Black hole spectroscopy from Loop Quantum Gravity models...Black hole spectroscopy from Loop Quantum...

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HAL Id: hal-01141854 https://hal.archives-ouvertes.fr/hal-01141854 Submitted on 14 Apr 2015 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Black hole spectroscopy from Loop Quantum Gravity models A. Barrau, Xiangyu Cao, Karim Noui, Alejandro Perez To cite this version: A. Barrau, Xiangyu Cao, Karim Noui, Alejandro Perez. Black hole spectroscopy from Loop Quantum Gravity models. Physical Review D, American Physical Society, 2015, pp.124046 10.1103/Phys- RevD.92.124046. hal-01141854

Transcript of Black hole spectroscopy from Loop Quantum Gravity models...Black hole spectroscopy from Loop Quantum...

  • HAL Id: hal-01141854https://hal.archives-ouvertes.fr/hal-01141854

    Submitted on 14 Apr 2015

    HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

    L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.

    Black hole spectroscopy from Loop Quantum Gravitymodels

    A. Barrau, Xiangyu Cao, Karim Noui, Alejandro Perez

    To cite this version:A. Barrau, Xiangyu Cao, Karim Noui, Alejandro Perez. Black hole spectroscopy from Loop QuantumGravity models. Physical Review D, American Physical Society, 2015, pp.124046 �10.1103/Phys-RevD.92.124046�. �hal-01141854�

    https://hal.archives-ouvertes.fr/hal-01141854https://hal.archives-ouvertes.fr

  • Black hole spectroscopyfrom Loop Quantum Gravity models

    Aurelien Barrau,1, ∗ Xiangyu Cao,2, † Karim Noui,3, 4, ‡ and Alejandro Perez5, 6, §

    1 Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS-IN2P353,avenue des Martyrs, 38026 Grenoble cedex, France

    2 Laboratoire de Physique Théorique et Modèles Statistiques, Université Paris-Sud, CNRS(UMR 8626),91405 Orsay, France

    3Laboratoire de Mathématiques et Physique Théorique, CNRS (UMR 7350),Fédération Denis Poisson,Université François Rabelais, Parc de Grandmont, 37200 Tours, France

    4Laboratoire APC – Astroparticule et Cosmologie,Université Denis Diderot Paris 7, 75013 Paris, France

    5Centre de Physique Thorique, CNRS (UMR 7332)6Aix Marseille Université and Université de Toulon, 13288 Marseille, France

    Using Monte Carlo simulations, we compute the integrated emission spectra of black holes in theframework of Loop Quantum Gravity (LQG). The black hole emission rates are governed by theentropy whose value, in recent holographic loop quantum gravity models, was shown to agree atleading order with the Bekenstein-Hawking entropy. Quantum corrections depend on the Barbero-Immirzi parameter γ. Starting with black holes of initial horizon area A ∼ 102 in Planck units, wepresent the spectra for different values of γ. Each spectrum clearly decomposes in two distinct parts:a continuous background which corresponds to the semi-classical stages of the evaporation and aseries of discrete peaks which constitutes a signature of the deep quantum structure of the blackhole. We show that γ has an effect on both parts that we analyze in details. Finally, we estimatethe number of black holes and the instrumental resolution required to experimentally distinguishbetween the considered models.

    PACS numbers:

    I. INTRODUCTION

    Loop quantum gravity (LQG) [1–3] proposes a descrip-tion of the fundamental degrees of freedom responsiblefor the black hole (BH) entropy (see [4] and referencestherein). According to the most recent results [5–7], thesefundamental excitations live on the horizon and are ele-ments of the Hilbert space of a Chern-Simons theory: thegauge group is SU(2), the canonical surface is a punc-tured two-sphere, and the level (coupling constant) k isproportional to the horizon area A in Planck units. Punc-tures are quanta of area associated with horizon-piercingedges of the spin-network-states that define the quantumstates of the bulk exterior geometry.

    As the quantization of a Chern-Simons theory witha compact gauge group is now well-understood (see [8–10] and references therein), the kinematical characteris-tics of a quantum black hole within the framework ofLQG are very well-defined. This enables the identifica-tion of the microstates for a black hole in equilibrium.The characterization of their number for a given macro-scopic horizon area A allows for the computation of BHentropy and leads to compatibility with the celebrated

    ∗Electronic address: [email protected]†Electronic address: [email protected]‡Electronic address: [email protected]§Electronic address: [email protected]

    Bekenstein-Hawking area law

    SBH =A

    4`2p. (1)

    Due to the dependence of the area spectrum onthe Immirzi parameter γ compatibility with Bekenstein-Hawking BH entropy required fixing the Immirzi param-eter to a special numerical value in early BH entropycalculations[11, 12]. More precisely in these models theBH entropy was shown to be given by

    S =γ0γ

    A

    4`2p, (2)

    where γ0 takes an order one numerical value dependingon the models [13, 14]. Compatibility with (1) was inter-preted as a constraint on the value of γ imposed by theexistence of the correct semiclassical regime. However,the Immirzi parameter is the coupling constant with atopological term in the action of gravity [15, 16] with noimpact in the classical equations of motion. Hence, thestrong dependence of the black holes entropy computa-tion on γ remained a controversial aspect. For furtherdiscussion of this and an exploration of the influence ofγ in the phase space structure of gravity see [17–19].

    A promising perspective on the issue of the γ depen-dence of BH entropy in LQG was recently put forwardthanks to the availability of the canonical ensemble for-mulation of the entropy calculation making use of thequasi-local description of black holes [20]. It was shownin [21] that the semi-classical thermodynamical behavior

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    of BHs can be recovered for all values of γ if one admitsthe existence of a non trivial (γ-dependent) chemical po-tential conjugate to the number of horizon punctures.The problem of the observability of the puncture num-ber is resolved by taking into account contributions tothe area degeneracy coming from the matter sector. Assuggested by the semiclassical description of vacuum fluc-tuations of non geometric degrees of freedom close to thehorizon one expects such contributions to degeneracy togrow exponentially with the BH area[22]. In [23] one pos-tulates such a phenomenological contribution and showsthat compatibility with the existence of large semiclassi-cal black holes implies that the matter sector saturatesthe holographic bound for a vanishing chemical poten-tial (the number of punctures drops out, in this way, ofthe list of macroscopic observable quantities). In sucha scenario, the entropy of large semiclassical black holescoincides with (1) to leading order, while the dependenceon the Immirzi parameter is shifted to sub-leading quan-tum corrections.

    A possible fundamental explanation the exponentialdegeneracy of the assumption made above was proposedin [24] (and developed further in [25]). In these works,the area degeneracy (viewed as an analytic function ofγ) is analytically continued from real γ to complex γand then evaluated at the special complex values γ = ±ito find that it grows asymptotically as exp(A/(4`2p)) forlarge areas. The result is striking in that these values ofthe Immirzi parameter are special in the connection for-mulations of gravity: they lead to the simplest covariantparametrization of the phase space of general relativityin terms of the so-called Ashtekar variables [26]. Theseresults suggest that the quantum theory, when defined interms of self dual variables, might automatically accountfor a holographic degeneracy of the area spectrum of theBH horizon.

    Given the present landscape of different models, andleaving aside theoretical reasons for preferring one overthe others, it would be interesting to test their differencesfrom an (idealized) observational viewpoint. The value ofblack hole entropy provides little information allowing forsuch a distinction as the models (either by fixing γ = γ0or by arguing for an additional exponential degeneracycoming from either semiclassical arguments or from an-alytic continuation to self dual variables) all coincide inreproducing (1) to leading order.

    The present work shows that a possible way of dis-tinguishing the models is to analyze out-of-equilibriumprocesses. More precisely, we describe the black holeradiation process using statistical mechanical techniquestogether with some assumptions about the dynamical be-havior of BHs close to equilibrium. In doing this we willbasically follow the ideas of [27] leading to the compu-tation of the integrated emission spectrum for a blackhole (under some assumptions which will be made preciselater). The simulated spectrography will allow for thestudy of deviations predicted by the old and new (holo-graphic) models of LQG from the semiclassical Hawk-

    ing process expectation. This is of special interest forthe holographic models of [23, 24] for which the effect ofγ-dependent sub-leading quantum corrections could bedetectable.

    The article is organized as follows. In section II werecall basic facts about black hole evaporation. In sectionIII we give a brief description of the different models forblack holes that we study in this paper. In section IV wepresent the numerical methods used for the simulations.Results are presented and discussed in section V. Wefinish with a discussion in section VI. From now on weconsider units such that G = c = ~ = 1. In particular,areas, lengths and energies are given in Planck units.

    II. EVAPORATION AND TRANSITION RATES

    According to the celebrated no-hair-theorem, station-ary (axisymmetric) electro-vacuum black holes (those ofastrophysical relevance) are described by members of theKerr-Newman family and are labelled by their ADMmass M , angular momentum J and electric charge Q.The dynamics of the system under small disturbances isdescribed by the first law of BH mechanics

    δM =κ

    8πδA+ ΩδJ + ΦδQ, (3)

    where A is the horizon area, κ its surface gravity, Φ is thehorizon electric potential and Q is the black hole electriccharge.

    Hawking’s semiclassical considerations [28] imply thatKerr-Newman black holes radiate particles according tothe Planck’s radiation law at temperature T = κ/(2π).More precisely, the mean number of particles N (ω,m, q)with momentum ka radiated at infinity and carrying en-ergy ω, angular momentum quantum number m andcharge q is given by

    〈N (ω,m, q)〉 = Γ(ω,m, q)exp

    (2πκ (ω − Ωm− qΦ)

    )± 1

    , (4)

    where ω = −kaξa and m = kaψa are the frequency andangular momentum of the mode at infinity, and ξa andψa are the Kerr-Newman stationarity and axi-symmetricKilling vectors respectively normalized at infinity. Thefunction Γ in (4) is the so-called greybody factor, andthe ± sign in the denominator depends on whether oneis considering radiation of bosons (+) or fermions (−).

    Assuming that when a wave packet carrying energy ω,angular momentum m and charge q is emitted the blackhole parameters are readjusted according to δM = ω,δJ = m and δQ = q, we can write, from the first law (3),the mean number of emitted particles as follows

    〈N 〉 = Γ(ω,m, q)eδA4 ± 1

    , (5)

    which would make the spectrum Planckian in terms ofthe “area quanta δA” if it was not for the distortion in-troduced by Γ(ω,m, q). This distortion has nothing to

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    do with the thermal properties of the horizon and ap-pears only when describing the radiation from the pointof view of observers at infinity: Γ(ω,m, q) encodes thebackscattering effects on modes propagating from thevicinity of the horizon out to infinity in the backgroundKerr-Newman space-time.

    A. The quasi-local point of view

    From the point of view of stationary observers, denotedO in what follows, at a proper distance `�M from thehorizon, one obtains

    〈N 〉 = Γ0eδA4 ± 1

    , (6)

    where Γ0 is a constant, and thus the spectrum becomesstrictly Planckian in area quanta. As shown in [20] the(global) first law (3) transforms into the following sim-plest local first law for O:

    δE =κ

    8πδA, (7)

    where δE is the local energy changes as measured by O.The variable κ denotes the local surface gravity corre-sponding to the local acceleration of the stationary ob-servers defined above, i.e. κ = ‖aO‖. The (Rindler like)near horizon geometry implies that κ = 1/`.

    Notice that κ is universal in the sense that it is inde-pendent of the macroscopic parameters defining the blackhole [21]. An important consequence of this is that onecan integrate the local first law to obtain a quasi-localenergy notion E associated with the (black hole) systemfrom the perspective of the observers O, namely

    E =A

    8π`. (8)

    Applying the first law (7) to the energy variation corre-sponds to the emission of a particle with four momentumka and local frequency ω ≡ kaua as measured by O (withfour velocity ua) implies that

    δA

    4=

    2πω

    κ, (9)

    which turns (5) into a Planckian spectrum at the Unruhtemperature

    T =κ

    2π=

    1

    2π`, (10)

    as measured by O. The frequency independence of Γ0in (6) in the quasi-local treatment follows from the scaleinvariance of the near horizon approximation1.

    1 In the spherically symmetric case, the Klein-Gordon equationsimplifies to a two dimensional wave equation in the transversalcoordinates t and r∗ (where r∗ is the tortoise coordinate) withan effective potential that vanishes at the horizon [29]. This canalso be shown in the rotating black hole case.

    B. Entropy and transition rates

    The integration of the first law, together with the areatheorem, leads to the thermodynamical entropy

    S =A

    4+ S0, (11)

    where S0 is an integration constant that cannot be fixedby a thermodynamical reasoning. In fact, as in any ther-modynamical system, entropy cannot be determined onlyby the use of the first law. It can either be measuredin an experimental setup (this was the initial way inwhich the concept was introduced) or calculated by us-ing statistical mechanical methods once a model for thefundamental building blocks of the system is available.For instance, by computing the microcanonical entropyS(A) = log(N (A)) where N (A) is the number of blackhole microstates compatible with the macroscopic hori-zon area A.

    When one takes into account the evaporation phe-nomenon, the black hole can no longer be considered asa system at thermodynamical equilibrium. However, fordynamical systems which remain close to equilibrium atany time during their evolution, the statistical entropyis still a well-defined quantity: we say that the systemis at a local thermodynamical equilibrium. This wouldbe the case of large black holes for which Hawking tem-perature is cold and the horizon area A evolves slowly.In such close to equilibrium scenarios, the micro statesof the horizon compatible with the instantaneous area Acan be assumed to be equally likely. Thus the probabilityP(A → A − δA) for the black hole of horizon area A toevaporate a piece of area δA is well approximated by therate of final to initial number of microstate states

    P(A→ A− δA) ∝ N (A− δA)N (A)

    = exp [−δS(A)]. (12)

    The proportionality factor is fixed by the requirementthat the probability density is normalized. Taking intoaccount backscattering and normalization issues, oneshows that the probability for a black hole to emit a par-ticle whose energy corresponds to the decrease of area δAis given by

    P(A→ A− δA) = Γ(A, δA) exp [−δS(A)]. (13)

    We will assume the BH to be of Schwarzschild type.Consequently, in the optical limit Mω � 1 the greybodyfactors Γ ≈M2ω2 independently of the particle’s spin. Inthe other limit Mω � 1 one has Γ ≈ A = 16πM2ω2 forscalar particles, Γ ≈ 2πM2ω2 for spin 1/2 particles, andΓ ≈ 129 AM

    2ω4 for photons [30]. Numerical limitationsconstrain us to study BHs of a maximum area of about102 Planck areas. Such BHs have a radius of about 2in Planck units which means that the local curvature attheir horizon is really Planckian. If we boldly stretch theassumption of the validity of the semiclassical descriptionof space-time outside the black hole, simple dimensional

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    analysis implies that most of the radiated particles fallinto the Mω < 1 regime.

    In order to isolate the quantum effects associated withvariations of the Barbero-Immirzi parameter it will beconvenient (in some cases) to assume Γ(A, δA) = Γ0 witha constant fixed from normalization issues only. Such achoice of Γ corresponds to neglecting backscattering inthe description of the emission process and could be re-garded as describing the physics of the local observersO placed close to the horizon. We will often use thisterminology in what follows. However, one has to keepin mind that while the quasi local perspective is veryuseful in thermodynamical investigations, the notion ofspectrography does not really make sense for O. Thereason for this is that quasi local observers perceive theirenvironment at thermal equilibrium at the Unruh tem-perature and are, in this sense, incapable of discerningparticles coming from the BH from others in the thermalbath.

    III. THREE MODELS OF BLACK HOLES

    According to (13), the evaporation process is com-pletely governed by the area spectrum of the black holeand its entropy. Here we briefly review the main modelsin the literature whose spectrography will be studied insection V.

    A. Models where γ must be fixed to γ0

    The micro canonical computations taking into accountonly the quantum geometry excitations with no extra de-generacy associated to non-geometric degrees of freedom(e.g. matter fields vacuum fluctuations) lead to the fol-lowing general expression for the entropy [31]

    S =γ0γ

    A

    4+ o(A), (14)

    where γ0 is a numerical factor of order one depending onthe details of the state counting [4]. The quantum correc-tions o(A) are usually logarithmic at the leading order.As explained in the introduction, compatibility with thefirst law imposes the constraint γ = γ0. The model onlymake sense for that special value of the Immirzi param-eter. There is therefore no interesting physics associatedwith varying γ.

    B. The holographic models

    The previous results correspond to the simplified situ-ation where purely gravitational degrees of freedom aretaken into account in the counting. This has been tra-ditionally the (rather artificial) play-ground for testingthe theory without worrying about the difficult issue of

    matter coupling. If one uses the qualitative behaviourof matter degeneracy suggested by standard QFT witha cut-off at the vicinity of the horizon (i.e. exponentialgrowth of vacuum entanglement in terms of the BH area),then the entropy becomes

    S =A

    4+

    √πA

    6γ+ o(√A). (15)

    The Immirzi parameter γ affects only quantum correc-tions. This result is in agreement with semiclassical ex-pectations [23]. Notice that the entropy has been com-puted using the quasi-local point of view, the second or-der corrections to the entropy are explored in [32].

    The presence of the√A correction can be understood

    directly in the microcanonical framework as follows. Inthe holographic model, the number of black hole mi-crostates is given by

    N (A) = exp(A4

    ) p(A), (16)

    where p(A) denotes the number of ways to write A as thefinite sum

    A = 8πγ∑j

    nj√j(j + 1), (17)

    where j are distinct half-integers and nj are integers.At the semi-classical limit (large A), the large spins jdominate and then p(A) is nothing but the number ofpartitions of the integer NA = A/(4πγ). Its asymptotichas been known for almost a century, namely

    log p(A) = π

    √2NA

    3+O(logNA) =

    √πA

    6γ+O(logA),

    which leads immediately to the quantum corrections ofthe entropy (15). In the present context one does notneed to fix the value of the Immirzi parameter and thereis interesting physics associated to its variations. We willstudy the effects of varying γ on the BH spectrographyin section V.

    As explained in the introduction, a fundamental un-derstanding of the holographic hypothesis in the mod-els leading to (15) might come from the recent results[24, 25, 33] that indicate a relationship between hologra-phy and LQG for complex Ashtekar variables (γ = ±i).Nonetheless, if in the holographic treatment the degener-acy originates from the matter fields degrees of freedom,it clearly has a (quantum) geometrical origin in the con-text of complex variables treatment of black holes. Onecan however argue that matter and geometry, which arealready intimately linked at the classical level, could giveexactly the same contribution to the black hole degener-acy (only does the point of view change).

    IV. NUMERICAL METHOD

    To study the evaporation process and to analyze theeffects of the different parameters, we have developed a

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    Monte Carlo simulation procedure. The simulation hasbeen adapted from the one used in [27]. The simula-tion starts at a given mass, whose value is high enoughto be well above the deep quantum regime but smallenough so that all states below this mass can be explic-itly calculated. In this paper, we start with black holesof area Ainit = 4πγmin × 70 with γmin = 0.2 which givesAin = 1.76 × 102 in Planck units, whatever the value ofγ. This corresponds to an initial BH mass of the orderof 2 Planck masses, i.e.

    MBH ≈ 2. (18)

    Starting with the same area enables us to make easycomparisons. 107 black hole evaporations are simulatedto produce each of the spectra shown in Figure 1.

    In each case, a standard-model particle is randomlyselected at each step, with a probability weighted byits number of internal degrees of freedom and the corre-sponding (spin-dependent) greybody factor. Only pho-tons, neutrinos, and charged leptons are finally kept forthe analysis. Quarks and gluons undergo fragmenta-tion and generate hadrons with wide energy distribu-tions. Were new particles beyond the standard mod-els to be emitted, this would not change drastically theresults presented here. Only the percentage of massemitted in the photon-neutrinos-charged-leptons chanelwould change and therefore the number of black holesrequired to reach a given confidence level. But the shapeof the spectra and the differences between the modelswould qualitatively remain the same.

    Once the particle type is selected, its energy is given bythe energy loss of the black hole, i.e. E = δM . From the(usual) point of view of an observer at infinity, the massM and the horizon area of the black hole are related byA = 16πM2, and then the energy of the emitted particlemeasured by such an observer is given by

    E = δM =1

    4√πδ(√

    A)≈ 1

    8√π

    δA√A, (19)

    where δA represents the variation of area due to the emis-sion of the particle. The last identity is true only for smallvariations δA compared to the original area A. The re-lation between the energy and δA is simpler from thequasi-local point of view. In that scheme, we saw thatthe energy of the black hole is directly proportional tothe horizon area according to (8). If we denote by E`the quasi-local energy of the emitted particle, with ` thedistance of the quasi-local observer to the horizon, weobtain immediately

    E` =δA

    8π`. (20)

    From a physical point of view, it is more satisfying toplot the spectrum as a function of E∞; but it is mucheasier to analyze the results when the spectrum is givenas a function of E`. For these reasons, we will give the

    two representations of the black hole spectrum.

    The probability of transition between two black holestates of different masses (or areas) is taken to be givenby the exponential of their entropy difference weightedby the greybody factor. Most probably, those greybodyfactors should receive quantum gravity corrections, es-pecially for the late stages of the evaporation. Thesecorrections may become very large and considering onlya semi-classical greybody factor may then lead to over-simplified physical predictions. The black hole can un-dergo a transition to any state having a lower mass. Inthe quantum gravity model we are considering, only dis-crete values of the area are possible, and the probabil-ity is driven by the entropy associated with the numberof states. To make comparisons with the semi-classicalblack hole evaporation, we have also computed numer-ically the Hawking spectrum which is governed by theclassical Bekenstein-Hawking area law.

    It should be noticed that, in quantum gravity models,the transition to the last state, that is M = 0, naturallyhappens. In the semi-classical Hawking case, however,the standard formula has to be slightly modified. TheHawking law naively applied would lead the black holeto emit, at the last stage, more energy than it has. Wehave therefore used a truncation and considered thatthe energy of the emitted particle is M each time theHawking spectrum would have led to E > M . This onlyaffects the very last emission.

    The Monte Carlo is by construction a random process.It means that simulating the same process in the sameconditions would lead to another spectrum, as in real life,due to the fundamentally quantum nature of the underly-ing physics. The spectra presented here are “mean” spec-tra corresponding to many evaporating black holes (107).As the number of emitted particles in the deep quantumgravity regime is quite small (generically less than 10) fora single black hole, obviously only a statistical analysiscan lead to a significant conclusion. To remain realis-tic, we have added an ‘experimental’ uncertainty on thereconstructed energy of the detected particles. If the res-olution was infinite, a single detection would immediatelyallow to distinguish between the Hawking case (that isa continuum of states) and the quantum gravity mod-els (the distinction between quantum gravity proposalshaving the same area eigenvalues but different number ofmicro-states is anyway more subtle). This is however notrealistic and we have taken into account a kind of reason-able detector uncertainty. The results presented dependon the number of black holes and on the experimentalresolution (see figure 6).

    V. RESULTS: BLACK HOLES SPECTRA

    We concentrate our analysis on the holographic modelwhere the number of microstates is exactly given by (16).

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    PlE

    0.2 0.4 0.6 0.8 1 1.2 1.40

    200

    400

    600

    800

    1000

    1200

    310×

    = .2γ

    = .3γ

    = .4γ

    Hawking

    plE0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    1

    10

    210

    310

    410

    510

    610

    =.2γ=.3γ=.4γ

    Hawking

    A∆0 5 10 15 20

    0

    500

    1000

    1500

    2000

    2500

    3000

    3500

    310×

    =.2γ=.3γ=.4γ

    Hawking

    A∆0 5 10 15 20 25 30 35 40 45

    1

    10

    210

    310

    410

    510

    610 =.2γ=.3γ=.4γ

    Hawking

    FIG. 1: Spectrum of a holographic black hole for different values of γ. The spectrum is represented in linear coordinates as afunction of E∞ in the graph at the top left; it is represented in logarithmic coordinates as a function of E∞ in the graph atthe top right; it is represented in linear coordinates as a function of ∆A in the graph at the bottom left; it is represented inlinear coordinates as a function of ∆A in the graph at the bottom right.

    At the semi-classical limit, the resulting entropy is (15).As the greybody factor is fixed to a constant, the proba-bility for the black hole to emit a particle associated withthe decrease δA of the horizon area is totally governed bythe entropy according to (12). In the quasi-local point ofview (20), such a probability amplitude can be expressedin terms of δA and is directly related to the black holespectrum because the energy of the particle E` is propor-tional to δA. From the point of view of an observer atinfinity, the spectrum stems from the probability P(A,E)which, as we will argue below, takes the form

    P(A,E) ∝ E2 exp[−δS(A)], (21)

    with the constant fixed by normalization issues.Using the Monte-Carlo simulation we have just de-

    scribed in the previous sections, we obtained:

    1. exact spectra with different values of γ of order10−1 (Fig. 1). We considered the following set{0.2; 0.3; 0.4} of values for γ. We have ploted thespectra in terms of δA (or equivalently in termsof the local energy E`) and also in terms of E∞.We have presented the results both in linear andlogarithmic representations.

    2. an accurate comparison of the spectra with the clas-

    sical Hawking spectrum.

    3. the smeared spectra which take into account an in-strumental resolution (Fig.5).

    Note that the choice of the values of γ are motivated bythe value for γ obtained from old models (γ ≈ 0.2). Sucha choice allows for an easier comparisons with previousresults.

    Let us start with some general observations. Figure1 shows a strong dependence of the spectrum on theBarbero-Immirzi parameter even though the leadingorder term of the black hole entropy (which mainlygoverns the evaporation process) is independent ofγ. This dependency, which is completely quantumgravitational in nature, originates from two aspects: γenters in the discretization of the area spectrum and γshows off in the sub-leading corrections to the entropy.Before going deeper into the explanation of the role ofγ, let us comment further on the general aspects of thespectrum.

    The spectrum of emission, as seen from infinity, sepa-rates into two rather distinct parts: a continuous back-ground whose amplitude is most important in the in-frared and a series of discrete peaks which go from the

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    infrared to the deep ultraviolet regime. This separationis apparent in the top two panels in figure 1 while it ishidden in the bottom panels because the range of thelocal emission plots in terms of ∆A has been chosen toillustrate the continuum region of the spectrum only. No-tice that the greybody contribution, the factor E2 in theprobability amplitude, is responsible for the peak in theemission spectrum as seen from infinity in the continuouslower energy region. As the greybody factors are trivialin the local framework we see a monotonic linear behav-ior in the logarithmic plot on the bottom right panel.Coming back to the bottom panels, we observe that thequantum peaks are mainly present in the deep ultravio-let regime. While the continuous part of the spectrumcorresponds to particles emitted at the early stage of theevaporation process, the peaks reveal the deep quantumstructure of the black hole corresponding to the lateststages of the evaporation when the discrete structure ofthe area is no longer negligible. These two regions of thespectrum exhibit structures that are dependent on theBarbero-Immirzi parameter. We will now describe thesefeatures in more detail.

    A. The continuous background: link to thesemi-classical evaporation

    In both the quasi-local and the infinity points of view,spectra have a continuum background which is exponen-tially decaying.

    1. The quasi-local point of view: analysis of Sp(∆A)

    This aspect is best observed in the logarithmic scalewhen the spectrum is viewed as a function of ∆A. Inthe infrared, the spectrum can be modeled, in a firstapproximation, as follows:

    Sp(∆A) ' C exp(−τ∆A), (22)

    where τ is the exponential decay rate and C is indepen-dent of ∆A. It is interesting to observe that τ dependson the Barbero-Immirzi parameter (and probably alsoon the normalization constant C but this aspect is lessinteresting). We clearly see that τ decreases with γ.

    All these observations are in fact easy to interpret. In-deed, in the infrared, the spectrum can be reasonablyapproximated by its semi-classical expression which isgiven by the probability rate P(A→ A−∆A). This ap-proximation immediately implies the following estimatefor τ :

    τ ' δSδA

    =1

    4

    (1 +

    √2π

    3γA

    ), (23)

    where we used the semi-classical expression for the en-tropy (15) up to the first quantum correction. Here Acan be identified with the initial black hole area, i.e.

    A = 1.76 × 102, as the continuous spectrum concernsthe early stage of the evaporation process when the blackhole area remains close to its initial value. The previousexpression for τ is consistent with the numerical simula-tions.

    We can go further in the analysis of Sp(∆A). We noticethat the exponential decay approximation is clearly bet-ter in the infrared than in the ultraviolet where, even ifthe statistical fluctuations are more important, the slopeτ seems to increase with ∆A: the spectrum is steeper forlarge ∆A (larger than 30 in Planck units) compared tosmall ∆A (between 0 and 10 Planck units). The previ-ous expression for τ (23) allows us to easily interpret thisfact. In the ultraviolet, the area A in (23) can indeed nolonger be identified with the initial area because we arefar from the earliest stages of the evaporation process. Inthat case, A has to be identified with the instantaneousarea of the black hole which is obviously smaller than theinitial one. The consequence is that the slope τ increasesbecause it scales as 1/

    √A. The more we go to the ultra-

    violet, the larger the slope is, exactly as observed in thenumerical simulations.

    Let us finish the analysis of Sp(∆A) with some remarksconcerning the comparison with the Hawking spectrum.We observe that the quantum and the Hawking spectraare closer one to the other in the infrared than in the ul-traviolet. The reason is simple and comes from the factthat quantum corrections to the entropy are no longernegligible in the late stages of the evaporation. From theexpression of the entropy, we see that the larger γ is, thesmaller quantum corrections are and then the better theapproximation to the Hawking spectrum in the ultravio-let is. This theoretical prediction is consistent with thenumerical calculations (see Figure 1).

    In the infrared, the situation is somehow the reverse.In that case, the quantum corrections to the BH entropyare negliguible, and thus cannot explain the differencesbetween the spectra with different γ. We observe that thesmaller γ the better the approximation to the Hawkingspectrum. This simply comes from the value of the areagap in the area spectrum of the black hole which is linearin γ. When γ is small, the area spectrum of the blackhole resembles a continuous spectrum and then we expectthe emission spectrum to be very close to the Hawkingone, as shown by the numerical simulation.

    2. The point of view at infinity: analysis of Sp(E)

    When the spectrum Sp(E) is viewed as a functionof E (which is relevant for the point of view of an ob-server at infinity), we observe that its continuous back-ground admits a maximum and the smallest energies aresuppressed. From a theoretical point of view, this isexplained because, near the maximum, the continuousbackground can be approximated by its semi-classical ex-

  • 8

    pression which leads to the following estimate2

    Sp(E) ' C̃E2 exp (−τ̃E) , (25)

    where the slope τ̃ is now given by

    τ̃ ' δSE' 2√πA+

    √8π2

    3γ. (26)

    As in the previous analysis, C̃ can be approximated by aconstant and A can be reasonably identified with the ini-tial area. From this result, we can immediately estimatethe energy Emax = 2/τ̃ for which the spectrum is maxi-mum together with the value of this maximum which isgiven by

    Sp(Emax) ' C̃(

    2

    eτ̃

    )2. (27)

    Therefore, we can predict that the maximum of the spec-trum increases with γ. This is exactly what we observe inthe numerical experiments. Such an analysis shows thatthe quantum corrections to the entropy manifest them-selves also in the infrared part of the spectrum, and notonly in the deep ultraviolet regime, what one would haveexpected. This aspect seems to us very interesting tounderline.

    2 As the initial mass of the BH Min ≈ 2 the continuous part of thespectrum falls in the regime Mω � 1 (putting all the units inthe previous criterion gives (M(~ω/c2))/M2p = (ME)/M2p < 1).In the Figure V A 2 below we show the exact distribution of Mωfor γ = 0.4. This means that we can use the low energy form ofthe form factors [30] from which we would get

    (NfafE2 +NphaphE

    4) exp (−τ̃E) ≈

    NfafE2 exp (−τ̃E) , (24)

    where Nf and Nph are the number of fermions and photon de-grees of freedom and af and aph are parameters of the sameorder. The photon term is subdominant as Nf > Nph in thestandard model.

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    1

    10

    210

    310

    410

    510

    610

    =0.4γ, ωM

    FIG. 2: Distribution of Mω for γ = 0.4 illustrating the factthat the continuous part of the spectrum is radiated in theIR regime Mω < 1.

    B. The quantum peaks: quantum nature ofspace-time

    In Figure 1, the linear plot (Sp(∆A) or Sp(E)) is themore convenient to analyze and interpret the propertiesof the quantum peaks in the black hole spectrum. Thesepeaks (above all the ones with a higher energy) are ex-pected to appear at the late stages of the black hole evap-oration process and therefore should correspond to thetransitions between low area states. In that respect, theyprovide us with a concrete signature of the deep quantumstructure of the black hole.

    More explicitly, each peak is produced by a transitionfrom a state i (of quantum area Ai) to a state j < i (ofquantum area Aj). Whether we consider the local pointof view or the point of view at infinity, the energy of thecorresponding emitted particle is given by

    E`ij =δAij8π`

    ∝ γ, (28)

    E∞ij =1

    4√π

    (√Ai −

    √Aj) ∝

    √γ, (29)

    and therefore it scales respectively as γ or√γ. This scal-

    ing can be immediately observed in the linear plot Sp(δA)and Sp(E). For instance, the three highest peaks in Sp(δ)correspond to the following approximate energies:

    γ 0.2 0.3 0.4

    δA(γ) 1.60 2.40 3.20

    δA/γ 0.80 0.80 0.80

    (30)

    Indeed, A(γ) = 4πγ(2√

    1× 3−√

    2× 4), which is the

    area difference between any configuration containing

  • 9

    (at least) two j = 12 punctures and the configurationobtained by replacing them by one j = 1 puncture.All other area–difference peaks can be identified to apuncture change or removal in the same manner.

    The same exercise for highest energy peaks can be doneand the observation of the scaling of the energy as

    √γ

    in the point of view at infinity can be checked. Nev-ertheless, since δE depends on δA and A, peaks areto be associated with transition between two configu-rations. For instance, the highest peaks correspond tothe transition (nj) = (2, 0, 0 . . . ) → (nj) = (0, 0, . . . )with E∞ =

    √γ31/42−1/2, which gives 0.42, 0.51, 0.59 for

    γ = 0.2, 0.3, 0.4 respectively.A more delicate effect of γ concerns the amplitude of

    the peaks. This effect is easier and more precise to studyin the quasi-local point of view where the amplitudesare given by Sp(E`ij). In fact, to take into account thenormalization issues, it is more judicious to study relativeamplitudes of the form Sp(E`ij)/Sp(E`0) where E0 is thereference energy corresponding to a given transition i0 toj0. One can for instance choose the highest energy peakas the reference. Such relative amplitudes depends on γaccording to

    Sp(E`ij)

    Sp(E`0)∝ exp[−2π`(E`ij − E`0)]. (31)

    As the energy scales as γ in the quasi-local point ofview, the relative amplitude is more suppressed for largerγ. This explains the observation that when γ increases,lower energy peaks have larger amplitudes. In the infi-nite viewpoint, although it is not a priori evident, weobserve that the presence of greybody factors do not al-ter the qualitative picture in the range of γ ∈ [0.2, 0.4]considered here. We leave the issue of how these featuresextrapolate to a larger range of γ for further studies,the main message here being that the effect of γ on thespectrum (in both its semi-classical and deep quantumregimes) is more than a simple dilation.

    C. Comparison with older models

    For completeness we here show in figures 3 and 4 theγ dependence of the integrated spectrum of the non-holographic models considered in previous literature [27].Notice that a comparison of different values of γ is only amathematical exercise as in the framework of such mod-els the Immirzi parameter must be fixed to the value thatis compatible with SBH = A/4 which in this case corre-sponds to γ ≈ 0.274 [4].

    A question of physical relevance is how many evapo-rating black holes would be necessary to distinguish be-tween the different models studied here and the Hawkingsemiclassical model. In order to do this in a quite re-alistic way, we introduced an uncertainty in the energymeasurement of emitted particles through a random er-ror. With this modification, the spectra in figure 1 gets

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

    50

    100

    150

    200

    250

    310×

    =.2γ=.3γ=.4γ

    FIG. 3: γ dependence of the integrated spectrum in the stan-dard LQG model (linear scale).

    old0.4Entries 1.12554e+07

    Mean 0.2898

    RMS 0.2997

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    1

    10

    210

    310

    410

    510

    old0.4Entries 1.12554e+07

    Mean 0.2898

    RMS 0.2997

    =.2γ=.3γ=.4γ

    FIG. 4: γ dependence of the integrated spectrum in the stan-dard LQG model (logarithmic scale).

    smeared as shown in figure 5. With such smeared spectrais now possible to compute how many detections wouldbe necessary to distinguish the different models. Resultsare shown in figure 6.

    In each case, if the number of black holes is highenough and/or the uncertainty small enough, it is possi-ble, through a Kolmogorov-Smirnov test, to statisticallydistinguish between models. Due to the structure andamplitude of the peaks, it is slightly easier to distinguishbetween the pure Hawking spectrum and the old LGQmodel than between the pure Hawking spectrum and theholographic one.

    VI. CONCLUSIONS

    The study presented in this article investigates theprecise emission from light quantum black holes. Start-ing with an initial horizon area A ∼ 102 in Planck units,

  • 10

    plE0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

    -710

    -610

    -510

    -410

    -310

    =.2γ=.3γ=.4γ

    Hawking

    FIG. 5: γ dependence of the integrated spectrum in the holo-graphic model with a simulated error of energy detection: aparticle of energy E has detected energy E + e, where e is anormal distribution with variance σ = 0.05E.

    relative error0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

    100

    BH

    5

    10

    15

    20

    25New vs. Hawking

    Old vs. Hawking

    FIG. 6: Number of observed events necessary to distinguishthe semiclassical Hawking model from LQG models, as a func-tion of the relative uncertainty of the apparatus.

    the spectra for different values of γ were computed,averaging over many realizations. The continuous

    background corresponding to the semi-classical stagesis complemented by discrete peaks associated with thedeep quantum regime. We have shown that the Barbero-Immirzi parameter has an important, and sometimessubtle, effect on both parts of the spectrum. Finally wehave calculated the number of black holes, for a givenexperimental uncertainty, required to experimentallydistinguish between the holographic model and thesemi-classical Hawking spectrum.

    The analysis presented in this article is more relevantat the conceptual level than at the strictly experimentallevel. It is more a “thought experiment”, useful tounderstand subtleties that the analytical investigationsdo not reveal easily, than a proposal for a real experi-ment. Is it however conceivable that evaporating blackholes can really be used to probe the models ? In theframework of a cosmological production of primordialblack holes, this is extremely unlikely (see [34] for areview). A very wide mass spectrum of primordial blackholes is now disfavoured as it would require either ahigh normalization of the primordial fluctuation powerspectrum (to produce a density contrast of order unity)or a blue tilted spectrum, both being excluded by CMBobservations. There are many other means for creatingprimordial black holes with a narrower mass spectrum.In any case, however, the emission spectrum roughly de-creases with the energy as E−α with α ranging betweenaround 1 below 100 MeV to around 3 above this scale[35, 36]. Unless a very strong boost phenomenon occursat late times, it is therefore basically impossible to seethe last stages without having first detected, at a muchhigher level, the semi-classical stage. A more promisingavenue would be associated with the production at col-liders. This however requires a low Planck scale so thatthe collision is trans-planckian and creates a black hole.This is not, in principle, incompatible with LQG butrelies on strong assumptions –such as extra-dimensions–that the theory precisely allows to avoid. At this stage,the observational detection is therefore unlikely andsimulations are mostly used as gedankenexperiments tounderstand better the detailed features of the model.

    [1] Carlo Rovelli. Quantum gravity. Cambridge UniversityPress (2004).

    [2] Thomas Thiemann. Modern canonical quantum generalrelativity. Cambridge, UK: Cambridge Univ. Pr. (2007),2007.

    [3] Abhay Ashtekar and Jerzy Lewandowski. Back-ground independent quantum gravity: A Status report.Class.Quant.Grav., 21:R53, 2004.

    [4] Fernando Barbero and Alejandro Perez. Quantum Ge-ometry and Black Holes. arXiv:1501.02963[gr-qc], 2015.

    [5] Jonathan Engle, Karim Noui, and Alejandro Perez. Blackhole entropy and SU(2) Chern-Simons theory. Phys. Rev.

    Lett., 105, 2010.[6] Jonathan Engle, Karim Noui, and Alejandro Perez. Black

    hole entropy from the SU(2)-invariant formulation oftype I isolated horizons. Phys. Rev. D, 82, 2010.

    [7] Jonathan Engle, Karim Noui, and Alejandro Perez. TheSU(2) black hole entropy revisited. JHEP, 05, 2011.

    [8] E. Buffenoir, K. Noui, and P. Roche. Hamiltonian quan-tization of Chern-Simons theory with SL(2,C) group.Class.Quant.Grav., 19:4953, 2002.

    [9] Karim Noui. Three Dimensional Loop Quantum Grav-ity: Particles and the Quantum Double. J.Math.Phys.,47:102501, 2006.

  • 11

    [10] Karim Noui. Three dimensional Loop Quantum Grav-ity: Towards a self-gravitating Quantum Field Theory.Class.Quant.Grav., 24:329–360, 2007.

    [11] Carlo Rovelli. Black hole entropy from loop quantumgravity. Phys.Rev.Lett., 77:3288–3291.

    [12] A. Ashtekar, J. Baez, A. Corichi, and Kirill Kras-nov. Quantum geometry and black hole entropy.Phys.Rev.Lett., 80:904–907.

    [13] Marcin Domaga la and Jerzy Lewandowski. Black-holeentropy from quantum geometry. Class. Quant. Grav.,21:5233–5243, 2004.

    [14] Amit Ghosh and P. Mitra. A Bound on the log correctionto the black hole area law. Phys.Rev., D71:027502.

    [15] Ghanashyam Date, Romesh K. Kaul, and Sandipan Sen-gupta. Topological Interpretation of Barbero-ImmirziParameter. Phys.Rev., D79:044008.

    [16] Danilo Jimenez Rezende and Alejandro Perez. 4dLorentzian Holst action with topological terms.Phys.Rev., D79:064026.

    [17] Marc Geiller and Karim Noui. A note on the Holst ac-tion, the time gauge, and the Barbero-Immirzi parame-ter. Gen.Rel.Grav., 45:1733–1760, 2013.

    [18] Jibril Ben Achour, Marc Geiller, Karim Noui, and ChaoYu. Testing the role of the Barbero-Immirzi parameterand the choice of connection in Loop Quantum Gravity.arXiv:1306.3241[gr-qc], 2013.

    [19] Jibril Ben Achour, Marc Geiller, Karim Noui, and ChaoYu. Spectra of geometric operators in three-dimensionalloop quantum gravity: From discrete to continuous.Phys.Rev., D89(6):064064, 2014.

    [20] Ernesto Frodden, Amit Ghosh, and Alejandro Perez.Quasilocal first law for black hole thermodynamics.Phys.Rev., D87:121503.

    [21] Amit Ghosh and Alejandro Perez. Black hole entropyand isolated horizons thermodynamics. Phys.Rev.Lett.,107:241301.

    [22] Sergey N. Solodukhin. Entanglement entropy of blackholes. Living Rev.Rel., 14:8, 2011.

    [23] Amit Ghosh, Karim Noui, and Alejandro Perez. Statis-tics, holography, and black hole entropy in loop quantumgravity. Phys.Rev., D89(8):084069, 2014.

    [24] Ernesto Frodden, Marc Geiller, Karim Noui, and Alejan-dro Perez. Black Hole Entropy from complex Ashtekarvariables. Europhys.Lett., 107:10005, 2014.

    [25] Jibril Ben Achour, Amaury Mouchet, and Karim Noui.Analytic Continuation of Black Hole Entropy in LoopQuantum Gravity. arXiv:1406.6021[gr-qc], 2014.

    [26] A. Ashtekar. New Variables for Classical and QuantumGravity. Phys.Rev.Lett., 57:2244–2247, 1986.

    [27] A. Barrau, T. Cailleteau, X. Cao, J. Diaz-Polo, andJ. Grain. Probing Loop Quantum Gravity with Evap-orating Black Holes. Phys.Rev.Lett., 107:251301, 2011.

    [28] S.W. Hawking. Particle Creation by Black Holes. Com-mun.Math.Phys., 43:199–220.

    [29] R. M. Wald. General Relativity. Chicago UniversityPress, Chicago, 1984.

    [30] Don N. Page. Particle Emission Rates from a Black Hole:Massless Particles from an Uncharged, Nonrotating Hole.Phys.Rev., D13:198–206, 1976.

    [31] Ivan Agullo, G.J. Fernando Barbero, Enrique F. Borja,Jacobo Diaz-Polo, and Eduardo J.S. Villasenor. TheCombinatorics of the SU(2) black hole entropy in loopquantum gravity. Phys.Rev., D80:084006, 2009.

    [32] Olivier Asin, Jibril Ben Achour, Marc Geiller, KarimNoui, and Alejandro Perez. Black holes as gases ofpunctures with a chemical potential: Bose-Einstein con-densation and logarithmic corrections to the entropy.Phys.Rev., D91:084005, 2015.

    [33] Ernesto Frodden, Marc Geiller, Karim Noui, and Alejan-dro Perez. Statistical Entropy of a BTZ Black Hole fromLoop Quantum Gravity. JHEP, 1305:139, 2013.

    [34] B.J. Carr, Kazunori Kohri, Yuuiti Sendouda, andJun’ichi Yokoyama. New cosmological constraints on pri-mordial black holes. Phys.Rev., D81:104019, 2010.

    [35] F. Halzen, E. Zas, J.H. MacGibbon, and T.C. Weekes.Gamma-rays and energetic particles from primordialblack holes. Nature, 353:807–815, 1991.

    [36] Aurelien Barrau, Gaelle Boudoul, Fiorenza Donato,David Maurin, Pierre Salati, et al. Anti-protons fromprimordial black holes. Astron.Astrophys., 388:676, 2002.