PPlanck star phenomenology
Aur´elien Barrau ∗ and Carlo Rovelli † Laboratoire de Physique Subatomique et de Cosmologie, Universit´e Grenoble-Alpes, CNRS-IN2P353,avenue des Martyrs, 38026 Grenoble cedex, France Aix Marseille Universit´e, CNRS, CPT, UMR 7332, 13288 Marseille, FranceUniversit´e de Toulon, CNRS, CPT, UMR 7332, 83957 La Garde, France. (Dated: March 6, 2018)It is possible that black holes hide a core of Planckian density, sustained by quantum-gravitationalpressure. As a black hole evaporates, the core remembers the initial mass and the final explosionoccurs at macroscopic scale. We investigate possible phenomenological consequences of this idea.Under several rough assumptions, we estimate that up to several short gamma-ray bursts per day,around 10 MeV, with isotropic distribution, can be expected coming from a region of a few hundredlight years around us.
I. THE MODEL
Recently, a new possible consequence of quantum grav-ity has been suggested [1]. The idea is grounded in arobust result of loop cosmology [2]: when matter reachesPlanck density, quantum gravity generates pressure suffi-cient to counterbalance weight. For a black hole, this im-plies that matter’s collapse can be stopped before the cen-tral singularity is formed: the event horizon is replacedby a “trapping” horizon [3] which resembles the standardhorizon locally, but from which matter can eventuallybounce out. Because of the huge time dilation inside thegravitational potential well of the star, the bounce is seenin extreme slow motion from the outside, appearing asa nearly stationary black hole. The core, called “Planckstar”, retains memory of the initial collapsed mass m i (because there is no reason for the metric of the core tobe fully determined by the area of the external evaporat-ing horizon) and the final exploding objects depends on m i and is much larger than Planckian [1]. The process isillustrated by the conformal diagram of Fig. 1.In particular, primordial black holes exploding todaymay produce a distinctive signal. The observability of aquantum gravitational phenomenon is made possible bythe amplification due to the large ratio of the black holelifetime (Hubble time t H ) over the Planck time [4].If this scenario is realised in nature, can the final ex-plosion of a primordial Planck star be observed? This isthe question we investigate here. II. DYNAMICS
As a first step, we evaluate the energy of the particlesemitted by the explosion of a primordial Planck star. Let m f = am i be the final mass reached by the black holebefore the dissipation of the horizon (at the point P in ∗ Electronic address: [email protected] † Electronic address: [email protected] P FIG. 1: Penrose diagram of a collapsing star. The dotted lineis the external boundary of the star. The shaded area is theregion where quantum gravity plays an important role. Thedark line represents the two trapping horizons: the externalevaporating one, and the internal expanding one. The low-est light-line is where the horizon of the black hole would bewithout evaporation. P is where the explosion happens. Thethin arrows indicate the Hawking radiation. The thick arrowis the signal studied in this paper. Figure 1). In [1], an argument based on information con-servation was given, pointing to the preferred value a ∼ √ , (2.1)where m i is the initial mass. This value contradicts theexpectation from the semiclassical approximation; it fol-lows from the hypothesis that the black hole informa-tion paradoxes might be resolved and firewalls avoided ifthe semiclassical approximation breaks down earlier thannaively expected, as a consequence of the strong quan-tum gravitational effects in the core and the fact thatthey alter the effective causal structure of the evolvingspacetime (the point P is in the causal past of the quan-tum gravitational region.) As shown by Hawking, non-rotating uncharged black holes emit particles with energyin the interval ( E, E + d E ) at a rate [5]d Nd E d t = Γ s h (cid:20) exp (cid:18) πGmE (cid:126) c (cid:19) − ( − s (cid:21) − (2.2) a r X i v : . [ g r- q c ] M a y er state of angular momentum and spin s . The absorp-tion coefficient Γ s , that is the probability that the parti-cle would be absorbed if it were incident in this state onthe black hole, is a function of E , m and s . By integrat-ing this expression it is straightforward to show that themass loss rate is given bydmd t = − f ( m ) m , (2.3)where f ( m ) accounts for the degrees of freedom of eachemitted particle. As times goes on, the mass decreases,the temperature increases and new types of particles be-come “available” to the black hole. Above each threshold, f ( m ) is given approximately by [6] f ( m ) ≈ (7 . α s =1 / + 3 . α s =1 ) × g s − , (2.4)where α s =1 / and α s =1 are the number of degrees offreedom (including spin, charge and color) of the emit-ted particles. If f ( m ) is assumed to be constant, e.g. f ( m ) = f ( m i ), the initial and final masses of a black holereaching its final stage today, that is in a Hubble time t H ,are easy to calculate. In the Planck star hypothesis, thefinal stage is reached when m = m f (cid:29) m P l . IntegratingEq. (2.3) leads to: m i = (cid:18) t H f ( m i )1 − a (cid:19) . (2.5)In practice, to account for the smooth evolution of f ( m ) when the different degrees of freedom open up, anumerical integration has to be carried out. This leads,for a = 1 / √
2, to m i ≈ . × g , (2.6)and m f ≈ . × g . (2.7)The value of m i is very close to the usual value m ∗ cor-responding to black holes requiring just the age of theUniverse to fully evaporate. This was expected as theprocess is explosive. The details may change dependingon the exact shape chosen for f ( m ).The value of the radius when m reaches m f is r f ≈ . × − cm. The size of the black hole is the onlyscale in the problem and therefore fixes the energy of theemitted particles in this last stage. We assume that allfundamental particles are emitted with the same energytaken at E burst = hc/ (2 r f ) ≈ . . (2.8)Of course, a more reliable model would be desirable butthis is the most natural hypothesis at this stage. eGammaEntries 1001464Mean 0.1068RMS 0.1116 E(GeV)0 0.1 0.2 0.3 0.4 0.5 N eGammaEntries 1001464Mean 0.1068RMS 0.1116 uubar channel: energy spectrum of photons eCM = 3.9 GeV, event = 100000 FIG. 2: Gamma-ray spectrum resulting from 10 u ¯ u jets (lin-ear scales). III. SINGLE EVENT DETECTION
We now study the signal that evaporating Planck starwould produce. From the phenomenological viewpoint, itis natural to focus on emitted gamma-rays: charged par-ticles undergo a diffusion process in the stochastic mag-netic fields and cannot be used to identify a single eventwhereas neutrinos are hard to detect. The importantfact is that most of the emitted gammas are not emit-ted at the energy E burst . Only those directly emittedwill have this energy. But assuming that the branchingratios are controlled, as in the Hawking process, by theinternal degrees of freedom, this represents only a smallfraction (1/34 of the emitted particles). Most gamma-rays will come from the decay of hadrons produced inthe jets of quarks, notably from neutral pions. E burst isalready much smaller than the Planck scale but the meanemission of emitted photons is even smaller.To simulate this process, we have used the ”LundMonte Carlo” PYTHIA code (with some scaling ap-proximations due the unusually low energy requiredfor this analysis). It contains theory and models fora number of physics aspects, including hard and softinteractions, parton distributions, initial- and final-stateparton showers, multiple interactions, fragmentationand decay. PYTHIA allowed us to generate the meanspectrum expected for secondary gamma-rays emittedby a Planck star reaching the end of its life. The mainpoint to notice is that the mean energy is of the order of0 . × E burst , that is in the tens of MeV range ratherthan in the GeV range. In addition, the multiplicity isquite high at around 10 photons per q ¯ q jet. Fig. 2 showsthe mean spectrum of photons resulting from 10 jets of3.9 GeV u ¯ u quarks.It is straightforward to estimate the total number ofparticles emitted m f /E burst and then the number ofphotons < N burst > emitted during the burst. As fora black hole radiating by the Hawking mechanism, weassume that the particles emitted during the bursts (thatis those with m < E burst ) are emitted proportionally to2heir number of internal degrees of freedom : gravity isdemocratic. The spectrum resulting from the emitted u, d, c, s quarks ( t and b are too heavy), gluons andphotons is shown on Fig. 3. The little peak on the rightcorresponds to directly emitted photons that are clearlysub-dominant. By also taking into account the emissionof neutrinos and leptons of all three families (leading tovirtually no gamma-rays and therefore being here a puremissing energy), we obtain < N burst > ≈ . × .The question of the maximum distance at which a sin-gle burst can be detected naturally arises. If one requiresto measure N mes photons in a detector of surface S , thisis simply given by R det = (cid:114) S < N burst > πN mes . (3.1)If we set, e.g. , N mes ≈
10 photons in a 1 m detector,this leads to R ≈
205 light-years. Otherwise stated,the “single event” detection of exploding Planck stars is local . The maximum distance at which such an eventcan be detected is just a few tens times the distance tothe nearest star. This is a tiny galactic patch around us.This already has an interesting consequence. Exceptif there is a significant dark matter clump within thissmall sphere – which is unlikely–, the signal is expectedto be isotropic as the halo is homogeneous at this scale.This is very different from most galactic signals usuallypeaked either in the galactic center direction, or thedirection of the motion of the solar system for directedsearches for dark matter. This means that the Planckstar signal could mimic a cosmological origin.Let us call the local density of dark matter ρ DM ∗ ≈ . ± . [7]. If Planck stars reaching m f wereto saturate the dark matter bound, and if they clusteras ordinary cold dark matter, their number within thedetectable horizon would be N maxdet = 4 πρ DM ∗ m f (cid:18) S < N burst > πN mes (cid:19) ≈ . × . (3.2)However, as their history is not that different from theone of standard primordial black holes with the sameinitial mass (of course the end of their lives is differentbut the total energy emitted remains the same) the usualconstraint Ω P BH < − for initial masses around 10 g basically holds [8]. This leads to N det < πρ DM ∗ Ω P BH m f (cid:18) S < N burst > πN mes (cid:19) ≈ . × . (3.3)This number is still quite high. It shows that theindividual detection is far from being, in principle, outof reach. One can then estimate the number of events that canbe expected for a given observation time ∆ t . This corre-sponds to Planck stars that have masses between m f and m (∆ t ) at the beginning of the observation time, withinthe volume R < R det . In this case, m (∆ t ) is simply: m (∆ t ) = (cid:0) m f + 3 f ( m )∆ t (cid:1) . (3.4)This number n (∆ t ) is (estimated for a unit volume)given by n (∆ t ) = (cid:90) m (∆ t ) m f d n d m d m, (3.5)where d n/ d m is the differential mass spectrum ofPlanck stars today, still per unit volume. Importantly,the shape of this mass spectrum in the interestingregion is mostly independent of the initial shape.This is exactly true only in the limit m (cid:28) f ( m ) t H and constitutes a rough approximation here. To getorders of magnitude, we however assume this to becorrect. In this case, due to the dynamics of theevaporation, d n/ d m ∝ m . This can be straightfor-wardly seen by writing d n/ d m = d n/ d m i × d m i / d m ,where d n/ d m i is the initial mass spectrum andd m i / d m = m (3 f ( m ) t + m ) − / .If primordial black holes leading to Planck stars areformed through a kind of phase transition in the earlyuniverse, their mass spectrum can be very narrow. Inthe most extreme optimistic case, this would cover ex-actly the range of masses reaching m f in the observationtime window ∆ t . In that case, the number of observedexplosions would be N expl = N det . This is obviouslyunrealistic. On the other extreme, one can assume avery wide mass spectrum. As the primordial cosmologi-cal power spectrum P ( k ) ∝ k n is now known to be red( n <
1) whereas it would have had to be blue to producea sizable amount of primordial black holes by standardprocesses, the usual historical spectrumd n d m i = αm − − w w i , (3.6)where w = p/ρ is the equation of state of the Universeat the formation epoch, can only be taken as an approx-imation on a reduced mass interval. This is however notunrealistic in models like Starobinski’s broken scale in-variance.This leads, assuming that the formation occurred inthe radiation dominated era, to a contemporary spectrumreadingd n d m ∼ α (cid:104) m − Θ( m − m ∗ ) + m − ∗ m Θ( m ∗ − m ) (cid:105) . (3.7)The number of expected ”events” during an observingtime ∆ t is given by N (∆ t ) = (cid:82) m (∆ t ) m f d n d m d m (cid:82) m max m f d n d m d m Ω P BH N maxdet Ω sr , (3.8)3 Gamma
Entries 318043Mean 0.09656RMS 0.3454
E [GeV] -2 -1
10 1 10 ] - [ G e V d E d N -2 -1 eGamma Entries 318043Mean 0.09656RMS 0.3454 energy spectrum of photons
FIG. 3: Full spectrum of gamma-rays emitted by a decayingPlanck star (log scales). where m max is the maximum mass up to which we as-sume the mass spectrum to be given by Eq. (3.7) andΩ sr is the solid angle acceptance of the considered detec-tor. Here, the value of Ω P BH is not easy to constrain.An upper limit can be taken conservatively at 10 − . Tofix orders of magnitude, if we set m max = m ∗ and a den-sity of a few percents of the maximum allowed density,that is Ω P BH ∼ − , this leads to one event per day.Detection is not hopeless. IV. VERY SHORT GAMMA-RAY BURSTS ANDDIFFUSE EMISSION
Could it be that those events have already beendetected? In particular : could they be associated withsome gamma-ray bursts (GRBs)? There are two mainclasses of GRBs : the long ones and the short ones. Thelong ones are quite well understood and are associatedwith the deaths of massive stars and have no link withthis study. Our model for the end of Planck stars doesnot allow for the calculation of any kind of light curve.This would be far beyond the current development. But,obviously, the time-scale is expected to be short.If Planck star explosions are to be associated withsome of the measured GRBs, this would obviously bewith short gamma-ray bursts (SGRBs) [9]. This couldmake sense for several reasons. Firstly, SGRBs arethe less well understood. In particular, the redshiftsare not measured for a large fraction of them, whichmeans that they could, in principle, be of local origin.Secondly, SGRBs are known to have a harder spectrumand some of them do indeed reach the energies thatwe have estimated in this work. Thirdly, a sub-class ofSGRB, the very short gamma ray bursts (VSGRBs), doexhibit an even harder spectrum and can be assumed tooriginate from a different mechanism as the SGRB timedistribution seems to be bimodal [10].Nothing can be concluded at this stage but it is worth noticing that there are indications that some VSGRBsare compatible with a Planck star origin. The fact thattheir angular distribution favors a cosmological origin isin fact also fully compatible with a local bubble origin.Another important question to address is the possiblediffuse emission from Planck stars. Not the single eventdetection but the integrated signal over huge distances.In that case, no time signature can be expected but theglobal spectrum can retain some specific characteristics.Two effects are competing. Let us consider a Planckstar exploding at a redshift z . Because, to be detectednow, it has exploded in the past, it means that theamount of cosmic time required for it to reach its finalmass was smaller than in our vicinity. Its initial andfinal masses were therefore smaller. This means that themean energy of emitted particles was higher (and thetotal amount smaller) than for Planck stars explodingwithin the Galaxy. But just because of the redshift, themeasured energy is smaller than the emitted one by afactor (1 + z ). In practice, this second effect slightlydominates over the first one. For example, a Planck starexploding at z = 3 emits photons with a mean energyhigher than around us by a factor 1.9. This energy isthen redshifted by a factor 4. Fig. 4 shows the samespectrum than in Fig. 3 but for a z = 3 Planck star (itis not just a rescaling of the previous one as the energyof jets changes).When considering a diffuse signal with very small ab-sorption effects, like gamma-rays in this range, the inte-gration effect is drastic. For each shell, the number ofdetected photons is inversely proportional to the squaredshell distance because of the solid angle effect. But thenumber of sources per shell is proportional to the squareddistance. The two effects compensate (modulo the slightenergy variation mentioned in the previous paragraph)and each shell contributes the same. The neat flux istherefore fully determined by the cutoff. We leave thedetailed study for future works as, in this case, it wouldbe mandatory to take into account not only the explo-sion spectrum but also the last stages of the Hawkingspectrum due to the evaporation before the explosion.It can however be easily concluded that the signal canin principal be detected as the mean density required toproduced a sizable amount of gamma-rays is very small. V. CONCLUSION AND PROSPECTS
We have shown that the detection of individual ex-plosions of Planck stars is not impossible and we haveestablished the main spectral characteristic of the sig-nal. Quantum gravity might show up in the tens of MeVrange. We have estimated the order of magnitudes forthe expected frequency of events: in some cases, it mightnot be small. Several approximations about the dynam-ics of the evaporation can be improved. The details of4
Gamma
Entries 425453Mean 0.03381RMS 0.1401
E [GeV] -2 -1
10 1 ] - [ G e V d E d N -2 -1 eGamma Entries 425453Mean 0.03381RMS 0.1401 energy spectrum of photons
FIG. 4: Full spectrum of gamma-rays emitted by a decayingPlanck star at z = 3 (log scales). the explosion remain to be investigated. The shape ofthe diffuse integrated signal, and more specifically its po-tential specific signature allowing to distinguish it fromstandard primordial black holes, requires a full numericalanalysis. It could also be interesting to investigate theemission of charged articles, in particular positrons andantiprotons (some interesting threshold effets could beexpected). As well known, the much smaller horizon canbe compensated by the large galactic confinement effect. Acknowledgments
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