Constraining Primordial Black Holes with Dwarf Galaxy Heating
Philip Lu, Volodymyr Takhistov, Graciela B. Gelmini, Kohei Hayashi, Yoshiyuki Inoue, Alexander Kusenko
IIPMU20-0076, RIKEN-iTHEMS-Report-20
Constraining Primordial Black Holes with Dwarf Galaxy Heating
Philip Lu, ∗ Volodymyr Takhistov, † Graciela B. Gelmini, ‡ KoheiHayashi,
2, 3, § Yoshiyuki Inoue,
4, 5, ¶ and Alexander Kusenko
1, 5, ∗∗ Department of Physics and Astronomy, University of California, Los AngelesLos Angeles, California, 90095-1547, USA Institute for Cosmic Ray Research, The University of Tokyo, Kashiwa 277-8582, Japan Astronomical Institute, Tohoku University, Sendai 980-8582, Japan Interdisciplinary Theoretical & Mathematical Science Program (iTHEMS),RIKEN, 2-1 Hirosawa, Saitama 351-0198, Japan Kavli Institute for the Physics and Mathematics of the Universe (WPI), UTIASThe University of Tokyo, Kashiwa, Chiba 277-8583, Japan (Dated: July 7, 2020)Black holes formed in the early universe, prior to the formation of stars, can exist as dark matterand also contribute to the black hole merger events observed in gravitational waves. We set a newlimit on the abundance of primordial black holes (PBHs) by considering interactions of PBHs withthe interstellar medium, which result in the heating of gas. We examine generic heating mechanisms,including emission from the accretion disk, dynamical friction, and disk outflows. Using the datafrom the Leo T dwarf galaxy, we set a new cosmology-independent limit on the abundance of PBHsin the mass range O (1) M (cid:12) − M (cid:12) . Primordial black holes (PBHs) can form in the earlyUniverse through a variety of mechanisms and can ac-count for all or part of the dark matter (DM) (e.g. [1–25]). PBHs surviving until present can span many ordersof magnitude in mass, from 10 g to well over 10 M (cid:12) ,and they can account for the entirety of the DM in themass window ∼ − − − M (cid:12) , where there are noobservational constraints [26–28]. PBHs with sublunarmasses can play a role in the synthesis of heavy ele-ments, production of positrons, as well as other astro-physical phenomena [29–31]. PBHs with larger massescan account for some of the gravitational wave eventsdetected by LIGO [32–34] as well as seed supermassiveblack holes [35–37]. The mass window of 10 − M (cid:12) is particularly interesting in connection with signals ob-served by LIGO [37–43]. While a variety of constraintsexist for this PBH mass range (see Ref. [44] for review),they often rely on multiple assumptions and are subjectto significant uncertainties.In this work, we set new constraints on PBH abun-dance based on the lack of gas heating from PBH inter-actions with the interstellar medium (ISM). We considerseveral generic heating mechanisms, including dynami-cal friction, accretion disk emission as well as mass out-flows/winds from accretion disk. We then apply our anal-ysis to dwarf DM-rich galaxies, focusing on Leo T. Leo Tis a transitional object between a dwarf irregular galaxyand a dwarf spheroidal galaxy that has been well studiedand modeled theoretically. It has the desired properties,such as a low baryon velocity dispersion, making it a sen-sitive probe of PBH heating. While constraints on BHsinteracting with surrounding stars have been extensivelydiscussed [45–47], gas heating has not been considered indetail. Other constraints focused on the X-ray emission,but not the heating of the surrounding gas [48, 49]. ISM heating has been used to constrain particle DM candi-dates [50–52], which have different heating mechanismswith a different velocity dependence compared to PBHs.The accretion of gas onto freely-floating BHs has beenanalyzed in Ref. [53] and applied to PBHs in Ref. [49].Bondi-Hoyle-Lyttleton accretion results in the mass ac-cretion rate of [23, 54, 55]˙ M = 4 πr B ˜ vρ = 4 πG M nµm p ˜ v , (1)where M is the PBH mass, r B = GM/ ˜ v is the Bondiradius, µ is the mean molecular weight, n is the ISM gasnumber density, m p is the proton mass and ˜ v ≡ ( v + c s ) / . Here, v is the PBH speed relative to the gasand c s is the temperature-dependent sound speed in gas,which we take to be c s ∼
10 km/s [49].The accretion rate can be related to the bolometricemission luminosity as L = (cid:15) ( ˙ M ) ˙ M , with a scaling ofa radiative efficiency (cid:15) ( ˙ M ) describing different accretionregimes. The Eddington accretion rate, assuming a char-acteristic radiative efficiency of (cid:15) = 0 . , is defined interms of the Eddington luminosity ˙ M Edd = L Edd /(cid:15) c .A convenient parameter for characterizing the accretionflow is ˙ m = ˙ M/ ˙ M Edd .With a sufficient angular momentum, the infalling gascan form an accretion disk around the BH. The angularmomentum necessary for a disk formation can be sup-plied by perturbations in the density or the velocity ofthe accreting gas. For a Schwarzschild BH, the inner The radiative efficiency (cid:15) can vary from 0.057 for a non-rotating Schwarzschild BH to 0.42 for an extremal Kerr BH (seee.g. Ref. [56]). a r X i v : . [ a s t r o - ph . C O ] J u l radius of the disk is taken to be the innermost stable cir-cular orbit (ISCO) of a test particle r ISCO = 3 r s , where r s = 2 GM/c is the Schwarzchild radius. Following thearguments of Refs. [49, 53], we have confirmed that anaccretion disk always forms for our parameters of inter-est.If PBHs constitute a fraction f PBH of the DM, the totalnumber of PBHs of mass M within a volume V is N PBH ( M ) = f PBH ρ DM VM , (2)where ρ DM is the DM density, assumed to be approx-imately constant. We assume a monochromatic PBHmass function for definiteness and for presenting our re-sults in the form of a differential exclusion plot. The ve-locity of PBHs contributing to the DM can be describedby a Maxwell-Boltzmann distribution f v ( v ) = r π v σ v exp (cid:18) − v σ v (cid:19) , (3)where σ v is the velocity dispersion in a given system.A distribution in gas number density f n ( n ) can also beintroduced, as in Refs. [49, 53].For a gas system in thermal equilibrium, the totalamount of heating by PBHs of mass M is H tot ( M ) = N PBH ( M ) H ( M ) (4)= Z n max n min Z v max v min dndv df n dn df v dv H ( M, n, v ) , where df n /dn is the gas density distribution, df v /dv isthe PBH relative speed distribution and H ( M, n, v ) isthe amount of heat deposited into the system from a sin-gle PBH. Here, H represents the cumulative contributionfrom all heating processes. For photon emission and out-flows we perform an additional integration to treat theabsorption efficiency. For gas of approximately constantdensity, one can replace df n /dn by a delta function.First, we consider gas heating due to photon emissionfrom accretion. Emission in the X-ray band generallyconstitutes the dominant contribution and it becomesmore efficient at high mass accretion rates.Photon emission from accretion depends on the accre-tion flow. To characterize the accretion flow, we fol-low the scheme outlined in Ref. [57] and assume thatthe accretion flow results in a (geometrically) thin diskfor ˙ m > ˙ M = 0 . α . The thin α -disk is the so-calledstandard disk [58], where α ∼ . : synchrotron radiation, inverse Compton scat-tering and bremsstrahlung. To describe the ADAF spec-trum, we employ approximate analytic expressions ob-tained in Ref. [62] in combination with the updated val-ues for the phenomenological input parameters consistentwith recent numerical simulations and observations [57].We take the ratio of direct viscous heating to electronsand ions δ = 0 . − .
5, and the ratio of gas pressure tototal pressure β = 10 / m ∼
1, because suchhigh accretion rates are not achieved for PBH masses andgas densities that we discuss.Emitted photons heat the ISM. Hydrogen gas is op-tically thin to radiation below the ionization thresholdof E i = 13 . E i .If the medium is optically thick, the photons areabsorbed, and most of their energy is deposited asheat. For absorption of photons with E > E i , weuse the photo-ionization cross-section [63, 64] σ ( E ) = σ y − (cid:16) y (cid:17) − , where y = E/E , E = 1 / E i and σ = 6 . × − cm . The optical depth of a gas sys-tem of size l and density n is τ ( n, E ) = σ ( E ) nl . Above30 eV, we use the combined attenuation length data fromFig. (32.16) of Ref. [65]. The resulting heating power is H phot ( M, n, v ) = Z E max E i L ν ( M, n, v ) (cid:0) − e − τ (cid:1) dν , (5)where L ν ( ν ) is the luminosity for the corresponding pho-ton emission process. For both the ADAF and thin diskregimes, the emission spectrum is exponentially decreas-ing at high energies, and we evaluate the integral up tothe maximum energy E max = ∞ .The second contribution to gas heating that we con-sider is dynamical friction due to gravitational interac-tions of traversing PBHs with the surrounding medium.Dynamical friction can be described as work done by the“gravitational drag” force F dyn (see e.g. [66, 67]). The We neglect additional possible contribution of the synchrotronradiation from non-thermal electrons, which is present in ADAFmodels of Sgr A ∗ [61]. resulting power deposited as heat is H dyn = F dyn v = − πG M ρv I , (6)where G is the gravitational constant, ρ is the gas den-sity and I is a velocity-dependent geometrical factor thatdiffers if the medium is collisionless or not [66, 67]. Wehave confirmed that the effect of the dynamical frictionon the PBH velocity is small.As a third heating component, mass outflows (winds)composed of protons can also contribute and they are ex-pected to be significant for hot accretion flows [57]. Incontrast to jets , the outflows are not highly relativis-tic and cover a wider angular distribution. The out-flows reduce the accretion rate at smaller radii and canbe approximately modelled by a self-similar power-lawform [68] ˙ M out ( r ) = ˙ M in ( r out ) (cid:18) rr out (cid:19) s , (7)where r out is the outer radius and the real index s ,0 ≤ s < s in the range0 . − . r out over a wide range of values, from 100 r s [70] to r B [71].The resulting outgoing wind has a velocity that is a frac-tion f k ’ . − . v ( r ) ’ f k p GM/r .We note that additional considerations regarding detailsof accretion may reduce emission efficiency (e.g. feed-back), but we do not expect this to be very significant.To evaluate how much energy is deposited into the gassystem from streaming outflow protons, we convolve theproton emission with the heat generated per proton ∆ E .The total heat deposited in the gas system is H out = Z r out r in ∆ Eµm p d ˙ M out dr dr , (8)where ∆ E = Z dEdx dx ’ min( E, nS ( E ) r max ) (9)takes into account energy losses due to the proton stop-ping power dE/dx = nS ( E ) adopted from Ref. [74] (seetheir Fig. 9). Here, r max is taken to be the size of the gassystem.We demonstrate our analysis by applying it to dwarfgalaxies, focusing on the Leo T dwarf galaxy. We stress, As jets are typically associated with Kerr black holes, they wouldrequire a separate treatment and we do not consider them here. however, that our methods are general and can be readilyapplied to other systems as well. To constrain the PBHmass fraction f PBH , we consider the balance between theheating and cooling processes of the gas system. Ourapproach to set the limits is similar to that used for par-ticle DM [50–52], but the heating mechanisms and thepreferred gas systems are different in our case.For simplicity, we ignore the contribution of naturalheating sources (e.g. stellar radiation), and hence ourbounds are conservative. Requiring thermal equilibrium,we only consider gas systems that are expected to beapproximately stable on sufficiently long timescales τ sys .Hence, the characteristic time over which the gas sys-tem remains steady must be greater than the coolingtimescale of the gas τ therm , i.e. τ sys (cid:29) τ therm = 3 nkT / C ,where k is the Boltzmann constant and ˙ C is the gas cool-ing rate per volume.Gas temperature exchange is a complex process, and adetailed analysis involving a full chemistry network canbe performed using numerical methods [82]. For the pa-rameters of interest, we employ approximate results ob-tained in Ref. [52]. For hydrogen gas, the cooling rate isgiven by ˙ C = n [Fe/H] Λ( T ) , (10)where [Fe/H] ≡ log ( n Fe /n H ) gas − log ( n Fe /n H ) Sun isthe metallicity, and Λ( T ) ∝ [Fe / H] is the cooling func-tion. Fitting numerically to the results of Ref. [82] li-brary, one can obtain Λ( T ) = 2 . × − T . , valid for300 K < T < H tot = N PBH H ( M ) = f PBH ρ DM V H ( M ) /M given by Eq. (4),where H ( M ) is the average heat generated from one PBHof mass M , should be less than the total cooling ˙ CV .This yields a condition on the PBH abundance that weuse to set our limits: f PBH < f bound = M ˙ Cρ DM H ( M ) . (11)We note that gas heating can be used to set a limiton the PBH abundance only if it is statistically likelyfor the gas system to harbor PBHs. If the PBH num-ber density is so low that, on average, a gas sys-tem of the size r sys contains fewer than one PBH, i.e., f PBH ρ DM (4 πr / /M <
1, such a system cannot beused for our purposes. We, therefore, set a limit only aslong as f bound > M πr ρ DM . (12)The gas in the inner region of Leo T, at a radius r (cid:46)
350 pc from its center, is dominated by atomic hydrogen,while the gas outside is highly ionized [83]. Since the freeelectrons in the ionized region cool very efficiently [82],we limit our analysis to the central region of Leo T. From − M ( M (cid:12) ) − − − − − f a ll o w e d − M ( M (cid:12) ) − − − − − f a ll o w e d P XRB DFI Ly- α S LSSX/R
FIG. 1.
Left:
Constraints from Leo T on the fraction of DM in PBHs, for a monochromatic mass function, derived fromconsiderations of only photon emission (red), dynamical friction (green), mass outflows (blue), as well as combined heating(dashed black). The reach of the constraints is bounded by the diagonal (solid black) line from the condition of Eq. (12). Theuncertainty in the emission and outflow input parameters leads to the uncertainty in the corresponding constraint (upper andlower dashed black lines).
Right:
Constraints from the Leo T dwarf galaxy on the PBH gas heating are shown in blue. Thelight blue shaded band denotes the variation in the PBH emission parameters. Other existing constraints are shown by dashedlines, including Icarus [75] (I) in purple, Planck [76] (P) in yellow, X-ray binaries [49] (XRB) in green, dynamical friction ofhalo objects [45] (DF) in red, Lyman- α [77] (Ly- α ) in maroon, combined bounds from the survival of astrophysical systems inEridanus II [78], Segue 1 [79], and disruption of wide binaries [80] (S) shown in magenta, large scale structure [21] (LSS) incyan, and X-ray/radio [81] (X/R) in brown. the model of Ref. [83], the hydrogen gas density is foundto vary from ∼ . − in the center to ∼ .
03 cm − at r = 350 pc. Both the cooling and heating rates scaleroughly as n , so we approximate the gas density to bea constant n = 0 .
07 cm − in the inner region. Similarly,the DM mass density drops from ρ DM ’ / cm atthe center to 2 GeV / cm at r = 350 pc, which we ap-proximate to be as a constant value of 3 GeV / cm . Thehydrogen gas has a dominant non-rotating warm com-ponent with a velocity dispersion of σ g = 6 . / s and T ’ σ v = σ g . The soundspeed is taken to be c s = 9 km / s from the adiabaticformula with T ’ nr sys = 7 . × cm − .We adopt the gas metallicity to approximately follow thestellar one , [Fe/H] ’ − C = 2 . × − erg cm − s − .In Fig. 1 we display the resulting limits from gas heat-ing in Leo T on PBHs contributing to DM, along with This is accurate to factor of few. other existing constraints.In summary, we have presented a new constraint on theabundance of PBHs in the intermediate ∼ − M (cid:12) mass range, which is of great interest in connection withthe LIGO gravitational waves events, as well as the lackof early seeds for supermassive black holes. PBH inter-actions with ISM result in the heating of gas, which wehave used to set the limit. We considered several genericheating mechanisms, including the photon emission fromaccretion, dynamical friction, and mass outflows/winds.Applied to the Leo T dwarf galaxy, our analyses yielda new constraint in a broad range of the PBH masses, M PBH ∼ O (1) M (cid:12) − M (cid:12) . This is a novel type ofa constraint, which was not previously considered forPBHs. Unlike some existing constraints, our limit doesnot depend on the cosmological history, which makes ita robust, independent test of PBHs in the intermediate-massive PBH mass-range. Our analysis can be readilyapplied to other systems. Acknowledgements. - The work of G.B.G., A.K., V.T.and P.L. was supported in part by the U.S. Departmentof Energy (DOE) Grant No. DE-SC0009937. This workwas supported in part by the MEXT Grant-in-Aid forScientific Research on Innovative Areas (No. 20H01895for K.H.). Y.I. is supported by JSPS KAKENHI GrantNumber JP18H05458, JP19K14772, program of Lead-ing Initiative for Excellent Young Researchers, MEXT,Japan, and RIKEN iTHEMS Program. A.K. and Y.I.are also supported by the World Premier InternationalResearch Center Initiative (WPI), MEXT, Japan. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] ∗∗ [email protected][1] Y. B. Zel’dovich and I. D. Novikov, Sov. Astron. , 602(1967).[2] S. Hawking, Mon. Not. Roy. Astron. Soc. , 75 (1971).[3] B. J. Carr and S. W. Hawking, Mon. Not. Roy. Astron.Soc. , 399 (1974).[4] J. Garcia-Bellido, A. D. Linde, and D. Wands, Phys.Rev. D54 , 6040 (1996), arXiv:astro-ph/9605094 [astro-ph].[5] M. Yu. Khlopov, Res. Astron. Astrophys. , 495 (2010),arXiv:0801.0116 [astro-ph].[6] P. H. Frampton, M. Kawasaki, F. Takahashi, and T. T.Yanagida, JCAP , 023 (2010), arXiv:1001.2308[hep-ph].[7] M. Kawasaki, A. Kusenko, Y. Tada, and T. T. Yanagida,Phys. Rev. D94 , 083523 (2016), arXiv:1606.07631 [astro-ph.CO].[8] B. Carr, F. Kuhnel, and M. Sandstad, Phys. Rev.
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