Constraints on Primordial Black Holes
Abstract
We update the constraints on the fraction of the Universe going into primordial black holes (PBHs) over the mass range . Those smaller than would have evaporated by now due to Hawking radiation, so their abundance at formation is constrained by the effects of evaporated particles on big bang nucleosynthesis, the cosmic microwave background (CMB), the Galactic and extragalactic ray and cosmic ray backgrounds and the possible generation of stable Planck mass relics. PBHs larger than are subject to a variety of constraints associated with gravitational lensing, dynamical effects, influence on largescale structure, accretion and gravitational waves. We discuss the constraints on both the initial collapse fraction and the current fraction of the cold dark matter in PBHs at each mass scale but stress that many of the constraints are associated with observational or theoretical uncertainties and some are no longer applicable. We also consider indirect constraints associated with the amplitude of the primordial density fluctuations, such as secondorder tensor perturbations and distortions arising from the effect of acoustic reheating on the CMB, but these only apply if PBHs are created from the high peaks of nearly Gaussian fluctuations. Finally we discuss how the constraints are modified if the PBHs have an extended mass function, this being relevant if PBHs provide some combination of the dark matter, the LIGO/Virgo coalescences and the seeds for cosmic structure.
Contents
 I Introduction
 II Constraints on evaporating PBHs

III Constraints on nonevaporated PBHs
 III.1 Evaporation Constraints
 III.2 Lensing Constraints

III.3 Dynamical Constraints
 III.3.1 Collisions
 III.3.2 Neutron stars and white dwarfs
 III.3.3 Wide binaries
 III.3.4 Disruption of globular clusters and dwarf galaxies
 III.3.5 Disc heating
 III.3.6 Tidal streams
 III.3.7 Dynamical friction effect on halo objects
 III.3.8 Disruption and tidal distortion of galaxies in clusters
 III.3.9 Intergalactic PBHs
 III.4 Cosmic Structure Constraints
 III.5 Accretion and distortion constraints
 III.6 Gravitational Wave Constraints
 IV Constraints for monochromatic and extended PBH mass functions
 V Conclusion
I Introduction
i.1 Overview
Primordial black holes (PBHs) have been a source of intense interest for more than 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 but this discovery is generally recognised as one of the key developments in 20th century physics and Hawking was only led to it through contemplating the properties of PBHs. Indeed, those smaller than about would have evaporated by now but could still have many interesting cosmological consequences.
PBHs larger than are unaffected by Hawking radiation but have also attracted interest because of the possibility that they provide the dark matter which comprises of the critical density Ade et al. (2016), an idea that goes back to the earliest days of PBH research Chapline (1975). Since PBHs formed in the radiationdominated era, they are not subject to the wellknown big bang nucleosynthesis (BBN) constraint that baryons can have at most of the critical density Cyburt et al. (2003). They should therefore be classed as nonbaryonic and from a dynamical perspective they behave like any other cold dark matter (CDM) candidate. There is still no compelling evidence that PBHs provide the dark matter, but nor is there for any of the more traditional CDM candidates.
Despite the lack of evidence for them, PBHs have been invoked to explain numerous cosmological features. For example, evaporating PBHs have been invoked to explain 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 Okele and Rees (1980), the reionization of the pregalactic medium Belotsky and Kirillov (2015) and some shortperiod gammaray bursts Cline et al. (1997). Nonevaporating PBHs – even if they do not provide the dark matter – have been invoked to explain lensing effects, the heating of the stars in our Galactic disc, the seeds for the supermassive black holes in galactic nuclei Bean and Magueijo (2002), the generation of largescale structure through Poisson fluctuations Afshordi et al. (2003), effects on the thermal and ionization history of the Universe Ricotti et al. (2008) and – most recently – the LIGO/Virgo gravitational wave bursts Abbott et al. (2016a, b, c, 2018a). Indeed, if the PBHs have an extended mass function, they might explain a multitude of cosmic conundra Carr et al. (2019).
There are usually other possible explanations for these features, so there is still no definitive evidence for PBHs. Nevertheless, studying each of these effects allows one to place interesting constraints on the number of PBHs of mass and this in turn places constraints on the cosmological models which would generate them. We will be discussing all these constraints in this review. For evaporating PBHs these are usually expressed as constraints on the fraction of the universe collapsing into PBHs at their formation epoch, denoted as . Indeed, this must be less than over the entire mass range . For nonevaporating ones they are most usefully expressed as constraints on the fraction of the dark matter in PBHs, denoted as . Despite the hope of many people that PBHs could provide the dark matter, it was already clear a decade ago that there are only a few mass windows where this is possible Barrau et al. (2004): the asteroid range (), the sublunar range () and the intermediatemass range (). Since then, interest in PBHs as dark matter has increased – primarily because of the LIGO/Virgo events – and the lowest and highest mass windows have narrowed and perhaps been excluded entirely. On the other hand, the middle mass window has opened due to a weakening of some constraints. In practice, PBHs are expected to have an extended mass function but whether this makes it easier or more difficult for them to provide the dark matter has been the subject of some dispute.
In this review, we discuss all the PBH constraints, summarizing our results as excluded regions in the and planes. Similar diagrams have been produced by numerous authors – indeed almost every paper on PBHs now includes such a diagram – and we should particularly draw attention to the recent review of PBHs by Sasaki et al. Sasaki et al. (2018). However, we hope this review will be more comprehensive and uptodate than the others. For example, Ref. Sasaki et al. (2018) puts more emphasis on gravitational waves but does not discuss the limits on evaporating PBHs.
We should stress at the outset that the limits are constantly changing as a result of both observational and theoretical developments, so our claim for comprehensiveness may be shortlived. We will also discuss all claimed limits – even those which are no longer believed – because this is historically illuminating. Even wrong calculations are worth recording because this may avoid their being repeated in the future. In fact, all of the limits come with certain caveats and few can be regarded as 100% secure. (For example, the evaporation constraints assume the validity of Hawking radiation, even though there is still no direct observational evidence for this.) While we emphasize which limits have gone away or are questionable, we do not attempt to specify the confidence levels precisely, this sometimes being a rather contentious task anyway.
Since our primary purpose is the constraints, we will only include a brief discussion of the PBH formation mechanisms and there will little attempt to describe the more positive aspects of PBHs (i.e. the numerous ways in which they can probe the early Universe, high energy physics or even quantum gravity). However, this broader range of topics is covered in two other forthcoming reviews: one on PBHs and dark matter (Carr & Kuhnel, in preparation) and another on all aspects of PBHs (Carr, in preparation). So all these works should be regarded as complementary.
i.2 PBH formation
Black holes with a wide range of masses could have formed in the early Universe as a result of the great compression associated with the big bang Hawking (1971); Carr and Hawking (1974). A comparison of the cosmological density at a time after the big bang with the density associated with a black hole of mass suggests that such PBHs would have a mass of order
(1) 
This roughly corresponds to the Hubble mass at time . PBHs could thus 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 could never be smaller than about . The high density of the early Universe is a necessary but not sufficient condition for PBH formation. One possibility is that they formed from large inhomogeneities – either primordial, in the sense that they were fed into the initial conditions of the universe, or arising spontaneously in an initially smooth universe through quantum effects during an inflationary epoch. Another possibility is that some sort of phase transition may have enhanced PBH formation from primordial inhomogeneities or triggered it even if there were none. We now briefly discuss these various formation scenarios.
i.2.1 Collapse from inhomogeneities
Many PBH formation scenarios depend on the development of inhomogeneities of some kind. Overdense regions could then stop expanding and recollapse Carr (1975); Nadezhin et al. (1978); Bicknell and Henriksen (1979). Whatever the source of the fluctuations, they would need to be large in order to ensure that the overdense region can collapse against the pressure. The overdensity when the region enters the horizon must exceed some critical value , which a simple analytic argument suggests is around in the radiation era Carr (1975). More precise numerical and analytic calculations indicate values in the range to , depending on the spatial profile of the perturbed region Musco et al. (2005, 2009); Harada et al. (2013); Musco (2019); Germani and Musco (2019). Shibata and Sasaki Shibata and Sasaki (1999) describe the formation condition using the compaction function, whose maximum may provide a better criterion for collapse, and in this case the critical value lies in the small range between and , depending on the shape of the perturbed region Harada et al. (2015); Escrivà et al. (2019). Nakama et al. Nakama et al. (2014) performed extensive numerical calculations of PBH formation, starting with initial profiles characterized by five parameters but concluding that just two of them are important: the radial integral of curvature profile in the central region and the size of the transition region. The former takes a single critical value of when the transition region is small enough. The most natural source of the fluctuations would be quantum effects during inflation. Although any PBHs formed before the end of inflation (i.e. with mass exceeding about ) will be diluted exponentially, the inflationary fluctuations themselves could generate PBHs and in most scenarios the PBHs would form immediately after reheating. In the simplest (single scalar field) scenario, the inflationary fluctuations are expected to have a powerlaw form and – since the fluctuations are only on the CMB scale – they would need to be “blue” for PBH formation. However, the observed fluctuations are red, so one needs a more complicated scenario, in which one either has running of the spectral index or some feature in the power spectrum at the PBH scale. Numerous studies have been made of such scenarios (e.g. Ivanov et al. (1994); Yokoyama (1997); GarcíaBellido et al. (1996); Yokoyama (1998a, b); Kawasaki and Yanagida (1999); Kanazawa et al. (2000); Chongchitnan and Efstathiou (2007); Saito et al. (2008); Alabidi and Kohri (2009); Kohri et al. (2013, 2008); Motohashi and Hu (2017)) but we will not discuss them further here.
i.2.2 Soft equation of state
Whatever the source of the inhomogeneities, PBH formation would be enhanced if some phase transitions led to a sudden reduction in the pressure – for example, if the early Universe went through a dustlike phase at early times as a result of being dominated by nonrelativistic 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). In such cases, the effect of pressure in stopping collapse is unimportant and the probability of PBH formation just depends upon the fraction of regions which are sufficiently spherical to undergo collapse. For a given spectrum of primordial fluctuations, this means that there may just be a narrow mass range – associated with the period of the soft equation of state – in which the PBHs form. Recently Kokubu et al. Kokubu et al. (2018) have investigated PBH formation in a matterdominated era and identified the threshold for black hole formation by considering the finite speed of propagation for information. They obtain a collapse fraction which is larger by an order of magnitude than the one derived in earlier work Khlopov and Polnarev (1980); Polnarev and Khlopov (1981, 1982), assuming instantaneous propagation of information.
i.2.3 Critical collapse
When the density perturbation approaches the threshold value required for PBH formation, a critical phenomenon occurs in which the black hole mass scales as and therefore extends down to arbitrarily small scales Niemeyer and Jedamzik (1998, 1999). Here the exponent is independent of the density profile and just depends on the equation of state ( in the radiation case). Most of the density is in PBHs with the horizon mass but they also have a low mass tail with a power law Yokoyama (1998c); Green and Liddle (1999); Kribs et al. (1999). Because is sensitive to the equation of state (), even a slight reduction in can enhance PBH production. For example, this may happen at the Quantum Chromodynamics (QCD) era Jedamzik (1997); Widerin and Schmid (1998); Jedamzik and Niemeyer (1999) and applying the critical collapse analysis then predicts the mass function very precisely Byrnes et al. (2018).
i.2.4 Collapse of cosmic loops
In the cosmic string scenario, one expects some strings to selfintersect 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. Observations imply that must be less than of order . However, as discussed by many authors Hawking (1989); Polnarev and Zembowicz (1991); Hansen et al. (2000); Hogan (1984); Nagasawa (2005); Honma and Minakata (1991), there is still 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. Black holes might also form through the collapse of string necklaces Matsuda (2006); Lake et al. (2009).
i.2.5 Bubble collisions
Bubbles of broken symmetry might arise at any spontaneously broken symmetry epoch and many 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); La and Steinhardt (1989); Moss (1994); Konoplich et al. (1998, 1999). 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 Grand Unified Theory (GUT) epoch would have a mass of , those forming at the electroweak unification epoch would have a mass of , and those forming at the QCD (quarkhadron) phase transition would have mass of around . The production of PBHs from bubble collisions at the end of 1st order inflation has also been studied.
i.2.6 Collapse of scalar field
Cotner et al. have argued that a scalar condensate can form in the early Universe and collapse into Qballs before decaying Cotner and Kusenko (2017a). If the Qballs dominate the energy density for some period, the statistical fluctuations in their number density can lead to PBH formation Cotner and Kusenko (2017b). For a general charged scalar field, this can generate PBHs over the mass range allowed by observational constraints and with sufficient abundance to account for the dark matter and the LIGO observations. If the scalar field is associated with supersymmetry, the mass range must be below . The fragmentation of the inflaton into oscillons can also lead to PBH production, plausibly in the sublunar range Cotner et al. (2018, 2019).
i.2.7 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 Rubin et al. (2000, 2001); Dokuchaev et al. (2005). These PBHs would have a small mass for a thermal phase transition but they could be much larger if one invoked a nonequilibrium scenario. Indeed, they could then span a wide range of masses, with a fractal structure of smaller PBHs clustered around larger ones 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. (2019).
In some of these scenarios, the PBH mass spectrum is expected to be narrow and centred around the mass given by Eq. (1) with corresponding to the time at which the PBH scale reenters the horizon in the inflationary model or to the time of the relevant cosmological phase transition otherwise. In this case, one expects a monochromatic mass spectrum. However, there are some circumstances in which the spectrum would be extended and this means that the constraint on one massscale would also imply a constraint on neighbouring scales. For example, we have seen that PBHs may be much smaller than the horizon size if they form as a result of critical phenomena or during a matterdominated phase and in these cases their spectrum could extend well below the horizon mass. If the PBHs form from inflationary fluctuations, they will generically have a lognormal mass function Dolgov and Silk (1993). It used to be claimed that a PBH could not be much larger than the value given by Eq. (1) at formation else it would be a separate closed universe rather than part of our universe Carr and Hawking (1974). This interpretation is misleading because the PBH mass necessarily goes to zero when the size of the overdensity region becomes too large Kopp et al. (2011) but there is still an effective upper limit to the mass of PBH forming at a given epoch and this is of order the horizon size Harada and Carr (2005). Another point is that if the PBHs with close to the mass of those evaporating at the present epoch have a spread of masses , one would expect evaporation to lead to a residual spectrum with for Page and Hawking (1976).
i.3 Mass and density fraction of PBHs
In the following discussion, we assume that the standard Lambda Cold Dark Matter (CDM) model applies, with the age of the Universe being and the Hubble parameter being with Akrami et al. (2018). We also put . The Friedmann equation implies that the density and temperature during the radiation era are given by
(2) 
where counts the number of relativistic degrees of freedom. This can be integrated to give
(3) 
where and are normalised to their values at the start of the BBN epoch. Since we are only considering PBHs which form during the radiation era (the ones generated before inflation being diluted to negligible density), the initial PBH mass is related to the particle horizon mass by
(4) 
Here the penultimate expression applies exactly in a radiation Universe (for which ) and is a numerical factor (somewhat below ) which depends on the details of gravitational collapse. For much of the following discussion, we will assume that the PBHs all have the same mass or at least a mass width no larger than . This simplifies the analysis considerably and suffices providing we only require limits on the PBH abundance at particular values of . If the PBHs have an extended mass function, as is more plausible, the analysis of the constraints is more complicated and we discuss this case later.
Assuming adiabatic cosmic expansion after PBH formation, the ratio of the PBH number density to the entropy density, , is conserved. Using the relation , the fraction of the Universe’s mass in PBHs at their formation time is then related to their number density during the radiation era by
(5) 
where the subscript “” indicates values at the epoch of PBH formation and we have assumed today. is now normalised to the value of at around since it does not increase much before that in the Standard Model and most PBHs are likely to form before then. We can also express this as
(6) 
where is the current density parameter of the PBHs. This can also be obtained using the relation
(7) 
where is the density parameter of the cosmic microwave background (CMB) and we have used Eq. (1). The factor can be understood as arising because the radiation density scales as , whereas the PBH density scales as .
Since always appears in combination with , it is convenient to define a new parameter
(8) 
where and can be specified very precisely but is rather uncertain. Note that the relationship between and must be modified if the universe ever deviates from the standard radiationdominated behaviour – for example, if there is a dustlike stage for some extended early period or a second inflationary phase or if there are extra dimensions Sendouda et al. (2006) or if the gravitational constant ever varies Barrow (1992); Barrow and Carr (1996); Harada et al. (2002).
Any limit on places a constraint on or . For nonevaporating PBHs with , one constraint comes from requiring that be less than the cold dark matter (CDM) density, with Akrami et al. (2018), so the upper limit is . Much stronger constraints are associated with PBHs smaller than since these would have evaporated by now. For example, the ray limit implies and this is the strongest constraints on over all mass ranges. Other ones are associated with the generation of entropy, modifications to the cosmological production of light elements and the cosmic microwave background (CMB) anisotropies. There are also constraints below based on the (uncertain) assumption that evaporating PBHs leave stable Planck mass relics, an issue which is discussed later.
The constraints on were first brought together by Novikov et al. Novikov et al. (1979) 40 years ago. Besides the entropy, BBN, ray background and density limit, they included a strong constraint above associated with the upper limit on the CMB dipole anisotropy. An updated version of this diagram appeared in Ref. Carr et al. (1994) about 15 years later but was essentially the same, apart from the omission of the dipole constraint (which was outside the considered mass range) and the addition of the ‘relics’ constraint. Because of their historical interest, both diagrams are shown in Fig. 1. Subsequently, this diagram has frequently been revised as the relevant effects have been studied in greater detail. For example, Josan et al. Josan et al. (2009) produced a comprehensive version a decade ago and we also produced a version around that time in Ref. Carr et al. (2010a) (henceforth CKSY), covering both the evaporating and nonevaporating PBHs. However, there have been many further developments since then, on both the observational and theoretical front. An updated review of the constraints as of a few years ago can be found in Ref. Carr et al. (2016a) and the present discussion might be regarded as another (albeit more comprehensive) update. The important qualitative point of all such diagrams is that the value of must be tiny over every mass range, even if the PBH density is large today, so any cosmological model which would entail an appreciable fraction of the Universe going into PBHs is immediately excluded.
i.4 Evaporation of PBHs
The realization that PBHs might be small prompted Hawking to study their quantum properties. This led to his famous discovery Hawking (1975) that black holes radiate thermally with a temperature
(9) 
so they evaporate completely on a timescale
(10) 
More precise expressions are given below but this implies that only PBHs smaller than about would have evaporated by the present epoch, so Eq. (1) implies that this effect could be important only for ones which formed before . Since PBHs with a mass of around would be producing photons with energy of order at the present epoch, the observational limit on the ray background intensity at immediately implies that their density could not exceed about times the critical density Page and Hawking (1976). This suggests that there is little chance of detecting their final explosive phase at the present epoch, at least in the Standard Model of particle physics. Nevertheless, the ray background limit does not preclude PBHs having important cosmological effects Carr (1976).
i.4.1 Mass loss and evaporation timescale
From Eq. (9) the temperature of a black hole with mass is
(11) 
This assumes that the hole has no charge or angular momentum, since charge and angular momentum are usually assumed to be lost through quantum emission on a shorter timescale than the mass, although this may fail at the Planck scale Lehmann et al. (2019). The emission is not exactly blackbody but depends upon the spin and charge of the emitted particle, the average energy for neutrinos, electrons and photons being , and , respectively Page (1976).
The mass loss rate of an evaporating black hole can be expressed as
(12) 
Here is a measure of the number of emitted particle species, normalised to unity for a black hole with , this emitting only particles which are (effectively) massless: photons, three generations of neutrinos and antineutrinos, and gravitons. The contribution of each relativistic degree of freedom to is MacGibbon (1991)
(13)  
Holes in the mass range emit electrons but not muons, while those in the range also emit muons, which subsequently decay into electrons and neutrinos. The latter range is relevant for the PBHs which are completing their evaporation at the present epoch.
Once falls to around , a black hole can also begin to emit hadrons. However, hadrons are composite particles made up of quarks held together by gluons. For temperatures exceeding the QCD confinement scale, , one would expect these fundamental particles to be emitted rather than composite particles. Only pions would be light enough to be emitted below . Above this temperature, the particles radiated can be regarded as asymptotically free, leading to the emission of quarks and gluons MacGibbon and Webber (1990). Since there are quark degrees of freedom per flavour and gluon degrees of freedom, one would expect the emission rate (i.e., the value of ) to increase suddenly once the QCD temperature is reached. If one includes just , and quarks and gluons, Eq. (13) implies that their contribution to is , compared to the preQCD value of about . Thus the value of roughly quadruples, although there will be a further increase in at somewhat higher temperatures due to the emission of the heavier quarks. After their emission, quarks and gluons fragment into further quarks and gluons until they cluster into the observable hadrons when they have travelled a distance . This is much larger than the size of the hole, so gravitational effects can be neglected.
If we sum up the contributions from all the particles in the Standard Model up to , corresponding to , this gives . Integrating the mass loss rate over time then gives a lifetime
(14) 
The mass of a PBH evaporating at time after the big bang is then
(15) 
The critical mass for which equals the age of the Universe is denoted by . For the currently favoured age of , one finds
(16) 
where the last step assumes , the value associated with the temperature . At this temperature muons and some pions are emitted, so the value of accounts for this. Although QCD effects are initially small for PBHs with , only contributing a few percent, it should be noted that they become important once falls to
(17) 
since the peak energy becomes comparable to then. This means that an appreciable fraction of the timeintegrated emission from the PBHs evaporating at the present epoch goes into quark and gluon jet products.
It should be stressed that the above analysis is not exact because the value of in Eq. (15) should really be the weighted average of over the lifetime of the black hole. The more precise calculations of MacGibbon MacGibbon (1991); MacGibbon et al. (2008) give the slightly smaller value . However, the weighted average is well approximated by unless one is close to a particle mass threshold. For example, since the lifetime of a black hole of mass is roughly that of an black hole, one expects the value of to be overestimated by a few percent. This explains the small difference from MacGibbon’s calculation.
i.4.2 Primary and secondary emission
Particles injected from PBHs have two components: the primary component, which is the direct Hawking emission, and the secondary component, which comes from the decay of gauge bosons or heavy leptons and the hadrons produced by fragmentation of primary quarks and gluons. For example, the photon spectrum can be written as
(18) 
with similar expressions for other particles. In order to treat QCD fragmentation, CKSY use the PYTHIA code (version 6), a Monte Carlo event generator constructed to fit hadron fragmentation for centreofmass energies . The pioneering work of MacGibbon and Webber used the HERWIG code but obtained similar results MacGibbon and Webber (1990).
The spectrum of secondary photons is peaked around , because it is dominated by the decay of soft neutral pions which are practically at rest. The peak flux can be expressed as
(19) 
where is the fraction of the jet energy going into neutral pions of energy . This is of order and fairly independent of jet energy. If we assume that most of the primary particles have the average energy , the last factor becomes . Thus the energy dependence of Eq. (19) comes entirely from the factor and is proportional to the Hawking temperature. The emission rates of primary and secondary photons for four typical temperatures are shown in Fig. 2.
It should be noted that the timeintegrated ratio of the secondary flux to the primary flux increases rapidly once goes below . This is because a black hole with will emit quarks efficiently once its mass gets down to the value given by Eq. (17) and this corresponds to an appreciable fraction of its original mass. On the other hand, a PBH with somewhat larger initial mass, , will today have a mass Carr et al. (2010a)
(20) 
Here we have assumed , which should be a good approximation for since the value of only changes slowly above the QCD threshold. However, falls below for and if we assume that jumps discontinuously from to at this mass, then Eq. (20) must be reduced by a factor . The fact that this happens only for means that the fraction of the black hole mass going into secondaries falls off sharply above . The ratio of the secondary to primary peak energies and the ratio of the timeintegrated fluxes are shown in Fig. 3. Relation (20) is important if one is considering the effects of PBHs evaporating at the present epoch (e.g. the local ray background generated by PBHs in the Galactic halo).
Ii Constraints on evaporating PBHs
In this section we discuss the constraints on evaporating PBHs. The results are summarised in Fig. 4, which is an update of Fig. 6 of Ref. Carr et al. (2010a). The important point is that the BBN and photon background limits are the most stringent ones over almost the entire mass range . There is just a small range where the CMB anisotropy damping limit dominates and another range where the Galactic positron limit dominates. Obviously none of these constraints would apply if there were no Hawking radiation, the only constraint then coming from the condition . In this context, Raidal et al. have suggested that large primordial curvature fluctuations could collapse into horizonless exotic compact objects (ECOs) instead of PBHs Raidal et al. (2018a). In this case, they either do not evaporate at all or they do so much more slowly, so only the dynamical constraints apply and this opens up a much larger parameter space.
ii.1 Big bang nucleosynthesis
PBHs with and have a lifetime and therefore evaporate at the epoch of cosmological nucleosynthesis. The effect of these evaporations on BBN has been a subject of longstanding interest. Injection of highenergy neutrinos and antineutrinos Vainer and Naselskii (1978) changes the weak interaction freezeout time and hence the neutrontoproton ratio at the onset of BBN, which changes production. Since PBHs with evaporated during or after BBN, the baryontoentropy ratio at nucleosynthesis would be increased, resulting in overproduction of and underproduction of Miyama and Sato (1978). Emission of highenergy nucleons and antinucleons Zel’dovich et al. (1977) increases the primordial deuterium abundance due to capture of free neutrons by protons and spallation of . The emission of photons by PBHs with dissociates the deuterons produced in nucleosynthesis Lindley (1980). The limits associated with these effects are shown in Fig. 1.
Observational data on both the light element abundances and the neutron lifetime have changed since these early papers. Much more significant, however, have been developments in our understanding of the fragmentation of quark and gluon jets from PBHs into hadrons. Most of the hadrons created decay almost instantaneously compared to the timescale of nucleosynthesis but longlived ones (such as pions, kaons and nucleons) remain long enough in the ambient medium to leave an observable signature on BBN. These effects were first discussed by Kohri and Yokoyama Kohri and Yokoyama (1999) for the relatively low mass PBHs evaporating in the early stages of BBN but the analysis has now been extended to incorporate the effects of heavier PBHs evaporating after BBN Carr et al. (2010a), the hadrons and high energy photons from these PBHs further dissociating synthesised light elements.
ii.1.1 Modern studies
High energy particles emitted by PBHs modify the standard BBN scenario in three different ways: (1) high energy mesons and antinucleons induce extra interconversion between background protons and neutrons even after the weak interaction has frozen out in the background Universe; (2) high energy hadrons dissociate light elements synthesised in BBN, thereby reducing and increasing , , , and ; (3) high energy photons generated in the cascade further dissociate to increase the abundance of lighter elements even more.
The PBH constraints depend on three parameters: the initial baryontophoton ratio , the PBH initial mass or (equivalently) its lifetime , and the initial PBH number density normalised to the entropy density, . From Eq. (5) this is related to the initial mass fraction by
(21) 
The parameters , and all depend on but we suppose a monochromatic mass function in what follows. The initial baryontophoton ratio is set to the present one , after allowing for entropy production from PBH evaporations and photon heating due to annihilations.
Figure 5 summarises the results of these calculations (see CKSY for further details). PBHs with lifetime smaller than are free from BBN constraints because they evaporate well before weak freezeout and leave no trace. PBHs with and lifetime are constrained by the extra interconversion between protons and neutrons due to emitted mesons and antinucleons, which increases the freezeout ratio as well as the final abundance. For , corresponding to , hadrodissociation processes become important and the debris deuterons and nonthermally produced put strong constraints on . Finally, for , corresponding to , energetic neutrons decay before inducing hadrodissociation. Instead, photodissociation processes are operative and the most stringent constraint comes from overproduction of . However, even these effects become insignificant after .
These computations involved MonteCarlo simulations and included experimental and observational uncertainties in the baryon number and reaction and decay rates to obtain error bars in the light element abundances. These bounds should therefore be much more conservative than the ones obtained without these uncertainities. We adopted the upper bound on the abundance of rather than since it is more reliable. Because is both produced and destroyed in lowmass stars, the observed local value of does not imply an upper bound on the primordial component Kawasaki et al. (2005a). However, the observed in the solar system should give a reasonable upper bound on the primordial value because this increases with cosmic time Kawasaki et al. (2005b). In addition, we included both hadronic and radiative emissions; these can occur together and cancel each other, which reduces the constraint overlapping regions. Reference Acharya and Khatri (2020) claims stronger BBN constraints on evaporating PBHs but does not consider these effects.
For comparison, Fig. 5 shows the much weaker constraint imposed by the entropy production from evaporating PBHs Miyama and Sato (1978). These are labelled by the value of , which is the ratio of the entropy density after and before PBH evaporations. We also show as a broken line the BBN limits obtained earlier by Kohri and Yokoyama Kohri and Yokoyama (1999). The helium limit is weaker because the helium abundance is now known to be smaller, while the deuterium limit is stronger because hadrodissociation of helium produces more deuterium.
ii.1.2 Lithium7
Overproduction of by a factor is the most serious issue with the standard cosmological model Coc (2017). Astrophysical solutions have so far failed. Particle decay models reduce lithium via neutron injection but generically overproduce Coc et al. (2014), for which there are increasingly precise constraints. Therefore more exotic particle decays – involving neutrontriggered destruction of , followed by ray destruction of excess – have been suggested, notably those associated with leptophilic metastable massive particle of mass Goudelis et al. (2016). However, PBHs evaporating at might provide a plausible alternative. For example, PBHs with an extended mass function around could provide an early injection of neutrons, followed by soft rays. This combination could destroy some at , avoiding overproduction of , and subsequently destroy excess at .
ii.2 Cosmic microwave background
ii.2.1 Generation of entropy
The effects of PBH evaporations on the CMB (Cosmic Microwave Background) were first analysed by Zel’dovich and Starobinskii Zel’dovich et al. (1977). They pointed out that photons from PBHs smaller than are emitted sufficiently early to be completely thermalised and merely contribute to the photontobaryon ratio. The requirement that this does not exceed the observed ratio of around leads to a limit
(22) 
so only PBHs below could generate all of the CMB. This limit is not shown in Fig. 4 because it is very weak.
ii.2.2 CMB spectral distortions
Zel’dovich and Starobinskii Zel’dovich et al. (1977) also noted that photons from PBHs in the range , although partially thermalised, will produce noticeable distortions in the CMB spectrum unless
(23) 
this corresponding to the fraction of the density in PBHs being less than unity at evaporation. In the intermediate mass range, , there is a transition from limit (22) to the much stronger limit (23). Subsequently the form of these distortions has been analysed in greater detail. If an appreciable number of photons are emitted after the freezeout of doubleCompton scattering (), corresponding to , the distribution of the CMB photons develops a nonzero chemical potential, leading to a distortion. On the other hand, if the photons are emitted after the freezeout of the singleCompton scattering (), corresponding to , the distribution is modified by a distortion. These constraints were first calculated in the context of decaying particle models Hu and Silk (1993). In the PBH context, recent calculations of Tashiro and Sugiyama Tashiro and Sugiyama (2008) show that the CMB distortion constraints are of order for some range of . The precise form of the constraints is shown by the brown lines in Fig. 4. We note that they are weaker than the BBN constraint but stronger than the constraint given by Eq. (23).
There are also CMB distortion constraints associated with the accretion of larger nonevaporating black holes and PBH evaporations could be potentially constrained by their effect on the form of the recombination lines in the CMB spectrum Sunyaev and Chluba (2009), just as in the annihilating dark matter scenario Chluba (2010); Slatyer et al. (2009).
ii.2.3 CMB anisotropies
Another constraint on PBHs evaporating after the time of recombination is associated with the damping of smallscale CMB anisotropies. The limit can be obtained by modifying an equivalent calculation for decaying particles, as described by Zhang et al. Zhang et al. (2007). Their constraint can be written in the form
(24) 
where is the decay rate, which corresponds to in our case, and is the fraction of the CDM in PBHs, which is simply related to , times the fraction of the emitted energy which goes into heating the matter. The last factor, which includes the effects of the electrons and positrons as well as the photons, will be denoted by and depends on the redshift MacGibbon and Carr (1991); Chen and Kamionkowski (2004). Most of the heating will be associated with the electrons and positrons; they are initially degraded by inverse Compton scattering off the CMB photons but after scattering have an energy
(25) 
where is the Lorentz factor and in the last expression we assume that the mass of a PBH evaporating at redshift in the matterdominated era is . Since this energy is always above the ionization threshold for hydrogen (), we can assume that the heating of the electrons and positrons is efficient before reionization. Using Eq. (5) for and Eq. (14) for , one can now express Eq. (24) as a limit on . In the mass range of interest, the rather complicated cubic expression in can be fitted by the approximation
(26) 
where is the fraction of emission which comes out in electrons and positrons. Here the lower mass limit corresponds to black holes evaporating at recombination and the upper one to those evaporating at a redshift Fan et al. (2006), after which the ionisation ensures the opacity is too low for the emitted electrons and positrons to heat the matter.
Equation (26) is stronger than all the other available limits in this mass range but had not been pointed out before CKSY. Recently it has been studied more carefully by Poulin et al. Poulin et al. (2017a), who compute CMB anisotropy constraints on electromagnetic energy injection over a large range of timescales. They apply their formalism for PBHs with mass , showing that the constraints are comparable to the ray background ones and dominate below around . Stöcker et al. Stöcker et al. (2018) have followed up on this work, using the Boltzmann code CLASS. Their bounds are several orders of magnitude stronger than the EGB limit in the range and exclude PBHs with a monochromatic mass distribution in the range from containing all of the dark matter. Future CMB or 21 cm experiments could improve these limits models even more Lucca et al. (2019). Poulter et al. Poulter et al. (2019) have extended these constraints to the case in which the PBHs have an extended mass function.
ii.3 Cosmic ray and ray backgrounds
ii.3.1 Extragalactic ray background
One of the earliest works that applied the theory of black hole evaporation to astrophysics was carried out by Page and Hawking Page and Hawking (1976). They used the diffuse EGB observations to constrain the mean cosmological number density of PBHs which are completing their evaporation at the present epoch to be less than . This corresponds to an upper limit on of around . The limit was subsequently refined by MacGibbon and Carr MacGibbon and Carr (1991) who considered how it is modified by including quark and gluon emission and inferred , corresponding to a upper limit of . Later they used EGRET observations to derive a slightly stronger limit Carr and MacGibbon (1998) or a upper limit of . Using the modern value of gives and this corresponds to from Eq. (6). They also inferred from the form of the ray spectrum that PBHs could not provide the dominant contribution to the background.
The photon emission has a primary and secondary component and these are calculated according to the prescription of Sec. I.4.2. The relative magnitude of these two components is sensitive to the PBH mass and this affects the associated limit. In order to determine the present background spectrum of particles generated by PBH evaporations, we must integrate over the lifetime of the black holes, allowing for the fact that particles generated in earlier cosmological epochs will be redshifted in energy by now.
If the PBHs all have the same initial mass , and if we approximate the number of emitted photons in the energy bin by ), then the emission rate per volume at cosmological time is
(27) 
where the dependence of just reflects the evaporation. Since the photon energy and density are redshifted by factors and , respectively, the present number density of photons with energy is
(28) 
where corresponds to the earliest time at which the photons freely propagate and is the current PBH number density for or the number density they would have now had they not evaporated for . The photon flux is
(29) 
The calculated presentday fluxes of primary and secondary photons are shown in Fig. 6, where the number density for each has the maximum value consistent with the observations.
Note that the highest energy photons are associated with PBHs of mass . Photons from PBHs with are at lower energies because they are cooler, while photons from PBHs with are at lower energies because (although initially hotter) they are redshifted. The spectral shape depends on the mass and can be easily understood. PBHs with have a rather sharp peak, well approximated by the instantaneous blackbody emission of the primary photons, while holes with have an falloff for due to the final phases of evaporation MacGibbon (1991).
The relevant observations come from HEAO 1 and other balloon observations in the range, COMPTEL in the range, EGRET in the range and FermiLAT in the range. All the observations are shown in Fig. 6. The origin of the diffuse Xray and ray backgrounds is thought to be primarily distant astrophysical sources, such as blazars, and in principle one should remove these contributions before calculating the PBH constraints. This is the strategy adopted by Barrau et al. Barrau et al. (2003a), who thereby obtain a limit . CKSY do not attempt such a subtraction, so their constraints on may be overly conservative.
In order to analyse the spectra of photons emitted from PBHs, different treatments are needed for PBHs with initial masses below and above . We saw in Sec. I.4.2 that PBHs with can never emit secondary photons at the present epoch, whereas those with will do so once falls below . One can use simple analytical arguments to derive the form of the primary and secondary peak fluxes. The observed Xray and ray spectra correspond to where lies between and . For , the limit is determined by the secondary flux and one can write the upper bound on as
(30) 
For , secondary photons are not emitted and one obtains a limit
(31) 
These dependences explain qualitatively the slopes in Fig. 7. The limit bottoms out at but we note that it strengthens by a factor of below the mass because of secondary emission. There is also a narrow band in which PBHs have not yet completed their evaporation even though their current mass is below the mass at which quark and gluon jets are emitted. From Eq. (5), the associated limit on the density parameter is . Note that this is a factor of 3 stronger than the limit given in CKSY.
We stress that the discontinuity in the EGB constraint at the mass is entirely a consequence of the (probably unrealistic) assumption that the PBHs have a monochromatic mass function and even a tiny mass width would suffice to smear this out. Indeed, many derivations of the EGB constraint  including the original Page–Hawking calculation  assume that the PBHs have a powerlaw mass function. In this case, the discontinuity is removed and the stronger () limit pertains. Nevertheless, the discontinuity is still interesting in principle and the issue of nonmonochromaticity will be even more important when we consider the Galactic background constraint.
Finally, we determine the mass range over which the ray constraint applies. Since photons emitted at sufficiently early times cannot propagate freely, there is a minimum mass below which the above constraint is inapplicable. The dominant interactions between rays and the background Universe in the relevant energy range are pairproduction off hydrogen and helium nuclei. For the opacity appropriate for a hydrogen and helium mix Page and Hawking (1976), the redshift below which there is free propagation is given by MacGibbon and Carr (1991)
(32) 
with the nucleon density parameter being normalised to the modern value. The condition then gives
(33) 
The limit is therefore extended down to this mass in Fig. 7. It goes above the density constraint for .
Ballesteros et al. Ballesteros et al. (2019) have recently improved this bound and extended its mass range by better modeling of the combined AGN and blazar emission in the MeV range. They also estimate the constraints from any future Xray experiment capable of identifying a significantly larger number of astrophysical sources contributing to the diffuse background in this energy range. Note that Arbey et al. Arbey et al. (2019) have extended the EGB constraint on PBHs with masses from a monochromatic distribution of Schwarzschild black holes to an extended distribution of rotating Kerr black holes, showing that the lower part of this mass window can be closed for nearextremal black holes. See also Ref. Fukuda and Nakayama (2019) for constraints on spinning PBHs through superradiance.
ii.3.2 Galactic ray background
If PBHs of mass are clustered inside our own Galactic halo, as expected, then there should also be a Galactic ray background and, since this would be anisotropic, it should be separable from the extragalactic background. The ratio of the anisotropic to isotropic intensity depends on the Galactic longitude and latitude, the Galactic core radius and the halo flattening. Some time ago Wright Wright (1996) claimed that such a halo background had been detected in EGRET observations between and Sreekumar et al. (1998) and attributed this to PBHs. His detailed fit to the data, subtracting various other known components, required the PBH clustering factor to be , comparable to that expected, and the local PBH explosion rate to be . A later analysis of EGRET data between and , assuming a variety of distributions for the PBHs, was given by Lehoucq et al. Lehoucq et al. (2009). In the isothermal model, which gives the most conservative limit, the Galactic ray background requires . They claimed that this corresponds to , which from Eq. (6) implies , a factor of 30 above the extragalactic background constraint (30). Lehoucq et al. themselves claimed that it corresponds to but this is because they use an old and rather inaccurate formula relating and .
It should be stressed that the Lehoucq et al. analysis does not constrain PBHs of initial mass because these no longer exist. Rather it constrains PBHs of current mass and, from Eq. (20) with , this corresponds to an initial mass of . However, as shown below, this value of does not correspond to the strongest limit on . This contrasts to the situation with the extragalactic background, where the strongest constraint on comes from the timeintegrated contribution of the black holes. There would also be a Galactic contribution from PBHs which were slightly smaller than but sufficiently distant for their emitted particles to have only just reached us; since the lighttravel time across the Galaxy is , this corresponds to PBHs initially smaller than by , so this extra contribution is negligible.
To examine this issue more carefully, we note that Eq. (20) implies that the emission from PBHs with initial mass currently peaks at an energy for , which is in the range for . (Note that secondary emission can be neglected in this regime, this only being important for .) So these black holes correspond to the “tail” population discussed in Sec. II.3.1 and have a mass function . This means that the interpretation of the Galactic ray limit is sensitive to the nonmonochromaticity in the PBH mass function, so we must distinguish between the various situations described later in Sec. IV. In particular, we need to distinguish between mass functions which centre at and some higher value of . In the present context, we assume that the mass function is narrow, so that can be defined as the integral over the entire mass width.
The peak energy is above for , so the ray band is in the Rayleigh–Jeans region. The flux of an individual hole scales as in this regime. Although this flux must be weighted by the number density of the holes, for , this factor is necessarily balanced by the ratio of the mass widths at formation and now, so the limit on scales as . For , the current number flux of photons from each PBH scales as , so the effective limit on scales as . The observed ray band enters the Wien part of the spectrum for , so the limit on weakens exponentially for . (Although not mentioned in Ref. Lehoucq et al. (2009), one could in principle get a stronger limit in this mass regime from observations at energies lower than .) Hence the largest contribution to the Galactic background and the strongest constraint on comes from PBHs with and has the form indicated in Fig. 4.
Subsequently, the problem was studied in much greater detail by CKSY2 Carr et al. (2016b). To go beyond the Lehoucq et al. analysis, we included several important effects. First, we distinguished between the initial mass and the current mass , this being given by
(34) 
where
(35) 
is the value of corresponding to the mass at which secondary emission eventually becomes important. Second, we distinguished between primary and secondary emission. The ratio of the energies at which the spectra peak is
(36) 
while the flux ratio is
(37) 
Third, we distinguished between initial mass function and the current one,
(38) 
the main GRB contribution coming from the low mass tail.
Following Lehoucq et al., we took the halo density profile to have the form
(39) 
with a set of bestfit parameters . We described the directional dependence with the function
(40) 
with being our distance from the edge of the halo. We then compared the predicted intensity with FermiLAT observations and inferred constraints on and . We show the constraint for a monochromatic mass function in Fig. 8.
Note that the depth of the limit in Fig. 8 is sensitive to the width of the PBH mass function and becomes arbitrarily small as . However, a very small value of is almost certainly unphysical and the extragalactic and Galactic limits are comparable for .
ii.3.3 Galactic positrons and antiprotons
The evaporation of PBHs with is expected to inject subGeV electrons and positrons into the Galaxy. These particles are shielded by the solar magnetic field for Earthbound detectors, but not for VOYAGER1, which is now beyond the heliopause. Boudad and Cirelli Boudaud and Cirelli (2019) use its data to constrain the PBH dark matter fraction in the Galaxy and, as indicated in Fig. 4, find for . Their limits are based on local Galactic measurements and thus complement those derived from cosmological observations.
Since the ratio of antiprotons to protons in cosmic rays is less than over the energy range , whereas PBHs should produce them in equal numbers, PBHs could only contribute appreciably to the antiprotons Carr (1976); Turner (1982); Kiraly et al. (1981). It is usually assumed that the observed antiprotons are secondary particles, produced by spallation of the interstellar medium by primary cosmic rays. However, the spectrum of secondary antiprotons should show a steep cutoff at kinetic energies below , whereas the spectrum of PBH antiprotons should continue down to . Also any primary antiproton fraction should tend to at low energies. Both these features provide a distinctive signature of any PBH contribution.
The black hole temperature must be much larger than to generate antiprotons, so the local cosmic ray flux from PBHs should be dominated by the ones just entering their explosive phase at the present epoch. Such PBHs should be clustered inside our halo, so any charged particles emitted will have their flux enhanced relative to the extragalactic spectra by a factor which depends upon the halo concentration factor and the time for which particles are trapped inside the halo by the Galactic magnetic field. This time is rather uncertain and also energydependent. At one expects roughly for electrons or positrons and for protons or antiprotons Carr and MacGibbon (1998).
MacGibbon and Carr MacGibbon and Carr (1991) originally calculated the PBH density required to explain the interstellar antiproton flux at and found a value somewhat larger than the density associated with the ray limit. After the BESS balloon experiment measured the antiproton flux below Yoshimura et al. (1995), Maki et al. Maki et al. (1996) tried to fit this in the PBH scenario by using Monte Carlo simulations of cosmic ray propagation. They found that the local PBHproduced antiproton flux is mainly due to PBHs exploding within a few kpc and used the observational data to infer a limit on the local PBH explosion rate of . A more recent attempt to fit the antiproton data came from Barrau et al. Barrau et al. (2003b), who compared observations by BESS95 Yoshimura et al. (1995), BESS98 Orito et al. (2000), CAPRICE Boezio et al. (2001) and AMS Jacholkowska (2007) with the spectrum from evaporating PBHs. According to their analysis, PBHs with would be numerous enough to explain the observations. However, these results are based on the assumption that the PBHs have a spherically symmetric isothermal profile with a core radius of . A different clustering assumption would lead to a different constraint on .
PBHs might also be detected by their antideuteron flux. Barrau et al. Barrau et al. (2003c) argue that AMS and GAPS would be able to detect the antideuterons from PBH explosions if their local density were as large as and , respectively. If a null result were maintained up to these levels, it would imply