Primordial Kerr Black Holes
PPrimordial Kerr Black Holes
Alexandre Arbey, ๐,๐,๐, โ Jรฉrรฉmy Au๏ฌnger ๐ and Joseph Silk ๐,๐, ๐ ๐ Universitรฉ de Lyon, Universitรฉ Claude Bernard Lyon 1, CNRS/IN2P3, Institut de Physique des 2 In๏ฌnis deLyon, UMR 5822, F-69622, Villeurbanne, France ๐ Institut Universitaire de France, 103 boulevard Saint-Michel, 75005 Paris, France ๐ Theoretical Physics Department, CERN, CH-1211 Geneva 23, Switzerland ๐ Sorbonne Universitรฉ, CNRS, UMR 7095, Institut dโAstrophysique de Paris, 98 bis bd Arago, 75014 Paris,France ๐ Department of Physics and Astronomy, Johns Hopkins University, Baltimore MD 2218, USA ๐ Beecroft Institute of Particle Astrophysics and Cosmology, University of Oxford, Oxford OX14BN, UK
E-mail: [email protected], [email protected],[email protected]
Primordial Black Holes (PBHs) are appealing candidates for dark matter in the universe but areseverely constrained by theoretical and observational constraints. We will focus on the Hawkingevaporation limits extended to Kerr black holes. In particular, we will discuss the possibility todistinguish between black holes of primordial and of stellar origins based on the Thorne limit ontheir spin. We will also review the isotropic extragalactic gamma ray background constraints andshow that the โwindowโ in which PBHs can constitute all of the dark matter depends strongly onthe PBH spin. Finally, we will consider the possibility that the so-called Planet 9 is a primordialblack hole. โ Speaker ยฉ Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ a s t r o - ph . C O ] D ec rimordial Kerr Black Holes Alexandre Arbey
1. Introduction
Black holes (BHs) are currently under scrutiny, and are particularly studied via the observationsof gravitational waves (GWs) emitted during the mergers of black holes. Primordial black holes(PBHs) are speci๏ฌc cases of black holes which have been generated in the early Universe. They areknown to be potential dark matter candidates, and span a broad range of masses [1]. Assuming thatthe formation of PBHs occurred at the rate of one maximal mass BH per Hubble volume, the massof PBHs formed at the cosmological time ๐ก is given by ๐ PBH โผ ๐ Planck ร ๐ก ๐ก Planck โผ g ร ๐ก ( s ) . (1)In particular, PBHs formed at Planck time have the Planck mass, PBHs formed at ๐ก โผ โ s havea mass of ๐ โผ g, and PBHs formed at 1 s have ๐ โผ ๐ (cid:12) .Predictions for the spins of PBHs are highly model-dependent, and there exist two classes ofmodels mainly leading either to non-spinning PBHs (e.g. [2]) or to maximally spinning PBHs (forexample [3]). In the following, we will assume that PBHs can have any spin.
2. Black hole spins
Spinning black holes are described by the Kerr metric ๐ ๐๐ = (cid:0) ๐๐๐ก โ ๐ sin ๐๐๐ (cid:1) ฮฮฃ โ (cid:18) ๐๐ ฮ + ๐๐ (cid:19) ฮฃ โ (cid:0) ( ๐ + ๐ ) ๐๐ โ ๐ ๐๐๐ก (cid:1) sin ๐ ฮฃ , (2)where ๐ is the BH mass, ๐ฝ its angular momentum, ๐ = ๐ฝ / ๐๐ , ฮฃ = ๐ + ๐ cos ๐ , ฮ = ๐ โ ๐ ๐ ๐ + ๐ , ๐ ๐ = ๐บ ๐ / ๐ . The horizon of Kerr BHs exists but is deformed and ๏ฌattened in comparison toSchwarzschild BHs. One generally refers to the reduced spin of BHs ๐ โ = ๐ฝ / ๐ (in natural unitswhere ๐ = โ = ๐ โ = BlackHawk [5], which incorporates precise calculations of the so-calledgreybody factors and decay and hadronization of the produced emitted particles, and which wasused to produce the following ๏ฌgures. In particular, we precisely computed the emission spectra ofparticles of spins 0, 1, 2 and 1/2 by PBHs are shown in Figure 2 for di๏ฌerent reduced spins. Onecan see that the spin can strongly a๏ฌect the emission spectra.The emission of Hawking radiation results in a decrease of the energy mass of BHs, whichundergo a vanishing process. The lifetime of BHs is shown in Figure 3, where one can see thatPBHs with masses below 10 g have already vanished since their production in the early Universe,and that the existence of a spin does not strongly a๏ฌect the lifetime.In Figure 4, we show the evolutions of the mass and spin of Kerr BHs with time for di๏ฌerentvalues of their initial spin. We also compute the spin that PBHs can reach today depending ontheir initial mass and spin, and conclude that PBHs with initial reduced spins above 0.999 can stillhave today a spin above the Thorne limit, as long as their inital mass is above 2 ร g. This is2 rimordial Kerr Black Holes Alexandre Arbey
Figure 1:
Temperature of primordial black holes as a function of the black hole mass, for di๏ฌerent valuesof the reduced spin ๐ โ . Horizontal lines show limits on Hawking temperature from QCD considerations andfrom cosmic ray positrons. an important result since the Thorne limit corresponds to the maximal value that stellar BHs areexpected to have [7], and therefore the detection of many BHs with spins above the Thorne limitwould be an indication towards a primordial origin.
3. Detectability of Hawking radiation
We now consider the possibility to detect PBHs via their Hawking radiation. For this, we assessthe impact of Hawking radiation of PBHs on the extragalactic gamma-ray background (EGRB)radiation, and derive constraints on the PBHs from observational data. A detailed description ofthe analysis can be found in Ref. [8].In Figure 5, we show the constraints on the fraction of the PBH density relative to the darkmatter density as a function of the PBH mass, for di๏ฌerent initial spins. We consider di๏ฌerent
Figure 2:
Fluxes of emission ๐ โก ๐ ๐ / ๐๐ก๐๐ธ (in GeV โ ยท s โ by a single black hole of spin-0, spin-1, spin-2and spin-1/2 particles as a function of 2 ๐ธ ๐ , where ๐ is the black hole mass and ๐ธ the particle energy, fordi๏ฌerent reduced spins ๐ โ . rimordial Kerr Black Holes Alexandre Arbey M i (g)10 t B H ( s ) age of the Universe a โi = 0.0000a โi = 0.9000a โi = 0.9999 10 โ4 โ3 โ2 โ1 โi t B H / t S M i = 10 gM i = 10 gM i = 10 g Figure 3:
Left: lifetime of primordial black holes as a function of their initial mass, for di๏ฌerent values ofthe initial reduced spin ๐ โ ๐ . Right: Lifetimes of 10 and 10 g black holes as functions of ๐ โ ๐ , relative to thelifetime of a non-spinning black hole of equal mass. (From [6].) distributions for the PBH masses: monochromatic spectrum (Dirac distribution), and log-normaldistribution with variance ๐ = .
1, 0.5 and 1. The results show that the strongest constraint isobtained for masses about 10 g, and can extend up to 10 g. The initial distribution a๏ฌects theresults, as well as the initial spin of PBHs.It is interesting to notice that if gravity is mediated by a graviton particle, Hawking radiationcan contain applies to gravitons, which results in the fact that PBHs emit gravitons, which can beinterpreted in terms of GWs. The background of GWs emitted by PBHs which are still present todayis however too weak to be observed, even with the future generations of GW detectors. However,contrary to radiation, GWs emitted at the beginning of the Universe can still be observable today.In Figure 6 we show the density of GWs emitted at the beginning of the Universe by PBHs whichhave vanished today, as a function of their frequency, for di๏ฌerent initial masses of PBHs. ForPBHs of 10 g, the GW density can reach the sensitivity of the current GW experiments, but withfrequencies around 10 Hz, which would necessitate extremely small and precise GW detectors. S M / M i a โi = 0.0000a โi = 0.9000a โi = 0.9999 a โ M i (g)10 โ4 โ3 โ2 โ1 โ a โ t o d a y Thorne = 0.998a โi = 0.9922a โi = 0.9994a โi = 0.9999 Figure 4:
Left: evolution of the mass (solid line) and spin (dotted line) as functions of time, for di๏ฌerentinitial reduced spin ๐ โ ๐ . Right: Current values of the reduced spins of black holes with reduced initial spin 0,0.99 and 0.9999, as functions of the initial black hole mass. The Thorne line delimits the maximal spin ofblack holes of stellar origin. (From [6].) rimordial Kerr Black Holes Alexandre Arbey
Figure 5:
In the plane of the fraction of PBHs relative to dark matter vs. PBH mass, regions excluded byEGRB radiation observations, for di๏ฌerent values of the initial spin, and for di๏ฌerent distributions of thePBH masses. The grey line corresponds to the exclusion limit derived in Ref. [1]. (From [8].)
4. P9, a black hole in the Solar System?
Concordant observations seem to point towards the existence of a ninth planet in the SolarSystem, at more than 400 AU from the Sun. However, since no planet has been observed withconventional telescopes, it has been proposed in Ref. [9] that it could instead be a PBH, named P9.Should this hypothesis be true, such a PBH in the Solar System would constitute a wonderfullaboratory for studying black hole properties and quantum gravity e๏ฌects. We have shown inRef. [10] that it would be impossible to detect Hawking radiation of P9 from Earth, and not evenwith a probe travelling towards P9, because of the blackbody radiation from the Cosmic MicrowaveBackground (CMB). However, a satellite probe orbiting P9 would allow the possibility of detection
Figure 6:
Density of gravitational waves emitted by evaporated PBHs at the beginning of the Universe asa function of their frequency, for di๏ฌerent initial masses of PBHs. We assume a cosmological scenariowith an in๏ฌation period of 60 e-folds. The limits of BBO, eLISA and eLIGO/adVirgo are superimposed forcomparison. rimordial Kerr Black Holes Alexandre Arbey โ4 โ3 โ2 โ1 ฮฝ (GHz)10 โ21 โ20 โ19 โ18 โ17 โ16 โ15 โ14 โ13 โ12 โ11 โ10 โ9 โ8 โ7 d P / d ฮฝ ( W ยท G H z โ ) BB model, T = 2.725KCOBE/FIRASground basedballoonCyanogenCOBE/DMR10 โ4 โ3 โ2 โ1 M = 5M โ , a โ = 0M = 5M โ , a โ = 0.99M = 10M โ , a โ = 0M = 10M โ , a โ = 0.99 r (m)10 โ37 โ36 โ35 โ34 โ33 โ32 โ31 โ30 โ29 โ28 โ27 โ26 โ25 F ( W ยท m โ ) M = 5M โ , a โ = 0M = 5M โ , a โ = 0.99M = 10M โ , a โ = 0M = 10M โ , a โ = 0.99 Figure 7:
Left: Hawking radiation spectrum of P9 as a function of the frequency. The CMB emissionspectrum is shown for comparison. Right: Flux received by a satellite as a function of the orbital radius. and study of Hawking radiation, since the BH horizon would screen radiation from the CMB.In Figure 7 we show the Hawking radiation spectrum for photons, for di๏ฌerent values of themass and spin of P9. This reveals that the maximum emissivity is close to the GHz, making possiblethe use the precision antenna technology developed for wireless internet on Earth. We also plot theelectromagnetic ๏ฌux received by a satellite from the Hawking radiation of P9, as a function of thedistance from P9.
5. Summary
Primordial black holes are potential candidates for dark matter. Scenarios of formation do notimpose strong constraints on the spins of PBHs, which are therefore rather unconstrained. We havediscussed the possibility of having nearly extremal spin PBHs and studied constraints from EGRBradiation. We have also considered P9 as a potential PBH in the Solar System. This could thereforeconstitute an ideal lab to study a PBH.
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