Cosmic rays from primordial black holes and constraints on the early universe

Physics Reports 307 (1998) 141—154

Cosmic rays from primordial black holes and constraints on the early universe B.J. Carr *, J.H. MacGibbon
School of Mathematical Sciences, Queen Mary & Westfield College, Mile End Road, London E1 4NS, UK
Code SN3, NASA Johnson Space Center, Houston, Texas 77058, USA

Abstract The constraints on the number of evaporating primordial black holes imposed by observations of the cosmological gamma-ray background do not exclude their making a significant contribution to the Galactic flux of cosmic ray photons, electrons, positrons and antiprotons. Even if this contribution is small, cosmic ray data place important limits on the number of evaporating black holes and thereby on models of the early Universe. Evaporating black holes are unlikely to be detectable in their final explosive phase unless new physics is invoked at the QCD phase transition.  1998 Published by Elsevier Science B.V. All rights reserved. PACS: 98.80.Cq; 97.60.Lf; 98.70.Vc Keywords: Cosmic rays; Primordial black holes; Early universe

1. Introduction It is well known that primordial black holes (PBHs) could have formed in the early Universe [25,66]. A simple comparison of the cosmological density at time t with the density associated with a black hole shows that PBHs forming at time t would have of order the horizon mass:






ct t M (t)+ +10 g. & G 10\ s

(1)

PBHs could thus span an enormous mass range: those formed at the Planck time (10\ s) would have the Planck mass (10\ g), whereas those formed at 1 s would be as large as 10M , >

* Corresponding author. E-mail: [email protected]. 0370-1573/98/$ — see front matter  1998 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 8 ) 0 0 0 3 9 - 8

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comparable to the mass of the holes thought to reside in galactic nuclei. PBHs would most naturally form from initial inhomogeneities but they might also form through other mechanisms at a cosmological phase transition [8]. The realization that small PBHs might exist prompted Hawking to study their quantum properties. This led to his famous discovery [26] that black holes radiate thermally with a temperature 10\(M/M )\ K and evaporate completely on a timescale 10(M/M ) yr. Indeed only > > black holes of primordial origin could be small enough for this effect to be important. Despite the conceptual importance of this result, it was bad news for PBH enthusiasts. For since PBHs with a mass of 10 g, which evaporate at the present epoch, would have a temperature of order 100 MeV, the observational limit on the c-ray background density at 100 MeV immediately implied that the density of such holes could not exceed 10\ times the critical density [53]. Not only did this render PBHs unlikely dark matter candidates, it also implied that there was little chance of detecting black hole explosions at the present epoch [55]. Despite this conclusion, it was realized that PBH evaporations could still have interesting cosmological consequences. In particular, they might generate the microwave background [67] or modify the standard cosmological nucleosynthesis scenario [39,48,50,59] or contribute to the cosmic baryon asymmetry [3]. There was also interest in whether PBH evaporations could account for the electrons and positrons observed in cosmic rays [9] or the annihilation-line radiation coming from the Galactic centre [51] or the unexpectedly high fraction of antiprotons in cosmic rays [35,62]. Renewed efforts were also made to look for black hole explosions after the realization that — due to the interstellar magnetic field — these might appear as radio rather than c-ray bursts [58]. Even if PBHs had none of these consequences, studying such effects led to strong upper limits on how many PBHs could ever have formed and thereby constrained models of the early Universe [10]. Later many of these calculations had to be modified when it was realized that the usual assumption that particles are emitted with a black-body spectrum as soon as the temperature of the hole exceeds their rest mass is too simplistic. If one adopts the conventional view that all particles are composed of a small number of fundamental point-like constituents (quarks and leptons), it would seem natural to assume that it is these fundamental particles rather than the composite ones which are emitted directly once the temperature goes above the QCD confinement scale of 250 MeV. One can therefore envisage a black hole as emitting relativistic quark and gluon jets which subsequently fragment into hadrons. On a somewhat longer timescale these hadrons themselves decay into photons, neutrinos, gravitons, electrons, positrons, protons and antiprotons. The results of such a calculation are very different from a simple direct emission calculation [41,42]. In an earlier paper we considered how these effects modify the cosmological consequences of evaporating PBHs and, in particular, their contribution to cosmic rays ([43]; MC). However, in recent years much more cosmic ray data has accrued, so it is timely to update these calculations. The results are described more fully elsewhere [12]. The plan of the present paper is as follows. In Section 2 we show why limits on the number of PBHs constrain models of the early Universe. In Section 3 we review the physics of black hole evaporation with particular emphasis on processes above the QCD temperature. In Section 4 we make a detailed comparison with the cosmic ray data. In Section 5 we consider the possibility of detecting PBH explosions and in Section 6 we draw some general conclusions.

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2. Constraints on the early Universe One of the most important reasons for studying the cosmological effects of PBHs is that it enables one to place limits on the spectrum of density fluctuations in the early Universe. This is because, if the PBHs form directly from density perturbations, the fraction of regions undergoing collapse at any epoch is determined by the root-mean-square amplitude e of the fluctuations entering the horizon at that epoch and the equation of state p"co (0(c(1). One usually expects a radiation equation of state (c"1/3) in the early Universe. In order to collapse against the pressure, an overdense region must be larger than the Jeans length at maximum expansion and this is just (c times the horizon size. This implies that the density fluctuation must exceed c at the horizon epoch, so — providing the fluctuations have a Gaussian distribution and are spherically symmetric — one can infer that the fraction of regions of mass M which collapse is [8]





c , b(M)&e(M) exp ! 2e(M)

(2)

where e(M) is the value of e when the horizon mass is M. The PBHs can have an extended mass spectrum only if the fluctuations are scale-invariant (i.e. with e independent of M). In some situations, Eq. (2) would fail. For example, PBHs would form more easily if the equation of state of the Universe were ever soft (c;1). This might apply if there was a phase transition which channelled the mass of the Universe into non-relativistic particles or which temporally reduced the pressure. In particular, this might happen at the quark—hadron era [34]. In this case, only those regions which are sufficiently spherically symmetric at maximum expansion can undergo collapse; the dependence of b on e would then be weaker than indicated by Eq. (2) but there would still be a unique relationship between the two parameters [36]. The fluctuations required to make the PBHs may either be primordial or they may arise spontaneously at some epoch. One natural source of fluctuations would be inflation [37,49] and, in this context, e(M) depends implicitly on the inflationary potential [11,13,21,23,33,57,64]. Recently Bullock and Primack [6] have questioned the Gaussian assumption in the inflationary context, so that Eq. (2) may not apply, but they still find that b depends very sensitively on e. Some formation mechanisms for PBHs do not depend on having primordial fluctuations at all. For example, at any spontaneously broken symmetry epoch, PBHs might form through the collisions of bubbles of broken symmetry [17,29,38]. PBHs might also form spontaneously through the collapse of cosmic strings [7,22,28,44,54]. In these cases b(M) depends not on e(M) but on other cosmological parameters, such the bubble formation rate or the string mass-per-length. In all these scenarios, the current density parameter X associated with PBHs which form at . & a redshift z or time t is related to b by







t \ M \ +10b , X "bX (1#z)+10b . & 0 s 10g

(3)

where X +10\ is the density of the microwave background and we have used Eq. (1). The (1#z) 0 factor arises because the radiation density scales as (1#z), whereas the PBH density scales as (1#z). Any limit on X (M) therefore places a tight constraint on b(M) and the constraints are . & summarized in Fig. 1. The constraint for non-evaporating mass ranges above 10 g comes from

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Fig. 1. Constraints on b(M).

requiring X (1. Much stronger constraints are associated with any PBHs which were small . & enough to have evaporated by now. For example, the constraints below 10 g assume that evaporating PBHs leave stable Planck mass relics, in which case these relics are required to have less than the critical density [4,13,40]. The constraints are discussed in detail by Carr et al. [13] but here we just wish to emphasize that the strongest one is the cosmic ray limit associated with PBHs evaporating currently. The constraints on b(M) can be converted into constraints on e(M) using Eq. (2) and these are shown in Fig. 2.

3. Evaporation of primordial black holes A black hole of mass M will emit particles in the energy range (Q, Q#dQ) at a rate [27]


 

C dQ Q \ exp $1 dNQ " , 2p

¹

(4)

where ¹ is the black hole temperature, C is the absorption probability and the # and ! signs refer to fermions and bosons respectively. This assumes that the hole has no charge or angular momentum. This is a reasonable assumption since charge and angular momentum will also be lost through quantum emission but on a shorter timescale that the mass [52]. C goes roughly like

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Fig. 2. Constraints on e(M).

Q¹\, though it also depends on the spin of the particle and decreases with increasing spin, so a black hole radiates almost like a black-body. The temperature is given by




¹+10




M \ M \ K+ GeV . g 10 g

(5)

This means that it loses mass at a rate MQ "!5;10M\f (M) g s\ ,

(6)

where the factor f (M) depends on the number of particle species which are light enough to be emitted by a hole of mass M, so the lifetime is q(M)"6;10\f (M)\M s .

(7)

The factor f is normalized to be 1 for holes larger than 10 g and such holes are only able to emit “massless” particles like photons, neutrinos and gravitons. Holes in the mass range 10 g(M(10 g are also able to emit electrons, while those in the range 10 g(M(10 g emit muons which subsequently decay into electrons and neutrinos. The latter range includes, in particular, the critical mass for which q equals the age of the Universe. This can be shown to M*"4.4;10h\  g where h is the Hubble parameter in units of 100 and we have assumed that the total density parameter is 1 [42].

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Once M falls below 10 g, 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 of K "250—300 GeV, one would therefore expect these fundamental /!" particles to be emitted rather than composite particles. Only pions would be light enough to be emitted below K . Since there are 12 quark degrees of freedom per flavour and 16 gluon degrees /!" of freedom, one would also expect the emission rate (i.e. the value of f ) to increase dramatically once the QCD temperature is reached. The physics of quark and gluon emission from black holes is simplified by a number of factors. Firstly, since the spectrum peaks at an energy of about 5¹, Eq. (5) implies that most of the emitted particles have a wavelength j+2.5M (in units with G"k"c"1), so they have a size comparable to the hole. Secondly, one can show that the time between emissions is Dq+20j, which means that short range interactions between successively emitted particles can be neglected. Thirdly, the condition ¹'K implies that Dq is much less than K\ +10\ cm (the characteristic strong /!" /!" interaction range) and this means that the particles are also unaffected by gluon interactions. The implication of these three conditions is that one can regard the black hole as emitting quark and gluon jets of the kind produced in collider events. The jets will decay into hadrons over a distance which is always much larger than M, so gravitational effects can be neglected. The hadrons may then generate other particles through weak and electomagnetic decays. To find the final spectra of stable particles emitted from a black hole, one must convolve the Hawking emission spectrum given by Eq. (4) with the jet fragmentation function. The fragmentation function has an upper cut-off at Q, a lower cut-off and peak around the hadron mass, and an E\ Bremmstrahlung tail in between. The convolution then gives the instantaneous emission spectrum shown in Fig. 3 for a ¹"1 GeV black hole [42]. The direct emission just corresponds to the small bumps on the right. All the particle spectra show a peak at 100 MeV due to pion decays; the electrons and neutrinos also have peaks at 1 MeV due to neutron decays.

4. Cosmic rays from PBHs In order to determine the present day background spectrum of particles generated by PBH evaporations, one must first integrate over the lifetime of each hole of mass M and then over the PBH mass spectrum [41]. In doing so, one must allow for the fact that smaller holes will evaporate at an earlier cosmological epoch, so the particles they generate will be redshifted in energy by the present epoch. If the holes are uniformly distributed throughout the Universe, the background spectra should have the form indicated in Fig. 4. All the spectra have rather similar shapes: an E\ fall-off for E'100 MeV due to the final phases of evaporation at the present epoch and an E\ tail for E(100 MeV due to the fragmentation of jets produced at the present and earlier epochs. Note that the E\ tail generally masks any effect associated with the PBH mass spectrum (cf. [9]). The situation is more complicated if the PBHs evaporating at the present epoch are clustered inside our own Galactic halo (as is most likely). In this case, any charged particles emitted after the epoch of galaxy formation (i.e. from PBHs only somewhat smaller than M*) will have their flux enhanced relative to the photon spectra by a factor m which depends upon the halo concentration factor and the time for which particles are trapped inside the halo by the Galactic magnetic field.

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Fig. 3. Instantaneous emission from a 1 GeV black hole.

MC assume that the particles are uniformly distributed throughout a halo of radius R and infer  q o q R \ X \  +10h\     , (8) m"   q o q 10 kpc 0.1     where X is the density parameter associated with halos. The ratio of the leakage time q to the    age of the Galaxy q is rather uncertain and also energy-dependent. At 100 MeV we take q to be    about 10 yr for electrons or positrons (m&10) and 10yr for protons or antiprotons (m&10). The postgalactic contribution of charged particles is shown in Fig. 5. For comparison with the observed cosmic ray spectra, one needs to determine the amplitude of the spectra at 100 MeV. This is because the observed fluxes all have slopes between E\ and E\, so the strongest constraints come from measurements at 100 MeV. The amplitudes all scale with X and are found to be (MC) . & 1.5;10\hX GeV\ cm\ (c) . & dF (e>, e\) (9) " 9.5;10\hX (m/10) GeV\ cm\ . & dE 4.5;10\hX (m/10) GeV\ cm\ (p, p ) . . & We now use the observed cosmic ray spectra to constrain X . . &














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4.1. Gamma-rays Our earlier c-ray background constraint was based on a comparison with the observations of Fichtel et al. [19]:






E \ !  dF A"1.1($0.2);10\ cm\ GeV\ 100 MeV dE

(10)

between 35 and 175 MeV; this led to an upper limit (MC) X 4(7.6$2.6);10\h\. (11) . & Indeed the comparison suggested that PBH emission might even be the dominant contribution above 50 MeV. However, more recent EGRET observations [61] give a background of






dF E \ !  A"7.3($0.7);10\ cm\ GeV\ dE 100 MeV

(12)

between 30 MeV and 120 GeV. This leads to a slightly stronger upper limit X 4(5.1$1.3);10\h\ (13) . & and the form of the spectrum no longer suggests that PBHs provide the dominant contribution. If PBHs are clustered inside our own Galactic halo, then there should also be a Galactic c-ray background and, since this would be anisotropic, it should be separable from the extragalactic background. Wright [63] has shown that the ratio of the anisotropic to isotropic intensity is





I 3j (R )H R  "g(l, b, R /R , q)     ,   I 4cj  

Fig. 4. Spectrum of particles from uniformly distributed PBHs.

(14)

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Fig. 5. Spectrum of charged particles from PBHs in our own halo to be (MC).

where j and j are the halo and background emissivities. The function g depends on Galactic   longitude (l) and latitude (b), the ratio of the core radius (R ) to our Galactocentric radius (R ), and   the halo flattening (q). A detailed fit to the EGRET data, subtracting various other known components, gives 3j (R )H R    "0.4—2.5 . 4cj 

(15)

Note that this assumes the isotropic intensity given by Fichtel et al. [19] and replacing this with the Shreekumar et al. [61] intensity increases the ratio by 1.5. Eq. (15) requires the PBH clustering factor to be (2!12);10h\. This is comparable with the expected local density enhancement, given by Eq. (8) without the (q /q ) factor, providing X is in the range 0.04 to 0.2, which     is plausible. Recently Dixon [18] also claims to have detected diffuse halo emission from EGRET. 4.2. Electrons and positrons There is now extensive data on the spectra of cosmic ray electrons and positrons between 100 MeV and 100 GeV. The positron fraction in this range has been summarized by Barwick et al. [2] and is shown in Fig. 6. In the simplest picture all positrons, together with an equal number of electrons, are secondary particles which are generated through the decay of pions created in the collisions between protons and interstellar matter. The remaining electrons are supposed to be produced by primary cosmic ray sources like supernovae. However, it is not clear that measurements of the positron fraction support this picture. Between 5 and 10 GeV there seems to be an

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Fig. 6. Data on positron fraction from Barwick et al. [2].

increase in the positron fraction, in contrast with the predicted decrease of the “leaky box” model. Below 500 MeV (more relevant to the PBH scenario), the measured positron fraction is 0.3, while the “leaky box” model prediction is 0.1. One may therefore need to invoke a primary source of electrons and positrons. PBH evaporations are one such source since they naturally emit electrons and positrons in equal numbers. MC calculated the PBH density required to explain the interstellar positron flux at 300 MeV inferred by Ramaty and Westergaard [56]. However, they used a rather simplistic model in which the positrons were assumed to be spread uniformly throughout the Galactic halo. It is probably more appropriate to assume that the positrons come from PBHs within a few kiloparsecs, in which case the limit becomes





q \ X  . X K(1.7!2.1);10\   . & 10 yr 0.1

(16)

This is comparable with the c-ray limit (13). An updated version of this limit, incorporating the new data shown in Fig. 6, is given by Carr and MacGibbon [12]. However, it must be stressed that the inconsistencies of the “leaky box” model may just reflect inadequacies in the propagation model or insufficient allowance for solar modulation effects (which depend on the sign of the charge). One should therefore not interpret Eq. (16) as positive evidence for PBHs. 4.3. Antiprotons Since the ratio of antiprotons to protons in cosmic rays is less than 10\ over the energy range 100 MeV—10 GeV, whereas PBHs should produce them in equal numbers, PBHs could only contribute appreciably to the antiprotons. It is usually assumed that the observed antiproton cosmic rays are secondary particles, produced by spallation of the interstellar medium by primary

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Fig. 7. Comparison of PBH emission and antiproton data from Maki et al. [46].

cosmic rays. However, the spectrum of secondary antiprotons should show a steep cut-off at kinetic energies below 2 GeV, whereas the spectrum of PBH antiprotons should increase down to 0.2 GeV, so this provides a distinct signature [35]. MC calculated the PBH density required to explain the interstellar antiproton flux at 1 GeV estimated by Ip and Axford [32]. However, this calculation is prone to the same criticism (regarding the assumed cosmic ray distribution) as their positron one. Making the equivalent correction gives a limit





q \ X  X K(1.6—3);10\   . & 10 yr 0.1

(17)

which is somewhat stronger than the c-ray limit. More recent data on the antiproton flux below 0.5 GeV comes from the BESS balloon experiment [65] and Maki et al. [46] have tried to fit these data in the PBH scenario. They model the Galaxy as a cylindrical diffusing halo of diameter 40 kpc and thickness 4—8 kpc and then using Monte Carlo simulations of cosmic ray propagation. In contrast to MC, they find that most of the antiprotons come from PBHs within a few kiloparsecs of the solar neighbourhood. A comparison with the data in Fig. 7 shows that there is no positive evidence for PBHs (i.e. there is no tendency for the positron fraction to tend to 0.5 at low energies). However, they require the fraction of the local halo density in PBHs to be less than 3;10\ and this is stronger than the c-ray background limit. However, Maki et al. do not allow for the fact that solar modulation will affect protons and antiprotons differently. Mitsui et al. [47] have pointed out that a key test of the PBH hypothesis will arise during the solar minimum period. This is because the flux of primary antiprotons should be enhanced then, while that of the secondary antiprotons should be little affected.

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5. PBH explosions One of the most striking observational consequences of PBH evaporations would be their final explosive phase. However, in the standard particle physics picture, where the number of elementary particle species never exceeds around 100, the likelihood of detecting such explosions is very low. Indeed, in this case, observations only place an upper limit on the explosion rate of R(5;10 pc\ yr\ [1,60]. This compares to Wright’s c-ray halo limit of R(0.3 pc\ yr\ and the Maki et al. antiproton limit of R(0.02 pc\ yr\. However, the physics at the QCD phase transition is still uncertain and the prospects of detecting explosions would be improved in less conventional particle physics models. For example, in a Hagedorn-type picture, where the number of particle species exponentiates at the quark—hadron temperature, the limit is strengthened to R(0.05 pc\ yr\ [20]. Cline and colleagues have argued that one might expect the formation of a QCD fireball at this temperature [14] and this might even explain some of the short period (100 ms) c-ray bursts observed by BATSE [15]. Although this proposal is speculative, it has the attraction of making testable predictions (e.g. the hardness ratio should increase as the duration of the burst decreases and the spatial distribution should be Euclidean since the bursts are local). A rather different way of producing a c-ray burst is to assume that the outgoing charged particles form a plasma due to turbulent magnetic field effects at sufficiently high temperatures [5]. Some people have emphasized the possibility of detecting very high energy cosmic rays from PBHs using air shower techniques [16,24]. However, recently these efforts have been set back by the claim of Heckler [30] that QED interactions could produce an optically thick photosphere once the black hole temperature exceeds ¹ "45 GeV. In this case, the mean photon energy is   reduced to m (¹ /¹ ), which is well below ¹ , so the number of high energy photons is much  &   & reduced. He has proposed that a similar effect may operate at even lower temperatures due to QCD effects [31]. However, these arguments should not be regarded as definitive: MacGibbon et al. [45] claim that Heckler has not included Lorentz factors correctly in going from the black hole frame to the centre-of-mass frame of the interacting particles; in their calculation QED interactions are never important.

6. Conclusions We have seen that PBH evaporations could contribute significantly to the Galactic flux of cosmic rays photons, electrons, positrons and antiprotons at energies of around 100 MeV. Indeed it is striking that the PBH density required to explain the fluxes of these particles are all comparable with the c-ray background limits. On the other hand, the evidence that PBH evaporations produce cosmic rays is far from conclusive: there is some uncertainty in interpreting the charged particle data (e.g. due to solar modulation effects) and there are anyway other sources of primary cosmic rays at these energies (e.g. decaying wimps). Therefore the most conservative approach is to use the cosmic ray data to constrain the number of evaporating PBHs and we have seen that this in turn places important constraints on models of the early Universe (including inflationary secarios). PBHs therefore provide a unique probe of the earliest moments of the Big Bang and even their non-existence provides vital cosmological information.

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