Gas Heating from Spinning and Non-Spinning Evaporating Primordial Black Holes
CCERN-TH-2020-159
Gas Heating from Spinning and Non-Spinning Evaporating Primordial Black Holes
Ranjan Laha, ∗ Philip Lu, † and Volodymyr Takhistov ‡ Theoretical Physics Department, CERN, 1211 Geneva, Switzerland Department of Physics and Astronomy, University of California, Los AngelesLos Angeles, California, 90095-1547, USA (Dated: October 15, 2020)Primordial black holes (PBHs) from the early Universe constitute a viable dark matter (DM)candidate and can span many orders of magnitude in mass. Light PBHs with masses around 10 gcontribute to DM and will efficiently evaporate through Hawking radiation at present time, leadingto a slew of observable signatures. The emission will deposit energy and heat in the surroundinginterstellar medium. We revisit the constraints from dwarf galaxy heating by evaporating non-spinning PBHs and find that conservative constraints from Leo T dwarf galaxy are significantlyweaker than previously suggested. Furthermore, we analyse gas heating from spinning evaporatingPBHs. The resulting limits on PBH DM abundance are found to be stronger for evaporating spinningPBHs than for non-spinning PBHs. I. INTRODUCTION
Primordial black holes (PBHs), formed in the earlyUniverse prior to any galaxies and stars, are a viablecandidate for DM (e.g., [1–31]). Depending on forma-tion, PBHs surviving until the present can span manyorders of magnitude in mass, from ∼ g to well over10 M (cid:12) . They can account for the entirety of the DMin the mass window ∼ − − − M (cid:12) , where thereare no observational constraints [32–38]. While signifi-cant attention has been devoted to larger mass PBHs, ithas been realized recently that light PBHs can result ina larger variety of observable signatures than previouslythought and is thus ripe for further exploration.Light PBHs with mass (cid:46) − M (cid:12) existing at presenttime will be evaporating and copiously emitting parti-cles through Hawking radiation [39]. Non-rotating PBHswith masses below 2 . × − M (cid:12) have lifetimes smallerthan the age of the Universe and thus do not contributeto DM abundance [40, 41]. Particle emission from cur-rently evaporating PBHs produces a variety of signatures,providing insight into this region of PBH DM parameterspace. Leading constraints on light PBHs have been ob-tained from observations of photon flux [42–45], cosmicmicrowave background [46–49], electron and positron cos-mic rays [50], 511 keV gamma-ray line [35, 51–56], as wellas neutrinos [35].Usually, PBHs are assumed to be non-rotating(Schwarzschild) [57–59]. However, PBHs can be formed with significant spin (Kerr BHs) [22–26, 61, 62]. BHspin will affect the Hawking radiation, generally in-creasing the emission while favoring particles with largerspin [39, 41, 63, 64]. Furthermore, the mass limit of ∼ . × − M (cid:12) for PBHs below which their lifetime is ∗ [email protected] † [email protected] ‡ [email protected] Heavier PBHs can also efficiently acquire spin via accretion [60]. smaller than the age of the Universe varies by a factor of ∼ II. EVAPORATING BLACK HOLE EMISSION
An un-charged rotating (Kerr) PBH radiates at atemperature given by [40, 41, 63, 73–75] T PBH = 14 πGM
PBH (cid:16) p − a ∗ p − a ∗ (cid:17) , (1) BHs with mass below (cid:46) M (cid:12) are expected to rapidly lose anyaccumulated charge due to Schwinger pair production [71, 72]. a r X i v : . [ a s t r o - ph . C O ] O c t M PBH (g) H ( G e V / s ) a ∗ = 0 M PBH (g) H ( G e V / s ) a ∗ = 0 . M PBH (g) H ( G e V / s ) a ∗ = 0 . FIG. 1. Emission components from evaporating PBHs contributing to gas heating in Leo T, assuming PBHs are non-rotatingwith a ∗ = 0 [Left] , PBHs are spinning with a ∗ = 0 . [Middle] , and PBHs approaching the Kerr BH limit with spin a ∗ = 0 . [Right] . Contributions from primary photons (green line) and electrons/positrons (red line) as well as secondary photons (bluedashed line) and electrons/positrons (orange dashed line) arising from QCD jets and the primary spectrum, are shown. Thedifference between the primary and the secondary spectra is (cid:46)
15% in all cases. where G is the gravitational constant, M PBH and a ∗ = J PBH / ( GM ) are the PBH mass and reduced spinKerr parameter, for a PBH with angular momentum J PBH . In the limit a ∗ →
0, Eq. (1) reduces to the usualHawking evaporation temperature of a SchwarzschildBH, T ’ . (cid:0) M PBH / g (cid:1) − . The temperatureis seen to be significantly diminished for a Kerr BH inthe limit a ∗ → E ’ . T PBH [75]. At lower BH masses, secondary emissionchannels due to quark and gluon QCD jets become rele-vant.For primary emission, the number of particles, N i ,emitted per unit energy per unit time is given by[40, 41, 63, 73–75] d N i dtdE = 12 π X dof Γ i ( E, M
PBH , a ∗ ) e E /T PBH ± , (2)where the greybody factor Γ i ( E, M
PBH , a ∗ ) encodes theprobability that the emitted particle overcomes the grav-itational well of the BH, E is the total energy of a par-ticle when taking BH rotation into account, the ± signsare for fermions and bosons, respectively, and summa-tion is over considered degrees of freedom. Secondaryemission of particles from QCD jets can be computednumerically [76].For our study we generate the PBH emission spectrumfor each particle species using BlackHawk code [77]. Results of numerical computation have been verified againstsemi-analytical formulas [40, 41, 63, 73].
III. TARGET SYSTEM: LEO T
DM-rich dwarf galaxies represent favorable environ-ments to investigate the effects of PBH heating due tointeractions with gas. Throughout this work, we focuson the well-modelled Leo T dwarf galaxy as our targetsystem due to its desirable cooling, gas, and DM proper-ties. We stress, however, that our analysis is general andcan be readily extended to other gas systems of interest .To describe Leo T, we follow the model of Refs. [78–80] for the DM density, neutral hydrogen (HI) gas dis-tribution, and ionization fraction. The hydrogen gasin the inner r s <
350 pc of Leo T system is largelyun-ionized [70, 81]. We consider only this central re-gion, employing the average HI gas density of n H =0 .
07 cm − , ionization fraction x e = 0 .
01 and DM densityof 1 .
75 GeV cm − [78]. Hence, the gas column densitycan be estimated as n H r s = 9 . × cm − and themass column density as m H n H r s = 1 . × − g cm − .The velocity dispersion of the HI gas in this region σ v = 6 . T ’ IV. GAS HEATING BY EVAPORATING PBHS
As proposed in Ref. [69], accretion emission from heav-ier PBHs will deposit energy and heat the gas in sur-rounding interstellar medium. It was subsequently sug-gested that emitted particles from evaporating lightPBHs can also deposit energy and heat surrounding gas We estimate that heating of Milky Way gas clouds leads toweaker bounds. M PBH (g) − − − − f P B H M PBH (g) − − − − f P B H VIGRBS ICMB MeV
FIG. 2. Constraints from Leo T on the fraction of DM PBHs, f PBH , for a monochromatic PBH mass function. [Left]
Resultsfor non-rotating PBHs with spin a ∗ = 0 (black solid line), PBHs with spin a ∗ = 0 . a ∗ = 0 . [Right] Overlay of our results with existing constraints on non-spinning PBHs from
Voyager-1 detection of positrons and using propagation model B without background (“V”, shaded red) [50],
Planck cosmic microwavebackground (“CMB”, shaded brown) [46], isotropic gamma-ray background (“IGRB”, shaded green) [36, 42, 43],
INTEGRAL
511 keV emission line for the isothermal DM profile with 1.5 kpc positron annihilation region (“I”, shaded blue) [35, 55, 56],
Super-Kamiokande neutrinos (“S”, shaded orange) [35], as well as
INTEGRAL
Galactic Center MeV flux (“MeV”, shadedmagenta) [45]. The constraint marked “I” and “MeV” are shown till the lowest PBH masses as displayed in Refs. [56] and [45]respectively. [70]. Below, we revisit gas heating due to non-rotatingevaporating PBHs with an improved treatment. We alsoextend our study of gas heating to emission from rotatingevaporating PBHs.For our PBH masses of interest, both photon as wellas electron/positron emission channels from evaporat-ing PBHs can contribute to heating. In Ref. [70], pho-ton heating contribution has been assumed negligibledue to power law scaling of photo-electric cross-section σ PE ∝ E − . when photon energies are above keV. How-ever, for photon energies around MeV that are typicalto our study, the cumulative photon interaction cross-section levels out, primarily due to Compton scatter-ing contribution (see Fig. 33.19 of Ref. [84]). The aver-age heating rate due to photon emission of PBH of mass M PBH and spin a ∗ is given by [69] H γ ( M PBH , a ∗ ) = Z ∞ f h ( E ) E d N γ dtdE (cid:0) − e − τ (cid:1) dE , (3)where f h ( E ) ∼ O (1) is the fraction of photon energyloss deposited as heat, and τ = m H n H r s /λ is the opticaldepth of gas in terms of the absorption length λ . Wetake the cumulative photon absorption length from Ref.[84]. We assume that the photon deposits heat similarlyto electrons of the same energy. Hence, we approximatethe fraction of energy deposited as heat to be similarto that of electrons, f h ( E ) = 0 .
367 + 0 . / ( E − m e )) . [70, 85–87], where m e is the electron mass. Theefficiency of photon heating is rather poor, with the heat deposited within Leo T from characteristic MeV photonswith λ ’
10 g/cm being only ∼ − fraction of thephoton energy.Analogously to the photon case, heating due to PBHelectron/positron emission can be stated as H e ( M PBH , a ∗ ) = 2 Z ∞ m e f h ( E )[ E − m e ] d N e dtdE (cid:0) − e − τ (cid:1) dE , (4)where factor of 2 comes from summing contributions ofelectrons and positrons, f h ( E ) is taken as before and thefactor (1 − e − τ ) accounts for the gas system’s opticalthickness. When the system is not optically thick, op-tical depth can be written in terms of stopping power, S ( E ), as τ ’ m H n H r s S ( E ) /E . For the electron stoppingpower on hydrogen gas, we use NIST database [88]. Forcharacteristic MeV electrons with S ( E ) ’ /g,only ∼ − fraction of the electron energy is depositedas heat in Leo T.The suppression of gas heating in Eq. (4) frompositrons and electrons, (1 − e − τ ), due to optical thick-ness of the gas system was not accounted for in the studyof Ref. [70], effectively assuming that the gas is fully op-tically thick (i.e., τ (cid:38) H ( M PBH , a ∗ ) for Leo T, including contributionsof primary photons and electrons/positrons as well assecondary photons and electrons/positrons arising fromQCD jets. Electrons/positrons are seen to provide thedominant contribution to heating rate within a broadrange of parameter space of interest. Photons provide asub-dominant contribution to the heating, but could infact dominate in the regimes T (cid:28) m e (where electronemission is heavily suppressed).We further analyze heating from spinning PBHs, dis-playing results for a ∗ = 0 . a ∗ = 0 . a ∗ = 0and a ∗ = 0 . a ∗ →
1, the pattern of PBH emis-sion and hence heating contributions changes. The emis-sion tends to be higher for spinning PBHs and for highlyspinning PBHs, photons can become dominant at smallerPBH masses, as they are produced in greater abundancethan electrons [63].
V. COOLING AND THERMAL BALANCE
The thermal balance of heating from PBHs and gascooling allows us to constrain the PBH abundance withLeo T [69, 70]. We ignore the possible additional con-tributions of natural heating sources, resulting in moreconservative bounds.Gas temperature exchange is a complex process and adetailed analysis involving a full chemistry network canbe performed using numerical methods [90]. For the pa-rameters of interest, we employ the approximate gas cool-ing rate results obtained in Ref. [81]. The cooling rateper unit volume of the hydrogen gas is given by˙ C = n H [Fe/H] Λ( T ) , (5)where [Fe/H] ≡ log ( n Fe /n H ) gas − log ( n Fe /n H ) Sun isthe metallicity, and Λ( T ) is the cooling function. Weobtain Λ( T ) = 2 . × − T . , valid for 300 K 75 GeV cm − , the total num-ber of PBHs residing in Leo T is N PBH = (cid:16) πr s (cid:17) f PBH ρ DM M PBH . (6)We take the average density such that the N PBH is the same as that obtained while integrating overthe DM profile. Requiring the total generated heat, N PBH H ( M PBH , a ∗ ), to be less than the total cooling ratein the central region of Leo T, yields the constraint [69] f PBH < f bound = M PBH ˙ Cρ DM H ( M PBH , a ∗ ) . (7)In Fig. 2, we display our resulting constraints fromPBH gas heating along with other existing limits, assum-ing a monochromatic PBH mass-function. Our resultscan be readily extended for other PBH mass-functions.Spinning PBHs are seen to induce stronger limits thannon-spinning PBHs. Our results are several orders ofmagnitude below the results suggested by the analysisof Ref. [70], which can be attributed primarily to notaccounting for the optical thickness of gas as describedabove. Furthermore, we have extended the constraintsto smaller PBHs masses.A better understanding of the standard astrophysicalheating rate in Leo T can substantially improve this limit.Similarly, discovery of more DM dominated dwarf galaxysystems and a good understanding of heating and coolingrates inside them can even lead to discovery of low-massPBHs via this technique. VI. CONCLUSIONS Light PBHs, with masses (cid:46) g, contributing toDM will significantly emit particles via Hawking radia-tion depositing energy and heat in the surrounding gas.We have studied gas heating due to spinning and non-spinning PBHs, focusing on the dwarf galaxy Leo T. Adetailed, conservative, and proper treatment of heatingresults in presented limits being significantly weaker thanpreviously claimed. We find that limits from spinningevaporating PBHs are stronger than for the non-spinningcase. ACKNOWLEDGMENTS We thank Jeremy Auffinger, Hyungjin Kim, Alexan-der Kusenko, and Anupam Ray for comments and dis-cussions. The work of P.L. and V.T. was supported bythe U.S. Department of Energy (DOE) Grant No. DE-SC0009937. R.L. thanks CERN theory group for sup-port. [1] Y. B. Zel’dovich and I. D. Novikov, The Hypothesis ofCores Retarded during Expansion and the HotCosmological Model , Sov. Astron. (1967) 602.[2] S. Hawking, Gravitationally collapsed objects of verylow mass , Mon. Not. Roy. Astron. Soc. (1971) 75.[3] B. J. Carr and S. W. Hawking, Black holes in the earlyUniverse , Mon. Not. Roy. Astron. Soc. (1974) 399.[4] G. F. Chapline, Cosmological effects of primordial blackholes , Nature (1975) 251.[5] P. Meszaros, Primeval black holes and galaxy formation , Astron. Astrophys. (1975) 5.[6] B. J. Carr, The Primordial black hole mass spectrum , Astrophys. J. (1975) 1.[7] J. Garcia-Bellido, A. D. Linde and D. 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