Primordial Black Holes as Dark Matter

Bernard Carr Department of Physics and Astronomy, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom    Florian Kühnel The Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, AlbaNova, SE–106 91 Stockholm, Sweden    Marit Sandstad Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE–106 91 Stockholm, Sweden
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The possibility that the dark matter comprises primordial black holes (PBHs) is considered, with particular emphasis on the currently allowed mass windows at g, g and . The Planck mass relics of smaller evaporating PBHs are also considered. All relevant constraints (lensing, dynamical, large-scale structure and accretion) are reviewed and various effects necessary for a precise calculation of the PBH abundance (non-Gaussianity, non-sphericity, critical collapse and merging) are accounted for. It is difficult to put all the dark matter in PBHs if their mass function is monochromatic but this is still possible if the mass function is extended, as expected in many scenarios. A novel procedure for confronting observational constraints with an extended PBH mass spectrum is therefore introduced. This applies for arbitrary constraints and a wide range of PBH formation models, and allows us to identify which model-independent conclusions can be drawn from constraints over all mass ranges. We focus particularly on PBHs generated by inflation, pointing out which effects in the formation process influence the mapping from the inflationary power spectrum to the PBH mass function. We then apply our scheme to two specific inflationary models in which PBHs provide the dark matter. The possibility that the dark matter is in intermediate-mass PBHs of is of special interest in view of the recent detection of black-hole mergers by LIGO. The possibility of Planck relics is also intriguing but virtually untestable.

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I Introduction

Primordial black holes (PBHs) have been a source of intense 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 has not yet been confirmed experimentally and there remain major conceptual puzzles associated with the process, with Hawking himself still grappling with these Hawking et al. (2016). Nevertheless, this discovery 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 may not exist!

PBHs smaller than about would have evaporated by now with many interesting cosmological consequences. 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) and Galactic Lehoucq et al. (2009) -ray backgrounds, antimatter in cosmic rays Barrau (2000), the annihilation line radiation from the Galactic centre Bambi et al. (2008), the reionisation of the pregalactic medium Belotsky and Kirillov (2015) and some short-period -ray bursts Cline et al. (1997). For more comprehensive references, see recent articles by Khlopov Khlopov (2010) and Carr et al. Carr et al. (2010a) and the book by Calmet et al. Calmet et al. (2014). However, there are usually other possible explanations for these features, so there is no definitive evidence for evaporating PBHs.

Attention has therefore shifted to the PBHs larger than , which are unaffected by Hawking radiation. Such PBHs might have various astrophysical consequences, such as providing seeds for the supermassive black holes in galactic nuclei Bean and Magueijo (2002), the generation of large-scale structure through Poisson fluctuations Afshordi et al. (2003) and important effects on the thermal and ionisation history of the Universe Ricotti et al. (2008). For a recent review, in which a particular PBH-producing model is shown to solve these and several other observational problems, see Ref. Dolgov (2016). But perhaps the most exciting possibility — and the main focus of this paper — is that they could provide the dark matter which comprises of the critical density, 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 (BBNS) constraint that baryons can have at most of the critical density Cyburt et al. (2003). They should therefore be classed as non-baryonic and from a dynamical perspective they behave like any other form of cold dark matter (CDM).

There is still no compelling evidence that PBHs provide the dark matter, but nor is there for any of the more traditional CDM candidates. One favored candidate is a Weakly Interacting Massive Particle (WIMP), such as the lightest supersymmetric particle Jungman et al. (1996) or the axion Preskill et al. (1983), but 30 years of accelerator experiments and direct dark-matter searches have not confirmed the existence of these particles Di Valentino et al. (2014). One should not be too deterred by this — after all, the existence of gravitational waves was predicted 100 years ago, the first searches began nearly 50 years ago Weber (1969) and they were only finally detected by LIGO a few months ago Abbott et al. (2016a). Nevertheless, even some theorists have become pessimistic about WIMPs Frampton (2016), so this does encourage the search for alternative candidates.

There was a flurry of excitement around the PBH dark-matter hypothesis in the 1990s, when the Massive Astrophysical Compact Halo Object (MACHO) microlensing results Alcock et al. (1997) suggested that the dark matter could be in compact objects of mass since alternative MACHO candidates could be excluded and PBHs of this mass might naturally form at the quark-hadron phase transition at Jedamzik (1998). Subsequently, however, it was shown that such objects could comprise only of the dark matter and indeed the entire mass range to was excluded from providing the dark matter Tisserand et al. (2007). At one point there were claims to have discovered a critical density of PBHs through the microlensing of quasars Hawkins (1993) but this claim was met with scepticism Zackrisson et al. (2003) and would seem to be incompatible with other lensing constraints. Also femtolensing of -ray bursts excluded PBHs Nemiroff et al. (2001), microlensing of quasars constrained PBHs Dalcanton et al. (1994) and millilensing of compact radio sources excluded PBHs Wilkinson et al. (2001) from explaining the dark matter. Dynamical constraints associated with the tidal disruption of globular clusters, the heating of the Galactic disc and the dragging of halo objects into the Galactic nucleus by dynamical friction excluded PBHs in the mass range above Carr and Sakellariadou (1999).

About a decade ago, these lensing and dynamical constraints appeared to allow three mass ranges in which PBHs could provide the dark matter Barrau et al. (2004): the subatomic-size range (), the sublunar mass range () and what is sometimes termed the intermediate-mass black hole (IMBH) range ().111This term is commonly used to describe black holes intermediate between those which derive from the collapse of ordinary stars and the supermassive ones which derive from general relativistic instability, perhaps the remnants of a first generation of Population III stars larger than . Here we use it in a more extended sense to include the black holes detected by LIGO. The lowest range may now be excluded by Galactic -ray observations Carr et al. (2016) and the middle range — although the first to be proposed as a PBH dark-matter candidate Chapline (1975) — is under tension because such PBHs would be captured by stars, whose neutron star or white-dwarf remnants would subsequently be destroyed by accretion Capela et al. (2013a). One problem with PBHs in the IMBH range is that such objects would disrupt wide binaries in the Galactic disc. It was originally claimed that this would exclude objects above Quinn et al. (2009) but more recent studies may reduce this mass Monroy-Rodríguez and Allen (2014), so the narrow window between the microlensing and wide-binary bounds is shrinking. Nevertheless, this suggestion is topical because PBHs in the IMBH range could naturally arise in the inflationary scenario Frampton et al. (2010) and might also explain the sort of massive black-hole mergers observed by LIGO Bird et al. (2016). The suggestion that LIGO could detect gravitational waves from a population of IMBHs comprising the dark matter was originally proposed in the context of the Population III “VMO” scenario by Bond & Carr Bond and Carr (1984). This is now regarded as unlikely, since the precursor stars would be baryonic and therefore subject to the BBNS constraint, but the same possibility applies for IMBHs of primordial origin.

Most of the PBH dark-matter proposals assume that the mass function of the black holes is very narrow (ie. nearly monochromatic). However, this is unrealistic and in most scenarios one would expect the mass function to be extended. In particular, this arises if they form with the low mass tail expected in critical collapse Niemeyer and Jedamzik (1998). Indeed, it has been claimed that this would allow PBHs somewhat above to contribute to both the dark matter and the -ray background Yokoyama (1998a). However, this assumes that the “bare” PBH mass function (ie. without the low-mass tail) has a monochromatic form and recently it has been realised that the tail could have a wider variety of forms if one drops this assumption Kuhnel et al. (2016). There are also many scenarios (eg. PBH formation from the collapse of cosmic strings) in which even the bare mass function may be extended.

This raises two interesting questions: (1) Is there still a mass window in which PBHs could provide all of the dark matter without violating the bounds in other mass ranges? (2) If there is no mass scale at which PBHs could provide all the dark matter for a nearly monochromatic mass function, could they still provide it by being spread out in mass? In this paper we will show how to address these questions for both a specific extended mass function and for the more general situation. As far as we are aware, this issue has not been discussed in the literature before and we will apply this methodology to the three mass ranges mentioned above. There are subtleties involved when applying differential limits to models with extended mass distributions, especially when the experimental bounds come without a mention of the bin size or when different limits using different bin sizes are combined. In order to make model-independent statements, we also discuss which physical effects need to be taken into account in confronting a model capable of yielding a significant PBH abundance with relevant constraints. This includes critical collapse, non-sphericity and non-Gaussianity, all of which we investigate quantitatively for two specific inflationary models. We also discuss qualitatively some other extensions of the standard model. In principle, this approach could constrain the primordial curvature perturbations even if PBHs are excluded as dark-matter candidates Josan et al. (2009).

The plan of this paper is as follows: In Sec. II we review the PBH formation mechanisms. In Sec. III we give a more detailed description of two inflationary models for PBH formation, later used to demonstrate our methodology. In Sec. IV we consider some issues which are important in going from the initial curvature or density power spectrum to the PBH mass function. Many of these issues are not fully understood, but they may have a large impact on the final mass function, so their proper treatment is crucial in drawing conclusions about inflationary models from PBH constraints. In Sec. V we review the constraints for PBHs in the non-evaporating mass ranges above g, concentrating particularly on the weakly constrained region around . In Sec. VI we explore how an extended mass function can still contain all the dark matter. In order to make model-independent exclusions, we develop a methodology for applying arbitrary constraints to any form of extended mass function. In Sec. VII we discuss the new opportunities offered by gravitational-wave astronomy and the possible implications of the LIGO events. In Sec. VIII we summarise our results and outline their implications for future PBH searches.

Ii Introduction to PBH 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 role Hawking (1971); Carr and Hawking (1974), yielding a rough connection between the PBH mass and the horizon mass at formation:


Hence PBHs could span an enormous mass range: those formed at the Planck time () would have the Planck mass (), whereas those formed at would be as large as , comparable to the mass of the holes thought to reside in galactic nuclei. By contrast, black holes forming at the present epoch (eg. in the final stages of stellar evolution) could never be smaller than about . In some circumstances PBHs may form over an extended period, corresponding to a wide range of masses. Even if they form at a single epoch, their mass mass spectrum could still extend much below the horizon mass due to “critical phenomena” Gundlach (1999, 2003); Niemeyer and Jedamzik (1998, 1999); Shibata and Sasaki (1999); Musco et al. (2005, 2009); Musco and Miller (2013); Kuhnel et al. (2016), although most of the PBH density would still be in the most massive ones. We return to these points in Sec. IV.

ii.1 Formation Mechanisms

The high density of the early Universe is a necessary but not sufficient condition for PBH formation. One possibility is that there were large primordial inhomogeneities, so that overdense regions could stop expanding and recollapse. In this context, Eq. (1) can be replaced by the more precise relationship Carr et al. (2010a)


Here is a numerical factor which depends on the details of gravitational collapse. A simple analytical calculation suggests that it is around during the radiation era Carr (1975), although the first hydrodynamical calculations gave a somewhat smaller value Nadezhin et al. (1978). The favoured value has subsequently fluctuated as people have performed more sophisticated computations but now seems to have settled at a value of around Green et al. (2004).

It has been claimed that a PBH cannot be much larger than the value given by Eq. (1) at formation, else it would be a separate closed Universe rather than a part of our Universe Carr and Hawking (1974); Harada and Carr (2005). While there is a separate-Universe scale and Eq. (1) does indeed give an upper limit on the PBH mass, the original argument is not correct because the PBH mass necessarily goes to zero on the separate-Universe scale Kopp et al. (2011); Carr and Harada (2015). However, the effective value of in Eq. (2) could exceed in some circumstances. In particular, if a PBH grows as a result of accretion, its final mass could well be larger than the horizon mass at formation.

As discussed in numerous papers, the quantum fluctuations arising in various inflationary scenarios are a possible source of PBHs. In some of these scenarios the fluctuations generated by inflation are “blue” (i.e. decrease with increasing scale) and this means that the PBHs form shortly after reheating Carr and Lidsey (1993); Carr et al. (1994); Leach et al. (2000); Kohri et al. (2007). Others involve some form of “designer” inflation, in which the power spectrum of the fluctuations — and hence PBH production — peaks on some scale Hodges and Blumenthal (1990); Ivanov et al. (1994); Yokoyama (1997, 1998b, 1998c); Kawasaki and Yanagida (1999); Yokoyama (1999); Easther and Parry (2000); Kanazawa et al. (2000); Blais et al. (2003, 2002); Barrau et al. (2003); Chongchitnan and Efstathiou (2007); Nozari (2007); Saito et al. (2008); Lyth and Wands (2002); Lyth et al. (2003); Kasuya and Kawasaki (2009); Kohri et al. (2013); Kawasaki et al. (2013); Josan and Green (2010); Belotsky et al. (2014); Clesse (2011); Kodama et al. (2011); Bugaev and Klimai (2009a); Cheng et al. (2016). In other scenarios, the fluctuations have a “running index”, so that the amplitude increases on smaller scales but not according to a simple power law Garcia-Bellido et al. (1996); Randall et al. (1996); Stewart (1997a, b); Lidsey et al. (2002); Easther et al. (2004); Lyth et al. (2006); Kohri et al. (2008); Bugaev and Klimai (2008); Alabidi and Kohri (2009); Leach et al. (2000); Drees and Erfani (2012a, 2011, b); Kuhnel et al. (2016). PBH formation may also occur due to some sort of parametric resonance effect before reheating Taruya (1999); Bassett and Tsujikawa (2001); Green and Malik (2001); Finelli and Khlebnikov (2001); Kawaguchi et al. (2008); Kawasaki et al. (2007); Frampton et al. (2010). In this case, the fluctuations tend to peak on a scale associated with reheating. This is usually very small but several scenarios involve a secondary inflationary phase which boosts this scale into the macroscopic domain. Recently there has been a lot of interest in the formation of intermediate-mass PBHs in the “waterfall” scenario Frampton et al. (2010); Bugaev and Klimai (2011a); Clesse and García-Bellido (2015); Kawasaki and Tada (2015) and the generation of PBH dark matter in supergravity inflation models is discussed in Ref. Kawasaki et al. (2016). It has been claimed Young and Byrnes (2015) that any multiple-field inflationary model which generates enough PBHs to explain the dark matter is ruled out because it also generates an unacceptably large isocurvature perturbation due to the inherent non-Gaussianities in these models. We will discuss this in more detail in Sec. IV.

Whatever the source of the inhomogeneities, PBH formation would be enhanced if there was a sudden reduction in the pressure — for example, at the QCD era Jedamzik (1997); Widerin and Schmid (1998); Jedamzik and Niemeyer (1999) — or if the early Universe went through a dustlike phase at early times as a result of either being dominated by non-relativistic particles for a period Khlopov and Polnarev (1980); Polnarev and Khlopov (1981, 1982) or undergoing slow reheating after inflation Khlopov et al. (1985); Carr et al. (1994). Another possibility is that PBHs might have formed spontaneously at some sort of phase transition, even if there were no prior inhomogeneities, for example from bubble collisions Crawford and Schramm (1982); Hawking et al. (1982); Kodama et al. (1982); La and Steinhardt (1989); Moss (1994); Konoplich et al. (1998, 1999) or from the collapse of cosmic strings Hogan (1984); Hawking (1989); Polnarev and Zembowicz (1991); Garriga and Sakellariadou (1993); Caldwell and Casper (1996); Cheng and Li (1996); MacGibbon et al. (1998); Hansen et al. (2000); Nagasawa (2005), necklaces Matsuda (2006); Lake et al. (2009) or domain walls Berezin et al. (1983); Caldwell et al. (1996); Khlopov et al. (2000); Rubin et al. (2000, 2001); Dokuchaev et al. (2005). Braneworld scenarios with a modified-gravity scale of may lead to the production of lunar-mass PBHs Inoue and Tanaka (2003).

ii.2 Collapse Fraction

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 .The current density parameter (in units of the critical density) associated with unevaporated PBHs which form at a redshift or time is roughly related to by Carr (1975)


where is the density parameter of the cosmic microwave background (CMB) and we have used Eq. (1). The factor arises because the radiation density scales as  , whereas the PBH density scales as  . Any limit on therefore places a constraint on  . The parameter must be interpreted with care for PBHs which have already evaporated, since they no longer contribute to the cosmological density. Note that Eq. (3) assumes that the PBHs form in the radiation-dominated era, in which case is necessarily small.

We can determine the relationship (3) more precisely for the standard CDM model, in which the age of the Universe is , the Hubble parameter is Ade et al. (2015a) and the time of photon decoupling is Hinshaw et al. (2009). If the PBHs have a monochromatic mass function, then the fraction of the Universe’s mass in PBHs at their formation time is related to their number density at and by Carr et al. (2010a)


where we have used Eq. (2) and is the number of relativistic degrees of freedom at PBH formation. is normalised to its value at around since it does not increase much before that in the Standard Model and that is the period in which most PBHs are likely to form. The current density parameter for PBHs which have not yet evaporated is therefore


which is a more precise form of Eq. (3). Since always appears in combination with  , we follow Ref. Carr et al. (2010a) in defining a new parameter


where and can be specified very precisely but is rather uncertain.

An immediate constraint on comes from the limit on the CDM density parameter, with , so the upper limit is Dunkley et al. (2009). This implies


However, this relationship must be modified if the Universe ever deviates from the standard radiation-dominated behaviour. The expression for may also be modified in some mass ranges if there is a second inflationary phase Green and Liddle (1997) or if there is a period when the gravitational constant varies Barrow and Carr (1996) or there are extra dimensions Sendouda et al. (2006).

Any proposed model of PBH formation must be confronted with constraints in the mass range where the predicted PBH mass function peaks. These constraints are discussed in Sec. V and expressed in terms of the ratio of the current PBH mass density to that of the CDM density:


where we assume . We can also write this as


where is the PBH mass fraction at matter-radiation equality. This procedure will be applied in Sec. IV to two specific models, the axion-like curvaton model and running-mass inflation (specified in detail in the next section). We will also demonstrate the influence of critical collapse, non-sphericity and non-Gaussianity on the PBH dark-matter fraction.

ii.3 Extended Versus Monochromatic Mass Functions

As regards the representation of constraints for extended mass functions, one approach is to integrate the differential mass function over a mass window of width at each , giving the continuous function


Here can be interpreted as the number density of PBHs in the mass range (). One can then define the quantities


which correspond to the mass density and dark-matter fraction, respectively, in the same mass range. This is equivalent to breaking the mass up into bins and has the advantage that one can immediately see where most of the mass is. If one knows the expected mass function, one can plot or in the same figure as the constraints to see which one is strongest. Alternatively, one can define , and as integrated values for PBHs with mass larger or smaller than . However, these are only simply related to the functions defined above for a power-law spectrum.

The above representations are problematic if the width of the mass function is less then . Indeed, one might define an extended mass function as one with a width larger than , in which case we have seen that one can always specify an effective value at each mass-scale. The situation for monochronatic mass functions is generally more complicated, although it is straightforward if the mass function is a delta function (ie. exactly monochromatic). The problem arises if it is nearly monochromatic (ie. with width ). This is discussed in more detail in Ref. Carr et al. (2016).

Although a precisely monochromatic mass spectrum is clearly unphysical, one would only expect the mass function to be very extended if the PBHs formed from exactly scale-invariant density fluctuations Carr (1975) or from the collapse of cosmic strings Hawking (1989). In this case, one has


This is not expected in the inflationary scenario but in most circumstances the spectrum would still be extended enough to have interesting observational consequences, since the constraint on one mass-scale may also imply a constraint on neighbouring scales. We have mentioned that the monochromatic assumption fails badly if PBHs form through critical collapse and the way in which this modifies the form of has been discussed by Yokoyama Yokoyama (1998a). This will be discussed in more detail in Sec. IV.

Iii Specific Models

For a large fraction of PBH formation scenarios, an extended feature in the primordial density power spectrum is generic. This leads to a non-monochromatic PBH mass spectrum. As demonstrated in the next section, even an initially peaked spectrum of density perturbation will acquire a significant broadening. In the light of the recent detection of merging black holes in the intermediate-mass range by the LIGO and Virgo collaboration Abbott et al. (2016a, b), we consider below some models which are capable of producing PBHs in this or one of the other two possible mass intervals. We look first at running-mass inflation and the axion-like curvaton model. These are chosen because their parameters can be tuned so as to give a peak in PBH production in any of the three ranges we would like to investigate. They also have the advantage of not being ruled out by non-Gaussianity effects. We will determine the mass functions explicitly in Sec. IV and confront them with recent observational bounds in Sec. VI. We will also briefly review scale-invariant mass functions.

iii.1 Running-Mass Inflation

PBH formation in the running-mass model Stewart (1997a, b) has been intensively studied in Refs. Drees and Erfani (2012a, 2011, b); see also Ref. Leach et al. (2000) for a discussion of constraints and Ref. Kuhnel et al. (2016) for an investigation of critical collapse in these models. Perhaps the simplest realisation of this is the inflationary potential


where is the scalar field and is a constant. There exists a plethora of embeddings of this model in various frameworks, such as hybrid inflation Linde (1994), which lead to different functions . These yield distinct expressions for the primordial density power spectra whose variance can be recast into the general form Drees and Erfani (2011)


where the spectral indices and are given by


with real parameters , , and .

As the spectral index and amplitude of the primordial power spectrum at the pivot scale have been measured Komatsu et al. (2011); Ade et al. (2014, 2015a) to be and , respectively, models without running cannot produce an appreciable PBH abundance. Furthermore, with the measurement of Ade et al. (2015a), running alone cannot give sufficient increase of the power spectrum at early times. One needs to include at least a running-of-running term, this being subject only to the weak constraint Ade et al. (2014, 2015a). In order to avoid overproduction of PBHs on the smallest scales, a running-of-running-of-running parameter is also needed, so a minimal viable model has all three parameters , and .

iii.2 Axion-Curvaton Inflation

The original curvaton scenario was introduced by Lyth et al. Lyth and Wands (2002); Lyth et al. (2003). The model we investigate here is a variant of this and was introduced by Kasuya and Kawasaki Kasuya and Kawasaki (2009) (cf. Kohri et al. (2013); Kawasaki et al. (2013)). It describes a curvaton moving in an axion or natural inflation-type potential. For a recent study of PBH production in this model, including critical collapse, see Ref. Kuhnel et al. (2016).

In this model, the inflaton is the modulus and the curvaton is related to the phase of a complex superfield . In practice, the inflaton rolls down a potential of the form


where is the Hubble rate and is a constant derived from combinations of parameters in supergravity theory. Because of its large mass, the inflaton rolls fast towards its minimum . After this, the curvaton becomes well-defined as and this becomes the primary degree of freedom of the superfield. The curvaton is assumed to move in an axion-like potential, similar to that of natural inflation Freese et al. (1990),


where the last equality holds when is close to its minimum at and the curvaton mass is . The particular shape of this potential, which preserves the shift-symmetry peculiar to axions, is what makes this curvaton axion-like.

The power spectrum of primordial perturbations is generated by the combined effect of the inflaton and the curvaton,


The first term is dominant on large scales (small ) and the second on small scales (large ). The inflaton perturbation is assumed to yield a near scale-invariant spectrum with , in accordance with CMB observations Komatsu et al. (2011); Ade et al. (2014, 2015a). This contribution should dominate up to at least . We define as the crossing scale at which the curvaton and inflaton contributions to the power spectrum are equal, and as the scale at which the inflaton reaches its minimum, , so that the curvaton becomes well-defined. and are the horizon masses when these scales cross the horizon. PBHs cannot form before these horizon-crossing times, because the perturbations are too small when , and no curvaton perturbations exist for . Here can be found explicitly from the parameters of the theory and has the value


where is the curvaton spectral index and is the number of radiative effective degrees of freedom at the scale . Throughout our considerations, we will follow Ref. Kawasaki et al. (2013) in assuming and .

PBHs cannot form from the inflationary density perturbations, as these are constrained by CMB observations. By contrast, when the curvaton power spectrum becomes dominant, it can have much more power and still evade the CMB bounds, allowing the production of large PBHs. However, the curvaton perturbations are assumed not to collapse to PBHs before the inflaton has decayed to standard-model particles. Hence PBHs can only form with the minimum mass , these being produced at or after the curvaton decay time. The exact value for the decay time — and hence the minimum mass  — is not known, but it should be smaller than the horizon mass at BBNS (), in order not to interfere with this process, and smaller than to yield PBH production. In Ref. Kawasaki et al. (2013), and are considered, so we will do the same here.

It can be shown that the variance of the density power spectrum due to the curvaton perturbations in a model with an axion-like curvaton is Kawasaki et al. (2013)


for a horizon mass . For , we assume the curvaton power spectrum which can transform into PBHs is zero. Due to inhomogeneities of the curvaton decay, this is not strictly true. However, as in Ref. Kawasaki et al. (2013), we will take this to be a reasonable approximation. The curvaton spectral index is controlled by the parameter :


By setting , we can obtain a sufficiently blue power spectrum of curvature perturbations for the curvaton to produce PBHs at some scale without violating the CMB constraints. The minimum mass , defined by the decay time of the curvaton, protects the model from overproducing PBHs at very small scales in spite of the blue power spectrum. The functions and are given by


which are the lower incomplete gamma function and the exponential integral, respectively.

iii.3 Scale-invariant Mass Functions

For a scale-invariant PBH mass function (ie. with independent of ), one has and so the largest contribution to the dark-matter density comes from the smallest holes. One therefore needs to specify the lower mass cut-off and then check that the implied value of on scales above does not violate any of the other PBH constraints. For , the strongest constraint is likely to come from the -ray background limit Carr et al. (2010a).

Cosmic strings produce PBHs with a scale-invariant-mass function, with the lower cut-off being associated with the symmetry-breaking scale Hogan (1984); Hawking (1989); Polnarev and Zembowicz (1991); Garriga and Sakellariadou (1993); Caldwell and Casper (1996); Cheng and Li (1996); MacGibbon et al. (1998); Hansen et al. (2000); Nagasawa (2005). Since decreases with increasing and from the -ray background limit, such PBHs cannot provide the dark matter unless exceeds g, which seems implausible. On the other hand, if evaporating black holes leave stable Planck-mass relics, these might also contribute to the dark matter. The discussion in Sec. VI.D shows that the -ray background limit excludes relics from providing all of the dark matter unless .

Iv Extended mass functions: criticality, non-sphericity and non-Gaussianity

The simplest model of PBH formation assumes that the mass spectrum is monochromatic — with mass comparable to the horizon mass at formation — and that the PBHs derive from the collapse of overdensities which are spherical and have a Gaussian distribution. Given the large uncertainties in the PBH formation process and the plethora of models for it, this naïve approach has been adopted in many papers. This includes Ref. Carr et al. (2010a), which discusses the numerous constraints on the PBH abundance as a function of mass on the assumption that the mass spectrum has a width of order . As the bounds on the allowed PBH density at each epoch have become more refined, and since PBHs of intermediate mass may even have been observed Bird et al. (2016), a more precise treatment of the formation process is necessary. In this section we therefore go beyond the usual assumptions and attempt a more realistic treatment.

Figure 1: Effect of critical collapse on the fraction as a function of black-hole mass in units of solar mass for a nearly monochromatic mass function (left panel), and for axion-like curvaton (red) as well as running-mass inflation (blue) (right panel). The latter models are specified in Sec. III. Solid lines show the PBH abundances for the horizon-mass collapse estimate, dashed lines show the same models with critical collapse included.

iv.1 Monochromaticity

The monochromatic assumption is a good starting point if the spread in mass is narrow enough, but this is not very likely for most inflationary models which produce PBHs Clesse and García-Bellido (2015); Kohri et al. (2013); Drees and Erfani (2011). As reviewed below, although models exist with a narrow spectrum, such as the axion curvaton model Kawasaki et al. (2013) or some phase transition models Kodama et al. (1982); Jedamzik and Niemeyer (1999); Khlopov et al. (2000), more realistic treatments — involving critical collapse — yield extended mass functions. This applies even if the PBHs derive from a very narrow feature in the original power spectrum, as illustrated in Fig. 1. This can lead to the misinterpretation of observational constraints, since these are mostly derived for monochromatic mass functions. We shall discuss how to treat constraints for extended mass functions in Sec. VI (cf. a related discussion in Ref. Carr et al. (2016)).

iv.2 Critical Collapse

Early research assumed that a sufficiently large overdensity re-enters the horizon and collapses to a black hole of order the horizon mass almost immediately. However, under the assumption of spherical symmetry, it has been shown Choptuik (1993); Koike et al. (1995); Niemeyer and Jedamzik (1999); Gundlach (1999, 2003) that the functional dependence of the PBH mass on and follows the critical scaling relation


for . The constant , the threshold and the critical exponent all depend on the nature of the fluid containing the overdensity at horizon-crossing Musco and Miller (2013). Careful numerical work Musco et al. (2005, 2009); Musco and Miller (2013) has confirmed the scaling law (23). In particular, Fig. 1 of Ref. Musco et al. (2009) suggests that it applies over more than orders of magnitude in density contrast.

Soon after the first studies of critical collapse Choptuik (1993); Koike et al. (1995), its application to PBH formation was studied and incorporated in concrete models Niemeyer (1998); Green and Liddle (1999). The conclusion was that the horizon-mass approximation was still reasonably good. However, this conclusion depended on the assumption that the mass function would otherwise be monochromatic. As shown in Kuhnel et al. (2016) 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.

Regarding Eq. (23), it has been shown that the critical exponent is independent of the perturbation profile Neilsen and Choptuik (2000); Musco and Miller (2013), though and may depend on this. Throughout this work we shall follow the pioneering work of Ref. Carr (1975) and later Ref. Green et al. (2004) in applying the Press–Schechter formalism Press and Schechter (1974) for spherical collapse. As a first approximation, we assume a Gaussian perturbation profile,


which accords well with current CMB measurements Ade et al. (2015b). As detailed below, this should be generalized to include simple non-Gaussian profiles, as even the slight non-Gaussianities permitted by observation may alter the final PBH abundance Young and Byrnes (2013, 2015). The quantity is the variance of the primordial power spectrum of density perturbations generated by the model of inflation. In radiation-dominated models, which are the focus of this paper, repeated studies have shown that Koike et al. (1995); Niemeyer and Jedamzik (1999); Musco et al. (2005, 2009); Musco and Miller (2013) and Musco et al. (2005, 2009); Musco and Miller (2013). In accordance with Ref. Niemeyer and Jedamzik (1998), we set .

As mentioned in Sec. II, a convenient measure of how many PBHs are produced is the ratio of the PBH energy density to the total energy density at PBH formation. Using the Press–Schechter formalism, we can express this as


where we assume . We have numerically confirmed the validity of this approximation for our purposes but some subtleties are involved here. These concern the validity of the Press–Schechter formalism, the use of the density rather than curvature power spectrum, and the upper integration limit. We discuss these points more thoroughly below but none of them changes the main signatures of critical collapse, which are the broadening, lowering and shifting. Nevertheless, these small effects should be accounted for in obtaining precise constraints on inflationary models.

Following Ref. Niemeyer and Jedamzik (1998), we next derive the PBH initial mass function . We define this as the black-hole number per normalised mass interval , where , within each collapsing horizon:222Note the slight difference in the definition of the initial mass function compared to the one in Ref. Niemeyer and Jedamzik (1998). The latter is expressed in terms of a logarithmic derivative, whereas Eq. (26) involves an ordinary derivative, corresponding to an extra inverse power of on the r.h.s.


Here the factor is required to normalise so that . In deriving Eq. (26), we have used the Gaussian profile (24) for the amplitude of the fluctuations, the last relation holding for . When we apply the critical-collapse scenario in the subsequent sections, we will follow the procedure outlined at the end of Sec. II in Ref. Kuhnel et al. (2016). In particular, given a certain horizon mass, this means that we must consider how the PBH mass distribution is spread around the initial mass function . Each contribution has to be evolved from the time of PBH formation to the time of radiation-matter equality, so that the dark-matter fraction , given by Eq. (9), can be evaluated by summing over the individual contributions from each horizon mass.

Examples of how critical collapse affects the abundance and mass distribution of PBHs can be seen in Fig. 1. The left panel shows the application of the above scheme for critical collapse to a nearly monochromatic feature in the initial density perturbations. As can be seen, the resulting mass function is far from monochromatic and yields PBHs over a wide range of masses. In the right panel of Fig. 1 our scheme has been applied to initial perturbation spectra from two inflationary models, the axion-like curvaton model and a running-mass model, the details of which have been discussed in Sec. III. In all cases, the change in shape and mass range due to critical collapse is clearly visible. This is particularly true for the axion-like curvaton model or the critical-collapse version of the monochromatic function where the slope for masses smaller then its initial peak, is entirely due to critical collapse. Fig. 7 of Ref. Carr et al. (2016) shows the same behaviour, ie. the power-law tails towards lower masses are equivalent.

iv.3 Non-sphericity

The above results rely on the assumption of spherical collapse. The inclusion of non-sphericity is significantly more complicated and has not been subject to extensive numerical studies of the kind in Ref. Musco and Miller (2013). Inspired by related work on gravitational collapse in the context of galactic halo formation. where it has been known for a long time (cf. Sheth et al. (2001)) that non-zero ellipticity leads to possibly large effects, Ref. Kuhnel and Sandstad (2016) shows that this also holds for PBH mass spectra. One essential consequence is that the threshold value is increased and can generically be approximated as


with being the threshold value for spherical collapse and the amplitude of the density power spectrum at the given scale. In Ref. Sheth et al. (2001) the above result was derived and numerically confirmed for a limited class of cosmologies, mostly relevant to structure formation, where and were found to be and , respectively. In particular, this does not include the case of ellipsoidal collapse in a radiation-dominated model, which is most relevant for PBH formation.

In Ref. Kuhnel and Sandstad (2016) it was argued that a relation of the form of Eq. (27) should hold for ellipsoidal gravitational collapses in arbitrary environments. Schematically, the argument goes as follows: The collapse starts along the smallest axis and thereafter the longer axes collapses faster than linearly Bond and Myers (1996). The mass dependence of the overdensity suggests that the density perturbation in the primary collapsing sphere — with radius equal to the shortest axis — will be smaller by . Here accounts for the difference in mass of a sphere and an ellipsoid. By considering Gaussian-distributed overdensities, it can be shown that the expectation values for the shape of overdensities are Doroshkevich (1970); Bardeen et al. (1986); Bond and Myers (1996)


where is the ellipticity and the prolateness, which runs from in the maximally prolate case to in the maximally oblate case. Since the collapse is initiated along the shortest axis, it may be compared to that of the largest sphere contained within it. The volume of the ellipsoid is then . Taking the ellipsoid to be of uniform density, combined with the demand that the density threshold should be exceeded in the enclosed sphere, leads to an increase in mass . As the density contrast associated with a given mass roughly scales as in the PBH case Carr (1975), to first order in the ellipticity this leads to Eq. (27) with and .

In more realistic situations, these values will not be exact and a thorough numerical investigation is needed to precisely determine the change of the threshold for fully relativistic non-spherical collapse. In particular, the above derivation assumes a uniform density in the ellipsoid, whereas a density profile with higher density in the central regions seems more realistic. This should lead to a less pronounced effect. However, the effect will always be an increase in the threshold, leading to a general suppression of the mass spectrum.

The left panel of Fig. 2 shows the effect of non-sphericity on the PBH mass fraction , which is given as a function of black-hole mass for the axion-like curvaton model (red) and the running-mass model (blue). (These will be discussed in detail in Sec. VI.) For both models, the solid lines involve only critical collapse, while the dotted lines also include the effect of ellipticity according to our naïve model with and . One can see a significant global shift downwards. This general behaviour is expected because non-spherical effects raise the formation threshold, making it harder for PBHs to from.

Although the exact amount of suppression due to non-sphericity is not known, the functional form of the mass spectrum is essentially unchanged. Also there is degeneracy with the effects of other parameters. Therefore we will not include this effect explicitly when comparing models with observational constraints in Sec. VI. Nevertheless, if one wants to use PBH constraints on concrete inflationary theories, the suppression due to non-sphericity should be properly accounted for. In fact, for precise constraints, numerical relativistic modelling of ellipticity is required.

Note also that if the overdensities are non-Gaussian, the ellipticity is no longer given by Eq. (28) and one should pay attention to the interplay between these two effects. We will not take this into account here, as an exact knowledge of the non-Gaussianities is needed, but the main effect will again be to change the amount by which the amplitude is shifted. These uncertainties will hence be degenerate with uncertainties in the ellipticity effects.

iv.4 Non-Gaussianity

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, 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. (2015a), this leads to a constraint on the non-Gaussianity parameters for a PBH-producing theory of roughly . 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. Ref. Chisholm (2006) claims this isocurvature production is generic to PBHs, since they represent very large perturbations. However, the analysis was probably not appropriate for PBHs from a generic source. It is important to note that these constraints can change somewhat if the non-Gaussianities are non-local or non-scale-invariant. They are also weakly dependent on the PBH mass, so care should be taken in making definite statements about particular theories when the magnitude of the non-Gaussianities lies close to the bound (see Ref. Young and Byrnes (2015) for details).

Even if PBHs are produced in the multi-field models, they do not give isocurvature modes on the CMB scale if the CMB and PBH scale are sufficiently decoupled (so that one effectively has a single-field model on the CMB scale). This is because the CMB-scale isocurvature modes are caused by the non-Gaussian correlation between the CMB and PBH scales. Although the non-Gaussian constraints , apply in the isocurvature case, these parameters should be evaluated as a correlation between the CMB and PBH scales and this is generally unrelated to the values of and on the CMB scale Tada and Yokoyama (2015).

In order to be realistic, non-Gaussianities should be taken properly into account when considering a model for PBH production. If a certain model with a manifest inflationary origin is considered, the non-Gaussianity parameters should first be obtained. If their values are higher than the above bound, the model is already excluded as a producer of PBH dark matter. In fact, for a realistic treatment of PBH dark-matter production from an inflationary model, this should be the first constraint to consider, as no further investigation of the model is necessary if the non-Gaussianity is too large. If it falls below this limit, it should still be taken into account when calculating abundances. Examples of how this is done in practice can be found in Refs. Young and Byrnes (2013, 2015).

We show an example of the effect of non-Gaussianity in the right panel of Fig. 2 for the axion-like curvaton (red) and running-mass inflation (blue) models (to be specified in Sec. VI). Again, the solid curves include only critical collapse, while the dot-dashed curves are for . Here, the lower and upper curves correspond to the plus and minus signs, respectively. We have checked that the inclusion of does not have a large effect. The chosen values are of course not precisely accurate for these models; rather they demonstrate the qualitative effect of non-Gaussianity five times larger than the allowed values. In general, the effect is similar to that seen for ellipticity in the left panel of Fig. 2. However, for reasonable values of the non-Gaussianity, the effect is much smaller.

To obtain more precise results, the full nature of the non-Gaussianity should be accounted for. In this work, however, rather than focussing on particular models, we will consider the possibility of non-constrained windows for PBHs to comprise all of the dark matter. We will therefore neglect non-Gaussian effects in our subsequent analysis. More importantly, although not visible directly in our plots, constraints from non-Gaussianity-induced isocurvature must also be considered. This excludes at the outset the production of PBH dark matter in multi-field models. However, the two models discussed in this paper are not affected by this claim: our running-mass model is not multi-field and our axion-curvaton model does not produce curvaton fluctuations on the CMB scale.

Figure 2: Fraction as a function of black-hole mass in units of solar mass for axion-like curvaton (red) as well as running-mass inflation (blue); the models are specified in Sec. VI. Left panel: Effect of non-sphericity (dotted lines), with parameters and (see Eq. (27)). Right panel: Effect of non-Gaussianity (dot-dashed curves), where we chose (lower curve) and (upper curve). In both cases the parameter choices are made for illustrative purpose (see main text for details). Critical collapse is assumed throughout the plots.

iv.5 Miscellaneous Caveats

In addition to the issues mentioned above, some more technical issues concerning the PBH mass spectrum expected from inflationary models have been discussed in the literature. However, none of them are expected to lead to effects which are quantitatively large.

First, we have chosen to use the Press–Schechter formalism for obtaining the mass spectrum from the perturbations. Alternatively, one could calculate the mass fraction using peaks theory Bardeen et al. (1986). Recently, there has been some discussion Young et al. (2014) of whether these two formalisms predict different values for . If precise constraints on inflationary models are to be obtained from PBH production, this issue should be resolved. However, the signature of this effect would not be a shift or broadening, so the critical collapse effects would be distinguishable from this. In addition, the difference will presumably be much less than the uncertainty in the non-spherical collapse situation. As the issue is currently unresolved, we use Press–Schechter here but attention should be paid to this in the future.

A second (related) subtlety concerns the cloud-in-cloud problem Jedamzik (1995), which involes the overcounting of small PBHs contained in larger PBHs. This would lead to suppression at the low-mass end of the spectrum. This might counteract the effect of critical collapse but would not occur for spectra deriving from very localised features in the perturbation spectrum. How to account for this is not settled and it is better addressed using peaks theory. Here we will ignore this issue but for precise constraints it should be dealt with properly.

Third, there is the claim Carr and Hawking (1974) that an overdense region represents a separate closed Universe rather than a part of our Universe if exceeds . In integral (25) we have extended the upper integration above , in contrast to what was done in Ref. Niemeyer and Jedamzik (1998). However, Ref. Kopp et al. (2011) claims that there is no separate-Universe constraint. This is because the meaning of the density perturbation needs to be specified very carefully on large scales: necessarily goes to zero on the separate-univese scale, even though the curvature perturbation diverges. A subsequent discussion Carr and Harada (2015) agrees with this conclusion but stresses that the separate-Universe scale is still interesting because it relates to the maximum mass of a PBH forming at any epoch. In any case, the integrand for large values of is so small that this does not make much difference in practice.

Fourth, there are in principle two choices of power spectra from the inflationary models: the curvature power spectrum and the density power spectrum . We choose the latter as this seems to be more accurate for an in-depth discussion (cf. Young et al. (2014)). This might also link with the separate-Universe and cloud-in-cloud issues Kopp et al. (2011); Harada et al. (2013). Furthermore, Young et al. Young et al. (2014) have argued that using instead of to calculate the PBH abundance may yield errors due to the spurious influence of super-horizon modes.

Fifth, when we evolve our PBH densities through the radiation-dominated epoch, we use a simplified model of cosmic expansion, assuming complete radiation-domination until matter-radiation equality. A more refined treatment should be applied if we are trying to exclude an inflationary model on account of the overproduction of dark matter. However, for the purposes of this paper, this assumption has very little impact. Also this type of modelling would be problematic if one produced more PBHs than there is cold dark matter, as this would change the time of matter-radiation equality, leading to other problems. Careful consideration of this effect may be needed when considering otherwise unconstrained models for the production of evaporating PBHs.

Sixth, once produced, PBHs not only lose mass through Hawking radiation but can also grow by accreting matter and/or radiation or by merging with other PBHs. While Hawking radiation is completely negligible for intermediate-mass PBHs, their growth can be very important in the matter-dominated epoch Carr and Hawking (1974); Bicknell and Henriksen (1978a, b). For instance, it has been conjectured that PBHs with mass of could provide seeds for the supermassive black holes of up to in the centers of galaxies Greene (2012). However, this involves a growth of many orders of magnitude and careful numerical integration is required to study this, allowing for the dilution of the PBHs due to cosmic expansion and the merger of the smaller ones originating from critical collapse. The clustering of PBHs will also have significant effects on their merger rates Mészáros (1975); Carr (1975); Meszaros (1980). In particular, Chisholm Chisholm (2006) showed that the clustering would produce an inherent isocurvature perturbation and used this to constrain the viability of PBHs as dark matter. Later he studied the effect of clustering on mergers Chisholm (2011) and found that these could dominate over evaporation, causing PBHs with mass below to combine and form heavier long-lived black holes rather than evaporating. So far, no compelling study of this effect has been carried out for a realistic mass spectrum, so we will not include it in our discussion below.

V Summary of constraints on monochromatic non-evaporated black holes

We now review the various constraints associated with PBHs which are too large to have evaporated yet, updating the equivalent discussion which appeared in Carr et al. Carr et al. (2010a). All the limits assume that PBHs cluster in the Galactic halo in the same way as other forms of CDM. In this case, the fraction of the halo in PBHs is related to by Eq. (8). Our limits on are summarised in Fig. 3, which is an updated version of Fig. 8 of Ref. Carr et al. (2010a). A list of approximate formulae for these limits is given in Tab. 1. Both Fig. 3 and Tab. 1 are intended merely as an overview and are not exact. A more precise discussion can be found in the original references. Many of the constraints depend on other physical parameters, not shown explicitly. In general, we show only the most stringent constraints in each mass range, although constraints are sometimes omitted when they are contentious. Further details of these limits and similar figures can be found in other papers: for example, Tab. 1 of Josan et al. Josan et al. (2009), Fig. 4 of Mack et al. Mack et al. (2007), Fig. 9 of Ricotti et al. Ricotti et al. (2008), Fig. 1 of Capela et al. Capela et al. (2013a) and Fig. 1 of Clesse & Garcia-Bellido Clesse and García-Bellido (2016). We group the limits by type and discuss those within each type in order of increasing mass. Since we are also interested in the mass ranges for which the dark-matter fraction is small, where possible we express each limit in terms of an analytic function over some mass range. We do not treat Planck-mass relics, since the only constraint on these is that they must have less than the CDM density, but we do discuss them further in Sec. VI.

Figure 3: Constraints on for a variety of evaporation (magenta), dynamical (red), lensing (cyan), large-scale structure (green) and accretion (orange) effects associated with PBHs. The effects are extragalactic -rays from evaporation (EG) Carr et al. (2010a), femtolensing of -ray bursts (F) Barnacka et al. (2012), white-dwarf explosions (WD) Graham et al. (2015), neutron-star capture (NS) Capela et al. (2013a), Kepler microlensing of stars (K) Griest et al. (2014), MACHO/EROS/OGLE microlensing of stars (ML) Tisserand et al. (2007); Calchi Novati et al. (2013) and quasar microlensing (broken line) (ML) Mediavilla et al. (2009), survival of a star cluster in Eridanus II (E) Brandt (2016), wide-binary disruption (WB) Quinn et al. (2009), dynamical friction on halo objects (DF) Carr and Sakellariadou (1999), millilensing of quasars (mLQ) Wilkinson et al. (2001), generation of large-scale structure through Poisson fluctuations (LSS) Afshordi et al. (2003), and accretion effects (WMAP, FIRAS) Ricotti et al. (2008). Only the strongest constraint is usually included in each mass range, but the accretion limits are shown with broken lines since they are are highly model-dependent. Where a constraint depends on some extra parameter which is not well-known, we use a typical value. Most constraints cut off at high due to the incredulity limit. See the original references for more accurate forms of these constraints.

v.1 Evaporation Constraints

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 less than  g Carr et al. (2016). PBHs with could still be relevant to the dark-matter problem, although there is a strong constraint on from observations of the extragalactic -ray background Page and Hawking (1976). Those 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 , there is no jet emission.

For , one can neglect the change of mass altogether and the time-integrated spectrum of photons from each PBH is just obtained by multiplying the instantaneous spectrum by the age of the Universe . From Ref. Carr et al. (2010a) this gives


This peaks at with a value independent of . The number of background photons per unit energy per unit volume from all the PBHs is obtained by integrating over the mass function:


where and specify the mass limits. For a monochromatic mass function, this gives


and the associated intensity is


with units . This peaks at with a value . The observed extragalactic intensity is where lies between (the value favoured in Ref. Sreekumar et al. (1998)) and (the value favoured in Ref. Strong et al. (2004)). Hence putting gives Carr et al. (2010a)


In Fig. 3 we plot this constraint for . The Galactic -ray background constraint could give a stronger limit Carr et al. (2016) but this requires the mass function to be extended and depends sensitively on its form, so we do not discuss it here. The reionising effects of g PBHs might also be associated with interesting constraints Belotsky and Kirillov (2015).

v.2 Lensing Constraints

Constraints on MACHOs with very low come from the femtolensing of -ray bursts. Assuming the bursts are at a redshift , early studies Marani et al. (1999); Nemiroff et al. (2001) excluded in the mass range but more recent work Barnacka et al. (2012) gives a limit which can be approximated as


The precise form of this limit is shown is Fig. 3.

Microlensing observations of stars in the Large and Small Magellanic Clouds probe the fraction of the Galactic halo in MACHOs of a certain mass range Paczynski (1986). The optical depth of the halo towards LMC and SMC, defined as the probability that any given star is amplified by at least at a given time, is related to the fraction by


for the S halo model Alcock et al. (2000). Although the initial motivation for microlensing surveys was to search for brown dwarfs with , the possibility that the halo is dominated by these objects was soon ruled out by the MACHO experiment Alcock et al. (1998). However, MACHO observed events and claimed that these were consistent with compact objects of contributing of the halo mass Alcock et al. (2000). This raised the possibility that some of the halo dark matter could be PBHs formed at the QCD phase transition Jedamzik (1997); Widerin and Schmid (1998); Jedamzik and Niemeyer (1999). However, later studies suggested that the halo contribution of PBHs could be at most 10% Hamadache et al. (2006). The EROS experiment obtained more stringent constraints by arguing that some of the MACHO events were due to self-lensing or halo clumpiness Tisserand et al. (2007) and excluded MACHOs from dominating the halo. Combining the earlier MACHO Alcock et al. (2001) results with the EROS-I and EROS-II results extended the upper bound to Tisserand et al. (2007). The constraints from MACHO and EROS about a decade ago may be summarised as follows:


Similar limits were obtained by the POINT-AGAPE collaboration, which detected microlensing events in a survey of the Andromeda galaxy Calchi-Novati et al. (2005). Since then further limits have come from the OGLE experiment. The OGLE-II data Wyrzykowski et al. (2009); Novati et al. (2009); Wyrzykowski et al. (2010) yielded somewhat weaker constraints but data from OGLE-III Wyrzykowski et al. (2011a) and OGLE-IV Wyrzykowski et al. (2011b) gave stronger results for the high mass range:


We include this limit in Fig. 3, and Tab. 1 but stress that it depends on some unidentified detections being attributed to self-lensing. Later (comparable) constraints combining EROS and OGLE data were presented in Ref. Calchi Novati et al. (2013). Recently Kepler data has improved the limits considerably in the low mass range Griest et al. (2013, 2014):


It should be stressed that many papers give microlensing limits on but it is not easy to combine these limits because they use different confidence levels. Also one must distinguish between limits based on positive detections and null detections. The only positive detection in the high mass range comes from Dong et al. Dong et al. (2007).

Early studies of the microlensing of quasars Dalcanton et al. (1994) seemed to exclude all the dark matter being in objects with . However, this limit does not apply in the CDM picture and so is not shown in Fig. 3. More recent studies of quasar microlensing suggest a limit Mediavilla et al. (2009)


However, this limit might not apply in the CDM picture, and furthermore the paper states only three data points, so the limit is shown as a dashed line in Fig. 3. In this context, Hawkins Hawkins (1993) once claimed evidence for a critical density of jupiter-mass objects from observations of quasar microlensing and associated these with PBHs formed at the quark-hadron transition. However, the status of his observations is no longer clear Zackrisson et al. (2003), so this is not included in Fig. 3. Millilensing of compact radio sources Wilkinson et al. (2001) gives a limit which can be approximated as


Though weaker than other constraints in this mass range, we include this limit in Fig. 3 and Tab. 1. The lensing of fast radio bursts could imply strong constraints in the range above but these are not shown in Fig. 3, since they are only potential limits Muñoz et al. (2016).

v.3 Dynamical Constraints

The effects of PBH collisions on astronomical objects — including the Earth Jackson and Ryan (1973) — have been a subject of long-standing interest Carr and Sakellariadou (1999). For example, Zhilyaev Zhilyaev (2007) has suggested that collisions with stars could produce -ray bursts and Khriplovich et al. Khriplovich et al. (2008) have examined whether terrestrial collisions could be detected acoustically. Gravitational-wave observatories in space might detect the dynamical effects of PBHs. For example, eLISA could detect PBHs in the mass range by measuring the gravitational impulse induced by any nearby passing one Adams and Bloom (2004); Seto and Cooray (2004). However, we do not show these constraints in Fig. 3 since they are only potential.

Roncadelli et al. Roncadelli et al. (2009) have suggested that halo PBHs could be captured and swallowed by stars in the Galactic disk. The stars would eventually be accreted by the holes, producing a lot of radiation and a population of subsolar black holes which could only be of primordial origin. They argue that every disc star would contain such a black hole if the dark matter were in PBHs smaller than  g and the following analytic argument Carr et al. (2010a) gives the form of the constraint. Since the time-scale on which a star captures a PBH scales as , requiring this to exceed the age of the Galactic disc implies


which corresponds to a lower limit on the mass of objects providing the dark matter. A similar analysis of the collisions of PBHs with main-sequence stars, red-giant cores, white dwarfs and neutron stars by Abramowicz et al. Abramowicz et al. (2009) suggests that collisions are too rare for or produce too little power to be detectable for g. However, in a related argument, Capela et al. have constrained PBHs as dark-matter candidates by considering their capture by white dwarfs Capela et al. (2013b) and neutron stars Capela et al. (2013a). The survival of these objects implies a limit which can be approximated as


This is similar to Eq. (41) at the high-mass end, the upper cut-off at  g corresponding to the condition . There is also a lower cut-off at  g because PBHs lighter than this will not have time to consume the neutron stars during the age of the Universe. This argument assumes that there is dark-matter at the centers of globular clusters and is sensitive to the dark-matter density there (taken to be  GeVcm). Pani & Loeb Pani and Loeb (2014) have argued that this excludes PBHs from providing the dark matter throughout the sublunar window, although this has been disputed Capela et al. (2014); Defillon et al. (2014). In fact, the dark-matter density is limited to much lower values than assumed above for particular globular clusters Ibata et al. (2013); Bradford et al. (2011).

Binary star systems with wide separation are vulnerable to disruption from encounters with MACHOs Bahcall et al. (1985); Weinberg et al. (1987). Observations of wide binaries in the Galaxy therefore constrain the abundance of halo PBHs. By comparing the results of simulations with observations, Yoo et al. Yoo et al. (2004) originally ruled out MACHOs with from providing the dark matter. However, a careful analysis by Quinn et al. Quinn et al. (2009) of the radial velocities of these binaries found that the widest-separation one was spurious, so that the constraint became


It flattens off above because the encounters are non-impulsive there. Although not shown in Fig. 3, more recent studies by Monroy-Rodriguez & Allen reduce the mass at which can be from to or even Monroy-Rodríguez and Allen (2014). The narrow window between the microlensing lower bound and the wide-binary upper bound is therefore shrinking and may even have been eliminated altogether (see Sec. VI).

A variety of dynamical constraints come into play at higher mass scales. These have been studied by Carr and Sakellariadou Carr and Sakellariadou (1999) and apply providing there is at least one PBH per galactic halo. This corresponds to the condition


which they term the “incredulity limit”. An argument similar to the binary disruption one shows that the survival of globular clusters against tidal disruption by passing PBHs gives a limit (not shown in Fig. 3)


although this depends sensitively on the mass and the radius of the cluster. The limit flattens off above because the encounter becomes non-impulsive (cf. the binary case). The upper limit of on the mass of objects dominating the halo is consistent with the numerical calculations of Moore Moore (1993). In a related limit, Brandt Brandt (2016) claims that a mass above is excluded by the fact that a star cluster near the centre of the dwarf galaxy Eridanus II has not been disrupted by halo objects. His constraint can be written as


where the density of the dark matter at the center of the galaxy is taken to be , the velocity dispersion there is taken to be , and the age of the star cluster is taken to be . The second expression in Eq. (46) was not included in Ref. Brandt (2016) but is the incredulity limit, corresponding to having one black hole for the dwarf galaxy.

Halo objects will overheat the stars in the Galactic disc unless one has Carr and Sakellariadou (1999)


where the lower expression is the incredulity limit. The upper limit of agrees with the more precise calculations by Lacey and Ostriker Lacey and Ostriker (1985), although they argued that black holes with could explain some features of disc heating. Constraint (47) bottoms out at with a value . Evidence for a similar effect may come from the claim of Totani Totani (2009) that elliptical galaxies are puffed up by dark halo objects of . These disk-heating limits are not shown in Fig. 3 because they are smaller than other limits in this mass range.

Another limit in this mass range arises because halo objects will be dragged into the nucleus of our own Galaxy by the dynamical friction of the spheroid stars and halo objects themselves (if they have an extended mass function), this leading to excessive nuclear mass unless Carr and Sakellariadou (1999)


The last expression is the incredulity limit and first three correspond to the drag being dominated by spheroid stars (low ), halo objects (high ) and some combination of the two (intermediate ). The limit bottoms out at with a value but is sensitive to the halo core radius . Also there is a caveat here in that holes drifting into the nucleus might be ejected by the slingshot mechanism if there is already a binary black hole there Hut and Rees (1992). This possibility was explored by Xu and Ostriker Xu and Ostriker (1994), who obtained an upper limit of .

Each of these dynamical constraints is subject to certain provisos but it is interesting that they all correspond to an upper limit on the mass of the objects which dominate the halo in the range , the binary-disruption limit being the strongest. This is particularly relevant for constraining models in which the dark matter is postulated to comprise IMBHs. Apart from the Galactic disc and elliptical galaxy heating arguments of Refs. Lacey and Ostriker (1985); Totani (2009), it must be stressed that none of these dynamical effects gives positive evidence for MACHOs. Furthermore, none of them requires the MACHOs to be PBHs. Indeed, they could equally well be clusters of smaller objects Carr and Lacey (1987); Belotsky et al. (2015a) or Ultra-Compact Mini-Halos (UCMHs) Bringmann et al. (2012). This is pertinent in light of the claim by Dokuchaev et al. Dokuchaev et al. (2005) and Chisholm Chisholm (2006) that PBHs could form in tight clusters, giving a local overdensity well in excess of that provided by the halo concentration alone. It is also important to note that the UCMH constraints on the density perturbations may be stronger than the PBH limits in the higher-mass range Bringmann et al. (2012). This is relevant if one wants to consider the effect of an extended mass function.

v.4 Large-Scale Structure Constraints

Sufficiently large PBHs could have important consequences for large-scale structure formation because of the Poisson fluctuations in their number density. This effect was first pointed out by Mészáros Mészáros (1975) and subsequently studied by various authors Freese et al. (1983); Carr (1977); Carr and Silk (1983). In particular, Afshordi et al. Afshordi et al. (2003) used observations of the Lyman- forest to obtain an upper limit of about on the mass of any PBHs which provide the dark matter. Although this conclusion was based on numerical simulations, Carr et al. Carr et al. (2010a) obtained this result analytically and extended it to the case where the PBHs only provide a fraction of the dark matter. Since the Poisson fluctuation in the number of PBHs on a mass-scale grows between the redshift of CDM domination () and the redshift at which Lyman- clouds are observed () by a factor , the clouds will bind too early unless


The lower expression corresponds to having at least one PBH per Lyman- mass, so the limit bottoms out at with a value . The data from SDSS are more extensive McDonald et al. (2006), so the limiting mass may now be reduced. A similar effect can allow clusters of large PBHs to evolve into the supermassive black holes in galactic nuclei Carr and Rees (1984); Düchting (2004); Khlopov et al. (2005); if one replaces with and with in the above analysis, the limiting mass in Eq. (49) is reduced to  .

Recently, Kashlinksy has been prompted by the LIGO observations to consider the effects of the Poisson fluctuations induced by a dark-matter population of black holes Kashlinsky (2016). This can be seen as a special case of the general analysis presented above. However, he adds an interesting new feature to the scenario by suggesting that the black holes might also lead to the cosmic infrared background (CIB) fluctuations detected by the Spitzer/Akari satellites Kashlinsky et al. (2015); Helgason et al. (2016). This is because the associated Poisson fluctuations would allow more abundant early collapsed halos than in the standard scenario. It has long been appreciated that the CIB and its fluctuations would be a crucial test of any scenario in which the dark matter comprises the black-hole remnants of Population III stars Bond et al. (1986), but in this case the PBHs are merely triggering high-redshift star formation and not generating the CIB directly. We do not attempt to derive constraints on the PBH scenario from the CIB observations, since many other astrophysical parameters are involved.

v.5 Accretion Constraints

There are good reasons for believing that PBHs cannot grow very much during the radiation-dominated era. Although a simple Bondi-type argument suggests that they could grow as fast as the horizon Zel’dovich and Novikov (1967), this does not account for the background cosmological expansion and a fully relativistic calculation shows that such self-similar growth is impossible Carr and Hawking (1974); Bicknell and Henriksen (1978a, b). Consequently there is very little growth during the radiation era. The only exception might be if the Universe were dominated by a “dark energy” fluid with , as in the quintessence scenario, since self-similar black-hole solutions do exist in this situation Harada et al. (2008); Maeda et al. (2008); Carr et al. (2010b). This may support the claim of Bean and Magueijo Bean and Magueijo (2002) that intermediate-mass PBHs might accrete quintessence efficiently enough to evolve into the SMBHs in galactic nuclei.

Even if PBHs cannot accrete appreciably in the radiation-dominated era, massive ones might still do so in the period after decoupling and the Bondi-type analysis should then apply. The associated accretion and emission of radiation could have a profound effect on the thermal history of the Universe, as first analysed by Carr Carr (1981). This possibility was investigated in more detail by Ricotti et al. Ricotti et al. (2008), who studied the effects of such accreting PBHs on the ionisation and temperature evolution of the Universe. The emitted X-rays would produce measurable effects in the spectrum and anisotropies of the CMB. Using FIRAS data to constrain the first and WMAP data to constrain the second, they improve the constraints on by several orders of magnitude for  . The WMAP limit can be approximated as


where the last expression is not included in Ref. Ricotti et al. (2008) but corresponds to having one PBH on the scale associated with the CMB anisotropies; for modes, this is . The FIRAS limit can be approximated as


Although these limits appear to exclude down to masses as low as , they are model-dependent (spherically symmetric Bondi accretion etc.) and therefore not as secure as the dynamical ones. In particular, they depend on the duty-cycle parameter; we assume a smaller value for this than Ref. Carr et al. (2010a), which is why our limits are somewhat weaker. Mack et al. Mack et al. (2007) have considered the growth of large PBHs through the capture of dark-matter halos and suggested that their accretion could give rise to ultra-luminous X-ray sources. The latter possibility has also been explored by Kawaguchi et al. Kawaguchi et al. (2008).

In Ref. Eroshenko (2016a) it is claimed that dark matter will cluster around PBHs from very early times, causing sharp density spikes. These would be observable as bright -ray sources from the annihilation of dark-matter particles in orbit around the PBHs. Very stringent constraints on are obtained using Fermi-LAT data Abdo et al. (2010) for . As this constraint depends on the assumption that the dark-matter density is dominated by WIMPs, we do not include it here. However, such PBH limits must be taken into account if they are to be used to constrain models of inflation.

Mass range Constraint Source