Origin and growth of primordial black holes
aa r X i v : . [ a s t r o - ph . H E ] J a n Origin and growth of primordial black holes
Krzysztof A. Meissner and Hermann Nicolai Faculty of Physics, University of WarsawPasteura 5, 02-093 Warsaw, Poland Max-Planck-Institut f¨ur Gravitationsphysik (Albert-Einstein-Institut)M¨uhlenberg 1, D-14476 Potsdam, Germany
In a previous paper we have argued that primordial black holes can arise from the formation andsubsequent gravitational collapse of bound states of stable supermassive elementary particles (grav-itinos) during the early radiation era. Here we offer a comprehensive picture, describing the evolutionand growth of the resulting mini-black holes through both the radiation and matter dominated phasesuntil the onset of inhomogeneities, by means of an exact metric solving Einstein’s equations. Weshow that, thanks to a special enhancement effect producing an effective horizon above the actualevent horizon, this process can explain the observed mass values of the earliest giant black holes.
In a previous paper [1] we have proposed a new mech-anism to explain the origin of supermassive black holesin the early Universe by means of the condensation ofsuperheavy elementary particles during the early radia-tion phase. Accordingly, the existence of primordial blackholes would be due to the gravitational collapse of suchbound states, shortly after their formation, to small blackholes, whose masses must lie above a critical value toavoid Hawking evaporation. The subsequent growth ofthese black holes during the radiation era is then mod-eled by the exact solution of Einstein’s equation derivedin [1], such that towards the end of the radiation erathe emerging macroscopic black holes can grow to nearlysolar mass objects.As we have explained in [1], superheavy gravitinos canserve as microscopic seeds for generating mini-black holesif their mass is sufficiently large so that their gravita-tional attraction exceeds the repulsive or attractive elec-tric forces between them. Furthermore, these seed par-ticles must be stable against decay into Standard Modelmatter. Although other kinds of particles with similarproperties might serve the same purpose, we have ar-gued in [1] that gravitinos are distinguished in view ofa possible unification of the fundamental interactions.This follows from the structure of the fermionic sectorof the N = 8 supermultiplet [2] (however, as we haveexplained there, the underlying theory must transcendmaximal N = 8 supergravity). Namely, identifying the48 non-Goldstino spin- fermions of the N = 8 super-multiplet with three generations of quarks and leptons(including right-chiral neutrinos) of the Standard Model,one is left with eight massive gravitinos with the proper-ties described in [1]. These properties are radically dif-ferent from those of the more familiar sterile gravitinosof low energy N = 1 supergravity models; in particular,unlike the latter, superheavy gravitinos do participate inStandard Model interactions.In this paper we discuss the further evolution of theseblack holes during the matter dominated phase, and showthat the proposed mechanism can indeed explain the ob-served mass values of supermassive black holes, as re-ported in [3]. Namely, in [1] we did not follow the evo- lution of the emergent macroscopic black holes beyondequilibrium time t eq into the matter dominated era, nordid we provide mass estimates for the large black holesthat emerge at the time of the formation of inhomo-geneities. In this paper we close this gap by offering amuch more comprehensive picture, modeling the growthof mini-black holes into giant black holes ‘from beginningto end’. The fact that this can be done by means of aclosed form metric solving the Einstein equations that en-compasses both the radiation and the matter dominatedphase is a main new result of the present paper.According to [1] the gravitino mass M g is hypothe-sized to lie between M BPS and M Pl , where the ‘BPS-mass’ M BPS is the mass for which the electrostatic re-pulsion between two (anti-)gravitinos of the same chargeequals their gravitational attraction. Here M Pl is the re-duced Planck mass ( ∼ . · − kg, corresponding to thePlanck time t Pl = 2 . · − s). For numerical estimateswe will take M BPS ∼ . · M Pl , so that0 . · M Pl < M g < M Pl (1)This ensures that the force remains attractive also be-tween gravitinos of the same electric charge. The min-imal seed mass for a primordial black hole in the earlyradiation phase is determined by asking the total energyof a bound system of N (anti-)gravitinos to be negative.For the minimum number of (anti-)gravitinos in a boundstate this leads to the estimate [1] N & (2)Importantly, the cosmic time t drops out in the derivationof this inequality, hence the value of N remains the samethroughout the radiation phase. To be sure, if the boundstate is meta-stable, the collapse can be delayed in sucha way that an even larger number N of (anti-)gravitinoscan accrue before gravitational collapse occurs, in whichcase the seed mass value M seed ∼ N M g could be evenlarger. The minimum mass of a black hole resulting fromgravitational collapse of such a bound state is therefore(we set c = 1 throughout) M seed ∼ M g ∼ kg ⇒ GM seed ∼ − m (3)where we assumed M g ∼ − kg. Now, a black hole ofsuch a small mass would be expected to decay immedi-ately by Hawking radiation: from the well known formulafor the lifetime of a black hole (see e.g. [4]) we have τ evap ( m ) = t Pl (cid:18) mM Pl (cid:19) (4)This is the result which would hold in empty space. How-ever, during the early radiation phase this is not the onlyprocess that must be taken into account, because of thepresence of extremely hot and dense radiation, which can‘feed’ black hole growth. The absorption of radiationthus provides a competing process which can stabilizethe black hole against Hawking decay, such that with theinitially extremely high temperatures of the radiation eramass accretion can overwhelm Hawking evaporation evenfor very small black holes . More precisely, for a black holeof given mass m the criterion for accretion to overcomethe rate for Hawking radiation reads T rad ( t ) > T Hawking ( m ) = ~ πGm (5)The break-even point is reached when the radiation tem-perature equals the Hawking temperature, at time t = t ( m ) when T rad ( t ) = T Hawking ( m ). For larger times t > t (and lower radiation temperatures) a black holeof mass m will decay. Imposing this equality, or alterna-tively using eqn.(26) of [1] we deduce the relevant massat time t , which gives m ( t ) ≃ M t Pl · G ρ rad ( t ) = 32 πM Gt Pl · t (6)When read from right to left this equation tells us whichis the latest time for a mini-black hole of given mass m to remain stable against Hawking decay during theradiation phase. This is the case for t < t ≡ t ( m ) ∝ m ,after which time the black hole will decay. Conversely,for a given time t any mini-black hole of initial massgreater than m ( t ) will be able to survive and can startgrowing, whereas those of smaller mass decay. With (3)as the reference value we thus take the initial mass to be ∼ M seed , and assume that the time range available forthe formation of such a mini-black hole is t min = 10 · t Pl ≃ − s < t < t max ≃ − s (7)During this time interval a black hole of initial mass(3) can survive and start growing by accreting radia-tion. While the upper bound is thus determined by set-ting t max ≡ t ( M seed ), the lower bound has been chosenmainly to stay clear of the quantum gravity regime anda possible inflationary phase.Once we have a stable mini-black hole we can study itsfurther evolution through the radiation phase by meansthe exact solution derived in [1], until matter starts todominate over radiation at time t ∼ t eq ∼ − M ⊙ . m ( t eq ) . − M ⊙ (8)However, the solution in [1] does not apply to the matterdominated phase. To investigate the further evolutionone would conventionally switch to a different descriptionby invoking the Eddington formula [3, 5] m ( t ) = M exp (cid:18) πGm p tǫcσ T (cid:19) ≃ M exp (cid:18) t
45 Myr (cid:19) (9)where m p is the proton mass, σ T is the Thompson crosssection, and ǫ is the fraction of the mass loss that isradiated away, which is usually taken as ǫ = 0 .
1. Taking M = 10 − M ⊙ as the initial value at time t = t eq , themass at t = 690 · yr comes out to be m (690 Myr) ≃ · M ⊙ , (10)less than the actually observed value ( ∼ · M ⊙ ) [3].Since the ‘blind’ application of the Eddington formuladoes not produce the desired order of magnitude, we herewant to proceed differently. This is because this for-mula was originally developed to describe the evolutionof luminous stars [5], and it is therefore rather doubtfulwhether one can use it in the present context. In particu-lar, its derivation relies on the Newtonian approximationand is based on a simple equilibrium condition, balancingthe rate of mass absorption against the luminosity of in-falling matter, where the luminosity is assumed to growlinearly with the mass of the black hole. Therefore, afully relativistic treatment by means of an exact solutionof Einstein’s equations seems preferable, even if it doesnot (yet) take into account matter self-interactions.To present this new solution we employ conformal co-ordinates, with conformal time η , instead of the cosmictime coordinate t used above. One main advantage ofthis coordinate choice is that the causal structure ofthe space-time is often easier to analyze (for the solu-tion to be presented below it is the same as that of theSchwarzschild solution). Secondly, we wish to exploit theremarkable fact that the use of conformal time allows usto exhibit a simple closed form solution that encompasses both the radiative and the matter dominated phase. Withconformal time η , the Friedmann equations read (for aspatially flat universe and vanishing Λ)˙ a = 8 πG ρa , a ¨ a − ˙ a = − πG ρ + 3 p ) a (11)where ˙ a ≡ da/dη , dt = a ( η ) dη . (12)The requisite solution is a ( η ) = Aη + B η ⇒ t = 12 Aη + 13 B η (13)The density and pressure following from (11) are8 πGρ ( η ) = 3 A a ( η ) + 12 B a ( η ) , πGp ( η ) = A a ( η ) (14)where, for our Universe (starting from nucleosynthesis) A = 2 . · − s − , B = 6 . · − s − . (15)These numbers can be calculated from known data up torescaling η → λη, A → λ − A, B → λ − / B, a → λ − a .The latter scale is conventionally fixed by setting a ( t ) =1, where t ≃ . · yr is the present time. At theequilibrium between radiation and matter we have [6] a ( η eq ) ≃ , t eq ≃ . · s (16)At the last scattering we have [6] a ( η LS ) ≃ , t LS ≃ . · s (17)and these numbers give (15).We now generalize the solution of [1] by substituting(13) into the metric ansatzd s = a ( η ) (cid:20) − ˜ C ( r )d η + d r ˜ C ( r ) + r dΩ (cid:21) (18)Here the a priori unknown function ˜ C ( r ) is uniquely fixedby imposing two requirements. For the limiting case ofpure radiation ( B = 0) the trace of the Einstein tensorresulting from (18) must vanish: T µµ = 0 ⇒ ( r ˜ C ) ′′ ! = 2[1]. In the other limiting case of pure matter ( A = 0),the pressure must vanish: p = 0 ⇒ ( r ˜ C ) ′ ! = 1. Imposingboth requirements leads to the unique solution˜ C ( r ) ≡ C ( r ) := 1 − G m r (19)and this choice will be used in the following. The newand essential feature here is that the metric (18) allowsus to evolve the black hole through both the radiativeand matter dominated periods, with a smooth transitionbetween the two.In (19) we use a different font for the fixed mass pa-rameter because m is not the physical mass, unlike m ( t )above. This is most easily seen by replacing G m r → G m a ( η ) ra ( η ) ≡ G m a ( η ) r phys ⇒ m ( η ) = m a ( η ) (20)Using (3), (7) and the above relation with η min = 10 − sand η max = 10 s, as well as G m min = GM seed /a max and G m max = GM seed /a min we get G m min ∼ · − m , G m max ∼ · m (21) For the metric ansatz (18) with C ( r ) from (19) the non-vanishing components of the Einstein tensor, hence theassociated energy momentum tensor, are given by:8 πG T ηη = 3 ˙ a a = 3( A + 2 B η ) ( Aη + B η ) πG T rη = 2 G m r C ( r ) · ˙ aa = 2 G m r C ( r ) · A + 2 B ηAη + B η πGT rr = ˙ a − a ¨ aa C ( r ) = 1 C ( r ) · A ( Aη + B η ) (22)together with [10] T θθ = C ( r ) r T rr , T ϕϕ = sin θ T θθ (23)To endow (22) with physical meaning, we must inter-pret the r.h.s. in terms of physical sources of energy andmomentum. To this aim we rewrite the energy momen-tum tensor in the form [7] T µν = pg µν + ( p + ρ ) u µ u ν − u µ q ν − u ν q µ (24)neglecting higher derivatives in u µ and matter self-interactions (viscosity, etc. ). For the density and pres-sure to match between (24) and (22) we must includean extra inverse factor C ( r ) in comparison with (14) toaccount for the curvature8 πGρ ( η, r ) = 1 C ( r ) (cid:18) A a ( η ) + 12 B a ( η ) (cid:19) πGp ( η, r ) = 1 C ( r ) A a ( η ) (25)again with a ( η ) from (13). The 4-velocity is [11] u µ = − a ( η ) C ( r ) / (cid:0) C ( r ) cosh ξ , sinh ξ , , (cid:1) (26)while the heat flow vector is given by8 πGq µ = − G m ˙ a ( η ) r C ( r ) / a ( η ) (cid:0) C ( r ) sinh ξ , cosh ξ , , u µ u µ = − u µ q µ = 0. Theparameter ξ = ξ ( η, r ) > ξ = G m ηr · (cid:18) − B η A + 3 AB η + 3 B η (cid:19) (28)The signs in (26) and (27) are chosen such that for thecontravariant components of the 4-velocity we have u η > u r <
0, hence inward flow of matter. (Choosing theopposite sign for the components of u µ would correspondto a shrinking white hole.)To keep ξ real and finite we must demand tanh ξ < ξ ∼ G m ηr for B η ≪ A (radiation) ∼ G m ηr for B η ≫ A (matter) (29)The representation (24) is valid as long as all quantitiesremain real and finite. This requires r > O (1) G m η ,with a strictly positive O (1) prefactor. When r reachesthe value for which tanh ξ = 1 the components of u µ and q µ diverge, and the expansion (24) breaks down. Forthe external observer the average velocity of the infallingmatter then reaches the speed of light, so for all prac-tical purposes everything happening inside this shell isshielded from the outside (even though light rays canstill escape from this region, as long as r > G m ). As weare not concerned with O (1) factors here we define r H ( η ) := a ( η ) p G m η (30)and interpret the associated outward moving shell as an effective horizon (or ‘pseudo-horizon’) that lies above theactual event horizon; note that r H ( η ) is invariant underthe coordinate rescalings mentioned after (15). Physi-cally, we expect the matter inside the shell r phys . r H ( η )to be rapidly sucked up into the black hole, once the out-side region r phys > r H ( η ) gets depleted of ‘fuel’ due tothe formation of inhomogeneities. The extra matter in-side the shell r phys . r H ( η ) thus enhances the growthsubstantially, beyond the linear growth with the scalefactor implied by (20).At the onset of inhomogeneities, we must stop usingthe metric (18) because the growth of the black hole getsdecoupled from the growth of the scale factor a ( η ), afterwhich the black hole evolves in a more standard fashionby much slower accretion (for this reason there is also nopoint in extending the metric ansatz (18) into the presentepoch, which is dominated by Dark Energy). To estimateits mass we take the value of r H at that particular timeto define an effective Schwarzschild radius, thus equatingthe mass with the maximum energy that can possibly fit inside a shell of radius r H . This approximation is justifiednot only because of the apparent divergent kinetic energyof the infalling matter near r H , but also because of thestrong increase of the density and pressure inside thisshell, which is due to the extra factor C − ( r ) in (25).The relevant time t at which to evaluate r H ( t ) ≡ r H ( η ( t )) lies well after decoupling, since the inhomo-geneities in the CMB are still tiny, of order O (10 − ).Rather, we take t inhom ≃ yr ≃ . · s, which isthe time when the first stars are born [8]. This corre-sponds to η inhom ≃ . · s ⇒ a ( η inhom ) ≃ . r S ( M ⊙ ) = 3 km wecan calculate the range of possible black hole masses at t ∼
100 Myr as10 M ⊙ . m BH . · M ⊙ (31)consistent with observations [3]. To reach such large massvalues the replacement of G m by √ G m η in (30), as ad-vocated in this paper, is evidently of crucial importance. Acknowledgments:
K.A.M. was partially supportedby the Polish National Science Center grant. The work ofH.N. has received funding from the European ResearchCouncil (ERC) under the European Union’s Horizon2020 research and innovation programme (grant agree-ment No 740209). The title of this paper was inspiredby a philosophical treatise by one of the first author’sancestors [9]. [1] K.A. Meissner and H. Nicolai, Phys.Rev.
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The origin and growth of Plato’s logic ,Longmans, Green and Co., London New York and Bom-bay, (1897)[10] We take this opportunity to correct two misprints in [1]:the extra factor of C in (23) below is missing in (46)there. Furthermore, in eqn.(50) of [1] it should read8 πGp ( η, r ) = rA η ( r − Gm )[11] There is a second solution with the same ρ and p , but u r = q ηη