Constraints on Stupendously Large Black Holes

Bernard Carr,1, 2, ∗ Florian Kuhnel,3, † and Luca Visinelli4, ‡

1School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London E1 4NS, UK2Research Center for the Early Universe, University of Tokyo, Tokyo 113-0033, Japan

3Arnold Sommerfeld Center, Ludwig-Maximilians-Universitat, Theresienstraße 37, 80333 Munchen, Germany4Gravitation Astroparticle Physics Amsterdam (GRAPPA),

Institute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics,University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands

(Dated: Thursday 10th September, 2020, 12:35am)

We consider the observational constraints on stupendously large black holes (SLABs) in the massrange M & 1011M. These have attracted little attention hitherto and we are aware of no publishedconstraints on a SLAB population in the range (1012 – 1018)M. However, there is already evidencefor black holes of up to nearly 1011M in galactic nuclei, so it is conceivable that SLABs exist andthey may even have been seeded by primordial black holes. We focus on limits associated with (i)dynamical and lensing effects, (ii) the generation of background radiation through the accretion ofgas during the pregalactic epoch, and (iii) the gamma-ray emission from the annihilation of thehalo of weakly interacting massive particles (WIMPs) expected to form around each SLAB if theseprovide the dark matter. Finally, we comment on the constraints on the mass of ultra-light bosonsfrom future measurements of the mass and spin of SLABs.

I. INTRODUCTION

Black holes (BHs) are a key prediction of general rel-ativity. There are a plethora of observations indicatingtheir existence in the solar [1] or intermediate-mass [2]range. In particular, the existence of binary black holesin the mass range (10 – 50)M has been demonstratedby the detection of gravitational waves from inspirallingbinaries [3].

There is also evidence for supermassive black holes(SMBHs) at the centres of galaxies [4], including Sagit-tarius A∗ at the centre of our own galaxy, with a massof 4 × 106M [5]. Recently, the imaging of the shadowcreated by M87∗, the SMBH at the centre of the giantelliptical galaxy M87 with a mass of 6.5 × 109M, hasbeen reported by the Event Horizon Telescope [6]. TheSMBHs in galactic nuclei span a huge mass range, ex-tending up to nearly 1011M [7]. The current heaviestBH is associated with the quasar TON 618 and has amass of 7×1010M [8], while the second heaviest, at thecentre of the galaxy IC 1101, has a mass inferred fromits radio emission of 4× 1010M [9].

This raises the issue of whether there could be evenlarger BHs in galactic nuclei and whether indeed thereis any natural upper limit to the mass of a SMBH. Wewill not review the extensive literature on this topic herebut this can be found in the pioneering paper of Natara-jan and Treister [10], who investigated the possibility of“ultra-massive” black holes, defined to be ones larger

∗ Electronic address: B.J.Carr@qmul.ac.uk† Electronic address: kuhnel@kth.se‡ Electronic address: l.visinelli@uva.nl

than 5 × 109M. They argued that their existence isimplied by the observed relationship between the SMBHmass and the bulge luminosity of the host galaxy. It isalso a natural implication of the accretion and mergerof smaller SMBHs, so one expects a few such objects incentral galaxies in clusters. However, they concluded onthe basis of observational and theoretical considerationsand the known local SMBH mass function that there isa natural upper limit on the mass of ∼ 1011M.

More recently, based on the assumption that SMBHsare hosted by disc galaxies, King [11] has derived an up-per limit of 5 × 1010M (or 3 × 1011M for maximalprograde spin) above which they cannot grow throughluminous accretion of gas, the precise value depending onthe properties of the host galaxy. He points out that theassociated Eddington luminosity is close to the largestobserved AGN bolometric luminosity. Black holes canstill grow above this mass by non-luminous means, suchas mergers, but they cannot become luminous again. Arelated upper limit has been obtained by Yazdi and Af-shordi [12]. Both arguments are based on the Toomreinstability of α-disks [13]: if the SMBH is too large, itceases to grow through accretion because the disk be-comes too massive and fragments under self-gravity. Thisleads to a strict upper limit on the mass of SMBHs as afunction of cosmic time and spin.

We describe these models in more detail in Sec. II. Allthe estimates of the maximum SMBH mass are in roughagreement with the observations but they are dependenton details of the accretion models. They also leave openthe possibility that more massive SMBHs could exist evenif they cannot be luminous and this provides the motiva-tion for the present paper.

We will describe BHs larger than 1011M (i.e. largerthan the SMBHs currently observed in galactic nuclei) as

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“Stupendously Large Black Holes” or “SLABs”, this ex-tending well beyond the ultra-massive range of Ref. [10].While the most natural assumption is that SLABs rep-resent the high-mass tail of the population of SMBHs ingalactic nuclei, we also consider the possibility that theycould exist in intergalactic space, in which case some ofthe above arguments can be circumvented. There is nocurrent evidence for such objects but their existence onlyentails a small extrapolation from the available data, so itseems surprising that this possibility has been neglectedin the literature. The purpose of this paper is to con-sider some of their observational consequences: in partic-ular, their dynamical and lensing effects, the influence oftheir accretion-generated luminosity on the thermal his-tory of the Universe, and the annihilation of the WIMPsexpected to form a halo around them.

Any intergalactic SLABs must have formed indepen-dently of galaxies and would probably need to be pri-mordial in origin. It has long been argued that primor-dial black holes (PBHs) could have formed in the earlyradiation-dominated Universe [14–16]. They are not usu-ally expected to be as large as SLABs but in principletheir mass could be anything up to the horizon mass atthe time of matter-radiation equality, which is of the or-der of 1017M. One fairly generic scenario in which thePBH mass function extends up to the SLAB range hasbeen suggested by Vilenkin and colleagues [17–22].

In this context, it is important to stress that even theSMBHs in galactic nuclei could conceivably have beenseeded by PBHs. The conventional assumption is thatthe SMBHs in galactic nuclei form as a result of dynam-ical processes after the galaxies themselves [23]. Specificscenarios are discussed in Sec. II. However, observationsof quasars at redshifts z & 6 suggest that SMBHs largerthan 109M were already present when the Universe wasless than a billion years old [24]. Generating BHs thislarge so early, while certainly possible [25], is challeng-ing. Therefore it is interesting to consider the possibil-ity that the SMBHs in galaxies formed before galaxies,in which case they could be primordial. However, thePBHs would inevitably have grown enormously throughaccretion since matter-radiation equality, which is whythey should only be regarded as seeding the SMBHs.

If SLABs are of primordial origin, this raises an inter-esting link with the suggestion that PBHs could providethe dark matter (see Ref. [26] for a recent review). Al-though SLABs themselves clearly cannot do this, sincethey are too large to reside in galactic halos, it is possiblethat PBHs provide the dark matter in a much lower massrange. This is because PBHs formed in the radiation-dominated era and are therefore non-baryonic, circum-venting the usual bound on the density of baryonic mat-ter. For example, PBHs could have formed from quan-tum fluctuations which were produced and re-entered thehorizon during inflation [27–29]. Although the SLAB andDM populations might be distinct, they could be relatedif PBHs have an extended mass function.

While there is no definitive evidence that PBHs pro-vide the dark matter, there is a huge literature dis-cussing constraints on their contribution to the dark den-sity [30, 31], these being associated with a wide vari-ety of effects: quantum evaporation, gravitational lens-ing, dynamical effects, accretion and influence on cos-mic structures. These studies show that there are onlyfour mass windows in which PBHs could have an “ap-preciable” density: (A) the asteroid mass range with1016 < M/g < 1018; (B) the lunar mass range with1020 < M/g < 1024; and (C) the intermediate mass rangewith 10 < M/M < 102; (D) the stupendous mass rangewith 1012 < M/M < 1018. While the first three win-dows have been well studied, the last window — whichcorresponds to the SLAB range — has been almost com-pletely neglected.

The apparent lack of constraints on SLABs probablyjust reflects the fact that very little attention has beenpaid to their possible existence, except perhaps in galac-tic nuclei. Nevertheless, they could have striking ob-servational consequences and the purpose of this paperis to examine some of these. Even though they couldnot explain the dark matter, they could still represent asmoothly distributed dark intergalactic contribution, soit is interesting to know how large this could be.

If SLABs are pregalactic, a particularly interestingconstraint would be associated with their accretion ofpregalactic gas. This problem was originally studied inRef. [32], on the assumption that the BHs accrete at theBondi rate. However, this neglected the fact that steady-state Bondi is inappropriate for very large BHs becausethe Bondi timescale exceeds the cosmic expansion time.It is still not clear how to correct for this but we willattempt to address this issue. Subsequently, many otherauthors [33–35] have studied PBH accretion but theiranalyses only apply for masses below around 104M, inpart because of the failure of the steady-state assump-tion. However, it is clear that the accretion constraintsdo not suddenly cut off above this mass.

If the PBH dark-matter fraction is low, some other can-didate must dominate and the most studied is a weaklyinteracting massive particle (WIMP). Each PBH wouldthen provide a seed around which a halo of WIMPs wouldform, the gamma-rays from their annihilations then im-plying stringent bounds on combined dark-matter sce-narios [36–40]. A dark matter halo around BH bina-ries would also alter the expected gravitational-wave sig-nal [41, 42]. Previous work in this context has focussed onBHs in the mass ranges below 103M but in this paperwe will extend these arguments to the SLAB range. Ina separate paper [43], we discuss this problem in a moregeneral context, covering the entire mass range 10−12 –1012M and distinguishing between the Galactic and ex-tragalactic limits. Here we focus on the SLAB range,where only the extragalactic limit is relevant.

Regardless of their origin, SLABs could also play animportant role in the presence of light bosonic fields.

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Such bosons are expected to form condensates aroundthe holes and their quantum fluctuations would popu-late the quantum levels of the boson clouds [44]. If theBHs spin, a fraction of the rotational energy might betransferred to the surrounding boson cloud through su-perradiance [45, 46], even if the bosons only provide asmall fraction of the dark matter. This superradiant in-stability of rotating BHs has attracted increasing inter-est as a possible probe of light bosons [47]. However,this requires the BHs to spin appreciably, which is notexpected for PBHs [48–50] except in non-standard cos-mological scenarios [51, 52]. The effect of a boson cloudon the angular size of the shadow of M87∗ has also beenconsidered in Ref. [53], a lower bound on the mass of thelight boson being placed once the spin and the size of theBH shadow is measured with sufficient accuracy.

The plan of this paper is as follows. In Sec. II wediscuss how SLABs might form either primordially or ingalactic nuclei and clarify the relationship between thesetwo scenarios. In Sec. III we review the lensing bounds onSLABs and in Sec. IV we review the dynamical bounds.Most of these bounds have been reported before but wemake them more precise and correct some errors in pre-vious treatments. In Sec. V we derive accretion con-straints, developing some much earlier work but stress-ing that our approach faces some difficulties. In Sec. VIwe consider the constraints associated with the annihi-lation of WIMPs expected to form halos around SLABs,extending previous studies for a much lower PBH massrange. In Sec. VII we focus on constraints on the massof light bosons if the SLABs are rotating. Some generalconclusions are drawn in Sec. VIII.

II. FORMATION OF SLABS

Although we should stress at the outset that there isno current evidence for SLABs, the emphasis of this pa-per being on constraints, we will now discuss their possi-ble formation scenarios. There are two classes of model.The first — perhaps most natural — possibility is thatthey form in galactic nuclei, although we will see thatthere are both theoretical and observational reasons forbelieving that the mass function of such SMBHs doesnot extend beyond 1011M. Indeed, this is one reasonfor taking this to be the lower mass limit in our definitionof a SLAB. The second — more speculative — possibilityis that they form primordially (i.e. within the first fewminutes of the Universe). One does not usually envisagePBHs being in the SLAB range and there are argumentsthat their masses should not extend beyond ∼ 105M.However, there are some scenarios for producing PBHslarger than this and, as we will see, very massive oneswill inevitably grow enormously through accretion, soone must distinguish between the initial and final PBHmass. It is also possible that that the SMBHs in galacticnuclei were themselves seeded by PBHs, so the distinction

between the two classes of explanation is not clear-cut. Inprinciple, SMBHs could be located in intergalactic spacerather than galactic nuclei but one then needs to explainwhy such enormous objects do not act as seeds for galax-ies or clusters anyway.

A. Postgalactic SLABs in galactic nuclei

If the SMBHs in galactic nuclei are generated bydynamical processes after the formation of the galax-ies, there are two possible pathways: direct collapse toSMBHs [54] or super-Eddington growth [55]. Both ofthese have been explored in a series of papers by Agar-wal et al. [56–59] but they are not without difficulties. Inthe former case, the seeds are rare; in the latter case,they should be ultra-luminous and visible in deep X-ray surveys. Other scenarios include the direct collapseof gas clouds in the center of minihalos [60], the rapidaccretion experienced by a black hole moving inside astar cluster [61], and the formation out of supermassivestars [62, 63] and dark stars [64].

Whatever the scenario, consistency between X-ray andoptical data at high z and the observed SMBH mass func-tion at low z implies an upper limit on the mass. Thislimit was first studied by Natarajan and Treister [10],who pointed out that consistency requires the mass func-tion to steepen above around 109M, the UMBH scale,this being explained by a self-regulation mechanism thatlimits the mass. One expects a link between star for-mation and black hole fuelling and they discuss sev-eral mechanisms which explain the observed M–σ4 re-lation. The clearing out the nuclear region by the grow-ing black hole leads to an upper limit of ∼ 1010M andthe outflow from the accretion may also halt the flow,the mass limit in this case being similar but sensitive tothe halo spin and disc mass fraction. Extrapolating theM–σ4 relation to high σ suggests that UBMHs mightbe hosted by massive high-luminosity galaxies with largevelocity dispersions in the centres of clusters. This isalso a natural implication of the accretion and mergerof smaller SMBHs. The highest velocity dispersion cur-rently observed is ∼ 400 km/s and the limit would thenbe ∼ 1010M. However, for a cD (central Dominant)galaxy with a velocity dispersion of ∼ 700 km/s, it wouldbe ∼ 1011M.

The model of King [11] assumes that the SMBH issurrounded by an accretion disc which is self-gravitatingoutside some radius Rsg. Gas in the outer region thencools fast enough for self-gravity to lead to star forma-tion. Most of the gas initially outside Rsg either goes intothese stars or is expelled by them on a near-dynamicaltimescale. This implies that the outer radius of the accre-tion disc cannot exceed Rsg. However, the inner radiusmust be at least as large as the ISCO (innermost sta-ble circular orbit) and this precludes disc formation forSMBHs larger than some limit which depends on the ra-

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tio L/LED and the value of α. The limit is 5× 1010Mfor L = LED and α = 0.1. This applies for any SMBHs inquasars or AGN, since these must have accretion discs.For maximal prograde spin, it becomes 2.7 × 1011M,which is the absolute maximum for an accreting SMBH.At masses just below the limit, the disc luminosity of afield galaxy is likely to be below the Eddington limit andnot strong enough to trigger the feedback underlying theM–σ4 relation, so SMBHs close to the limit can evolveabove this.

The model of Yazdi and Niayesh [12] modifiesthe Shakura-Sunyaev model for radiation-pressure-dominated disks. They calculate the radius at whichPrad = Pgas as a function of M/MED for different valuesof M and the α parameter. The more efficient coolingthen keeps the disk in equipartition with Prad = Pgas

beyond this radius. As in the King model, self-gravitybecomes important in this region, so the matter clumpsthere and no longer accretes onto the SMBH. The largerthe mass, the closerRsg gets to the ISCO radius and therewill no longer be accretion when they become equal. Thissets a redshift-dependent upper limit to the SMBH mass.This lies in the range (2 – 5) × 1011M at z = 0.1 and(0.5 – 1.5)×1011M at z = 10 for non-rotating BHs butit is reduced for highly rotating BHs. Both their argu-ment and King’s are based on the Toomre instability ofα-disks [13].

B. Primordial SLABs in galactic nuclei

Generating a SMBH with a mass of 1010M by a red-shift of z = 7 (as observed) is challenging. It is thereforeinteresting to consider the possibility that the SMBHsformed before galaxies [65–68] and this leads to three pos-sible scenarios.

(i) The first possibility is that the PBHs underwentvery little accretion and were themselves supermassive,so that they can be directly identified with the SMBHs.In this case, they could also help to generate galaxiesthrough either the seed or Poisson effect, the fluctuationsgrowing by a factor ∼ 103 between the time of matter-radiation equality and galaxy formation [69]. This natu-rally explains the proportionality between the SMBH andgalaxy mass and also provides an early mode of galaxyformation that might be important for the reionisation ofthe Universe [70]. In this case, the galaxy mass functiondirectly reflects the PBH mass function and the lattermight conceivably extend to the SLAB scale.

(ii) The second possibility is that the PBHs had amore modest mass and then grew through Eddington-limited accretion. This scenario was first suggested byBean and Magueijo [65], although they overestimatedthe amount of accretion in the early radiation-dominatedphase. They argued that it needs a very narrow PBHmass function to reproduce the observed distribution ofSMBHs and Kawasaki et al. [71] suggested a specific in-

flationary scenario to account for this. However, mostof the accretion still occurs after decoupling, so it maybe difficult to distinguish this observationally from a sce-nario in which the original BHs are non-primordial. Inboth cases, one would expect a lot of radiation to havebeen generated and this would contribute to the observedX-ray background [72]. In a recent work, Nunes andPacucci [73] have analysed the most massive high-redshiftSMBHs and argued that the seed mass would need to beat least 104M to explain the observations. They findthat M > 1012M SLABs are theoretically possible butthat feedback and galaxy dynamics effects will have im-portant observational consequences above 1013M. It isunclear whether an object of 1017M would be obser-vationally compatible with other SMBH data. For ex-ample, its gravitational-wave signal would be out of therange even of LISA and its electromagnetic signal wouldbe uncertain.

(iii) The third possibility is that the PBHs had an evenmore modest mass and generated the SMBHs in galacticnuclei through the seed or Poisson effect. For example,to produce a SMBH of mass 108M by z ∼ 4, one re-quires M ∼ 105M for the seed effect or M ∼ 102Mfor the Poisson effect if the PBHs provide the dark mat-ter. To produce a SLAB of 1010M, one would requireM ∼ 107M for the seed effect or M ∼ 104M for thePoisson effect. However, this only produces a gravita-tionally bound region and one still has to explain howa region containing a central large PBH or a cluster ofintermediate mass PBHs can evolve to a single SMBH.Accretion and merging would clearly be important andonly some fraction of the bound region might end up inthe central BH.

C. Intergalactic primordial SLABs

Finally we consider the possibility that there is an in-tergalactic population of SLABs, not necessarily relatedto the SMBHs in galactic nuclei. In this case, they wouldalmost certainly need to be primordial. One scenario inwhich primordial SLABs form naturally has been sug-gested by Vilenkin and colleagues [17–22]. In this case,the PBHs are formed by bubbles nucleated during infla-tion and their mass function scales as f(M) ∝M−1/2, sothe value of f(M) at 1011M is uniquely determined bythe total PBH density. For example, if PBHs of 10Mprovide the dark matter, one would have f ∼ 10−5 inthe SLAB range. This mass function would also apply ifthe PBHs formed from the collapse of cosmic strings,although existing constraints on the string parameterwould exclude them providing the dark matter. Thepower spectrum invoked in the “thermal history” modelof Ref. [74] would also produce a significant SLAB den-sity, although the spectrum needs to be cut off at somemass-scale to be consistent with large-scale structure ob-servations.

5

Whatever the source of very massive PBHs, they wouldinevitably increase their mass by accretion of pregalac-tic gas [cf. case B(ii) above]. A simple Bondi accretionanalysis suggests that such holes go through an Edding-ton accretion phase which ends at some time tED afterthe Big Bang. During this phase, each BH doubles itsmass on the Salpeter timescale, tS ≈ 4 × 108 ε yr whereε is the luminosity efficiency [75]. This corresponds to

the age of the Universe at a redshift zS ≈ 40 (ε/0.1)2/3.Therefore, for ε = 0.1 one expects the mass of the BHto increase by a factor exp(tED/tS) ≈ exp[(40/zED)3/2],which is very large for zED 40. For example, thegrowth up to z = 7 would be of order 106. In Sec. V, wewill calculate the value of zED as a function of the PBHmass M and density parameter ΩPBH(M).

Understanding accretion is crucial in discussing theorigin of SLABs, since it is unlikely that PBHs couldbe in the SLAB mass range initially. This is becausethe formation of PBHs from primordial inhomogeneitiesrequires that the power spectrum of the curvature fluc-tuations be O(1), whereas observational constraints fromthe observed CMB anisotropies and spectral distortionsexcludes this above a mass of Mmax ∼ 1010M. Onetherefore needs a growth factor of at least MSLAB/Mmax.Since the initial mass may only be tiny fraction of thefinal mass, and since most of the mass increase occurs atthe end of the Eddington phase (i.e. long after matter-radiation equality), it is not so crucial whether the initialseed was primordial. There is also the semantic issue ofhow late a black hole can form and still be described asprimordial. One usually assumes that PBHs form in theradiation-dominated era but the crucial feature is thatthey form from O(1) density fluctuations rather than viathe fragmentation of much larger regions (like galaxies)during the matter-dominated era.

III. LENSING AND CMB CONSTRAINTS ONSLABS

If one has a population of compact objects with massM and density ΩC, then the probability P of one of themimage-doubling a source at redshift z ≈ 1 and the sepa-ration θs between the images are [76]

P ≈ (0.1− 0.2) ΩC , θs ≈ 5× 10−6 (M/M)1/2 arcsec .(1)

One can therefore use the upper limit on the frequency ofmacrolensing for different image separations to constrainΩC as a function of M . The usual approach is to derivea “detection volume”, defined as the volume between thesource and the observer within which the lens would needto lie to produce an observable effect. Limits are thenobtained by adding the detection volume for each sourceand comparing this to the volume per source expectedfor a given ΩC.

There have been several optical and radio surveys tosearch for multiply-imaged quasars [77]. In particular,Hewitt [78] used VLA observations to infer ΩC (1011 –1014M) < 0.4, Nemiroff [79] used optical QSO data toinfer ΩC(M > 109.9M) < 1 and ΩC(M > 1010.3M) <0.4, and Surdej et al. [80] used data on 469 highly lu-minous quasars to infer ΩC(1010 – 1012M) < 0.02.Most of the recent emphasis has been on pushing thesemacrolensing limits down to lower masses but in thepresent context we are interested in larger masses [81, 82].

SLABs would generate “cluster-like” anisotropies inthe CMB sky, with a lensing signal scaling proportionalto the mass. Any object with M > 1016M and locatednear the peak of the CMB lensing kernel (i.e. at redshiftsz = 0.5 – 10) should show up as a huge signal, possiblyvisible by eye in the SPT or ACT lensing maps. In the1014 – 1015M range, the issue is less clear, since usu-ally one only obtains good signal-to-noise by stacking theclusters (Horowitz, private communication).

The Vilenkin et al. scenario, in which the PBHs formfrom inflationary bubbles during, is particularly interest-ing in this context. There are no large density fluctu-ations outside the bubbles, so spectral distortions arelocalised in small angular patches on the sky. Theydiscuss observational constraints on the model due tospectral distortions and their calculations extend up to1020M. They find that the largest BH that one canexpect to find in the observable region has initial mass∼ 1014M [21, 22].

IV. DYNAMICAL CONSTRAINTS ON SLABS

The most clear-cut constraints on the PBH fractionfPBH in the mass range above 105M are dynamical andsummarised in Fig. 1. All of them have been discussedbefore, although we will need to refine some of the argu-ments for the SLAB range. Before reviewing the argu-ments, we point out an obvious lower limit on fPBH(M),which has been termed the “incredulity limit” [83]. ThePBH scenario is only interesting if there is at least onePBH in the relevant environment, be it a galactic haloor a cluster of galaxies or the entire observable Universe.This corresponds to the condition

fPBH(M) ≥ M

ME, (2)

where ME is the mass of the environment, taken to be1012M for a galactic halo, 1014M for a cluster ofgalaxies and 1022M for the observable Universe (i.e. themass within the current particle horizon). This specifiesthe right edge of some constraints shown in Fig. 1. How-ever, the scenario is not necessarily excluded beyond thisedge but merely irrelevant to the associated dark matterproblem. For example, it makes no sense to postulate thehalo being made of objects as big as the halo itself, so

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SLABs are clearly outside the galactic incredulity limit,though not necessarily the cluster one.

A. Dynamical friction limit

Halo BHs with M > 105M would be dragged intothe nucleus of the Galaxy by the dynamical friction ofvarious stellar populations. These holes can then mergeto form a single SMBH larger than the observed mass of4× 106M unless fPBH(M) is suitably constrained [83].We include this limit in Fig. 1 but do not give an explicitexpression for it since it is complicated. This is becausethere are different sources of friction and the limit de-pends on parameters such as the halo core radius. Alsoit may be weakened if black holes can be ejected fromthe Galactic nucleus by 3-body effects [84]. The limitbottoms out at M ∼ 107M with a value fPBH ∼ 10−4.

B. Destruction of galaxies in clusters

If the dark matter in clusters comprises SLABs, thengalaxies would be heated up and eventually destroyedby encounters with them. The associated limit can beexpressed as [83]

fPBH(M) <

(M/1010M)−1 (1010 < M/M < 1011) ,

0.1 (1011 < M/M < 1013) ,

(M/1014M) (1013 < M/M < 1014) ,

(3)although it depends sensitively on the mass and the ra-dius of the cluster. The last expression corresponds tothe incredulity limit of one SLAB per cluster. Below thisline, the SLABs are not confined to clusters and the tidaldistortion limit is weakened by factor of 100, correspond-ing to the ratio of the cluster density to the cosmologicalbackground density, and is uninteresting.

There is also a constraint from the lack of unexplainedtidal distortions of galaxies in clusters. If the fraction ofdistorted galaxies at any time is ∆g, this limit takes theform [83]

fPBH(M) < λ−3 ∆g

(ρg

ρDM

)≈ 20

(λ

3

)−3

∆g (4)

where λ ≈ 3 represents the difference between distortionand disruption. So one has an interesting limit providingthe fraction of unexplained distorted galaxies is less than5%. This limit only applies forM > λ−3Mg ≈ 3×109Msince otherwise the tidal distortion radius is smaller thanRg. In particular, Van den Bergh [85] concluded from ob-

servations of the Virgo cluster, which has a mass 1015Mand only 4 of 73 cluster members with unexplained tidaldistortions, that the dark matter could not be in the form

of compact object in the mass range (109 – 1013)M.Thisassumes that the distortion only persists during the en-counter itself, the tidal stretching being reversed as theSLAB recedes.

C. Large-scale structure limits

PBHs larger than 102M cannot provide dark matterbut Carr and Silk [69] have studied how such PBHs couldgenerate cosmic structures through the ‘seed’ or ‘Poisson’effect even if fPBH is small. If a region of mass M containsPBHs of mass M , the initial fluctuation is

δi ≈

M/M (seed) ,

(fPBHM/M)1/2 (Poisson) .(5)

If fPBH = 1, the Poisson effect dominates for all M ; iff 1, the seed effect dominates for M < M/fPBH . Ineither case, the fluctuation grows as z−1 from the redshiftof DM domination (zeq ≈ 4000), so the mass binding atredshift zB is

M ≈

4000M z−1

B (seed) ,

107 fPBHM z−2B (Poisson) .

(6)

This assumes the PBHs have a monochromatic massfunction. If it is extended, the situation is more compli-cated because the mass of the effective seed for a regionmay depend on the mass of that region [69].

Even if PBHs do not play a role in generating cosmicstructures, one can still place interesting upper limits onthe fraction of dark matter in them by requiring thatvarious types of structure do not form too early [69]. Forexample, if we apply the above argument to Milky-Way-type galaxies, assuming these must not bind before aredshift zB ∼ 3, we obtain

fPBH(M) <

(M/106M)−1 (106 < M/M . 109) ,

(M/1012M) (109 .M/M < 1012) .

(7)This limit bottoms out at M ∼ 109M with a valuefPBH ∼ 10−3. The first expression can be obtained byputting M ∼ 1012M and zB ∼ 3 in Eq. (6). The sec-ond expression corresponds to having just one PBH pergalaxy and is also the line above which the seed effectdominates the Poisson effect (fPBH < M/M). (Thislimit also arises because competition from other seedsimplies that the mass bound is at most f−1

PBHM .) Simi-lar constraints are associated with the first bound clouds,dwarf galaxies and clusters of galaxies, all the limits be-ing shown by the LSS line in Fig. 1.

There is also a Poisson constraint associated with ob-servations of the Lyman-α forest [86, 87], which hasa similar form to Eq. (7) but with the lower limit of

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M ∼ 106M reduced to M ∼ 102M. The constraintreaches a minimum value f ∼ 10−5 at 107M. However,it is unclear how to extend this limit to larger values ofM . The galactic incredulity limit need not apply becauseLyman-α forest observations would be affected even ifthere was less than one PBH per halo. If the limit wereflat beyond 107M, it would be strongest constraint inthe SLAB region of Fig. 1. However, the relationshipbetween the Poisson and seed effects is rather subtle inthis context. The latter can still be important below theincredulity limit but only influences a small fraction ofregions, so the interpretation of the observations is un-clear.

D. CMB dipole limit

If there were a population of huge intergalactic PBHswith density parameter ΩIG(M), which is the likely sit-uation with SLABs, each galaxy would have a peculiarvelocity due to its gravitational interaction with the near-est one [88]. If the objects were smoothly distributed, thetypical distance between them would be [31]

d ≈ 40 ΩIG(M)−1/3

(M

1016M

)1/3(h

0.7

)−2/3

Mpc ,

(8)where h is the Hubble constant in units of100 km s−1 Mpc−1, and this should also be the ex-pected distance of the nearest one. Over the age ofthe Universe t0, this should induce a peculiar velocityin the Milky Way of Vpec ≈ GM f(Ωm) t0/d

2 whereΩm ≈ 0.3 is the total matter density parameter (darkplus baryonic) and f(Ωm) ≈ Ω0.6

m . Since the CMB dipoleanisotropy shows that the peculiar velocity of the MilkyWay is only 400 km s−1, one infers

ΩIG <

(M

2× 1016M

)−1/2(t0

14 Gyr

)−3/2(h

0.7

)−2

.

(9)This scenario is interesting only if there is at least onesuch object within the observable Universe and this cor-responds to the lower limit

ΩIG(M) > 2× 10−8

(M

1016M

)(t0

14 Gyr

)−3(h

0.7

)−2

,

(10)where we have taken the horizon scale to be d ≈ 3 c t0 ≈10h−1 Gpc. This intersects Eq. (9) at

M ≈ 1.1× 1021

(t0

14 Gyr

)M , (11)

so this corresponds to the largest possible BH within thevisible Universe.

-

-

-

FIG. 1. SLAB constraints on fPBH(M) for a monochromaticmass function. Dynamical limits derive from halo dynami-cal friction (DF), galaxy tidal distortions (G) and the CMBdipole (CMB). Large-scale structure limits derive from requir-ing that various cosmological structures do not form earlierthan observed (LSS). Accretion limits come from X-ray bi-naries (XB). The cosmological incredulity limit (CIL) corre-sponds to one PBH within the cosmological horizon. Basedin part on Ref. [26].

V. ACCRETION CONSTRAINTS ON SLABS

Clearly there would be interesting constraints onSLABs due to their accretion at the present or recentepochs but the form of these constraints depends ontheir environment and is complicated. For example, ifthe SLABs fall within larger-scale cosmic structures, thiswill have important consequences for their velocity andspatial distribution. If they reside in galactic nuclei, oreven seed galaxies, they would accrete local gas and starsin a manner which has been studied in much previous lit-erature. If they reside outside galaxies, or even outsideclusters of galaxies, they would still accrete intergalacticgas, the consequences of which depend on the (some-what uncertain) state of the intergalactic medium. Herewe study the accretion of gas by SLABs before the for-mation of galaxies or other cosmic structures. They mayalso accrete dark matter in this period, this being rele-vant to the considerations of Sec. VI, but we neglect thatprocess here.

A. Pregalactic gas accretion

Our analysis is based on the Bondi accretion formulaand covers a wide range of epochs, although only PBHscould exist all the way back to matter-radiation equalityat teq = 2.4×1012 s or zeq ≈ 3400. Before teq, the sound-

speed is cs = c/√

3, where c is the speed of light, and onecan show that there is very little accretion [16]. After

8

teq, the accretion radius is increased (since cs falls belowc) and the accretion rate is larger. However, the problemis complicated because the BH luminosity will boost thematter temperature of the background Universe abovethe standard Friedmann value even if the PBH density issmall, thereby reducing the accretion.

Thus there are two distinct but related PBH con-straints: one associated with the effects on the Universe’sthermal history and the other with the generation ofbackground radiation. This problem was first studiedin Ref. [32]. Even though this work neglected some fac-tors and was later superseded by more detailed numer-ical investigations, we will use it here because it is theonly analysis which applies for very large PBHs. Theneglected factors mainly affect the time from which theanalysis described below can be applied.

We assume that each PBH accretes at the Bondi rate

M ≈ 1011

(M

M

)2 ( n

cm−3

)( T

104 K

)−3/2

g s−1 , (12)

where a dot indicates a derivative with respect to cosmictime t and n and T are the particle number density andtemperature of the gas at the black-hole accretion radius,

Ra ≈ 1014

(M

M

)(T

104 K

)−1

cm . (13)

For large values of M , we will find that the Bondi formulamay be inapplicable at early times and we return to thisissue later. Each PBH will initially be surrounded byan HII region of radius Rs. If Ra > Rs or if the wholeUniverse is ionised (so that the individual HII regionsmerge), the appropriate values of n and T are those inthe background Universe (n and T ). In this case, after

decoupling, M is epoch-independent so long as T hasits usual Friedmann behaviour (T ∝ z2). However, Mdecreases with time if T is boosted above the Friedmannvalue. If the individual HII regions have not merged andRa < Rs, the appropriate values for n and T are thosewhich pertain within the HII region. In this case, T isusually close to 104 K and pressure balance at the edgeof the region implies n ∼ n (T /104 K). This implies M ∝z5, so the accretion rate rapidly decreases in this phase.

We assume that accreted mass is converted into out-going radiation with constant efficiency ε, so that theassociated luminosity is

L = ε Mc2 , (14)

until this reaches the Eddington limit,

LED =4πGMmp

σT≈ 1038

(M

M

)erg s−1 , (15)

at which the Thomson drag of the outgoing radiation bal-ances the gravitational attraction of the hole. (Here σT

is the Thomson cross-section and mp is the proton mass.)

The assumption that ε is constant may be simplistic andmore sophisticated models allow it to be M -dependent.We also assume that the spectrum of emergent radiationis constant, extending up to an energy Emax = 10 η keV,with Ref. [32] considering models with η = 0.01, 1 and100. This assumption is also simplistic but allows ananalytic treatment of the problem.

We must distinguish between: (1) the local effect of aparticular BH at distances sufficiently small that it dom-inates the effects of the others; and (2) the combinedeffect of all the BHs on the mean conditions of the back-ground Universe. Both effects are very dependent onthe spectrum of the accretion-generated radiation. Asregards (1), the temperature in the HII region aroundeach PBH is somewhat smaller than 104 K, falling asθ ≈ (z/103)0.3, where θ is the temperature in units of104 K, because the temperature inside the HII region isdetermined by the balance between photoionisation heat-ing and inverse Compton cooling off the CMB photons.As regards (2), if the spectrum is hard, photons with en-ergy much above 10 eV will escape from the HII regionunimpeded, with most of the black-hole luminosity go-ing into background radiation or global heating of theUniverse through photoionisation when the backgroundionisation is low and Compton scattering off electronswhen it is high. The matter temperature would gener-ally be boosted well above its Friedmann value and, fora wide range of values for M and fPBH(M), the Universewould be reionised, which may itself be inconsistent withobservations.

Providing the Bondi formula applies, the analysis ofRef. [32] shows that a PBH will accrete at the Eddingtonlimit for some period after decoupling if

M > MED ≈ 103 ε−1 Ω−1g M , (16)

where Ωg ≈ 0.05 is the gas density parameter, and thiscondition certainly applies for SLABs. This phase willpersist until a redshift zED which depends upon M , ε,Ωg and ΩPBH and could be as late as galaxy formationfor large enough PBHs. So long as the PBHs radiateat the Eddington limit, which could be true even if theBondi formula is an overestimate, the evolution of thetemperature in the background Universe is as indicatedin Fig. 2.

The overall effect on the thermal history of the Uni-verse (i.e. going beyond the Eddington phase) for differ-ent (ΩPBH, M) domains is indicated in Fig. 3. Again thisassumes the Bondi formula applies. In domain (1), T isboosted above 104 K and the Eddington phase persistsuntil after the redshift [32]

z∗ ≈ 102 Ω1/3m (ηΩg)−2/3 (17)

at which most of the black-hole radiation goes into Comp-ton heating; T is boosted to Emax in the top right. Indomain (2), T is also boosted above 104 K but the Ed-dington phase ends before z∗. In domain (3), T is boosted

9

to 104 K but not above it because of the cooling of theCMB. In domain (4), T increases for a while but does notreach 104K, so the Universe is not re-ionised. In domain(5), T never increases during the Eddington phase butfollows the CMB temperature for a while (i.e. falls likez). We are mainly interested in domains (1) to (3) andin these the Eddington phase ends at a redshift [32]

zED ≈

103.8 (ΩPBH η)2/9 (MεΩg/M)−4/27 (1) ,

103.3 Ω1/6PBH (Mε/M)−1/9 Ω−5/18

g (2) ,

104.0 (MεΩg/M)−1/3 (3) .

(18)

Following Ref. [89], we now derive constraints on thePBH density by comparing the time-integrated emissionto the observed background intensity. In domain (1) themaximum contribution comes from the end of the Ed-dington phase (zmax = zED), which is after the epoch z∗;

in domain (2) it comes from the epoch zmax = z∗, some-what after the end of the Eddington phase; in domain(3) it also comes from this epoch but the backgroundtemperature never goes above 104 K and the backgroundlight limit turns out to be unimportant. The redshiftedtime-integrated energy production per PBH in the rele-vant domains is

E ≈

1046 (εΩg)10/27(ΩPBH η)−5/9 (M/M)

37/27erg (1) ,

1045.5 ε2/5 Ω3/5PBH η

−11/15 Ω4/15g (M/M)

7/5erg (2) .

(19)If η = 1, corresponding to Emax = 10 keV, then in do-main (1), where zmax ∼ (10 – 100), the radiation wouldreside in the range (0.1 – 1) keV where the observed back-ground radiation density is ΩR ∼ 10−7; in domain (2),where zmax ∼ 100, the radiation would presently resideat ∼ 100 eV where ΩR ∼ 10−6.5. The associated limit onthe PBH density parameter is then

ΩPBH <

(10 ε)−5/6(M/2× 105M

)−5/6η5/4

(Ωg/0.05

)−5/6(1) ,

(10 ε)−1(M/3× 104M

)−1η11/6

(Ωg/0.05

)7/6(2) .

(20)

There is a discontinuity at the 1/2 boundary because ofthe assumed jump in ΩR. Clearly this is unrealistic sinceΩR would vary continuously across the (fPBH,M) planein a more precise analysis. We therefore only show thedomain (1) limit in Fig. 3, as indicated by the upperpart of the blue line. This constraint depends on thevalidity of the Bondi formula with constant M , so wenow consider whether this is applicable.

Three feature could modify the above analysis. First,the accretion which generates the luminosity also in-creases the BH mass, so we need to consider the con-sequences of this. During the Eddington phase, eachBH doubles its mass on the Salpeter timescale, tS ≈4 × 108 ε yr [75], so M can only be regarded as constant

if tED < tS; this implies zED > zS ≈ 10 ε−2/3. FromEq. (18), this corresponds to (ΩPBH,M) values to theleft of the bold line in Fig. 3, which is given by

M <

1019 Ω−1

g ε7/2 (ΩPBH η)3/2M (1) ,

109 Ω−1g εM (3) .

(21)

To the right of the bold line, as may apply for SLABs,the PBH mass increases by a factor

M/Mi ≈ exp(tED/tS) ≈ exp[(0.1/ε)(40/zED)3/2

], (22)

where Mi is the initial mass of the PBH, so the previousanalysis is inconsistent in this region. It is therefore bestto regard M in Eq. (18) and Fig. 3 as the current massand accept that the above analysis only applies above

the bold line. However, one can analyse the problemmore carefully by using Eq. (22) to express Eq. (18) interms of Mi. One then finds a limiting initial mass abovewhich the Eddington phase extends right up to galaxyformation, when the model breaks down anyway.

The constraint in Eq. (20) does not apply below thebold line in Fig. 3 and it intersects this at

M ≈ 109 (10 ε)η3/2 Ω−1g M . (23)

However, the following argument gives its form for largerM . Since most of the final black-hole mass generatesradiation with efficiency ε, the current energy of theradiation produced is E(M) ≈ εMc2/z(M) where theredshift at which the radiation is emitted must satisfyz(M) < 10 ε−2/3. The current background radiation den-sity is therefore ΩR ≈ εΩPBHz(M)−1, so the constraintbecomes

ΩPBH < ε−1 ΩR zS ≈ 10−5 (10 ε)−5/3 . (24)

This limit is shown by the flat part of the blue line inFig. 3. This is equivalent to the well-known Soltan con-straint [72] from observations of the X-ray backgroundon the SMBHs that power quasars. A more precise cal-culation would be required to derive the exact transitionfrom the limit (20). The evolution of the PBH massand temperature after tED is complicated in this casebut limit (24) is independent of this.

The second problem is that the steady-state assump-

10

FIG. 2. This shows how the effect of PBH accretion on theevolution of the matter temperature T during the Eddingtonphase, while independent of the mass M , depends on the PBHdensity ΩPBH, from Ref. [32]. We assume that the energy ofthe accretion-generated photons is Emax = 10 keV and thatΩg = 0.05. For ΩPBH > 10−8, T will always deviate fromFriedmann behaviour and the whole Universe will be colli-sionally re-ionised when T reaches 104 K. For ΩPBH > 10−4,T never falls below 104 K (so the Universe does not go througha neutral phase at all after decoupling) and it will eventuallyrise above 104 K when the Compton heating of the generatedphotons exceeds the inverse Compton cooling of the CMB. Tflattens off when it reaches Tmax ∼ 108 K.

tion fails if the Bondi accretion timescale,

tB ≈ 1012

(M

104M

)(T

104 K

)−3/2

s , (25)

exceeds the cosmic expansion time (i.e. the Bondi for-mula is inapplicable at times earlier than tB). This isequivalent to the condition that the mass within the ac-cretion radius exceeds M , a problem already discussedin Ref. [32]. For Ra < Rs or in domain (3), one hasT ≈ 104 K, so Eq. (25) implies that the Bondi formulaapplies only at times later than 108 (M/M) s. In do-mains (1) and (2), T is increased, so the calculation of tBis more complicated. Ref. [32] argues that the large masswithin the accretion radius will complicate the dynamicsof the accretion flow but that there is no reason for sup-posing that M will be reduced relative to the Bondi rate.However, that conclusion is questionable and probably

-

-

-

-

-

-

FIG. 3. This shows how the effect of PBH accretion onthe evolution of the background matter temperature dependson the PBH mass and density, from Ref. [32]. We assumeε = 0.1, Ωg = 0.05 and Emax = 10 keV. The accretion rateexceeds the Eddington limit for some period after decouplingto the right of the line zED = 103 and the Eddington phasepersists throughout the pregalactic era to the right of the linezED = 10. In each domain the end of the Eddington phasetED depends on ΩPBH and M in a different way. In domains(1) and (2), T is boosted above 104 K by Compton heating;tED exceeds t∗ in domain (1) but it is less than it in domain(2). Note that T can attains the temperature of the hottestaccretion-generated photons above the line in the top right-hand corner of domain (1). In domain (3), T is boosted up to104 K but not above it and the whole Universe is re-ionised,with no neutral phase at decoupling at all for ΩPBH > 10−4.The ionised phase necessarily persists until galaxy formationin domains (1) and (2); it may also do so in parts of domain(3). In domain (4), the Universe is not re-ionized but there is aperiod in which T rises. In domain (5), T always falls but, fora period after decoupling, it stays at the CMB temperaturerather than falling like z2.

the accretion radius is reduced to the value within whichthe gas mass contained is comparable to M 1. In fact,the situation is analogous to that discussed in Ref. [12] inthe context of AGN accretion, where fragmentation intostars is assumed to reduce the accretion.

Because of the uncertainty, we now consider thesteady-state condition more carefully. Clearly Eq. (18)for zED applies only if this corresponds to a time laterthan tB at that epoch. Since Eq. (18) implies that

tED ∝ z−3/2ED increases more slowly than M , the steady-

state condition can only be satisfied if T increases above104 K, which requires that one be in domain (1) or (2).The temperature in these domains is determined by thebalance of Compton heating from accretion-generatedphotons and Compton cooling off the CMB and this

1 The neglect of the cosmic expansion also invalidates the use of theBondi formula during the radiation era and this hugely reducesthe expected PBH accretion [16]. The consequences in the matterera may be less significant but are still uncertain.

11

FIG. 4. The end of the Eddington phase tED (red), theBondi timescale at that epoch tB(tED) (blue) and the timeat which radiation drag becomes unimportant ta (green) asfunctions of M for ΩPBH = 0.1 (solid) and 10−5 (broken). TheBH increases it mass appreciably if tED exceeds the Salpetertimescale tS (black dotted). The steady-state Bondi formulais applicable for tED < tB(tED) and radiation drag can be ne-glected by the end of the Eddington phase for tED > ta, bothconditions applying for sufficiently small M .

gives [32]

T ≈

1016 ΩPBH Ω−1

g z−4 K (z > z∗) ,

1013 ΩPBH ηΩ−1/2m z−5/2 K (z < z∗) ,

(26)

where z∗ is defined by Eq. (17). Setting z = zED inthis expression then gives tB at tED as a function of Mand ΩPBH. This is plotted in Fig. 4 and compared withthe function tED(M) for particular values of ΩPBH. Thisshows that the Bondi formula is applicable at tED onlyfor M < 104M.

We conclude that the increase in the background tem-perature does not suffice to restore steady-state accre-tion before the end of the Eddington phase for SLABs.However, the implications of this remain unclear. Pos-sibly one might expect the solution to be described byself-similar infall instead [90]. Figure 4 also shows theSalpeter timescale tS, so the BH mass increases appre-ciably only where this falls below the tED line (i.e. onlyfor very large values of M).

The third problem is that accreting gas will have an in-ward velocity (vin) relative to the expanding backgroundof CMB photons and the Thomson drag of these photonswill inhibit accretion at sufficiently early times. If thedrag per particle (∼ ρR σT c vin where ρR is the radia-tion density) exceeds the gravitational attraction of thehole at the accretion radius (GMmp/R

2a), the effective

accretion radius will be reduced to

R∗ ≈(GMmp t

ρR σT c

)1/3

, (27)

where the drag and attraction balance. This implies that

accretion is reduced until the time at which R∗ reachesRa. If Ra < Rs or in domain (3), one can assume T ≈104 K and this time can be shown to be

ta ≈

1011 (M/M)3/8 Ω−1/2

m s (106 < M/M < 107) ,

1010 (M/M)6/11 Ω−4/11m s (M > 107M) ,

(28)The mass scales M ∼ 106M and 107M correspond tota ∼ 1013 s (decoupling) and ta ∼ 1014 s, respectively. Indomains (1) and (2), one must account for the tempera-ture increase and one finds

ta ≈

4× 103 (M/M)

3/13(ηΩPBH)

−9/26yr (1) ,

3× 104 (M/M)6/35

(Ωg/ΩPBH)9/35 yr (2) .

(29)This function is also shown in Fig. 4 and the expres-sion for the end of the Eddington phase is unaffected iftED > ta . We can see that radiation drag is alreadyunimportant at decoupling for M < 106M and thatit becomes unimportant before the end of the Edding-ton phase for all M . So the only effect of the drag is topostpone the onset of this phase.

B. More recent accretion studies

Later an improved numerical analysis of pregalacticPBH accretion was provided by Ricotti et al. [33]. Theyused a more realistic model for the efficiency parame-ter ε, allowed for the increased density in the dark haloexpected to form around each PBH and included the ef-fect of the velocity dispersion of the PBHs on the accre-tion in the period after cosmic structures start to form.They found much stronger accretion limits by consider-ing the effects of the emitted radiation on the spectrumand anisotropies of the CMB rather than the backgroundradiation itself. Using FIRAS data to constrain the first,they obtained a limit fPBH(M) < (M/M)−2 for 1M <M . 103M; using WMAP data to constrain the sec-ond, they obtained a limit fPBH(M) < (M/30M)−2

for 30M < M . 104M. The constraints flatten offabove the indicated masses but are taken to extend upto 108M. Although these limits appeared to excludefPBH = 1 down to masses as low as 1M, they werevery model-dependent and there was also a technical er-ror (an incorrect power of redshift) in the calculation.

This problem has been reconsidered by several groups,who argue that the limits are weaker than indicated inRef. [33]. Ali-Haımoud and Kamionkowski [35] calculatethe accretion on the assumption that it is suppressed byCompton drag and Compton cooling from CMB photonsand allowing for the PBH velocity relative to the back-ground gas. They find the spectral distortions are toosmall to be detected, while the anisotropy constraintsonly exclude fPBH = 1 above 102M. Horowitz [91] per-forms a similar analysis and gets an upper limit of 30M.

12

Poulin et al. [92, 93] argue that the spherical accretionapproximation probably breaks down, with an accretiondisk forming instead, and this affects the statistical prop-erties of the CMB anisotropies. Provided the disks formearly, these constraints exclude a monochromatic distri-bution of PBH with masses above 2M as the dominantform of dark matter. However, none of these analysesconsiders masses above 104M, which is why we havefocussed on the old analysis of Ref. [32].

VI. SLAB CONSTRAINTS FROM WIMPANNIHILATIONS

If PBHs do not constitute most of the dark matter,the question of the nature of the remaining part arises.In the following analysis, we assume a mixed DM sce-nario in which the PBHs are subdominant, i.e. fPBH ≡ρPBH/ρDM 1, where ρDM is the total observed DM en-ergy density. In more detail, the dominant dark-mattercomponent is taken to be a WIMP, whose abundance isset through a thermal mechanism, although our conclu-sions below hold in more general cases. We therefore as-sume a WIMP density ρχ ≡ fχ ρDM with fχ + fPBH = 1.One could envisage a scenario in which fPBH is very smallfor SLABs but close to unity in some other mass range,in which case both fSLAB and fχ could be small. Wediscuss such a scenario in an accompanying paper [43].

A. Structure of the dark-matter halos

We first consider the production of the WIMP num-ber density through thermal freeze-out. When the an-nihilation rate falls below the expansion rate of theUniverse, the number of WIMPs per comoving volumefreezes out. This occurs at a temperature given bykB TF ∼ mχ c

2/20 [94, 95], where mχ is the mass ofthe WIMP and kB is Boltzmann’s constant. Even af-ter freeze-out, the relativistic plasma and WIMP popu-lation keep exchanging energy and momentum until thescattering rate falls below the Hubble rate at kinetic de-coupling (KD) [96]. This leaves an imprint on the currentWIMP velocity dispersion, since the photon temperatureTγ scales as a−1 after KD, while the temperature of the

non-relativistic WIMPs scales as a−2. We use the follow-ing expression for the KD temperature [97]:

kB TKD =mχ c

2

Γ(3/4)

(gmχ

MPl

)1/4

, (30)

where g ≈ 10.9 for temperatures in the range 0.1 –10 MeV and Γ(3/4) ≈ 1.225. This expression coincides,within a numerical factor, with other definitions in theliterature [98]. The corresponding Hubble rate and timeare HKD and tKD = 1/(2HKD), respectively.

PBHs are formed prior to teq (i.e. during the radiation-dominated epoch) from the direct collapse of mildly non-linear perturbations. The PBH mass at KD is

MKD ≈ 300Mm5/4100 , (31)

where m100 = mχ c2/(100 GeV). After PBH formation,

the WIMPs will be gravitationally attracted to the PBHs,leading to the formation of surrounding halos. The struc-ture of these halos depends on the specific circumstancesand particle velocities. The fraction of WIMPs with lowvelocities remain gravitationally bound to the PBHs andform density spikes around them. For PBHs smallerthan MKD, the WIMP density is expected to be uni-form, since these PBHs have formed before kinetic de-coupling, when the WIMPs are still tightly coupled tothe plasma. WIMP accretion occurs during two differentperiods: (1) between the kinetic decoupling and teq; (2)through secondary accretion after teq. The halo mass isnever much more than the PBH mass in the first phasebut it can be much larger in the second phase. In bothcases, the WIMPs form a halo with a universal densityprofile ρχ(r) ∝ r−9/4 and this halo grows with time [90].We elaborate on this and the mass ranges involved below.

B. Formation of WIMP halo around PBH

WIMPs which are non-relativistic after freeze-out canform a gravitationally-bound halo around PBHs [99–101]immediately after kinetic decoupling [36–39]. Assumingcomoving entropy conservation, the WIMP energy den-sity at a time t < teq is

ρχ spike = fχρeq

2

(a

aeq

)−3

≈ fχρeq

2

(t

teq

)−3/2

, (32)

where ρeq = 3/(32πGt2eq). The second expression ne-glects any change in the entropy degrees of freedom. Inorder to find the extent of the DM profile around a PBH,we consider the turn-around point of the radial motion ofan orbiting particle, assuming the Newtonian equation,

r = −GMr2

+a

ar = −GM

r2− r

4 t2, (33)

where r is the distance of the particle from the PBH.The second expression holds for a radiation-dominateduniverse with a ∝ t1/2. The numerical solution for theturn-around radius obtained from Eq. (33) is well ap-proximated by [39]

rta(t) ≈[rg (c t)2

]1/3, (34)

where rg = 2GM/c2. This is just the condition that thetwo terms on the right-hand-side of Eq. (33) are com-parable. As explained in Ref. [39], Eq. (34) is just theevolving radius within which the cosmological mass is

13

comparable to the PBH mass since overdensities do notgrow during the radiation era.

Using Eqs. (32) and (34), it can be shown that thedensity profile of the WIMPs around the BH at time teq

corresponds to a spike with [39]

ρχ spike(r) = fχρeq

2

(rta(teq)

r

)9/4

= fχρeq

2

(M

M

)3/4 (ro

r

)9/4, (35)

where ro = (2GM t2eq)1/3 = 0.0193 pc. This just comes

from the cosmological density at the epoch when r is theturn-around radius. Eq. (35) only applies up to the ra-dius rta(teq) and the mass within this radius is compara-ble to M (as expected). This profile has been confirmedby numerical simulations for PBHs of 30M [39] but itshould also hold for more massive PBHs. In principle,the orbital motion of the WIMPs would influence theirdensity profile [36]. However, the WIMP kinetic energycan be neglected for the PBH masses relevant to this pa-per. In this regime, a detailed derivation of the WIMPdensity profile after KD leads to the density profile (35)multiplied by a concentration parameter αE ≈ 1.53 [39].

After matter-radiation equality, the mass gravitation-ally bound by the PBH grows according to

M(z) = M

(1 + zeq

1 + z

)(36)

and this is described as ‘secondary’ accretion [102].

Equivalently, the overdensity on a scale M is M/M atteq, so the mass binding at t is

M ∼M (t/teq)2/3 ∼M (ρeq/ρ)1/3, (37)

corresponding to radius

r ∼ (M/ρ)1/3 ∼M1/3 ρ1/9eq ρ−4/9 , (38)

which just gives ρ ∝ r−9/4. Therefore, at a given redshift,secondary infall and virialisation lead to a DM densityspike with the same radial dependence as Eq. (35). Thisis confirmed by the numerical calculations of Ref. [39].The accretion halts around the epoch of galaxy forma-tion, which we set at z? ∼ 10, because of the effects ofdynamical friction between DM halos and hierarchicalstructure formation.

The WIMP population inside the halo is consumed byself-annihilation [103]. This gives a maximum concentra-tion at redshift z of

ρχmax(z) = fχmχH(z)

〈σv〉, (39)

where 〈σv〉 is the velocity times the WIMP self-annihilation cross-section, averaged over the velocity dis-

tribution, and H(z) = H0 h(z) is the Hubble rate at red-shift z. Combining these results gives the WIMP profile

ρχ(r) = min[ρχmax(z), αE ρχ spike(r)

]. (40)

The extent of the plateau rcut is obtained by equatingthe two expressions in Eq. (40),

rcut(z) ≈ 21[m100 h(z)

]−4/9(

M

1010M

)1/3pc . (41)

C. WIMP annihilation rate around PBHs

We assume that WIMPs annihilate into StandardModel particles, in particular photons, through the s-wave channel, with no significant contribution from co-annihilation [104] or Sommerfeld enhancement [105].The thermal freeze-out mechanism fixes the numberof WIMPs in a comoving volume for a specific valueof the velocity-average WIMP annihilation cross-section〈σv〉 [94, 106, 107]. The value of 〈σv〉 is independenton the WIMP velocity distribution to lowest order inv/c and, for non-relativistic WIMPs, it is the samethroughout the history of the Universe. Here, we set〈σv〉 = 〈σv〉F/fχ, where 〈σv〉F ≈ 3×10−26 cm3 s−1 is thevalue of 〈σv〉 required for WIMPs with mχ & 10 GeV tobe produced at thermal freeze-out with fχ = 1. The scal-ing 〈σv〉 ∝ 1/fχ is expected within the standard freeze-out theory [108].

The annihilation rate is taken to be

Γ(z) ≡ 〈σv〉m2χ

∫dr 4πr2 ρ2

χ(r)

=4π 〈σv〉m2χ

ρ2χmax(z) r3

cut(z) , (42)

where the last expression assumes the DM density profilegiven by Eq. (40). The redshift dependence of the decayrate can be expressed as

Γ(z) = f5/3χ Γ0 [h(z)]2/3 , (43)

where

Γ0 ≈3

8

(α4

E 〈σv〉F ρeqH20

2m4χ

)1/3

M ≡ ΥM (44)

and Υ has units of g−1 s−1.

D. Extragalactic background flux

The extragalactic gamma-ray flux produced by thecollective annihilations around PBHs at all redshifts z

14

is [109]

dΦγdE dΩ

∣∣∣∣e.g.

=c

8π

∫ ∞0

dze−τE(z, E)

H(z)

dNγdE

∫dM Γ(z)

dn(M)

dM,

(45)where Γ(z) is the WIMP annihilation rate around PBHsat redshift z and τE is the optical depth back to thatredshift. We make the following assumptions.

• The spectrum of by-products from WIMP annihi-lation is obtained using the numerical package inRef. [110] (see also Ref. [111] for relevant updates).Integral (45) only depends on the PBH mass func-tion at z = 0, since the (1+z)3 volume factors can-cel out in the computation of Eq. (45) [110, 112].

• The optical depth τE in Eq. (45) results from var-ious processes [112, 113]: (i) photon-matter pairproduction; (ii) photon-photon scattering; (iii)photon-photon pair production. We adopt the op-tical depth obtained in Ref. [110]. We assumethat the curvature contribution is zero, in agree-ment with the prediction from inflation and CMBmeasurements [114, 115], although Refs. [116, 117]argue that the CMB data favours spatially closedmodels.

• We assume a flat FRW metric with the Hubble rate

h(z) =√

ΩΛ + Ωm (1 + z)3 + ΩR (1 + z)4 (46)

in units of H0, where we fix the values of the presentdensity parameters to be ΩR = 7×10−5, Ωm = 0.31and ΩΛ = 1 − Ωm − ΩR = 0.69. The PBH massfunction is normalised according to∫

dM Mdn(M, z)

dM≡ ρPBH(z) , (47)

where ρPBH(z) ≡ fPBH ρDM(z).

Inserting Eq. (44) into Eq. (45) and using the normal-isation in Eq. (47), we can eliminate the mass functionin the expression for the flux to obtain

Φγ = fPBH f5/3χ

ΥρDM

2H0Nγ(mχ) , (48)

where the average number of photons produced is

Nγ(mχ) ≡∫ ∞z?

dz

∫ mχ

Eth

dEdNγdE

e−τE(z, E)

[h(z)]1/3. (49)

The redshift integral is dominated by the range z .O(100), because of the sharp decline in the optical depthat large redshifts. A numerical fit to the WIMP massdependence of Eq. (49) with the results obtained from

the package in Ref. [110] leads to Nγ ≈ 220m0.22100 .

Eq. (48) is valid for all PBH mass distributions, in-cluding the monochromatic case [37] and the more re-

alistic extended case. For example, one has dn/dM ∝M−1/2 for PBHs formed from exactly scale-invariant den-sity fluctuations [118] or from the collapse of cosmicstrings [119] and a lognormal mass function for PBHsformed from a large class of inflationary PBH mod-els [120], such as the axion-curvaton model [121].

Comparing the integrated flux with the Fermi point-source sensitivity ΦFermi

100 MeV for fPBH 1 and fχ ≈ 1

yields the limit 2

fPBH .2 ΦFermi

100 MeV H0

ΥρDM Nγ(mχ)≈ 8× 10−12m1.11

100 . (50)

This limit intersects the extragalactic incredulitylimit (2) at a mass

M egIL =

2 ΦFermi100 MeV

ΥH20 Nγ(mχ)

≈ 2.5× 1010m1.11100 M , (51)

where the numerical expression accounts for the fit ofNγ(mχ) and we set 〈σv〉 = 〈σv〉F. This corresponds toan upper limit on the mass of a SLAB in our Universe.

E. Flux from nearest individual source

The gamma-ray background flux produced from dark-matter annihilation around an individual PBH is [124]

dΦγdE dΩ

=Γ

8π d2L

dNγdE

, (52)

where Γ is the DM decay rate around the BH and dL

is the distance of the PBH, which is necessarily extra-galactic in the SLAB case. The BH-halo system can bedetected within a distance

dL =

√ΓNγ(mχ)

2 ΦFermi100 MeV

, (53)

where the average number of photons resulting from theannihilation processes is

Nγ(mχ) =

∫ mχ

Eth

dEdNγdE

. (54)

We fit the numerical solution of Eq. (54) with the packagein Ref. [110] to obtain

Nγ ≈ 2.0 (mχ/GeV)0.32 . (55)

The ratio Nγ(mχ)/Nγ(mχ) can be estimated analyticallyby neglecting the E-dependence of the opacity and as-

2 This constraint could in principle be refined by performing a like-lihood analysis accounting for the differential energy spectrumfrom WIMP annihilation in each energy bin [122, 123].

15

INCREDULITYLIMIT

-

-

-

-

-

-

-

-

-

FIG. 5. Constraints on fPBH as a function of PBH mass.Results are shown for mχ = 10 GeV/c2 (dashed line), mχ =100 GeV/c2 (solid line) and mχ = 1 TeV/c2 (dotted line),setting 〈σv〉 = 3× 10−26 cm3/s. Also shown is the incredulitylimit (black dashed line)

suming h(z) ≈ Ω1/2m (1 + z)3/2. Then Eq. (49) implies

Nγ ≈ 50Nγ .

For a given value of M , we can compare Eq. (53)with the expected distance to the nearest BH, d ≈(M/ρBH)1/3. If BHs are primordial, ρBH = fPBH ρDM,so this leads to the constraint

fPBH .

(2 ΦFermi

100 MeV

ΥNγ(mχ)

)3/2M−1/2

ρDM. (56)

The BH is necessarily extragalactic in the SLAB case andthis analysis holds providing it is at a redshift z 1.The bound (56) is more stringent than the backgroundbound (50) only if M exceeds

M ≡ 2 ΦFermi100 MeV

ΥH20

N2γ (mχ)

N3γ (mχ)

= M egIL

[Nγ(mχ)

Nγ(mχ)

]3

, (57)

where M egIL is given in Eq. (51). However, the quan-

tity in square brackets is O(105), so M M egIL and

the nearest-source bound lies well outside the cosmolog-ical incredulity limit. Therefore the individual boundis never applicable for the range of WIMP masses con-sidered. Figure 5 shows the constraints on fPBH fordifferent WIMP masses: mχ = 10 GeV (dashed lines),mχ = 100 GeV (solid lines), mχ = 1 TeV (dotted lines).We extend the computation to a wider range of WIMPand BH masses in a follow-up paper [43].

VII. SLABS AND LIGHT BOSONS

Is it possible for spinning BHs to lose a portion of theirrotational energy via the interaction with an interferingboson wave of frequency ω < µΩBH, where ΩBH is theBH horizon frequency and µ is the azimuthal numberof the wave. When this criterion is satisfied, the outgo-ing wave extracts energy and angular momentum fromthe BH through the phenomenon of superradiance. Iflight bosonic fields exist in nature, they could accumulatearound rotating SMBHs and form a condensate, lead-ing to such superradiant instabilities [125]. A portion ofthe rotational energy of the SLAB might be dissipatedby the boson cloud via superradiance if the Comptonwavelength of the boson λC = h/(mφc), where h is thePlanck constant and mφ is its mass, is comparable tothe Schwarzschild radius of the SLAB. Interestingly, forM & 1010M, this condition is realised for an ultra-lightboson of mass mφ . 10−22 eV, which has important as-trophysical consequences [126, 127]. For example, thismechanism has been applied jointly with the observa-tions of the mass and spin of the SMBH M87∗ to placebounds on the mass of hypothetical light bosons [53]. Al-though PBHs are generally formed with a negligible spin,we expect SLABs to acquire a large momentum due tothe accretion mechanisms described in Sec. V. We dis-cuss the phenomenon for light spin-zero fields, althoughimportant consequences are also obtained for spin-onefields [128, 129] and tensor fields [130].

The dimensionless spin parameter is a∗ = Jc/(GM2),where J is the angular momentum of the SLAB. Whenthe angular velocity of the BH horizon is larger than theangular phase velocity ω of the wave, a population ofspin-zero bosons grows around a spinning BH [44, 47],

ω <µ

rS

a∗

1 +√

1− a2∗. (58)

The square root term ensures that the spin parameter ofa Kerr BH cannot exceed unity.

The leading mode of the superradiant bound state ofscalar bosons grows exponentially, Nµ ∝ exp(Γφt), at arate [131]

Γφ = a∗ r8g m

9φ/24 . (59)

For example, for an ultra-light axion of mass mφ =

10−22 eV and with a Compton wavelength comparablewith rg, the rate is Γφ ≈ 10−8 s−1.

Superradiance is disrupted over a characteristic BHtimescale τBH, related to the accretion timescale by

Γφ τBH & lnNµ . (60)

We take the characteristic BH timescale to be τBH ∼tS [131], where tS is the Salpeter timescale introduced inSec. II and we use an efficiency parameter ε ∼ 0.1 [132].The occupation number of the boson cloud for the az-

16

-

-

-

-

-

-

-

-

FIG. 6. Superradiance constraints on the mass mφ of a hy-pothetical boson as a function of BH mass M . Results areshown for the observed black-hole spin a∗ = 0.99 (solid line),a∗ = 0.2 (dashed line) and a∗ = 0.01 (dotted line).

imuthal number µ after the SLAB has spun down by avalue ∆a∗ is [129]

Nµ =GM2∆a∗

µ. (61)

If a BH with spin a∗ is observed, the condition inEq. (58) yields a lower bound on the mass of the lightboson, mφ ≈ ω, while the requirement of Eq. (60) thatsuperradiance has not depleted the spin of the BH by theamount ∆a∗ leads to an upper bound on mφ. Figure 6shows the region excluded by Eqs. (58) and (60). Weassume that the SLAB had an initial spin a∗ i ≈ 1 andevolved so that its spin today is 0.99 (solid line), 0.2 (dot-ted line) or 0.01 (dashed line). Since the Schwarzschildradius of a SLAB is considerably larger than that ofM87∗, observing these objects would lead to a constraintfor extremely light bosons with mass mφ 10−20 eV.

VIII. RESULTS AND DISCUSSION

In this work, we have examined the bounds on stupen-dously large BHs with M & 1011M, here referred toas SLABs. We have considered their possible formationmechanisms and assessed the limits coming from dynam-ical, lensing and accretion effects and from gamma-rayannihilation of WIMP dark-matter around PBHs.

We have assumed that the WIMP cross-section doesnot change during the evolution of the Universe. This isnot true if there is a light mediator that leads to a Som-merfeld enhancement of the WIMP annihilation [105].We have also assumed that the cross-section is fixed tothe value obtained at freeze-out 〈σv〉F in the standardcosmological model. However, its value might deviateconsiderably from this if there were an early period inwhich the cosmological density was dominated by mat-ter or some other exotic form of energy [133]. The ex-pected signal from WIMPs annihilating around a SLABalso needs to be reconsidered if the WIMP velocity dis-tribution plays a role in the computation of 〈σv〉, forexample when corrections of order (v/c)2 are to be takeninto account or when the annihilation does not proceedthrough an s-channel.

The expected gamma-ray flux from WIMP annihila-tion depends on the combination Φγ ∝ fPBH f

5/3χ , as

shown by Eq. (48). In this work, we have assumed thatWIMPs make up most of the DM, with PBHs contribut-ing a negligible fraction. However, this reasoning can beinverted to constrain the WIMP fraction fχ when PBHsform most the DM. We explore the consequences of thisin an accompanying paper [43].

In Sec. VII we have discussed the possible constraintson the mass of ultra-light bosons for a given SLAB spindue to superradiance effects. Although SMBHs nearly aslarge as SLABs are known to exist, they are considerablyfurther away than M87∗ or Sagittarius A∗, making thedetermination of their spin and their imaging more chal-lenging [8]. Furthermore, their accretion effects wouldmodify the size of the black-hole shadow with time. Weleave these consideration for future work.

Our discussion has not covered other possible particleDM candidates, like the sterile neutrino [134] or the QCDaxion [135]. If SLABs are present in the Universe, theywould provide a powerful tool for cosmological tests dueto their unique imprints. In fact, the stringent bound onthe fraction of PBHs given by Eq. (56) is based on thisassumption. However, our constraints cannot be appliedto models in which PBHs provide nearly all the DM.

ACKNOWLEDGMENTS

We thank Niayesh Afshordi, Yacine Ali-Haımoud, BenHorowitz, Priya Natarajan, Rafaek Nunes, Alex Vilenkinand Kumar Shwetketu Virbhadra and for helpful com-ments. F.K. Acknowledges hospitality and support fromthe Delta Institute for Theoretical Physics. L.V. ac-knowledges support from the NWO Physics Vrij Pro-gramme “The Hidden Universe of Weakly InteractingParticles” with project number 680.92.18.03 (NWO VrijeProgramma), which is (partly) financed by the Dutch Re-search Council (NWO).

17

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