Evaporation Spectrum of Black Holes from a Local Quantum Gravity Perspective
EEvaporation Spectrum of Black Holes from a Local Quantum Gravity Perspective
Aur´elien Barrau Laboratoire de Physique Subatomique et de Cosmologie, Universit´e Grenoble-Alpes, CNRS/IN2P353, avenue des Martyrs, 38026 Grenoble cedex, France (Dated: December 21, 2016)We revisit the hypothesis of a possible line structure in the Hawking evaporation spectrum of blackholes. Because of nonperturbative quantum gravity effects, this would take place arbitrarily far awayfrom the Planck mass. We show, based on a speculative but consistent hypothesis, that this naiveprediction might in fact hold in the specific context of loop quantum gravity. A small departurefrom the ideal case is expected for some low-spin transitions and could allow us to distinguish severalquantum gravity models. We also show that the effect is not washed out by the dynamics of theprocess, by existence of a mass spectrum up to a given width, or by the secondary componentinduced by the decay of neutral pions emitted during the time-integrated evaporation.
INTRODUCTION AND MODEL
The Hawking radiation of black holes [1], characterizedby a Planck law slightly modulated by grey-body factors[2], is one of the most robust predictions of quantum fieldtheory on a curved background. It is also the perfectphenomenon to investigate possible deviations from thesemiclassical dynamics due to nonperturbative quantumgravity effects.Bekenstein and Mukhanov [3] have suggested thatan interesting way to account for quantum gravity atthe effective level could be to assume that the area ofa black hole (BH) can only take values proportionalto a fundamental area assumed to be of the order ofthe Planck area. This has interesting consequences. Inparticular, this leads to the appearance of emission linesinstead of a continuous spectrum, as we will explain inmore details later. However, it was then shown thatin the specific setting of loop quantum gravity (LQG)the use of the actual eigenstates of the area operatordoes not lead to a Bekestein-Mukhanov-like spectrum[4]. The density of energy levels instead reads as ρ ( M ) ∼ exp(M (cid:112) π G /
3) which means that the spectrallines are virtually dense in frequency for large masses.The phenomenology of evaporating black holes in LQGhas, therefore, focused so far on the very last stages ofthe emission where lines are distinguishable. This istheoretically interesting but probably out of reach ofany reasonable phenomenological approach.We would like to revisit this conclusion, somewhat inthe line of [5]. Basically, what was assumed in virtu-ally all LQG studies (see, e.g. , [6, 7]) on evaporatingblack holes is that “something” independent from quan-tum gravity triggers the emission of a particle from theblack hole. The exact energy of this particle is deter-mined by one of the area eigenvalues of the hole. Theselected value is usually taken to be as close as possibleto the one favored by the semiclassical quantum process.Alternatively, one can just calculate the transition prob- abilities between black-hole states by weighting them byexp( S − S ) where S and S are the entropies associ-ated to the initial and final states. What is a black holein LQG (see, e.g. , [8–12]) ? It is basically an isolatedhorizon punctured by the edges of a spin network, thatis a graph with edges labeled by SU (2) representationsand nodes characterized by intertwiners. An edge withspin representation j carries an area of eigenvalue A j = 8 πγl P l (cid:112) j ( j + 1) , (1)where j is a half-integer and γ is the Barbero-Immirzi pa-rameter. A surface punctured by N edges has a spectrumgiven by A j = 8 πγl P l N (cid:88) n =1 (cid:112) j n ( j n + 1) , (2)where the sum is carried out over all intersections of theedges with the surface. Each state with spin j has a de-generacy (2 j + 1). We believe that there might be twoproblems with the usual view of the Hawking evapora-tion in this framework. When one considers the transi-tion from a state with mass M to a state with mass M which is, in general, very close to M if the black hole ismacroscopic, the actual final quantum state is completelydifferent from the initial one most of the time. Even ifthe masses typically differ by much less than the Planckmass (and the areas differ by approximately the Planckarea), the second quantum state corresponds to values ofthe spins that are in general completely different from theinitial state. Using the quasidense distribution of statesrequires a complete reassigning of the quantum numbersto each puncture for every single transition. This is intension with a quantum gravitational origin of the evap-oration process itself. If one considers the evaporationas due to a change of state of a given “elementary areacell” (or, more precisely, to the settling down of the BHfollowing this transition), there is no reason for all of theother elementary surfaces to change their quantum stateat the same time. This was considered in, for example,[13]. In addition, this raises an obvious second problem a r X i v : . [ g r- q c ] D ec about causality: how can a “far away” elementary cellknow the way it has to change to adjust to the others?Our hypothesis here is that each particle evaporated by ablack hole is basically due to the relaxation of the blackhole following a change of state of a single elementarycell. We call this a local quantum gravity process andinvestigate it in the well-developed framework of LQG.In principle, however, the idea is quite general. It shouldbe made clear that our hypothesis is nothing more thana reasonable alternative view that deserves to be studieduntil we have a fully dynamical quantum gravity descrip-tion of the process. Ideally, one would build a model inwhich loop quantum gravity is coupled directly to quan-tum electrodynamics and compute the actual changes inthe gravitational state upon emission of Hawking radia-tion. This is obviously beyond the scope of this study.The key point here is that a purely local change of stateof an “elementary cell”, with a small or moderate changein quantum number, is the fundamental quantum grav-ity process taking place. This does not inconsistentlyassume that local physics knows the global BH quanti-ties like temperature and mass. Once the quantum tran-sition has taken place, without any a priori knowledgeof the global picture, the BH relaxes through semiclassi-cal processes according to the energy made available bythe quantum transition. This automatically leads to aspectrum whose main classical properties agree with theHawking evaporation. Of course, in the future, it wouldbe important to investigate the settling down in a fullyconsistent way, going beyond the isolated horizon consid-ered here which is, by construction, stationary. LOW ENERGY CASE
Let us first consider the simpler case of a low-energyevaporation signal. It is easier for two reasons: first be-cause the dynamics can then be neglected and, second,because there is no secondary emission associated withdecaying particles in the sense that the black hole doesnot emit hadrons leading to gamma-rays.The main image is very simple and relies on the factthat the structure of a Schwarzshild black hole is suchthat (in Planck units) dA = 32 πM dM . If a quantumof energy E ∼ T , with T = 1 / (8 πM ), is emitted, itinduces a BH mass change dM = E . The area changewill then be dA ∼
4. This is the main interest of theBekenstein-Mukhanov hypothesis: assuming a discretearea spectrum with regularly spaced eigenvalues inducesa line structure in the energies of the evaporationspectrum, even arbitrarily far away from the Planckmass. The very same change of area dA (of order of thePlanck area) will indeed lead to a relative line separationin the spectrum dE/T which does not depend on themass. Following our hypothesis that the evaporation is due toa local change of state of a quantum of area, the spectrumwill still exhibit lines in LQG. However, the eigenvaluesof the area operator given by Eq. (1) are a bit moresubtle. In the large- j limit, one recovers a regular linespectrum but the first eigenvalues are not equally spaced.If the resulting line spectrum is to have, as expected, theHawking spectrum as an envelope, the favored transitionis always one between two states A j and A j − n with n a half-integer not much greater than unity: 95% of thetransitions will have n ≤ / e.g. ,[14]) claimed that the punctures are mostly “low-spin”ones. A black hole is then expected to have most ofhis j ’s close to 0. Transitions do exhibit generically aregular line structure. However, in some cases, largelycorresponding either to transitions between A j and A or between A j and A / , there will be a deviationwith respect to what would be expected from a regularline structure. This is what is shown in Fig. 1, whichdisplays the relative energy difference between some A j → A and A j → A / transitions and what wouldbe expected from the same transitions in the regularBekenstein-Mukhanov spectrum. The other way round,new models of holographic black holes were developed(see, e.g. , [15]). Here, one uses the qualitative behaviorof matter degeneracy suggested by standard QFT with acutoff at the vicinity of the horizon – i.e. , an exponentialgrowth of vacuum entanglement in terms of the BH area.In this case, large j ’s should dominate and the predictionis clearly that the line structure will be nearly perfect,as A j − A j − n is very close to n as soon as j is muchgreater than unity. This opens up a very interestingpossibility: not only should this effect allow to observequantum gravity features at high masses, but it shouldalso allow us to distinguish between BH models in LQG.An important question arises. Obviously, nearlyequally spaced area eigenvalues and regular jumps be-tween those values do not lead to the emission of quantaat the same energy as long as the evaporation goes on.When the black-hole area decreases by the same amount dA , the emitted energy varies like 1 /M . So, one shouldensure that the change of area during the evaporationdoes not destroy the very possibility of observing lines.Let us evaluate the energy shift between two successiveemissions associated with an identical area variation. Letus call pA the area variation induced by the transitionwhere p is a half-integer and A is the fundamental areaof order A P l . It is easy to show that the relative varia-tion of energy of the emitted quanta between consecutiveemissions is, at lower order,∆ EE ≈ p A A . (3)This ensures that as long as the area is much largerthan the Planck area, which is definitely the casefor macroscopic black holes, the change in energy isnegligible and the line structure can be observed ifit exists: the fact that the BH mass changes duringits evaporation does not wash out this interesting feature.In practice, however, evaporating black holes are low-mass black holes, at least when compared to a solar mass.This means that they must be primordial black holes(PBHs), except in some exotic low-Planck-scale modelswhere they could be formed by collisions of particles inthe contemporary Universe [16]. The modes of produc-tion of PBHs are hypothetical (see, e.g. , [17, 18]). Variousmechanisms have been considered. If we were to observea single evaporating black hole, we would not care aboutits origin as far as the phenomenon studied here is con-cerned. Let us estimate the maximum distance at whichthis can be efficiently measured. The lifetime of the blackhole is of order M , where M is its initial mass. It ismostly determined by the emission of quanta of energy E ∼ T ∼ /M . There are, therefore, roughly M quantaemitted in a time M , which means that the mean timebetween two emissions is of order M . The percentage ofemitted quanta reaching a detector of surface S is of order S/R if the PBH is at distance R . The correct criterionfor detection and identification of the signal consists ofrequiring a mean time ∆ t between two measured photonsfrom the same PBH to be smaller that a reference inter-val ∆ t (otherwise the signal is lost in the background).This leads to the requirement M R /S < ∆ t , that is, amaximum detectable distance of R max ≈ (cid:114) S ∆ t M . (4)The most interesting case is, however, the signal emittedby a distribution of PBHs, whose masses are necessar-ily not exactly the same. Does the global line structureremains? This is not obvious, as different masses willinduce different line energies and this might make thephenomenon experimentally invisible. By studying theenergy of an emitted quantum in a given transition fortwo different BH masses, one can see that the relativeenergy variation is actually given by ∆
M/M . It is aquite general prediction of PBH formation mechanismsthat their initial mass is roughly equal to the cosmolog-ical horizon mass at the formation time, M ∼ M H ∝ t .As t ∝ T − , where T is the temperature of the Universe, (cid:12)(cid:12)(cid:12)(cid:12) ∆ EE (cid:12)(cid:12)(cid:12)(cid:12) = 2 (cid:12)(cid:12)(cid:12)(cid:12) ∆ TT (cid:12)(cid:12)(cid:12)(cid:12) . (5)So if the relative change is to remain smaller than, say,10%, it is enough that the relative change in temperatureduring the formation period remains smaller than 5%.This is reasonable for a PBH production associated, forexample, with a phase transition (see, e.g. [19, 20]). j % FIG. 1. Relative difference in the emitted energy betweena purely regular line structure and the actual LQG linestructure (in the local point of view of this study) inthe A j → A transitions (upper curve) and A j → A / transitions (lower curve). HIGH ENERGY CASE
When the temperature of the black hole is greater thanthe QCD confinement scale (but still not much abovethe energies probed by accelerators), we conservativelyassume that quarks are emitted and fragmentate intosubsequent hadrons. This is a purely semiclassical pre-diction that does not rely on the underlying theory ofquantum gravity. Some of those hadrons are unstableand will eventually decay into gamma-rays [21, 22]. Mostgamma-rays emitted in this way come from the decay ofneutral pions. We call this the secondary component.The problem is that even if the quarks are emitted witha spectrum made of lines, the resulting gamma-rays willobviously be distributed according to a continuum andthe previously mentioned approach might not hold anylonger. In addition, the instantaneous spectrum emittedby a black hole at a given temperature contains manymore photons du to the secondary component than as-sociated with the primary component (direct emission).We have investigated this point into the details by us-ing the “Lund Monte Carlo” PYTHIA code (with somescaling approximations in the low-energy range) [23] todetermine the normalized differential fragmentation func-tions dg ( Q, E ) /dE , where Q is the quark energy and E is the photon energy. It takes into account a large num-ber of physics aspects, including hard and soft interac-tions, parton distributions, initial- and final-state partonshowers, multiple interactions, fragmentation and decay.With this tool, we derived an analytical fit for the result-ing fragmentation functions which describe the numberof gamma rays generated between E and E + dE by thedecay of the hadronization product of a quark with en-ergy between Q and Q + dQ . The secondary spectrum ofgamma-rays reads as d N γ dEdt = (cid:88) j (cid:90) ∞ Q = E α j Γ j ( Q, T ) (cid:16) e QT − ( − s j (cid:17) − (6) × dg jγ ( Q, E ) dE dQ, (7)where j is the type of quark and s j = 1 /
2. The time-integration of this spectrum can obtained by writing dN γ dE = (cid:90) M f M i d N γ dEdt dtdM dM. (8)The primary component of the instantaneous spectrumis a quasi-Planckian law (either a continuous one for theusual case or as the envelope of the lines for the quan-tum gravity case) but the secondary spectrum is morecomplicated. Mostly due to the decay of neutral pions, itcan be roughly approximated by a Cauchy distributionnear its maximum, and then an E − power law followedby an exponential cutoff around the initial quark energy.It is continuous even if the primary emission is discrete.The time-integrated signal associated with the primaryemission can easily be analytically shown to lead to adifferential spectrum scaling as E − . We have performedthe numerical integration of the secondary component.The very interesting point is that, as shown in Fig. 2,the time-integrated signal is nearly the same for bothcomponents. This is quite unexpected as the physics in-volved depends on the details of nongravitational pro-cesses (subtleties of the hadronization, cross sections fordecays into gamma-rays, etc.). At a given BH temper-ature – and, therefore, at a given mass – the numberof secondary photons is much higher than the numberof primary photons. However, the mean energy of thesecondary component is much smaller than for the pri-mary component. This means that the primary emissionwas peaked at this energy when the BH mass was higherand, due the dynamics of the process ( dM/dt ∝ − M − ),it has spent a “longer time” in this mass region (between M and M + dM ). Those phenomena compensate eachother and the neat result is that both components havethe same order of magnitude.
200 400 600 800 1000 Energy GeV10 - - - - - ( arbitrary units ) FIG. 2. Time integrated primary (upper curve) and sec-ondary (lower curve) gamma-ray emission from an evap-orating BH.The important consequence of this calculation is that,even in this case, the interesting quantum gravity linestructure remains, in principle, detectable. It is not “di-luted” in a huge continuous signal due to the secondarycomponent. The secondary component only induces areasonable “self-background” and as soon as the detec-tor resolution is better than the line spacing, the phe-nomenon is easy to identify, if it exists at all. The relativeenergy difference between lines can easily be shown to be∆
E/E ∼ πγ ∼
1. As a typical detector resolution is ofthe order of 10% − CONCLUSION
A quantum-gravity “local” perspective on the horizonstructure of black holes might lead to a new view of theHawking process: the evaporation would then be asso-ciated with field quanta emitted by the settling downof the black hole after the transition of a single “surfaceelement” between two area eigenstates. This is a specula-tive hypothesis that requires a more detailed theoreticalinvestigation. However, if correct, we have shown thatthis would lead to a line structure in the spectrum, evenfor masses arbitrarily larger than the Planck mass. Thisis not washed out by the fact that black holes might beformed over a nonvanishing interval of masses. It alsoremains during the dynamics of the process in the sensethat the energy variation between consecutive emissionsis very small when compared with the separation be-tween lines. More importantly it also remains visiblewhen the secondary component, associated with the de-cay of unstable hadrons, is also taken into account. Fi-nally, beyond being a “smoking gun” candidate probefor quantum gravity, this would open interesting per-spectives to discriminate between detailed loop quantumgravity models: high-spin models have a perfectly regu-lar line structure whereas low-spin models exhibit somedeviations with respect to the ideal case.
ACKNOWLEDGMENTS
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