Quantum Gravity Phenomenology with Primordial Black Holes

Home , Francesca Vidotto

FAST RADIO BURST

- see Jansen’s talk -

Short Observed width ≃ milliseconds No Long GRB associated Afterglow? Punctual No repetition (except for one source)

Enormous flux density Energy ≲ 1038 erg

Likely Extragalactic Dispersion Measure

104 event/day

Circular polarization

Quantum Black Holes Francesca Vidotto TRANSIENTS AT SKA

Quantum Black Holes Francesca Vidotto BH EXPLOSION Vidotto, Rovelli 1401.6562

quantum n

l region - l

n “Planck Star” t

Quantum Black Holes Francesca Vidotto WHERE are quantum effects?

r>2M r =2M r =0

- see Kavic’s talk -

Quantum Black Holes Francesca Vidotto WHEN are quantum effects?

For something quantum to happen, semiclassical approximation must fail. -2 Typically in quantum gravity: high curvature Curvature ~ (LP) Small effects can pile up: small probability per time unit gives a probable effect on a long time!

-1 Typically in quantum tunneling: Curvature × (time) ~ (LP) 1m TT 1 mr23 b ⇠ ⟹ the hole lifetime must be longer or of the order of ~ m2

Haggard, Rovelli 1407.0989

Black-to-White Tunnelling In the quantum world, things happen as soon as they can! Indications from a full Loop Quantum Gravity computations.

Chistodoulou, Rovelli, Speziale, Vilensky 1605.05268

Quantum Black Holes Francesca Vidotto PRIMORDIAL BLACK HOLES

All black holes are subject to quantum effects.

An explosion observed today, requires old black holes: primordial.

(Quantum) PBH dark matter:

Today, black holes smaller than m ( t ) t = t have already exploded. | H It decreases with time. ( but for later accretion/merging )

Caution with constraints! Constraints from Hawking evaporation do not apply. PBH should exists, but not necessarily constitute all DM.

Quantum Black Holes Francesca Vidotto EXPECTED SIGNALS

Barrau, Rovelli, Vidotto 1409.4031 fast process ( few milliseconds? ) the source disappears with the burst very compact object: big flux E = mc2 1.7 1047 erg ⇠ ⇥

tH 2Gm exploding now: m = 1.2 1023 kg R = 2 .02 cm r 4k ⇠ ⇥ c ⇠ 2Gm LOW ENERGY: size of the source ≈ wavelengthR = predicted & . 02 . 05 cm cm (?) c2 ⇠ HIGH ENERGY: energy of the particle liberated Tev ⇡

SYNCHROTRON EMISSION Kavic &al. 0801.4023

GRAVITATIONAL WAVES

Quantum Gravity Phenomenology Francesca Vidotto maximal extension of the Schwarzschild metric for a The energy (and amplitude) of the signal emitted in mass M. Region (III) is where quantum gravity becomes the quantum gravity model considered here remains non-negligible. open. As suggested in [11] and to remain general, we consider two possible signals of di↵erent origins. Importantly, by gluing together the di↵erent part of The first one, referred to as the low energy signal, the e↵ective metric and estimating the time needed for is determined by dimensional arguments. When the quantum e↵ects to happen, it was shown that the dura- bounce is completed, the black hole (more precisely the tion of the bounce should not be shorter than [10] emerging white hole) has a size (L 2M)determinedby maximal extension of the Schwarzschild metric for a The energy (and amplitude) of the signal emitted⇠ in mass M. Region (III) is where quantum gravity becomes the quantumits gravity mass M model. This considered is the main here scale remains of the problem and it 2 non-negligible. ⌧ =4kM , open.(1) As suggestedfixes an expectedin [11] and wavelength to remain for the general, emitted radiation: L. We assume that particles are emitted at the we consider two⇠ possible signals of di↵erent origins. Importantly, bywith gluingk> together0.05 a the dimensionless di↵erent part parameter. of The We first use one,prorata referred of totheir as number the low of energy internalsignal, degrees of freedom. (This is also the case for the Hawking spectrum at the the e↵ective metricPlanck and units estimating where theG time= ~ needed= c = for 1. Theis determined bounce by dimensional arguments. When the time is proportional to M 2 and not to M 3 as in the optical limit, i.e. when the greybody factors describ- quantum e↵ects to happen, it was shown that the dura- bounce is completed, the black hole (more precisely the Hawking process. As long as k remains small enough, ing the backscattering probability are spin-independent.) tion of the bounce should not be shorter than [10] emerging white hole) has a size (L 2M)determinedby the bounce time is much smaller than the Hawking its mass M. This is the main scale⇠ of the problem and it evaporation time and the evaporation can be considered The second signal, referred to as the high energy com- ⌧ =4kM2, (1) fixes an expected wavelength for the emitted radiation: as a dissipative correction that can be neglected in a ponent, has a very di↵erent origin. Consider the history L. We assume that particles are emitted at the first order approximation. ⇠ of the matter emerging from a white hole: it comes from with k>0.05 a dimensionless parameter. We use prorata of theirthe number bounce of of internal the matter degrees that of formed freedom. the black hole by Planck units where G = = c = 1. The bounce (This is also the case for the Hawking spectrum at the The phenomenology~ was investigated in [11] under the collapsing. In most scenarios there is a direct relation be- 2 3 optical limit, i.e. when the greybody factors describ- time is proportionalassumption to M thatandk takes not to itsM smallestas in possible the value, which tween the formation of a primordial black hole of mass M Hawking process.makes As long the bounce as k remains time as small short enough, as possible.ing The the aim backscatteringand the probability temperature are of spin-independent.) the Universe when it was formed the bounce timeof is the much present smaller article than is to thego beyond Hawking this first study in (see [14] for a review). M is given by the horizon mass evaporation time and the evaporation can be considered The second signal, referred to as the high energy com- two directions. First, we generalize the previous results MH : as a dissipativeby correction varying k that. The can only be condition neglected for in thea modelponent, to has be a very di↵erent origin. Consider the history first order approximation. of the matter emerging from a whiteM hole:M it comest. from (2) valid is that the bounce time remains (much) smaller ⇠ H ⇠ than the Hawking time. This assumption isthe supported bounce of the matter that formed the black hole by (Other more exotic models, e.g. collisions of cosmic The phenomenologyby the was “firewall investigated argument” in [11] presented under inthe [1]. Wecollapsing. study in In most scenarios there is a direct relation be- strings or collisions of bubbles associated with di↵erent assumption that detailk takes the its maximal smallest distance possible value, at which which a singletween black-hole the formation of a primordial black hole of mass M vacua, can lead to di↵erent masses at a given cosmic time. makes the bouncebounce time can as be short detected. as possible. Second, The we go aim beyondand this the “sin- temperature of the Universe when it was formed CHARACTERISTIC FEATURE We will not consider them in this study.) The cosmic of the present articlegle event is to detection” go beyond and this consider first study the di in↵use background(see [14] for a review). M is given by the horizon mass time t is related to the temperature of the Universe T by two directions. First,produced we generalize by a distribution the previous of bouncing results blackM holes.H : size of the source ≈ wavelength predicted & .05 cm by varying k. The only condition for the model to be 1 2 2 valid is that the bounce time remainsenergy (much) of smaller the particle liberated Tev M MH t.t 0.3g T , (2) (3) ⇡ ⇠ ⇠ ⇠ ⇤ than the Hawking time. This assumption is supported II. SINGLE EVENT DETECTION(Other more exoticwhere models,g 100e.g. is thecollisions number of of cosmic degrees of freedom. by the “firewall argument” presented in [1]. We study in ⇤ ⇠ strings or collisionsOnce ofk is bubbles fixed, M associated is fixed (by with⌧ ditH↵)erent and T is therefore detail the maximal distance at which a single black-hole ⇠ For detection purposes, we are interestedvacua, in black can leadknown. to di↵erent As masses the process at a given is time-symmetric, cosmic time. what comes bounce can be detected. Second, we go beyond this “sin- holes whose lifetime is less than the age of theWe Universe. will not considerout from them the in white this hole study.) should The be cosmic what went in the gle event detection” and consider the di↵use background For a primordial black hole detected today,time⌧ t=ist relatedH black to the hole, temperature re-emerging of at the the Universe same energy:T by a blackbody produced by a distribution of bouncing black holes. where tH is the Hubble time. This fixes the mass M, as spectrum at temperature T . Intuitively, the bouncing 1 a function of the parameter k (defined in the previous black hole plays2 the2 role of a “time machine” that sends t 0.3g T , (3) section) for black holes that can be observed. In all the primordial⇠ ⇤ universe radiation to the future: while the II. SINGLEcases considered, EVENT DETECTIONM is very small comparedwhere to a solarg 100surrounding is the number space of has degrees cooled of to freedom. 2.3K, the high-energy mass and therefore only primordial black holes possibly⇤ ⇠ radiation emerges from the white hole with its original Once k is fixed, M is fixed (by ⌧ tH ) and T is therefore For detectionformed purposes, in the we early are interested Universe are in interesting black known. from this As theenergy. process is time-symmetric,⇠ what comes holes whose lifetimepoint is of less view. than Although the age of no the primordial Universe. blackout hole from has the white hole should be what went in the been detected to date, various mechanism for their pro- When the parameter k is taken larger that its smallest For a primordial black hole detected today, ⌧ = tH black hole, re-emerging at the same energy: a blackbody duction shortly after the Big distant Bang signals have beenoriginated suggested in smallerpossible and hotter value, sources that is fixed for quantum e↵ects to be where tH is the Hubble time. This fixes the mass M, as spectrum at temperature T . Intuitively, the bouncing a function of the(see, parametere.g., [12]k for(defined an early in detailed the previous calculationblack and hole [13] playsimportant the role of enough a “time to machine” lead to a bounce, that sends the bounce time section) for blackfor holes a review). that can Although be observed. their number In all densitythe might primordial be universebecomes radiation larger for to a given the future: mass. while If this the time is assumed cases considered,wayM toois very small small for compared direct detection, to a solar the productionsurrounding of spaceto be has equal cooled to the to 2.3K, Hubble the time high-energy (or slightly less if we primordial black holes remainsQuantum Black a Holes quite generic prediction focus on black holes bouncingFrancesca Vidotto far away), this means mass and therefore only primordial black holes possibly radiation emerges from the white hole with its original of cosmological physics either directly from density that the mass has to be smaller. The resulting energy formed in the early Universe are interesting from this energy. perturbations –possibly enhanced by phase transitions– will be higher for both the low energy and the high point of view. Although no primordial black hole has or through exotic phenomena like collisions of cosmic energy signals, but for di↵erent reasons. In the first been detected to date, various mechanism for their pro- When the parameter k is taken larger that its smallest strings or bubbles of false vacua. case, because of the smaller size of the hole, leading duction shortly after the Big Bang have been suggested possible value,to that a smaller is fixed emitted for quantum wavelength. e↵ects In to the be second case, (see, e.g., [12] for an early detailed calculation and [13] important enough to lead to a bounce, the bounce time for a review). Although their number density might be becomes larger for a given mass. If this time is assumed way too small for direct detection, the production of to be equal2 to the Hubble time (or slightly less if we primordial black holes remains a quite generic prediction focus on black holes bouncing far away), this means of cosmological physics either directly from density that the mass has to be smaller. The resulting energy perturbations –possibly enhanced by phase transitions– will be higher for both the low energy and the high or through exotic phenomena like collisions of cosmic energy signals, but for di↵erent reasons. In the first strings or bubbles of false vacua. case, because of the smaller size of the hole, leading to a smaller emitted wavelength. In the second case,

2 maximal extension of the Schwarzschild metric for a The energy (and amplitude) of the signal emitted in mass M. Region (III) is where quantum gravity becomes the quantum gravity model considered here remains non-negligible. open. As suggested in [11] and to remain general, we consider two possible signals of di↵erent origins. Importantly, by gluing together the di↵erent part of The first one, referred to as the low energy signal, the e↵ective metric and estimating the time needed for is determined by dimensional arguments. When the quantum e↵ects to happen, it was shown that the dura- bounce is completed, the black hole (more precisely the tion of the bounce should not be shorter than [10] emerging white hole) has a size (L 2M)determinedby maximal extension of the Schwarzschild metric for a The energy (and amplitude) of the signal emitted⇠ in mass M. Region (III) is where quantum gravity becomes the quantumits gravity mass M model. This considered is the main here scale remains of the problem and it 2 non-negligible. ⌧ =4kM , open.(1) As suggestedfixes an expectedin [11] and wavelength to remain for the general, emitted radiation: L. We assume that particles are emitted at the we consider two⇠ possible signals of di↵erent origins. Importantly, bywith gluingk> together0.05 a the dimensionless di↵erent part parameter. of The We first use one,prorata referred of totheir as number the low of energy internalsignal, degrees of freedom. (This is also the case for the Hawking spectrum at the the e↵ective metricPlanck and units estimating where theG time= ~ needed= c = for 1. Theis determined bounce by dimensional arguments. When the time is proportional to M 2 and not to M 3 as in the optical limit, i.e. when the greybody factors describ- quantum e↵ects to happen, it was shown that the dura- bounce is completed, the black hole (more precisely the Hawking process. As long as k remains small enough, ing the backscattering probability are spin-independent.) tion of the bounce should not be shorter than [10] emerging white hole) has a size (L 2M)determinedby the bounce time is much smaller than the Hawking its mass M. This is the main scale⇠ of the problem and it evaporation time and the evaporation can be considered The second signal, referred to as the high energy com- ⌧ =4kM2, (1) fixes an expected wavelength for the emitted radiation: as a dissipative correction that can be neglected in a ponent, has a very di↵erent origin. Consider the history L. We assume that particles are emitted at the first order approximation. ⇠ of the matter emerging from a white hole: it comes from with k>0.05 a dimensionless parameter. We use prorata of theirthe number bounce of of internal the matter degrees that of formed freedom. the black hole by Planck units where G = = c = 1. The bounce (This is also the case for the Hawking spectrum at the The phenomenology~ was investigated in [11] under the collapsing. In most scenarios there is a direct relation be- 2 3 optical limit, i.e. when the greybody factors describ- time is proportionalassumption to M thatandk takes not to itsM smallestas in possible the value, which tween the formation of a primordial black hole of mass M Hawking process.makes As long the bounce as k remains time as small short enough, as possible.ing The the aim backscatteringand the probability temperature are of spin-independent.) the Universe when it was formed the bounce timeof is the much present smaller article than is to thego beyond Hawking this first study in (see [14] for a review). M is given by the horizon mass evaporation time and the evaporation can be considered The second signal, referred to as the high energy com- two directions. First, we generalize the previous results MH : as a dissipativeby correction varying k that. The can only be condition neglected for in thea modelponent, to has be a very di↵erent origin. Consider the history first order approximation. of the matter emerging from a whiteM hole:M it comest. from (2) valid is that the bounce time remains (much) smaller ⇠ H ⇠ than the Hawking time. This assumption isthe supported bounce of the matter that formed the black hole by (Other more exotic models, e.g. collisions of cosmic The phenomenologyby the was “firewall investigated argument” in [11] presented under inthe [1]. Wecollapsing. study in In most scenarios there is a direct relation be- strings or collisions of bubbles associated with di↵erent assumption that detailk takes the its maximal smallest distance possible value, at which which a singletween black-hole the formation of a primordial black hole of mass M vacua, can lead to di↵erent masses at a given cosmic time. makes the bouncebounce time can as be short detected. as possible. Second, The we go aim beyondand this the “sin- temperature of the Universe when it was formed CHARACTERISTIC FEATURE We will not consider them in this study.) The cosmic of the present articlegle event is to detection” go beyond and this consider first study the di in↵use background(see [14] for a review). M is given by the horizon mass time t is related to the temperature of the Universe T by two directions. First,produced we generalize by a distribution the previous of bouncing results blackM holes.H : size of the source ≈ wavelength predicted & .05 cm by varying k. The only condition for the model to be 1 2 2 valid is that the bounce time remainsenergy (much) of smaller the particle liberated Tev M MH t.t 0.3g T , (2) (3) ⇡ ⇠ ⇠ ⇠ ⇤ than the Hawking time. This assumption is supported II. SINGLE EVENT DETECTION(Other more exoticwhere models,g 100e.g. is thecollisions number of of cosmic degrees of freedom. by the “firewall argument” presented in [1]. We study in ⇤ ⇠ strings or collisionsOnce ofk is bubbles fixed, M associated is fixed (by with⌧ ditH↵)erent and T is therefore detail the maximal distance at which a single black-hole ⇠ For detection purposes, we are interestedvacua, in black can leadknown. to di↵erent As masses the process at a given is time-symmetric, cosmic time. what comes bounce can be detected. Second, we go beyond this “sin- holes whose lifetime is less than the age of theWe Universe. will not considerout from them the in white this hole study.) should The be cosmic what went in the gle event detection” and consider the di↵use background For a primordial black hole detected today,time⌧ t=ist relatedH black to the hole, temperature re-emerging of at the the Universe same energy:T by a blackbody produced by a distribution of bouncing black holes. where tH is the Hubble time. This fixes the mass M, as spectrum at temperature T . Intuitively, the bouncing 1 a function of the parameter k (defined in the previous black hole plays2 the2 role of a “time machine” that sends t 0.3g T , (3) section) for black holes that can be observed. In all the primordial⇠ ⇤ universe radiation to the future: while the II. SINGLEcases considered, EVENT DETECTIONM is very small comparedwhere to a solarg 100surrounding is the number space of has degrees cooled of to freedom. 2.3K, the high-energy mass and therefore only primordial black holes possibly⇤ ⇠ radiation emerges from the white hole with its original Once k is fixed, M is fixed (by ⌧ tH ) and T is therefore For detectionformed purposes, in the we early are interested Universe are in interesting black known. from this As theenergy. process is time-symmetric,⇠ what comes holes whose lifetimepoint is of less view. than Although the age of no the primordial Universe. blackout hole from has the white hole should be what went in the been detected to date, various mechanism for their pro- When the parameter k is taken larger that its smallest For a primordial black hole detected today, ⌧ = tH black hole, re-emerging at the same energy: a blackbody duction shortly after the Big distant Bang signals have beenoriginated suggested in smallerpossible and hotter value, sources that is fixed for quantum e↵ects to be where tH is the Hubble time. This fixes the mass M, as spectrum at temperature T . Intuitively, the bouncing a function of the(see, parametere.g., [12]k for(defined an early in detailed the previous calculationblack and hole [13] playsimportant the role of enough a “time to machine” lead to a bounce, that sends the bounce time section) for blackfor holes a review). that can Although be observed. their number In all densitythe might primordial be universebecomes radiation larger for to a given the future: mass. while If this the time is assumed cases considered,wayM toois very small small for compared direct detection, to a solar the productionsurrounding of spaceto be has equal cooled to the to 2.3K, Hubble the time high-energy (or slightly less if we primordial black holes remainsQuantum Black a Holes quite generic prediction focus on black holes bouncingFrancesca Vidotto far away), this means mass and therefore only primordial black holes possibly radiation emerges from the white hole with its original of cosmological physics either directly from density that the mass has to be smaller. The resulting energy formed in the early Universe are interesting from this energy. perturbations –possibly enhanced by phase transitions– will be higher for both the low energy and the high point of view. Although no primordial black hole has or through exotic phenomena like collisions of cosmic energy signals, but for di↵erent reasons. In the first been detected to date, various mechanism for their pro- When the parameter k is taken larger that its smallest strings or bubbles of false vacua. case, because of the smaller size of the hole, leading duction shortly after the Big Bang have been suggested possible value,to that a smaller is fixed emitted for quantum wavelength. e↵ects In to the be second case, (see, e.g., [12] for an early detailed calculation and [13] important enough to lead to a bounce, the bounce time for a review). Although their number density might be becomes larger for a given mass. If this time is assumed way too small for direct detection, the production of to be equal2 to the Hubble time (or slightly less if we primordial black holes remains a quite generic prediction focus on black holes bouncing far away), this means of cosmological physics either directly from density that the mass has to be smaller. The resulting energy perturbations –possibly enhanced by phase transitions– will be higher for both the low energy and the high or through exotic phenomena like collisions of cosmic energy signals, but for di↵erent reasons. In the first strings or bubbles of false vacua. case, because of the smaller size of the hole, leading to a smaller emitted wavelength. In the second case,

2 ENERGY-DISTANCE RELATION 4 by 1 1/2 other other 2Gm H0 1 ⌦⇤ (1 + z) sinh (z + 1) 3/2 obs =(1+z)emitted. ⟶ obs(6) 2 1/2 ⇠ c v ⌦M u6 k⌦⇤ " # u ✓ ◆ The specific redshift dependence of our model makes t 2 it possibly testable against other proposals. Obvi- ously, detecting such a signal from far away galaxies is challenging but this work might precisely motivate some experimental prospects for the next generation ofl gamma-ray satellites. taking into account the noise. There might be room for improvement. It is not redshift the resulting function The order of magnitude of the number of bouncing impossible that the time structure of the bounce could black holes in theis galactic very slowly center varying region required to account for the observed flux is 100 per second. The asso- lead to a characteristic time-scale of the event larger than ciated mass is negligible when compared to the expected dark matter density, even when integrated over a long the response time of the bolometer. In that case, a time interval. If the mass spectrum of primordial black FIG. 3. Best fit to the Fermi excess with bouncing black holes was known, which is not the case, it would in prin- specific analysis should allow for a dedicated search of holes. ciple be possible to fix the total mass associated with bouncing black holes. As a reasonable toy model, let us such events. We leave this study for a future work as assume that the mass spectrum is given by Barrau, Rovelli, Vidotto 1409.4031 z it requires astrophysical considerations beyond this first DISCRIMINATION WITH DARK MATTER AND 2 2 4 6 8 10 d N ↵ MASS SPECTRUM = pM . (7) dMdV investigation. An isotropic angular distribution of the Quantum Black Holes Francesca Vidotto The model presented in this work is unquestionably If the number of exploding black holes required to explain bursts, signifying their cosmological origin, could also be quite exotic when compared to astrophysical hypotheses. the data on a time intervalFIG.d⌧ is Nexp 1:, one White can estimate hole signal wavelength (unspecified units) as But the important point is than it can, in principle, be the mass variation associated, considered as an evidence for the model. In case many distinguished both from astrophysical explanations and a function of z. Notice the characteristic flattening at large d⌧ from other “beyond the standard model” scenarios. The dM = . (8) events were measured, it would be important to ensure distance:8kM the youth of the hole compensate for the redshift. reason for that is the redshift dependance. When look- ing at a galaxy at redshift z, the measured energy of With M0 the mass corresponding to a black hole explod- that there is no correlation with the mean cosmic-ray flux the signal emitted either by decaying WIMPS or by as- ing now, one then has trophysical objects will be E/(1 + z) if the rest-frame (varying with the solar activity) at the satellite location. M0+dM energy is E. But this is not true for the bouncing black ↵ Nexp = pM dM. (9) holes signal. The reason for this is that black holes that M The received signal is going to be corrected by standard Let us turn to something that has been observed. 0 have bounced far away and are observed now must have Z a smaller bouncing time and therefore a smaller mass. This allows, in principe, tocosmological determine p and therefore redshift. to However, signals coming form far- Fast Radio Bursts. Fast Radio Bursts are intense iso- Their emission energy – in the low energy channel we are normalize the spectrum. considering in this article – is therefore higher and this ther away were originated earlier, namely by younger, lated astrophysical radio signals with milliseconds dura- partly compensates for the redshift e↵ect. Following [9], we can write down the observed wavelength of the signal CONCLUSIONand therefore less massive, holes, giving a peculiar de- tion. A small number of these were initially detected from a host galaxy at redshift z, taking into account both the expansion of the universe and the change of bouncing Black holes could bouncecrease once they have of reached the the emitted wavelength with distance. The re- only at the Parkes radio telescope [39–41]. Observations time, as: “Planck star” stage. This is a well motivated quantum gravity idea. In this article,ceived we have shown wavelength, that this taking into account both the expan- from the Arecibo Observatory have confirmed the detec- 2Gm phenomenon could explain the GeV excess measured by BH (1 + z) (5) obs ⇠ c2 ⇥ the Fermi satellite. This wouldsion open of the fascinating the universe pos- and the change of time available for tion [42]. The frequency of these signals around 1.3 GHz, 1 1/2 sibility to observe (non perturbative) quantum gravity H0 1 ⌦⇤ 3/2 processes at energies 19 ordersthe of black magnitude hole below the to bounce, can be obtained folding (1)into 1/2 sinh (z + 1) , namely a wavelength v ⌦M u6 k⌦⇤ " # Planck scale. Interestingly the explanation we suggest is u ✓ ◆ t fully self consistant in the sensethe that standard the hadronic “noise” cosmological relation between redshift and where we have reinserted the Newton constant G and due to decaying pions remainsproper much below time. the observed A straightforward calculation gives the speed of light c; H0, ⌦⇤ and ⌦M being the Hubble background. Unquestionably, there are other – less exotic observed 20 cm. (7) constant, the cosmological constant, and the matter den- – ways to explain the Fermi excess. But the important sity. On the other hand, for other signals the measured point we have made is that there is specific redshift de- ⇠ wavelength this just related to the observed wavelength pendance of this model which, in principle, can2 leadGm to a obs (1 + z) (6) These signals are believed to be of extragalactic origin, ⇠ c2 ⇥ because the observed delay of the signal arrival time with 1 1/2 H0 1 ⌦⇤ frequency agrees well with the dispersion due to ionized sinh (z + 1) 3/2 . 1/2 medium as expected from a distant source. The total v ⌦M u6 k⌦⇤ " # u ✓ ◆ energy emitted in the radio is estimated to be of the t 38 where we have reinserted the Newton constant G and order 10 erg. The progenitors and physical nature of the Fast Radio Bursts are currently unknown [42]. the speed of light c while H0, ⌦⇤ and ⌦M are the Hub- ble constant, the cosmological constant, and the matter There are three orders of magnitude between the pre- density. This is a very slowly varying function of the dicted signal (5) and the observed signal (7). But the redshift. The e↵ect of the hole’s age almost compesates black-to-white hole transition model is still very rough. It for the red-shift. The signal, indeed, varies by less than disregards rotation, dissipative phenomena, anisotropies, an order of magnitude for redshifts up to the decoupling and other phenomena, and these could account for the time (z=1100). See Figure 1. discrepancy. If the redshift of the source can be estimated by using In particular, astrophysical black holes rotate: one may dispersion measures or by identifying a host galaxy, given expect the centrifugal force to lower the attraction and sucient statistics this flattening represents a decisive bring the lifetime of the hole down. This should allow signature of the phenomenon we are describing. larger black holes to explode today, and signals of larger Do we have experiments searching for these signals? wavelength. Also, we have not taken the astrophysics of There are detectors operating at such wavelengths, begin- the explosion into account. The total energy (3) avail- ning by the recently launched Herschel instrument. The able in the black hole is largely sucient –9 orders of 200 micron range can be observed both by PACS (two magnitude larger– than the total energy emitted in the bolometer arrays and two Ge:Ga photoconductor arrays) radio estimated by the astronomers. and SPIRE (a camera associated with a low to medium Given these uncertainties, the hypothesis that Fast Ra- resolution spectrometer). The predicted signal falls in be- dio Burst could originate from exploding white holes is tween PACS and SPIRE sensitivity zones. There is also a tempting and deserves to be explored. very high resolution heterodyne spectrometer, HIFI, on- High energy signal. When a black hole radiates by board Herschel, but this is not an imaging instrument, it the Hawking mechanism, its Schwarzschild radius is the observes a single pixel on the sky at a time. However, the only scale in the problem and the emitted radiation has bolometer technology makes detecting short white-hole a typical wavelength of this size. In the model we are bursts dicult. Cosmic rays cross the detectors very of- considering, the emitted particles do not come from the ten and induce glitches that are removed from the data. coupling of the event horizon with the vacuum quan- Were physical IR bursts due to bouncing black hole regis- tum fluctuations, but rather from the time-reversal of tered by the instrument, they would most probably have the phenomenon that formed (and filled) the black hole. been flagged and deleted, mimicking a mere cosmic ray Therefore the emitted signal is characterized by second MAXIMAL DISTANCE DISTANCE Barrau, Bolliet, Vidotto, Weimer 1507.1198

shorter lifetime — shorter wavelength

Low energy channel High energy channel

1024 1028

1022 26 Hubble radius 10 Galactic scale ] ] 1020 m m 24 [ [ 10 R R Hadron decay 1018 1022 Galactic scale Direct emission 1016 1020

k 1 104 108 1012 1016 1020 k 1 104 108 1012 1016 1020 E [eV] 2.7 2.7×102 2.7×104 2.7×106 2.7×108 E [eV] 3.6×1013 3.6×1014 3.6×1015 3.6×1016 3.6×1017

detection of arbitrarily far signals PBH: mass - temperature relation

better single-event detection different scaling

Quantum Gravity Phenomenology Francesca Vidotto 1 1 4 1 1 2 1 4 meas 2⇡ (1 + z) H0 1 ⌦⇤ 1 3 2 meas 2⇡ (1sinh +z) H0 (1 +1z) 2⌦⇤ . 3 (6) high 1 1 1/2 sinh (1 + z) 2 . (6) ⇠ k T high2 1 1 ⌦ 1/2 B (0.3g ) "6k⌦ 2" M ## ⇤ ⇠ kBT⇤ (0.3g ) ✓"6k⌦ ◆ " ⌦M ## ⇤ ⇤ ✓ ◆

This shows that althoughThis shows the that mean although wavelength the mean does wavelengthIt is worth does consideringIt is worth the consideringn(R) term the a bitn(R more) term in a bit more in decreases as a functiondecreases of k asin a both function cases, of itk doesin both not cases,detail. it does If not one denotesdetail. by If onedn denotesthe initial by dn di↵erentialthe initial di↵erential 1 1 dMdV dMdV follow the same generalfollow behavior. the same general It scales behavior. with k It2 scalesmass with spectrumk 2 ofmass primordial spectrum black of primordial holes per unit black volume, holes per unit volume, 1 1 for the low energyfor component the low energy and as componentk 4 for the and high as k 4itfor is possible the high to defineit is possiblen(R) as: to define n(R) as: energy one. energy one. M(t+t) dn M(t+t) dn n(R)= n(R)= dM, (8)dM, (8) The following conclusionsThe following can be conclusions drawn: can be drawn: dMdV M(t) dMdVZM(t) The low energy channel leads to a better single- Z The low energy channel• leads to a better single- • event detection thanevent the detectionhigh energy than thechannel.high energyleadingchannel. to leading to Although lower energy dilutes the signal in a Although lower energy dilutes the signal in a dn t higher astrophysicalhigher background, astrophysical this e↵ background,ect is over- this e↵ect is over- dn n(tR) , (9) n(R) , dMdV 8k (9) compensated by thecompensated larger amount by the of photons. larger amount of photons. ⇡ dMdV 8k ⇡ The di↵erence of maximal distances between the where the mass spectrum is evaluated for the mass cor- The di↵erence of• maximal distances between the where the mass spectrum is evaluated for theR mass cor- • low- and high energy channels decreases for higher responding toR a time (tH c ). If one assumes that pri- low- and high energy channels decreases for higher responding to a time (tH ). If one assumes that pri- values of k, i.e. for longer black-hole lifetimes. mordial black c holes are directly formed by the collapse values of k, i.e.INTEGRATED for longer black-hole EMISSION lifetimes. mordial black holesof density are directly fluctuations formed with by a the high-enough collapse density con- INTEGRATEDIn the low energy EMISSIONchannel, for the smallerof density values fluctuations with a high-enough density con- Barrau, Bolliet,trast Vidotto, Weimer in the 1507 early.1198 Universe, the initial mass spectrum is In the low energy• channel,of k, a single for the bounce smaller can be values detectedtrast arbitrary in the far early Universe, the initial mass spectrum is • of k, a single bounce can be detected arbitrary⌧ m far2 directly related to the equation of state of the Universe away in the Universe. ⇠ directly related toat the the equationformation of epoch. state It of is the given Universe by [18, 19]: away in the Universe. at the formation epoch. It is given by [18, 19]: Low energyIn all cases,channel the distances are large enoughHigh energy and ex-channel

• k=0.05 1+3w In all cases, the distances are large enough and ex- direct decayed dn 1 perimental detection is far from being hopeless. = ↵M 1+w , (10) • dn 1 1+3w perimental detection is far from being hopeless. = ↵M dMdV 1+w , (10) k=0.05 k=100 dMdV III. INTEGRATED EMISSION where w = p/⇢. In a matter-dominated universe the 1+3w III. INTEGRATED EMISSION where w = p/⇢.exponent In a matter-dominated 1 1+w takes universe the the value = 5/2. exponent The1 normalization1+3w takes⌘ the coe valuecient↵=will5 be/2. kept unknown In addition to the instantaneous spectrum emitteddirect+decayed by a 1+w enlarged The normalization⌘as it coe dependscient on↵ thewill details be kept of the unknown black hole formation In addition to thesingle instantaneous bouncing spectrum black hole, emitted it is interesting by a to consider the as it depends onmechanism. the details of For the a sizeableblack hole amount formation of primordial black single bouncing blackpossible hole, it di is↵ interestinguse background to consider due to the the integrated emis- k=10000 mechanism.k=109 Forholes a sizeable to form, amount the power of primordial spectrum black normalized on the possible di↵use backgroundsion of a duepopulation to the of integrated bouncing emis- black holes. Formally, holes to form, theCMB power needs spectrum to be boosted normalized at small on the scales. This can sion of a populationthe of number bouncing of measured black holes. photons Formally, detected per unit time, CMB needs to bebe achieved,boosted at for small example, scales. through This Staobinsky’s can broken the number of measuredunit energy photons and detected unit surface, per unit can time, be written as: be achieved, forscale example, invariance through (BSI) Staobinsky’s scenario. broken The idea is that the unit energy and unitdN surface,mes can be written as: mass spectrum takes a high enough value in the relevant = ind((1+z)E,R) n(R) Acc Absscale(E,R invariance)dR, (BSI) scenario. The idea is that the dN dEdtdS · · · masscharacteristic spectrum shape: takesrange distorted a whereas high black enough itbody is naturally value in the suppressed relevant at small masses mes Z by inflation and at large masses by the BSI hypothesis. = ind((1+z)E,R) n(R) Acc Abs(E,R)dR, range whereas(7) it is naturally suppressed at small masses dEdtdS where (·E,R)· denotes· the individual fluxdepends emitted on how muchWe will DM notare PBL study those questions here and just consider Z ind (7) by inflation and at large masses by the BSI hypothesis. by a single bouncing black hole at distance R and at the shape of the resulting emission, nor its normalisation where ind(E,R) denotes the individual flux emitted We will not study those questions here and just consider energyQuantumE Gravity, Acc Phenomenologyis the angular acceptance of the detector whichFrancesca depends Vidotto sensitively on the bounds of the mass by a single bouncingmultiplied black hole by its at e distanceciencyR (inand principle at thisthe shape is also of a the resultingspectrum, emission, that are highly nor its model-dependent. normalisation As this part energy E, Acc is thefunction angular of acceptanceE but this of will the be detector ignored here),whichAbs( dependsE,R) sensitivelyof the study on is thedevoted bounds to the of investigation the mass of the shape multiplied by its eisciency the absorption (in principle function, this and is alson(R) a is thespectrum, number that of areof highlythe signal, model-dependent. the y axis on the As figures this part are not normalized. function of E but thisblack will holes be ignoredbouncing here), at distanceAbs(E,RR per) unitof the time study and is devoted to the investigation of the shape is the absorption function,volume. and Then distance(R) is theR numberand the of redshiftof thez entering signal, the y axisFortunately, on the figures the results are not are normalized. weakly dependent upon black holes bouncingthe at above distance formulaR per are unit linked. time The and integration has to the shape of the mass spectrum. This is illustrated in volume. The distancebe carriedR and out the up redshift to cosmologicalz entering distancesFortunately, and it is theFig. results 5 where are di↵ weaklyerent hypothesis dependent for upon the exponent are the above formulatherefore are linked. necessary The integration to use exact has results to behindthe the shape linear of thedisplayed. mass spectrum. The resulting This is electromagnetic illustrated in spectrum is be carried out upapproximation. to cosmological The distances energy and is also it is correlatedFig. 5 with whereR di↵erentalmost hypothesis exactly the for same. the exponent Therefore are we only keep one therefore necessaryas to usethe exact distance results fixes behind the bounce the linear time of thedisplayed. black hole Thecase resulting ( = electromagnetic5/2, corresponding spectrum to w =1 is/3). The black approximation. Thewhich, energy subsequently, is also correlated fixes the emitted with R energy.almost exactly theholes same. are assumed Therefore to we be only uniformly keep distributedone in the as the distance fixes the bounce time of the black hole case ( = 5/2,Universe, corresponding which to isw a=1 meaningful/3). The hypothesis black as long as which, subsequently, fixes the emitted energy. holes are assumed to be uniformly distributed in the Universe, which5 is a meaningful hypothesis as long as

5 1 1 4 1 2 meas 2⇡ (1 + z) H0 1 ⌦⇤ 3 2 high 1 1 1/2 sinh (1 + z) . (6) ⇠ kBT (0.3g ) 2 "6k⌦ " ⌦M ## ⇤ ⇤ ✓ ◆

This shows that although the mean wavelength does It is worth considering the n(R) term a bit more in decreases as a function of k in both cases, it does not detail. If one denotes by dn the initial di↵erential 1 dMdV follow the same general behavior. It scales with k 2 mass spectrum of primordial black holes per unit volume, 1 for the low energy component and as k 4 for the high it is possible to define n(R) as: energy one. 1 M(t+t) 1 4 dn 1 2 meas 2⇡ (1 + z) H0 1 ⌦⇤ 3 n(R)= dM, (8) The following conclusions can be drawn: 2 high 1 1 1/2 sinh (1 + z) . M((6)t) dMdV ⇠ kBT (0.3g ) 2 "6k⌦ " ⌦M ## Z The low energy⇤ channel⇤ leads to✓ a better◆ single- • event detection than the high energy channel. leading to Although lower energy dilutes the signal in a 1 1 4 dn t 1 higher astrophysical2 background, this e↵ect is over- meas This2⇡ shows(1 + thatz) althoughH0 the1 mean⌦ wavelength⇤ does3 It is worth considering the n(R) term a bitn more(R) in , (9) 2 dMdV 8k high 1 1 1/2 sinh compensated(1 by + z the) larger. amount of photons. (6) dn ⇡ decreases⇠ kBT (0 as.3g a function) 2 of k in both⌦ cases,M it does not detail. If one denotes by dMdV the initial di↵erential "6k⌦⇤ "✓ ◆ 1## follow the same⇤ general behavior. It scales with k 2 mass spectrum of primordial black holes per unit volume, The di↵erence1 of maximal distances between the where the mass spectrum is evaluated for the mass cor- for the low energy component• and as k 4 for the high it is possible to define n(R) as: R low- and high energy channels decreases for higher responding to a time (tH c ). If one assumes that pri- energy one. values of k, i.e. for longer black-hole lifetimes. mordial black holes are directly formed by the collapse This shows that although the mean wavelength does It is worth considering the n(R) term a bit more inM(t+t) dn n(R)= of densitydM, fluctuations with(8) a high-enough density con- decreases as a function of k in bothThe following cases, it does conclusions not candetail. beIn drawn: the If onelowPBH denotes energy MASSchannel, by SPECTRUMdn forthe the initial smaller di↵erential values trastdMdV in the early Universe, the initial mass spectrum is 1 • of k, a singlePBH bounce MASS candMdV SPECTRUM be detected arbitraryZ farM(t) follow the same general behavior. It scales with k 2 mass spectrum of primordial black holes per unit volume, directly related to the equation of state of the Universe The low1 energy channel leadsaway to in a the better Universe. single- for the low energy component and• aseventk 4 detectionfor the high thanit the is possiblehigh energy to definechannel.n(R) as:leading to at the formation epoch. It is given by [18, 19]: energy one. Although lower energy dilutes the signal in a In all cases, the distancesM(t+t are) large enough and ex- higher astrophysical background,• this e↵ect is over- dn dn t dn 1 1+3w perimentaln( detectionR)= is far from beingdM, hopeless.n(R(8)) , =(9)↵M 1+w , (10) The following conclusions can be drawn:compensated by the larger amount of photons. dMdV ⇡ dMdV 8k dMdV ZM(t) The low energy channel leads to a better single- The di↵erence of maximal distances between the where the mass spectrum is evaluated for the mass cor- • event detection than the high energy channel. leading toIII. INTEGRATED EMISSION where w = p/⇢. In a matter-dominated universe the • low- and high energy channels decreasesLow for energy higher channelresponding to a time (t R ). If one assumes that1+3 pri-w Although lower energy dilutes the signal in a H exponentc 1 1+w takes the value = 5/2. values of k, i.e. for longer black-hole lifetimes. mordial black holes are directly formed by⌘ the collapse higher astrophysical background, this e↵ect is over- dn t The normalization coecient ↵ will be kept unknown In addition to the instantaneousn(R) of spectrum density, fluctuations emitted by(9) a with a high-enough density con- compensated by the larger amount of photons.1 ⇡ dMdV 8k as it depends on the details of the black hole formation behavior. It scalesIn the withlowk 2 energyfor thechannel, lowsingle energy bouncing for compo- the smaller black hole, values it is interesting to consider the • 1 trast in the early Universe, themechanism. initial mass For spectrum a sizeable is amount of primordial black nent and as k of4 fork, a the single high bounce energy one.The canpossible be detected following di↵use backgroundarbitrary far due to the integrated emis- The di↵erence of maximal distances between the where the mass spectrum is evaluateddirectly for related the mass to the cor- equationholes of to state form, of theDifferent the Universe power mass spectrum spectra normalized on the • conclusions canaway be drawn: in the Universe.sion of a population of bouncingR black holes. Formally, low- and high energy channels decreases for higher responding to a time (tH ).at If the one formation assumes that epoch. pri- It isCMB given needs by [18, to 19]:gives be boosted qualitatively at small same scales. This can the number of measured photons c detected per unit time, values of k, i.e. for longerThe black-holelow energyIn all cases, lifetimes.channel the leads distances to amordial better are large single- black enough holes and are ex- directly formed by the collapse be achieved, for example,diffuse emission through… Staobinsky’s broken • • unit energy and unit surface, can be written as: 1+3w event detectionperimental than the detectionhigh energy isof farchannel. density from being fluctuations Al- hopeless. with a high-enough densitydn con- scale1 invariance (BSI) scenario. The idea is that the In the low energy channel, for the smaller values = ↵M 1+w , (10) • though a lower energy dilutes thedNtrastmes signal in the in early a Universe, the initial mass spectrumdMdV is mass spectrum takes a high enough value in the relevant of k, a single bounce can be detected arbitrary far = ((1+z)E,R) n(R) Acc Abs(E,R)dR, higher astrophysical background, thisdEdtdSdirectly e↵ect is related over-ind to the equation of state of the Universe range whereas it is naturally suppressed at small masses away in the Universe. · where· w ·= p/⇢. In a matter-dominated universe the compensated byIII. the larger INTEGRATED amountat of thephotons. EMISSION formationZ epoch. It is given by [18, 19]: (7) by inflation and at large masses by the BSI hypothesis. exponent 1 1+3w takes the value = 5/2. In all cases, the distancesThe are di↵erence large enough of maximal and ex- distanceswhere betweenind(E,R the ) denotes the individual flux⌘ emitted 1+w We will not study those questions here and just consider • dn The1+3 normalizationw coecient ↵ will be kept unknown perimental detection• is far fromIn addition being hopeless. to the instantaneousby a single spectrum bouncing emitted black by a hole1 at1+ distancew R and at the shape of the resulting emission, nor its normalisation low- and high energy channels decreases for higher = ↵M as it depends, on the(10) details of the black hole formation valuessingle of k bouncing, i.e. for longer black black-holehole, itenergy is interesting lifetimes.E, Acc tois consider thedMdV angular the acceptance of the detector which depends sensitively on the bounds of the mass mechanism. For a sizeable amount of primordial black possible di↵use backgroundmultiplied due to the by integrated its eciency emis- (in principle this is also a spectrum, that are highly model-dependent. As this part where w = p/⇢. In a matter-dominatedholes to form, universe the power the spectrum normalized on the III. INTEGRATEDIn thesion EMISSIONlow of energy a populationchannel, of for bouncing thefunction smaller black of valuesE holes.but this Formally, will be ignored here), Abs(E,R) of the study is devoted to the investigation of the shape • of k, a single bounce can be detectedexponent arbitrary far 1 1+3w takesCMB the needs value to = be boosted5/2. at small scales. This can the number of measured photonsis the detected absorption per function, unit time,1+w and n(R) is the number of of the signal, the y axis on the figures are not normalized. away in the Universe. The normalization⌘ coecient ↵bewill achieved, be kept for unknown example, through Staobinsky’s broken In addition to the instantaneousunit spectrum energy and emitted unit by surface, a black can holes be written bouncingQuantum as:FIG. atGravity 5: distance LowPhenomenology energyR perchannel unit signal time calculated and for di↵erent Francesca Vidotto as it depends on themass details spectra. ofscale As the the black invariance mass hole spectrum formation (BSI) is not scenario. normalized, The the idea is that the single bouncing black hole, it is interesting to consider the volume. The distance R and the redshift z entering Fortunately, the results are weakly dependent upon In alldN cases, the distances are largemechanism. enough and ex- For aunits sizeable of the amounty axismass are spectrum of arbitrary. primordial takes black a high enough value in the relevant possible di↵use background• perimental due tomes the detection integrated is far emis- from beingthe above hopeless. formula are linked. The integration has to the shape of the mass spectrum. This is illustrated in = ind((1+z)E,Rholes) n(R to) Acc form,Abs the(E,R power)dR, spectrumrange whereas normalized it is on naturally the suppressed at small masses sion of a population of bouncingdEdtdS black holes. Formally, be carried· · out· up to cosmological distances and it is Fig. 5 where di↵erent hypothesis for the exponent are Z CMB needs to be boosted(7) atby small inflation scales. and This at large can masses by the BSI hypothesis. the number of measured photons detected per unit time, therefore necessary towhere use the exact mass results spectrum behind is evaluated the linear for thedisplayed. mass cor- The resulting electromagnetic spectrum is whereIII. INTEGRATEDind(E,R) denotes EMISSION thebe individual achieved, for flux example, emitted throughWe will Staobinsky’s not studyR thosebroken questions here and just consider unit energy and unit surface, can be written as: approximation. Theresponding energy is to also a time correlated (tH ). with TheR shapealmost of the mass exactly the same. Therefore we only keep one by a single bouncing black holescale at invariance distance (BSI)R and scenario. at the The shape idea of the isc resulting that the emission, nor its normalisation as the distance fixesspectrum the bounce obviously time depends of the black on the hole detailscase of the ( for-= 5/2, corresponding to w =1/3). The black dNmes In additionenergy to theE, instantaneousAcc is the angular spectrummass acceptance emitted spectrum by of a the takesmation detector a high mechanism enoughwhich value (see depends [39] in the for sensitively relevant a review on on PBHs the bounds and of the mass = ind((1+z)E,R) n(R) Acc Abs(E,R)dR, which, subsequently, fixes the emitted energy. holes are assumed to be uniformly distributed in the dEdtdS single bouncingmultiplied· black· · hole,by its it is e interestingciency (inrange to considerprinciple whereas the this it is isinflation). naturally also a As suppressedspectrum, an example, at that we small are shall masses highly assume model-dependent. that primor- As this part Z Universe, which is a meaningful hypothesis as long as possible difunction↵use background of E but due this to(7) will the be integratedby ignored inflation emis- here), andAbs atdial large(E,R black masses) holesof by the are the study directly BSI is hypothesis. formed devoted by to the the collapse investigation of of the shape where ind(E,R) denotession of the ais population individual the absorption of flux bouncing emitted function, black and holes.Wen will( Formally,R) not is studythe numberdensity those questions of fluctuationsof the here with signal, and a high-enoughjust the y consideraxis on density the figures contrast are not normalized. by a single bouncingthe black number holeblack of at measured holesdistance bouncing photonsR and detected at at distancethe per shape unitR per time, of unit the resulting timein the and early emission, Universe. nor The its normalisation initial mass spectrum5 is then energy E, Acc is the angularunit energy acceptancevolume. and unit The of surface, the distance detector can beR writtenandwhich the as: depends redshift sensitivelyz directlyentering related on the toFortunately, bounds the equation of the the of state results mass of arethe Universe weakly dependent upon multiplied by its eciency (inthe principle above formula this is also are linked.a spectrum, The integration that areat highly has the formation to model-dependent.the epoch. shape It of is As the given this mass by part [33, spectrum. 40]: This is illustrated in dNmes function of E but this will be=be ignored carriedind here),((1+ outz)AbsE,R up()E,R ton( cosmologicalR)) A(ofE) thef(E,R study distances)dR, is devoted and it to is the investigationFig. 5 where of di↵ theerent shape hypothesis for the exponent are dEdtdS · · · dn 1 1+3w is the absorption function, andthereforen(R) is necessary the number to use of exactof the results signal, behind the y theaxis linear on the figuresdisplayed. are not= ↵ TheM normalized. resulting 1+w , electromagnetic(14) spectrum is Z (11) dMdV black holes bouncing at distanceapproximation.R per unit time The and energy is also correlated with R almost exactly the same. Therefore we only keep one where ind(E,R) denotes the individual flux emitted volume. The distancebyR aand singleas the bouncingthe redshift distance blackz fixesentering hole the at bounce distanceFortunately, timeR and of at the the blackwhere results holew = arep/case weakly⇢. In( a= dependent matter-dominated5/2, corresponding upon universe to w the=1/3). The black the above formula are linked.which, The integration subsequently, has fixes to thethe emitted shape energy. of the massexponent spectrum. holes1 This are1+3 is assumedw illustratedtakes the to value bein uniformly = 5/2. distributed in the energy E, A(E) is the angular acceptance of the detector ⌘ 1+w be carried out up tomultiplied cosmological by its distances eciency, andf(E,R it is) isFig. the absorption 5 where di↵erentThe hypothesis normalizationUniverse, for coe the exponent whichcient ↵ iswill a meaningfulare be kept unknown hypothesis as long as therefore necessary to usefunction, exact and resultsn(R) behind is the number the linear of blackdisplayed. holes bouncing The resultingas it depends electromagnetic on the details spectrum of the black is hole formation approximation. Theat energy distance is alsoR per correlated unit time with and volume.R almost The distance exactly themechanism. same. Therefore For a sizeable we only amount keep of one primordial black as the distance fixes theR and bounce the time redshift of thez entering black hole the abovecase formula ( = are5/2, correspondingholes to form,5 to thew power=1/3). spectrum The black normalized on the which, subsequently, fixeslinked. the emitted The integration energy. has to be carriedholes out are up assumed to CMB to be needs uniformly to be boosted distributed at small in scales. the The formula cosmological distances and it is thereforeUniverse, necessary which to isgiven a meaningful above might hypothesis therefore as be long correct as only within a use exact results behind the linear approximation. The limited interval of masses. The idea is that the mass energy is also correlated with R as the distance fixes the spectrum takes a high enough value in the relevant bounce time of the black hole which, subsequently,5 fixes range whereas it is naturally suppressed at small masses the emitted energy. by inflation. We will neither study those questions here (focusing on the shape of the resulting emission), It is worth considering the n(R) term a bit more into nor the normalisation issues which depend sensitively dn on the bounds of the mass spectrum, that are highly the details. If one denotes by dMdV the initial di↵erential mass spectrum of primordial black holes per unit volume, model-dependent. As this part of the study is devoted it is possible to define n(R) as: to the investigation of the shape of the signal, the y axis on the figures are not normalized. As we show M(t+t) dn below, the shape of the signal is quite independent on n(R)= dM, (12) the shape of the mass spectrum, so Eq. 14 does not play dMdV ZM(t) any significant role for the spectra computer. leading to The results are indeed very weakly dependent upon dn t the shape of the mass spectrum. This is illustrated in n(R) , (13) ⇡ dMdV 8k Fig. 5 where di↵erent hypothesis for the exponent are

6 PBH SEARCHING MASS SPECTRUM PLANCK STARS WITH SKA

DIRECT DETECTION OF TRANSIENTS in the “low energy channel”: size of the source ≈ wavelength FLATTEN REDSHIFT CURVE: localisation required!

INTENSITY MAPPING: characteristic distorted BB spectrum

A theoretician advise: look toward the higher frequencies, not the lowers one!

SYNCHROTRON EMISSION

POSSIBLE EFFECTS ON STRUCTURE FORMATION

The lifetime of a black hole Francesca Vidotto REFERENCES

PUBLISHED: 2 MARCH 2017 | VOLUME: 1 | ARTICLE NUMBER: 0065 comment arXiv:1401.6562 Planck stars Planck stars as observational Carlo Rovelli, Francesca Vidotto probes of quantum gravity Int. J. Mod. Phys. D23 (2014) 12, 1442026 Carlo Rovelli A phenomenon recently studied in theoretical physics may hold considerable interest for astronomers: the explosive decay of primordial black holes through quantum tunnelling. Their detection would be of arXiv:1404.5821 major theoretical importance. Planck star phenomenology striking conclusion from the observations of the last decade is A that our Universe teems with black Aurelien Barrau, Carlo Rovelli. holes of widely diferent masses, spanning at least nine orders of magnitude — a conclusion reinforced by the recent detection Phys. Lett. B739 (2014) 405 of the merger of two black holes of 30 to 40 solar masses through the measurement of gravitational waves1. Black holes are stable according to classical general relativity, but (a.u.) obs there is theoretical consensus that they decay λ via quantum processes. Until recently, the arXiv: 1409.4031 only decay channel studied was Hawking evaporation2, a perturbative phenomenon probably too slow to have astrophysical Fast Radio Bursts and White Hole Signals signifcance: the evaporation time of a stellar black hole is 1050 Hubble times. Aurélien Barrau, Carlo Rovelli, Francesca Vidotto. What can bring black hole decay within potential observable reach is a diferent, 2 4 6 8 10 nonperturbative, quantum phenomenon: z Phys. Rev. D90 (2014) 12, 127503 tunnelling, the same phenomenon that

triggers nuclear decay in atoms. Te explosion Figure 1 | The observed wavelength, λobs, of the Planck star signal versus its redshift, z. The ﬂattened of a black hole out of its horizon is forbidden wavelength–distance relation (equation (2); solid line) for the radio component of the Planck star signal is by the classical Einstein equations, but compared with the standard redshift (dotted line). classical equations are violated by quantum tunnelling. Violation in a fnite spacetime region turns out to be sufcient for a black of light c — these are the quantities that set the Tis phenomenon is plausible on the arXiv: 1507.1198 hole to tunnel into a white hole and explode3. scale of quantum-gravitational phenomena. basis of our current understanding of Te phenomenon does not violate causality, At this stage the star, called a Planck star 4, gravity and quantum theory; the theoretical since it is the spacetime causal structure can still be much larger than the Planck uncertainty is on the scale of the decay time. Phenomenology of bouncing black holes itself that tunnels. Te physics governing length, but quantum efects are expected to If it is exponentially suppressed like generic the decay is not exotic — in fact, it is make gravity strongly repulsive5, triggering macroscopic quantum tunnelling, it has no in quantum gravity: a closer look conservative, involving just general relativity a bounce. Te phenomenon is similar to the astrophysical consequences either. But the and quantum mechanics: physically reliable ‘quantum pressure’ that prevents electrons dumping exponential factor may be balanced theories. However, lacking consensus on a from falling into an atomic nucleus. by the phase-space factor due to the large Aurélien Barrau, Boris Bolliet, Francesca Vidotto, theory of quantum gravity, current models Remarkably, because of the huge general- black hole entropy, and arguments have been are hypothetical. Nevertheless, detection and relativistic gravitational time dilation given3 indicating that the decay time could identifcation of these signals would represent involved, collapse and bounce can be fast be of the order of Celine Weimer the frst direct observation of a quantum- (milliseconds) in the proper time of the 2 gravitational phenomenon. collapsing matter, and extremely slow m τ ~ 2 tPlanck (1) JCAP 1602 (2016) no.02, 022 During the decay process, the (millions of years) in the external time. mPlanck gravitationally collapsing matter falls inside Tus the black holes we see in the sky can its horizon until its density reaches the Planck be bouncing Planck stars during their deep where m• is the mass of the black hole, and density, namely the quantity with dimensions bounce phase, observed in extreme slow tPlanck and mPlanck are the Planck time and of density determined by the Planck constant motion because of the very large gravitational mass. A detailed calculation of the black -h, the gravitational constant G and the speed time dilation. hole decay time from frst principles is

NATURE ASTRONOMY 1, 0065 (2017) | DOI: 10.1038/s41550-017-0065 | www.nature.com/natureastronomy 1

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Quantum Black Holes Francesca Vidotto