraction of the dark matter, this implies that the fraction of the dark matter in PBHs of mass larger than is
where , and is the mass scale which contains most of the dark matter. In a radiation-dominated era, the exponent in Equation (II.10) becomes .
II.2.3 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 at a phase transition in which the mass is channeled into non-relativistic particles Khlopov and Polnarev 1980; Polnarev and Khlopov 1982 or due to slow reheating after inflation Khlopov et al.; Carr et al. 1994. In a related context, Hidalgo et al. have recently studied Hidalgo et al. 2017 PBH formation in a dust-like scenario of an oscillating scalar field during an extended period of preheating. Since the value of in the above analysis is for , one might expect to increase logarithmically with . However, the analysis breaks down in this case because the Jeans length is much smaller than the particle horizon, so pressure is not the main inhibitor of collapse. Instead, collapse is prevented by deviations from spherical symmetry and the probability of PBH formation can be shown to be Khlopov and Polnarev 1980
This is in agreement with the recent analysis of Harada et al. Harada et al. 2016 and leads to a mass function
The lower limit is the horizon mass at the start of matter-dominance and the upper limit is the horizon mass when the regions which bind at the end of matter-dominance enter the horizon. This scenario has recently been studied in Reference Carr et al. 2017b.
II.2.4 Collapse from Inflationary Fluctuations
However, not all inflationary scenarios produce the mass function (II.14). Inomata et al. Inomata et al. 2017b propose a scenario which combines a broad mass function at low (to explain the dark matter) with a sharp one at high mass (to explain the LIGO events).
II.2.5 Quantum Diffusion
quantum effects are expected to be important whenever this quantity becomes of order one, i.e. . This is often the case for PBH formation, where recent investigations indicate an increase of the power spectrum and hence the PBH abundance Pattison et al. 2017. This quantum diffusion is inherently non-perturbative and so Kühnel & Freese Kühnel and Freese 2019 have developed a dedicated resummation technique in order to incorporate all higher-order corrections (see References Kühnel and Schwarz 2008; Kühnel and Schwarz 2009; Kühnel and Schwarz 2010 for an application of these techniques to stochastic inflation). Ezquiaga et al. have argued that quantum diffusion generically generates a high degree of non-Gaussianity Ezquiaga and García-Bellido 2018; Ezquiaga et al. 2020.
II.2.6 Critical Collapse
It is well known that black-hole formation is associated with critical phenomena Choptuik 1993 and various authors have applied this feature in investigations of PBH formation Koike et al. 1995; Niemeyer and Jedamzik 1998; Evans and Coleman 1994; Kühnel et al. 2016. The conclusion is that the mass function has an upper cut-off at around the horizon mass but there is also a low-mass tail Yokoyama 1998b. 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 Choptuik 1993
II.2.7 Collapse at the Quantum-Chromodynamics Phase Transition
II.2.8 Collapse of Cosmic Loops
In the cosmic string scenario, one expects some strings to self-intersect and form cosmic loops. A typical loop will be larger than its Schwarzschild radius by the factor , where is the string mass per unit length. If strings play a rôle in generating large-scale structure, must be of order . However, as discussed by many authors Hawking 1989; Polnarev and Zembowicz 1991; Garriga and Sakellariadou 1993; Caldwell and Casper 1996; MacGibbon et al. 1998; Jenkins and Sakellariadou 2020, there is always a small probability that a cosmic loop will get into a configuration in which every dimension lies within its Schwarzschild radius. This probability depends upon both and the string correlation scale. Note that the holes form with equal probability at every epoch, so they should have an extended mass spectrum with Hawking 1989
where is the ratio of the string length to the correlation scale. One expects and requires to avoid overproduction of PBHs.
II.2.9 Collapse through Bubble Collisions
Bubbles of broken symmetry might arise at any spontaneously broken symmetry epoch and various people have suggested that PBHs could form as a result of bubble collisions Crawford and Schramm 1982; Hawking et al. 1982; Kodama et al. 1982; Leach et al. 2000; Moss 1994; Kitajima and Takahashi 2020. However, this happens only if the bubble-formation rate per Hubble volume is finely tuned: if it is much larger than the Hubble rate, the entire Universe undergoes the phase transition immediately and there is not time to form black holes; if it is much less than the Hubble rate, the bubbles are very rare and never collide. The holes should have a mass of order the horizon mass at the phase transition, so PBHs forming at the GUT epoch would have a mass of g, those forming at the electroweak unification epoch would have a mass of g, and those forming at the QCD (quark-hadron) phase transition would have mass of around . There could also be wormhole production at a 1st-order phase transition Kodama et al. 1981; Maeda 1986. The production of PBHs from bubble collisions at the end of first-order inflation has been studied extensively in References Khlopov et al. 1998; Konoplich et al. 1999; Khlopov et al. 1999; Khlopov et al. 2000.
II.2.10 Collapse of Domain Walls
The collapse of sufficiently large closed domain walls produced at a 2nd-order phase transition in the vacuum state of a scalar field, such as might be associated with inflation, could lead to PBH formation Dokuchaev et al. 2005. These PBHs would have a small mass for a thermal phase transition with the usual equilibrium conditions. However, they could be much larger if one invoked a non-equilibrium scenario Rubin et al. 2001. Indeed, they could span a wide range of masses, with a fractal structure of smaller PBHs clustered around larger ones Khlopov et al. 1998; Konoplich et al. 1999; Khlopov et al. 1999; Khlopov et al. 2000. Vilenkin and colleagues have argued that bubbles formed during inflation would (depending on their size) form either black holes or baby universes connected to our Universe by wormholes Garriga et al. 2016; Deng et al. 2017. In this case, the PBH mass function would be very broad and extend to very high masses Deng and Vilenkin 2017; Liu et al. 2020.
II.3 Non-Gaussianity and Non-Sphericity
As PBHs form from the extreme high-density tail of the spectrum of fluctuations, their abundance is acutely sensitive to non-Gaussianities in the density-perturbation profile Young and Byrnes 2013; Bugaev and Klimai 2013. For certain models — such as the hybrid waterfall or simple curvaton models Bugaev and Klimai 2012; Bugaev and Klimai 2011a; Sasaki et al. 2006 — it has even been shown that no truncation of non-Gaussian parameters can be made to the model without changing the estimated PBH abundance Young and Byrnes 2013. However, non-Gaussianity induced PBH production can have serious consequences for the viability of PBH dark matter. PBHs produced with non-Gaussianity lead to isocurvature modes that could be detected in the CMB Young and Byrnes 2015; Tada and Yokoyama 2015. With the current Planck exclusion limits Ade et al. 2016, this argument implies that the non-Gaussianity parameters and for a PBH-producing theory are both less than . For theories like the curvaton and hybrid inflation models Linde 1994; Clesse and García-Bellido 2015, this leads to the immediate exclusion of PBH dark matter, as the isocurvature effects would be too large.
Non-sphericity has not yet been subject to extensive numerical studies of the kind in Reference Musco and Miller 2013 but non-zero ellipticity leads to possibly large effects on the PBH mass spectra as shown by Reference Kühnel and Sandstad 2016. Therein, the authors give an approximate analytical approximation for the collapse threshold, which will be larger than in the spherical case,
II.4 Multi-Spike Mass Functions
If PBHs are to explain phenomena on different mass scales, it is pertinent to consider the possibility that the PBH mass spectrum has several spikes. There are two known recent mechanisms for generating such spikes. The first has been proposed by Cai et al. Cai et al. 2018, who have discussed a new type of resonance effect which leads to prolific PBH formation. This arises because the sound-speed can oscillate in some inflationary scenarios, leading to parametric amplification of the curvature perturbation and hence a significant peak in the power spectrum of the density perturbations on some critical scale. The resonances are in narrow bands around certain harmonic frequencies with one of the peaks dominating. It turns out, one can easily get a peak of order unity. Although most PBHs form at the first peak, a small number will also form at subsequent peaks. The second mechanism for generating multi-spiked PBH mass spectra has recently been proposed by Carr & Kühnel Carr and Kühnel 2019 and has been demonstrated for most of the well-studied models of PBH formation. This mechanism relies on the choice of non-Bunch-Davies vacua, leading to oscillatory features in the inflationary power spectrum, which in turn generates oscillations in the PBH mass function with exponentially enhanced spikes.
III Constraints and Caveats
We now review the various constraints for PBHs which are too large to have evaporated completely by now, updating the equivalent discussion in References Carr et al. 2010 and Carr et al. 2016a. All the limits assume that PBHs cluster in the Galactic halo in the same way as other forms of CDM, unless they are so large that there is less than one per galaxy. Throughout this Section the PBHs are taken to have a monochromatic mass function, in the sense that they span a mass range . In this case, the fraction of the halo in PBHs is related to by Equation (II.3). Our limits on are summarised in Figure 1, which is based on Figure 10 of Reference Carr et al. 2020a, this providing a much more comprehensive review of the PBH constraints. Following Reference García-Bellido 2018, the constraints are also broken down according to the redshift of the relevant observations in Figure 2. The main constraints derive from PBH evaporations, various gravitational-lensing experiments, numerous dynamical effects and PBH accretion. Where there are several limits in the same mass range, we usually show only the most stringent one. It must be stressed that the constraints in Figures 1 and 2 have varying degrees of certainty and they all come with caveats. For some, the observations are well understood but there are uncertainties in the black-hole physics. For others, the observations themselves are not fully understood or depend upon additional astrophysical assumptions. The constraints may also depend on other physical parameters which are not shown explicitly. It is important to stress that some of the constraints can be circumvented if the PBHs have an extended mass function. Indeed, as discussed in Section V, this may be required if PBHs are to provide most of the dark matter.
A PBH of initial mass will evaporate through the emission of Hawking radiation on a timescale which is less than the present age of the Universe for below g Carr et al. 2016b. There is a strong constraint on from observations of the extragalactic -ray background Page and Hawking 1976. PBHs in the narrow band have not yet completed their evaporation but their current mass is below the mass at which quark and gluon jets are emitted. For , one can neglect the change of mass altogether and the time-integrated spectrum of photons from each PBH is obtained by multiplying the instantaneous spectrum by the age of the Universe . The instantaneous spectrum for primary (non-jet) photons is
where is the absorption cross-section for photons of energy , so this gives an intensity
This peaks at with a value , whereas the observed intensity is with between and , so putting gives Carr et al. 2010
We plot this constraint in Figure 1 for . The Galactic -ray background constraint could give a stronger limit Carr et al. 2016b but this depends sensitively on the form of the PBH mass function, so we do not discuss it here.
There are various other evaporation constraints in this mass range. Boudad and Cirelli Boudaud and Cirelli 2019 use positron data from Voyager 1 to constrain evaporating PBHs of mass g and obtain the bound . This complements the cosmological limit, as it is based on local Galactic measurements, and is also shown in Figure 1. Laha Laha 2019 and DeRocco and Graham DeRocco and Graham 2019 constrain – g PBHs using measurements of the keV annihilation line radiation from the Galactic centre. Other limits are associated with -ray and radio observations of the Galactic centre Laha et al. 2020; Chan and Lee 2020 and the ionising effect of – g PBHs Belotsky and Kirillov 2015.
III.2 Lensing Constraints
Constraints on MACHOs with very low have been claimed from the femtolensing of -ray bursts (GRBs). Assuming the bursts are at a redshift , early studies implied in the mass range – Marani et al. 1999; Nemiroff et al. 2001 and in the range – Barnacka et al. 2012. However, Katz et al. Katz et al. 2018 argue that most GRB sources are too large for these limits to apply, so we do not show them in Figure 1. Kepler data from observations of Galactic sources Griest et al. 2013; Griest et al. 2014 imply a limit in the planetary mass range: for . However, Niikura et al. Niikura et al. 2019a have carried out a seven-hour observation of M31 with the Subaru Hyper Suprime-Cam (HSC) to search for microlensing of stars by PBHs lying in the halo regions of the Milky Way and M31 and obtain the much more stringent bound for which is shown in Figure 1.
Recently Niikura et al. Niikura et al. 2019b have used data from a five-year OGLE survey of the Galactic bulge to place much stronger limits in the range , although they also claim some positive detections. The precise form of the EROS and OGLE limits are shown in Figure 1, while the possible detections are discussed in Section IV.
PBHs cause most lines of sight to be demagnified relative to the mean, with a long tail of high magnifications. Zumalacárregui and Seljak Zumalacarregui and Seljak 2018 have used the lack of lensing in type Ia supernovae (SNe) to constrain any PBH population, an approach that allows for the effects of large-scale structure and possible non-Gaussianity in the intrinsic SNe luminosity distribution. Using current JLA data, they derive a bound for , the finite size of SNe providing the lower limit, and this constraint is shown in Figure 1. García-Bellido & Clesse García-Bellido et al. 2018 argue that this limit can be weakened if the PBHs have an extended mass function or are clustered. There is some dispute about this but Figure 1 is only for a monochromatic mass function anyway.
The recent discovery of fast transient events in massive galaxy clusters is attributed to individual stars in giant arcs being highly magnified due to caustic crossing. Oguri et al. Oguri et al. 2018 argue that the particular event MACS J1149 excludes a high density of PBHs anywhere in the mass range because this would reduce the magnifications.
Early studies of the microlensing of quasars Dalcanton et al. 1994 seemed to exclude the possibility of all the dark matter being in objects with , although this limit preceded the CDM picture. More recent studies of quasar microlensing suggest a limit Mediavilla et al. 2009 for , although we argue in Section IV that these surveys may also provide positive evidence for PBHs. Millilensing of compact radio sources Wilkinson et al. 2001 gives a limit
Although weaker than the dynamical constraints in this mass range, and not included this in Figure 1, we mention it because it illustrates that lensing limits extend to very large values of .
III.3 Dynamical Constraints
The effects of collisions of planetary-mass PBHs on astronomical objects have been a subject of long-standing interest, although we do not show these constraints in Figure 1 because they are controversial. Roncadelli et al. Roncadelli et al. 2009 have suggested that halo PBHs could be captured and swallowed by stars in the Galactic disc. The stars would eventually be accreted by the holes, producing radiation and a population of subsolar black holes which could only be of primordial origin and this leads to a constraint g), corresponding to a lower limit on the mass. Capela et al. have constrained PBH dark matter by considering their capture by white dwarfs Capela et al. 2013a or neutron stars Capela et al. 2013b, while Pani and Loeb Pani and Loeb 2014 have argued that this excludes PBHs from providing the dark matter throughout the sublunar window. However, these limits have been disputed Defillon et al. 2014 because the dark-matter density in globular clusters is now known to be much lower values than assumed in these analyses Ibata et al. 2013. Graham et al. Graham et al. 2015 argue that the transit of a PBH through a white dwarf (WD) causes localised heating through dynamical friction and initiates runaway thermonuclear fusion, causing the WD to explode as a supernova. They claim that the shape of the observed WD distribution excludes – g PBHs from providing the dark matter and that – g ones are constrained by the observed supernova rate. However, these limits are inconsistent with hydrodynamical simulations of Montero-Camacho et al. Montero-Camacho et al. 2019, who conclude that this mass range is still allowed.
The three limits correspond to disruption by multiple encounters, one-off encounters and non-impulsive encounters, respectively. The fraction is thus constrained over the mass range
One can apply this argument to wide binaries in the Galaxy, which are particularly vulnerable to disruption by PBHs Bahcall et al. 1985; Weinberg et al. 1987. In the context of the original analysis of Reference Quinn et al. 2009, Equation (III.6) gives a constraint for , with representing the upper bound on the mass of PBHs which dominate the halo and being the mass at which the limit flattens off. Only the flat part of the constraint appears in Figure 1. However, the upper limit has been reduced to in later work Monroy-Rodríguez and Allen 2014, so the narrow window between the microlensing lower bound and the wide-binary upper bound is shrinking. On the other hand, Tian et al. Tian et al. 2020 have recently studied more than 4000 halo wide binaries in the Gaia survey and detected a break in their separation distribution, possibly indicative of PBHs with .
A similar argument for the survival of globular clusters against tidal disruption by passing PBHs gives a limit for , although this depends sensitively on the mass and the radius of the cluster Carr and Sakellariadou 1999. The upper limit of is consistent with the numerical calculations of Moore Moore 1993. In a related argument, Brandt Brandt 2016 infers an upper limit of from the fact that a star cluster near the centre of the dwarf galaxy Eridanus II has not been disrupted by halo objects. Koushiappas and Loeb Koushiappas and Loeb 2017 have also studied the effects of black holes on the dynamical evolution of dwarf galaxies. They find that mass segregation leads to a depletion of stars in the centres of such galaxies and the appearance of a ring in the projected stellar surface density profile. Using Segue 1 as an example, they exclude the possibility of more than of the dark matter being PBHs of around . One would also expect sufficiently large PBHs to disrupt Ultra-Faint Dwarf Galaxies and a recent study of UFDGs by Stegmann et al. Stegmann et al. 2020 appears to exclude PBHs in the mass range – from providing the dark matter. Only the Eridanus limit is shown in Figure 1.
Halo objects will overheat the stars in the Galactic disc unless one has for Lacey and Ostriker 1985. The incredulity limit, (corresponding one PBH per halo), takes over for and this is the only part appearing in Figure 1. Another limit in this mass range arises because halo objects will be dragged into the nucleus of the Galaxy by the dynamical friction of various stellar populations, and this process leads to excessive nuclear mass unless is constrained Carr and Sakellariadou 1999. As shown in Figure 1, this limit has a rather complicated form because there are different sources of friction and it also depends on parameters such as the halo core radius, but it bottoms out at with a value .
with the second expression corresponding to having one PBH per galaxy. This limit bottoms out at with a value . Similar constraints apply for the first bound clouds, dwarf galaxies and clusters of galaxies and the limits for all the systems are collected together in Figure 1. The Poisson effect also influences the distribution of the Lyman-alpha forest Afshordi et al. 2003; Murgia et al. 2019; the associated PBH constraint has a similar form to Equation (III.8) but could be much stronger, with the lower limit of being reduced to around .
III.4 Accretion Constraints
PBHs could have a large luminosity at early times due to accretion of background gas and this effect imposes strong constraints on their number density. However, the analysis of the problem is complicated because the black hole luminosity will generally boost the matter temperature of the background Universe well above the standard Friedmann value even if the PBH density is small, thereby reducing the accretion. Thus there are two distinct but related PBH constraints: one associated with the effects on the Universe’s thermal history and the other with the generation of background radiation. This problem was first studied in Reference Carr 1981 and we briefly review that analysis here. Even though this study was incomplete and later superseded by more detailed numerical investigations, we discuss it because it is the only analysis which applies for very large PBHs.
Reference Carr 1981 assumes that each PBH accretes at the Bondi rate Bondi 1952
where a dot indicates differentiation with respect to cosmic time and the appropriate values of and are those which pertain at the black-hole accretion radius:
If the accreted mass is converted into outgoing radiation with efficiency , the associated luminosity is
Reference Carr 1981 assumes that both and the spectrum of emergent radiation are constant. If the spectrum extends up to energy keV, the high-energy photons escape from the individual HII regions unimpeded, so most of the black-hole luminosity goes into background radiation or global heating of the Universe through photoionisation when the background ionisation is low and Compton scattering off electrons when it is high. Reference Carr 1981 also assumes that cannot exceed the Eddington luminosity,
and it is shown that a PBH will radiate at this limit for some period after decoupling providing
The effect on the thermal history of the Universe is then determined for different () domains. One has various possible behaviours: (1) is boosted above K, with the Universe being reionised, and possibly up to the temperature of the hottest accretion-generated photons; (2) is boosted to K but not above it because of the cooling of the CMB; (3) does not reach K, so the Universe is not re-ionised, but there is a period in which it increases; (4) never increases but follows the CMB temperature, falling like rather than , for a while; (5) never deviates from Friedmann behaviour.
This relates to the well-known Soltan constraint Soltan 1982 on the growth of the SMBHs that power quasars. The limit given by Equation (III.14) therefore flattens off at large values of .
One problem with the above analysis is that the steady-state Bondi formula fails if the accretion timescale,
exceeds the cosmic expansion time, with the solution being described by self-similar infall instead. For , this applies at decoupling and so one has to wait until the time given by Equation (III.16) for the Bondi formula to apply. Therefore the above analysis applies only if most of the radiation is generated after this time. Otherwise the background light limit is weakened.
Later an improved analysis was provided by Ricotti and colleagues Mack et al. 2007; Ricotti et al. 2008; Ricotti 2007. They used a more realistic model for the efficiency parameter , allowed for the increased density in the dark halo expected to form around each PBH and included the effect of the velocity dispersion of the PBHs on the accretion in the period after cosmic structures start to form. They found much stronger accretion limits by considering the effects of the emitted radiation on the spectrum and anisotropies of the CMB rather than the background radiation itself. Using FIRAS data to constrain the first, they obtained a limit for ; using WMAP data to constrain the second, they obtained a limit for . The constraints flatten off above the indicated masses but are taken to extend up to . Although these limits appeared to exclude down to masses as low as , they were very model-dependent and there was also a technical error (an incorrect power of redshift) in the calculation.
This problem has been reconsidered by several groups, who argue that the limits are weaker than indicated in Reference Ricotti et al. 2008. Ali-Haïmoud and Kamionkowski Ali-Haïmoud and Kamionkowski 2017 calculate the accretion on the assumption that it is suppressed by Compton drag and Compton cooling from CMB photons and 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, arXiv:1612.07264 [astro-ph.CO] 2016 performs a similar analysis and gets an upper limit of . Neither of these analyses includes the super-Eddington effects expected above some mass, and this should lead to a flattening of the constraint. Poulin et al. Poulin et al. 2017; Serpico et al. 2020 argue that the spherical accretion approximation probably breaks down, with an accretion disk forming instead, and this affects the statistical properties of the CMB anisotropies. Provided the disks form early, these constraints exclude a monochromatic distribution of PBH with masses above as the dominant form of dark matter. Since this is the strongest accretion constraint, it is the only one shown in Figure 1.
More direct constraints can be obtained by considering the emission of PBHs at the present epoch. For example, Gaggero et al. Gaggero et al. 2017 model the accretion of gas onto a population of massive PBHs in the Milky Way and compare the predicted radio and X-ray emission with observational data. The possibility that PBHs can provide all of the dark matter is excluded at level by a comparison with the VLA radio catalog and the Chandra X-ray catalog. Similar arguments have been made by Manshanden et al. Manshanden et al. 2019. PBH interactions with the interstellar medium should result in a significant X-ray flux, contributing to the observed number density of compact X-ray objects in galaxies. Inoue & Kusenko and Lu et al. Inoue and Kusenko 2017; Lu et al. 2020 use the data to constrain the PBH number density in the mass range from a few to and their limit is shown in Figure 1. However, De Luca et al. De Luca et al. 2020 have stressed that the change in the mass of PBHs due to accretion may modify the interpretation of the observational bounds on at the present epoch. In the mass range – this can raise existing upper limits by several orders of magnitude.
III.5 Cosmic Microwave Background Constraints
If PBHs form from the high- tail of Gaussian density fluctuations, as in the simplest scenario Carr 1975, then another interesting limit comes from the dissipation of these density fluctuations by Silk damping at a much later time. This process leads to a -distortion in the CMB spectrum Chluba et al. 2012 for , leading to an upper limit over the mass range . This limit was first given in Reference Carr and Lidsey 1993, based on a result in Reference Barrow and Coles 1991, but the limit on is now much stronger. There is also a distortion for .
This argument gives a very strong constraint on in the range Kohri et al. 2014 but the assumption that the fluctuations are Gaussian may be incorrect. For example, Nakama et al. Nakama et al. 2016 have proposed a “patch” model, in which the relationship between the background inhomogeneities and the overdensity in the tiny fraction of volumes collapsing to PBHs is modified, so that the -distortion constraint becomes much weaker. Recently, Nakama et al. Nakama et al. 2018 have used a phenomenological description of non-Gaussianity to calculate the -distortion constraints on , using the current FIRAS limit and the projected upper limit from PIXIE Abitbol et al. 2017. However, one would need huge non-Gaussianity to avoid the constraints in the mass range of . Another way out is to assume that the PBHs are initially smaller than the lower limit but undergo substantial accretion between the -distortion era and the time of matter-radiation equality. The constraint also implies a maximum mass for the PBHs which provide the dark matter if there is some theoretical limit on the steepness of the power spectrum Byrnes et al. 2019.
III.6 Gravitational-Wave Constraints
Interest in PBHs has intensified recently because of the detection of gravitational waves from coalescing black-hole binaries by LIGO/Virgo Abbott et al. 2016c; Abbott et al. 2016d; Abbott et al. 2016e; Abbott et al. 2019a. Even if these are not of primordial origin, the observations place important constraints on the number of PBHs. Indeed, the LIGO data had already placed weak constraints on such scenarios a decade ago Abbott et al. 2007. More recent LIGO/Virgo searches find no compact binary systems with component masses in the range – Abbott et al. 2018b. Neither black holes nor neutron stars are expected to form through normal stellar evolution below and one can infer for and for . A similar search from the second LIGO/Virgo run Abbott et al. 2019b found constraints on the mergers of and binaries corresponding to at most or of the dark matter, respectively. The possibility that the LIGO/Virgo events relate to PBHs is discussed further in Section IV.4.
A population of massive PBHs would be expected to generate a gravitational-wave background (GWB) Carr 1980 and this would be especially interesting if there were a population of binary black holes coalescing at the present epoch due to gravitational-radiation losses. Conversely, the non-observation of a GWB gives constraints on the fraction of dark matter in PBHs. As shown by Raidal et al. Raidal et al. 2017, even the early LIGO results gave strong limits in the range – and this limit is shown in Figure 1. A similar result was obtained by Wang et al. Wang et al. 2018. This constraint has been updated in more recent work, using both LIGO/Virgo data Raidal et al. 2019; Vaskonen and Veermae 2020 and pulsar-timing observations Chen et al. 2019. Bartolo et al. Bartolo et al. 2020 calculate the anisotropies and non-Gaussianity of such a stochastic GWB and conclude that PBHs could not provide all the dark matter if these were large.
A different type of gravitational-wave constraint on arises because of the large second-order tensor perturbations generated by the scalar perturbations which produce the PBHs Saito and Yokoyama 2009a. The associated frequency was originally given as but this estimate contained a numerical error Saito and Yokoyama 2009b and was later reduced by a factor of Bugaev and Klimai 2009. The limit on just relates to the amplitude of the density fluctuations at the horizon epoch and is of order . This effect has subsequently been studied by several other authors Assadullahi and Wands 2010; Bugaev and Klimai 2011b and limits from LIGO/Virgo and the Big Bang Observer (BBO) could potentially cover the mass range down to g. Conversely, one can use PBH limits to constrain a background of primordial gravitational waves Nakama and Suyama 2015; Nakama and Suyama 2016.
The robustness of the LIGO/Virgo bounds on PBHs depends on the accuracy with which the formation of PBH binaries in the early Universe can be described. Ballesteros et al. Ballesteros et al. 2018 revisit the standard estimate of the merger rate, focusing on the spatial distribution of nearest neighbours and the expected initial PBH clustering. They confirm the robustness of the previous results in the case of a narrow mass function, which constrains the PBH fraction of dark matter to be – .
Kühnel et al. Kühnel et al. 2018 investigate GW production by PBHs in the mass range – orbiting a supermassive black hole. While an individual object would be undetectable, the extended stochastic emission from a large number of such objects might be detectable. In particular, LISA could detect the extended emission from objects orbiting Sgr at the centre of the Milky Way if a dark-matter spike, analogous to the WIMP-spike predicted by Gondolo and Silk Gondolo and Silk 1999, forms there.
III.7 Interesting Mass Windows and Extended Mass Functions
Figure 1 shows that there are four mass windows (A, B, C, D) in which PBHs could have an “appreciable” density, which we somewhat arbitrarily take to mean , although this does not mean there is positive evidence for this. The cleanest window would seem to be A and many of the earlier constraints in this mass range have now been removed. Window C has received most attention, because of the LIGO/Virgo results, but it is challenging to put all the dark matter there because of the large number of constraints in this mass range.
A special comment is required about window D (i.e. the mass range ), since this has been almost completely neglected in previous literature. Obviously such stupendously large black holes (which we term “SLABs”) could not provide the dark matter in galactic halos, since they are too large to fit inside them (i.e. they violate the galactic incredulity limit). However, they might provide an intergalactic dark-matter component and the lack of constraints in this mass range may just reflect the fact that nobody has considered this possibility. While this proposal might seem exotic, we know there are black holes with masses up to nearly in galactic nuclei Shemmer et al. 2004, so it is conceivable that SLABs could represent the high-mass tail of such a population. Although PBHs are unlikely to be this large at formation, we saw in Section III.4 that they might increase their mass enormously before galaxy formation through accretion, so they could certainly seed SLABs. This possibility has motivated the study of such objects in Reference Carr et al. 2020b and this includes an update of the accretion limit mentioned in Section III.4 and a limit associated with WIMP annihilation, to be discussed in Section VI.1. These limits are not shown in Figure 1 since that is just a summary of previous literature.
The constraints shown in Figure 1 assume that the PBH mass function is quasi-monochromatic (i.e. with a width ). This is unrealistic and in most scenarios one would expect the mass function to be extended, possibly stretching over several decades of mass. A detailed assessment of this problem requires a knowledge of the expected PBH mass fraction, , and the maximum fraction allowed by the monochromatic constraint, . However, one cannot just plot for a given model in Figure 1 and infer that the model is allowed because it does not intersect . This problem is quite challenging and several different approaches have been suggested.
One approach is to assume that each constraint can be treated as a sequence of flat constraints by breaking it up into narrow mass bins Carr et al. 2016a but this is a complicated procedure and has been criticised by Green Green 2016. A more elegant approach, similar to Green’s, was proposed in Reference Carr et al. 2017a and also used in Reference Kühnel and Freese 2017. In this, one introduces the function
normalised so that the total fraction of the dark matter in PBHs is
The mass function is specified by the mean and variance of the distribution:
Two parameters should generally suffice locally (i.e. close to a peak), since these just correspond to the first two terms in a Taylor expansion. An astrophysical observable depending on the PBH abundance can generally be expanded as
where is the background contribution and the functions depend on the details of the underlying physics and the nature of the observation. If PBHs with different mass contribute independently to the observable, only the first two terms in Equation (III.21) need be considered. If a measurement puts an upper bound on the observable,
The maximum allowed fraction of dark matter in the PBHs is then
Combining Equations (III.21)–(III.24) then yields
In Reference Carr et al. 2017a this method is applied for various expected PBH mass functions, while Reference Kühnel and Freese 2017 performs a comprehensive analysis for the case in which the PBHs cover the mass range – . Generally the allowed mass range for fixed decreases with increasing width , thus ruling out the possibility of evading the constraints by simply extending the mass function. However, we stress that the situation could be more complicated than we have assumed above, with more than two parameters being required to describe the PBH mass function. For example, Hasegawa et al. Hasegawa and Kawasaki 2018 have proposed an inflationary scenario in the minimally supersymmetric standard model which generates both intermediate-mass PBHs to explain the LIGO/Virgo detections and lunar-mass PBHs to explain the dark matter. Section V considers a scenario in which the PBH mass function has four peaks, each associated with a particular cosmological conundrum.
IV Claimed Signatures
Most of the PBH literature has focussed on constraints on their contribution to the dark matter, as reviewed in the last Section. However, a number of papers have claimed positive evidence for them, the PBH masses required have been claimed to span this range over 16 orders of magnitude, from to . In particular, Reference Carr et al. 2019b summarises seven current observational conundra which may be explained by PBHs. The first three are associated with lensing effects: (1) microlensing events towards the Galactic bulge generated by planetary-mass objects Niikura et al. 2019b which are much more frequent than expected for free-floating planets; (2) microlensing of quasars Mediavilla et al. 2017, including ones that are so misaligned with the lensing galaxy that the probability of lensing by a star is very low; (3) the unexpectedly high number of microlensing events towards the Galactic bulge by dark objects in the ‘mass gap’ between and Wyrzykowski and Mandel 2020, where stellar evolution models fail to form black holes Brown et al. 1999. The next three are associated with accretion and dynamical effects: (4) unexplained correlations in the source-subtracted X-ray and cosmic infrared background fluctuations Cappelluti et al. 2013; (5) the non-observation of ultra-faint dwarf galaxies (UFDGs) below the critical radius associated with dynamical disruption by PBHs Clesse and García-Bellido 2018; (6) the unexplained correlation between the masses of galaxies and their central SMBHs. The final one is associated with gravitational-wave effects: (7) the observed mass and spin distributions for the coalescing black holes found by LIGO/Virgo Abbott et al. 2019a. There are additional observational problems which Silk has argued may be solved by PBHs in the intermediate mass range Silk 2017. We now discuss this evidence in more detail, discussing the conundra in order of increasing mass. In Section V we discuss how these conundra may have a unified explanation in the scenario proposed in Reference Carr et al. 2019b.
Observations of M31 by Niikura et al. Niikura et al. 2019a with the HSC/Subaru telescope have identified a single candidate microlensing event with mass in the range range . Kusenko et al. Kusenko et al. 2020 have argued that nucleation of false vacuum bubbles during inflation could produce PBHs with this mass. Niikura et al. also claim that data from the five-year OGLE survey of 2622 microlensing events in the Galactic bulge Niikura et al. 2019b have revealed six ultra-short ones attributable to planetary-mass objects between and . These would contribute about of the CDM, much more than expected for free-floating planets van Elteren et al. 2019, and compatible with the bump associated with the electro-weak phase transition in the best-fit PBH mass function of Reference Carr et al. 2019b.
The MACHO collaboration originally reported LMC microlensing events and claimed that these were consistent with compact objects of , compatible with PBHs formed at the QCD phase transition Alcock et al. 2000. Although they concluded that such objects could contribute only of the halo mass, the origin of these events is still a mystery and this limit is subject to several caveats. Calcino et al. Calcino et al. 2018 argue that the usual semi-isothermal sphere for our halo is no longer consistent with the Milky Way rotation curve. When the uncertainties in the shape of the halo are taken into account, they claim that the LMC microlensing constraints weaken for but tighten at lower masses. Hawkins Hawkins 2015 makes a similar point, arguing that low-mass Galactic halo models would relax the constraints and allow of the dark matter to be solar-mass PBHs. Several authors have claimed that PBHs could form in tight clusters, giving a local overdensity well in excess of that provided by the halo concentration alone Dokuchaev et al. 2005; Chisholm 2006, and that this increased overdensity may remove the microlensing constraint at – altogether, especially if the PBHs have a wide mass distribution.
OGLE has detected around long-duration microlensing events in the Galactic bulge, of which around have GAIA parallax measurements. This finding breaks the mass-distance degeneracy and implies that these events were probably generated by black holes Wyrzykowski and Mandel 2020. The event distribution implies a mass function peaking between and , which overlaps with the gap from to in which black holes are not expected to form as the endpoint of stellar evolution Brown et al. 1999. This implication is also consistent with the peak originating from the reduction of pressure at the QCD epoch Carr et al. 2019b.
Hawkins Hawkins 1993 originally claimed evidence for a critical density of Jupiter-mass PBHs from observations of quasar microlensing. However, his later analysis yielded a lower density (dark matter rather than critical) and a mass of around Hawkins 2007. Mediavilla et al. Mediavilla et al. 2017 have also found evidence for quasar microlensing, this indicating that % of the total mass is in compact objects in the mass range – . These events might be explained by intervening stars but in several cases the stellar region of the lensing galaxy is not aligned with the quasar, which suggests a different population of subsolar halo objects. Hawkins Hawkins 2020 has also argued that some quasar images are best explained as microlensing by PBHs distributed along the lines of sight to the quasars. The best-fit PBH mass function of Reference Carr et al. 2019b is consistent with these findings and requires in this mass range.
Recently Vedantham et al. Vedantham et al. 2017 have detected long-term radio variability in the light-curves of some active galactic nuclei (AGN). This is associated with a pair of strongly skewed peaks in the radio flux density and is observed over a broad frequency range. They propose that this arises from gravitational millilensing of relativistically moving features in the AGN jets, these features crossing the lensing caustics created by – subhalo condensates or black holes located within intervening galaxies.
IV.2 Dynamical
Lacey and Ostriker once argued that the observed puffing of the Galactic disc could be due to black holes of around Lacey and Ostriker 1985, older stars being heated more than younger ones. They claimed that this could explain the scaling of the velocity dispersion with age and the relative velocity dispersions in the radial, azimuthal and vertical directions, as well as the existence of a high-velocity tail of stars Ipser and Semenzato 1985. However, later measurements gave different velocity dispersions for older stars Carlberg et al. 1985; Strömgren 1987; Gomez et al. 1990 and it is now thought that heating by a combination of spiral density waves and giant molecular clouds may better fit the data Lacey 1991.
Fuller et al. Fuller et al. 2017 show that some -process elements can be produced by the interaction of PBHs with neutron stars if those in the mass range – have . When a PBH is captured by a rotating millisecond neutron star, the resulting spin-up ejects – of relatively cold neutron-rich material. This can also produce a kilonova-type afterglow and a fast radio burst. Abramowicz and Bejger Abramowicz et al. 2018 argue that collisions of neutron stars with PBHs of mass g may explain the millisecond durations and large luminosities of fast radio bursts.
IV.3 X-Ray/Infrared Background
As shown by Kashlinsky and his collaborators Cappelluti et al. 2013; Kashlinsky et al. 2005; Kashlinsky et al. 2018; Kashlinsky 2016, the spatial coherence of the X-ray and infrared source-subtracted backgrounds suggests that black holes are required. Although these need not be primordial, the level of the infrared background suggests an overabundance of high-redshift haloes and this could be explained by the Poisson effect discussed above if a significant fraction of the CDM comprises solar-mass PBHs. In these haloes, a few stars form and emit infrared radiation, while PBHs emit X-rays due to accretion. It is challenging to find other scenarios that naturally produce such features.
IV.4 LIGO/Virgo
It has long been appreciated that a key signature of PBHs would the gravitational waves generated by either their formation Carr 1980, although these would be hard to detect because of redshift effects, or their coalescences if the PBHs form binaries. Indeed, the detection of coalescing binary black holes was first discussed in Reference Bond and Carr 1984 in the context of Population III black holes and later in References Nakamura et al. 1997; Ioka et al. 1999 in the context of PBHs. However, the precise formation epoch of the holes is not crucial since the coalescence occurs much later. In either case, the black holes would be expected to cluster inside galactic halos and so the detection of the gravitational waves would provide a probe of the halo distribution Inoue and Tanaka 2003.
The suggestion that the dark matter could comprise PBHs has attracted much attention in recent years as a result of the LIGO/Virgo detections Abbott et al. 2016c; Abbott et al. 2016d. To date, events have been observed with component masses in the range – . After the first detection, Bird et al. Bird et al. 2016 claimed that the expected merger was compatible with the range – obtained by the LIGO analysis and this was supported by other studies Clesse and García-Bellido 2017; Blinnikov et al. 2016. On the other hand, Sasaki et al. Sasaki et al. 2016 argued that the lower limit on the merger rate would be in tension with the CMB distortion constraints if the PBHs provided all the dark matter, although one might avoid these constraints if the LIGO/Virgo back holes derive from the accretion and merger of smaller PBHs Clesse and García-Bellido 2017. Note that most of the observed coalesced black holes have effective spins compatible with zero. Although the statistical significance of this result is low Fernandez and Profumo 2019, it goes against a stellar binary origin Gerosa et al. 2018 but is a prediction of the PBH scenario García-Bellido 2017.
If the PBHs have an extended mass function, their density should peak at a lower-mass signal than the coalescence signal. For example, but in the range – for the mass distribution of Reference Carr et al. 2019b. Raidal et al. Raidal et al. 2017 have studied the production and merging of PBH binaries for an extended mass function and possible PBH clustering (cf. Dolgov et al. 2020). They show that PBHs can explain the LIGO/Virgo events without violating any current constraints if they have a lognormal mass function. Subsequent work Raidal et al. 2019; Vaskonen and Veermae 2020 has studied the formation and disruption of PBH binaries in more detail, using both analytical and numerical calculations for a general mass function. If PBHs make up just 10% of the dark matter, the analytic estimates are reliable and indicate that the constraint from the observed LIGO/Virgo rate is strongest in the mass range – , albeit weakened because of the suppression of mergers. Their general conclusion is that the LIGO/Virgo events can result from the mergers of PBHs but that such objects cannot provide all the dark matter unless the PBHs have an extended mass function.
Ali-Haïmoud et al. Ali-Haïmoud et al. 2017 have computed the probability distribution of orbital parameters for PBH binaries. Their analytic estimates indicate that the tidal field of halos and interactions with other PBHs, as well as dynamical friction by unbound standard dark-matter particles, do not provide a significant torque on PBH binaries. They also calculate the binary merger rate from gravitational capture in present-day halos. If binaries formed in the early Universe survive to the present time, as expected, they dominate the total PBH merger rate. Moreover, this merger rate would be above the current LIGO upper limit unless for – PBHs.
One of the mass ranges in which PBHs could provide the dark matter is around . If these PBHs are generated by enhanced scalar perturbations produced during inflation, their formation is inevitably accompanied by the generation of non-Gaussian gravitational waves with frequency peaked in the mHz range (the maximum sensitivity of LISA). Bartolo et al. Bartolo et al. 2019a; Bartolo et al. 2019b have studied whether LISA will be able to detect not only the gravitational-wave (GW) power spectrum but also the non-Gaussian three-point GW correlator (i.e. the bispectrum). However, they conclude that the inclusion of propagation effects suppresses the bispectrum. If PBHs with masses of – g are the dark matter, the corresponding GWs will be detectable by LISA, irrespective of the value of , and this has also been stressed by Cai et al. Cai et al. 2019.
IV.5 Arguments for Intermediate-Mass Primordial Black Holes
Silk has argued that intermediate-mass PBHs (IMPBHs) could be ubiquitous in early dwarf galaxies, being mostly passive today but active in their gas-rich past Silk 2017. This would be allowed by current AGN observations Kormendy and Ho 2013; Pardo et al. 2016; Baldassare et al. 2017 and early feedback from IMPBHs could provide a unified explanation for many dwarf galaxy anomalies. Besides providing a phase of early galaxy formation and seeds for SMBHs at high (discussed above), they could: (1) suppress the number of luminous dwarfs; (2) generate cores in dwarfs by dynamical heating; (3) resolve the “too big to fail” problem; (4) create bulgeless disks; (5) form ultra-faint dwarfs and ultra-diffuse galaxies; (6) reduce the baryon fraction in Milky-Way-type galaxies; (7) explain ultra-luminous X-ray sources in the outskirts of galaxies; (8) trigger star formation in dwarfs via AGN. As we will see in Section V.1, IMPBH production could be naturally triggered by the thermal history of the Universe Carr et al. 2019b. This would lead to other observational signatures: they would generate extreme-mass-ratio inspiral merger events detectable by LISA; they would tidally disrupt white dwarfs much more rapidly than main-sequence stars, leading to luminous flares and short time-scale nuclear transients Law-Smith et al. 2017; they would induce microlensing of extended radio sources Inoue and Chiba 2003; Inoue et al. 2013.
V Unified Primordial Black Hole Scenario
In this Section we describe a particular scenario in which PBHs naturally form with an extended mass function and provide a unified explanation of some of the conundra discussed above. The scenario is discussed in detail Reference Carr et al. 2019b and based on the idea that the thermal history of the Universe leads to dips in the sound-speed and therefore enhanced PBH formation at scales corresponding to the electroweak phase transition (), the QCD phase transition (), the pion-plateau () and annihilation (). This scenario requires that most of the dark matter is in PBHs formed at the QCD peak and is marginally consistent with the constraints discussed in Section III, even though this suggests that the QCD window cannot provide all the dark matter for a monochromatic PBH mass function.
Reheating at the end of inflation fills the Universe with radiation. In the standard model, it remains dominated by relativistic particles with an energy density decreasing as the fourth power of the temperature. As time increases, the number of relativistic degrees of freedom remains constant until around GeV, when the temperature of the Universe falls to the mass thresholds of the Standard Model particles. The first particle to become non-relativistic is the top quark at GeV, followed by the Higgs boson at GeV, the boson at GeV and the boson at GeV. At the QCD transition at around MeV, protons, neutrons and pions condense out of the free light quarks and gluons. A little later the pions become non-relativistic and then the muons, with annihilation and neutrino decoupling occur at around MeV.
Thus is exponentially sensitive to and the present CDM fraction for PBHs of mass is
There are many inflationary models and they predict a variety of shapes for . Some of them produce an extended plateau or dome-like feature in the power spectrum. For example, this applies for two-field models like hybrid inflation Clesse and García-Bellido 2015 and even some single-field models like Higgs inflation Ezquiaga et al. 2018; García-Bellido and Ruiz Morales 2017, although this may not apply for the minimal Higgs model Bezrukov et al. 2018. Instead of focussing on any specific scenario, Reference Carr et al. 2019b assumes a quasi-scale-invariant spectrum,
V.2 Resolving the Fine-Tuning Problem
The origin of the baryon asymmetry of the Universe (BAU) and the nature of dark matter are two of the most challenging problems in cosmology. The usual assumption is that high-energy physics generates the baryon asymmetry everywhere simultaneously via out-of-equilibrium particle decays or a first-order phase transition at very early times. García-Bellido et al. García-Bellido et al. 2019 propose a scenario in which the gravitational collapse of large inhomogeneities at the QCD epoch (invoked above) can resolve both these problems. The collapse to a PBH is induced by fluctuations of a light spectator scalar field and accompanied by the violent expulsion of surrounding material, which might be regarded as a sort of “primordial supernova”. This provides the ingredients for efficient baryogenesis around the collapsing regions, with the baryons subsequently propagating to the rest of the Universe, and naturally explains why the observed BAU is of order the PBH collapse fraction and why the baryons and dark matter have comparable densities.
We now discuss this proposal in more detail. The gravitational collapse of the mass within the QCD Hubble horizon can be extremely violent Musco and Miller 2013 with particles being driven out as a relativistically expanding shock-wave and acquiring energies a thousand times their rest mass from the gravitational potential energy released by the collapse. Such high density hot spots provide the out-of-equilibrium conditions required to generate a baryon asymmetry Sakharov 1967 through the well-known electroweak sphaleron transitions responsible for Higgs windings around the electroweak vacuum Asaka et al. 2004. In this process, the charge-parity (CP) symmetry violation of the Standard Model suffices to generate a local baryon-to-photon ratio of order one. The hot spots are separated by many horizon scales but the outgoing baryons propagate away from the hot spots at the speed of light and become homogeneously distributed well before BBN. The large initial local baryon asymmetry is thus diluted to the tiny observed global BAU.
The energy available for hot spot electroweak baryogenesis can be estimated as follows. Energy conservation implies that the change in kinetic energy due to the collapse of matter within the Hubble radius to the Schwarzschild radius of the PBH is
This proposal naturally links the PBH abundance to the baryon abundance and the BAU to the PBH collapse fraction (). The spectator field mechanism for producing the required curvature fluctuations also avoids the need for a fine-tuned peak in the power spectrum, which has long been considered a major drawback of PBH scenarios. One still needs fine-tuning of the mean field value to produce the observed values of and (i.e. ). However, the stochasticity of the field during inflation ensures that Hubble volumes exist with all possible field values and this means that one can explain the fine-tuning by invoking a single anthropic selection argument. The argument is discussed in Reference Carr et al. 2019a and depends on the fact that only a small fraction of patches will have the PBH and baryon abundance required for galaxies to form. In most patches the field is too far from the slow-roll region to produce either PBHs or baryons, leading to radiation universes without any dark matter or matter-antimatter asymmetry. In other (much rarer) patches, PBHs are produced too copiously, leading to rapid accretion of most of the baryons, as might have happened in ultra-faint dwarf galaxies. This anthropic selection effect may therefore explain the observed values of and .
VI Primordial Black Holes versus Particle Dark Matter
Presumably most particle physicists would prefer the dark matter to be elementary particles rather than PBHs, although there is still no direct evidence for this. However, even if this transpires to be the case, we have seen that PBHs could still play an important cosmological rôle, so we must distinguish between PBHs providing some dark matter and all of it. This also applies for the particle candidates. Nobody would now argue that neutrinos provide the dark matter but they still play a hugely important rôle in astrophysics. Therefore one should not necessarily regard PBHs and particles as rival candidates. Both could exist and we end by discussing two scenarios in this spirit. The first assumes that particles dominate the dark matter but that PBHs still provide an interesting interaction with them. The second involves the notion that evaporating black holes leave stable Planck-mass (or even sub-Planck-mass) relics, although such relics are in some sense more like particles than black holes.
Apart from the early formation of spikes around PBHs which are light enough to arise very early, WIMP accretion around heavier PBHs can also occur by secondary infall Bertschinger 1985. This leads to a different halo profile, yielding a constraint for the same values of and . While Adamek et al. Adamek et al. 2019 have derived this limit for solar-mass PBHs, the argument can be extended to much bigger masses, even up to the values associated with stupendously large black holes Carr et al. 2020b. The constraint at intermediate comes from the integrated effect of a population of such objects and is flat:
Figure 7 shows constraints on for WIMP masses of GeV, GeV and TeV. The falling part at low is associated with halos formed after dark-matter kinetic decoupling (when the kinetic energy of the WIMPs is important); the flat part is associated with halos formed by secondary infall at later times (when the kinetic energy can be neglected). No bound can be placed above the mass where the lines intersects the incredulity limit
VI.2 Planck-Mass Relics
One would now require the density to be less than , which strengthens the original limit by a factor of . The lower mass limit arises because PBHs generated before reheating are diluted exponentially. The upper mass limit arises because PBHs larger than this dominate the total density before they evaporate, in which case the final cosmological baryon-to-photon ratio is determined by the baryon-asymmetry associated with their emission. Limit (VI.4) still applies even if there is no inflationary period but then extends all the way down to the Planck mass.
It is usually assumed that such relics would be undetectable apart from their gravitational effects. However, Lehmann et al. Lehmann et al. 2019 have recently pointed out that they may carry electric charge, making them visible to terrestrial detectors. They evaluate constraints and detection prospects and show that this scenario, if not already ruled out by monopole searches, can be explored within the next decade with planned experiments.
VII Conclusions
While the study of PBHs has been a minority interest for most of the last 50 years, they have become the focus of increasing attention recently. This is strikingly reflected in the annual publication rate on the topic, which has now risen to several hundred. While the evidence for PBHs is far from conclusive, there is a growing appreciation of their many potential rôles in cosmology and astrophysics. This is why we have stressed the possible evidence for PBHs in this review rather than just the constraints.
PBHs have been invoked for three main purposes: (1) to explain the dark matter; (2) to generate the observed LIGO/Virgo coalescences; (3) to provide seeds for the SMBHs in galactic nuclei. The discussion in Section V suggests that they could also explain several other observational conundra, as well as alleviating some of the well-known problems of the CDM scenario. So PBHs could play an important cosmological rôle even if most of the dark matter transpires to be elementary particles.
As regards (1), there are only a few mass ranges in which PBHs could provide the dark matter. We have focused on the intermediate mass range , since this may be relevant to (2), but the sublunar range – g and the asteroid range – g have also been suggested. If the PBHs have a monochromatic mass function, the discussion in Section III suggests that only the lowest mass range is viable. However, the discussion in Section V indicates that this conclusion may not apply if they have an extended mass function.
As regards (2), while the possibility that the LIGO/Virgo sources could be PBHs is acknowledged by the gravitational-wave community, this is not the mainstream view. It is therefore important to stress that the next LIGO/Virgo runs should be able to test and possibly eliminate the PBH proposal. Indeed, it is remarkable that the three recent events GW190425, GW190814 and GW190521 fall precisely within regions (2), (4) and (5) of Figure 5. In any case, (2) does not require the PBHs to provide all the dark matter. If the PBHs have an extended mass function, the mass where the density peaks would be less than the mass which dominates the gravitational-wave signal.
As regards (3), there is no reason in principle why the maximum mass of a PBH should not be in the supermassive range, in which case it is almost inevitable that they could seed SMBHs and perhaps even galaxies themselves. The main issue is whether there are enough PBHs to do so but this only requires them to have a very low cosmological density. While the mainstream assumption is that galaxies form first, with the SMBHs forming in their nuclei through dynamical processes, this is not certain. A crucial question concerns the growth of such large black holes and this applies whether or not they are primordial.
Section V has described a scenario in which PBHs form with a bumpy mass function as a result of naturally occurring dips in the sound-speed at various cosmological epochs, this naturally explaining explain many cosmological conundra. This scenario also suggests that the cosmological baryon asymmetry may be generated by PBH formation at the QCD epoch, this naturally explaining the fine-tuning in the collapse fraction. This is not the mainstream view for the origin of the baryon asymmetry and this proposal require further investigation but this is a first attempt to address the much-neglected PBH fine-tuning problem. The possibility that evaporating PBHs leave stable relics opens up some of the mass range below g as a new world of compact dark-matter candidates waiting to be explored.