Neutron Stars Harboring a Primordial Black Hole: Maximum Survival Time
aa r X i v : . [ a s t r o - ph . H E ] J a n Neutron Stars Harboring a Primordial Black Hole: Maximum Survival Time
Thomas W. Baumgarte and Stuart L. Shapiro
2, 3 Department of Physics and Astronomy, Bowdoin College, Brunswick, ME 04011 Department of Physics, University of Illinois at Urbana-Champaign, IL 61801 Department of Astronomy and NCSA, University of Illinois at Urbana-Champaign, IL 61801
We explore in general relativity the survival time of neutron stars that host an endoparasitic,possibly primordial, black hole at their center. Corresponding to the minimum steady-state Bondiaccretion rate for adiabatic flow that we found earlier for stiff nuclear equations of state (EOSs),we derive analytically the maximum survival time after which the entire star will be consumed bythe black hole. We also show that this maximum survival time depends only weakly on the stiffnessfor polytropic EOSs with Γ ≥ /
3, so that this survival time assumes a nearly universal value thatdepends on the initial black hole mass alone. Establishing such a value is important for constrainingthe contribution of primordial black holes in the mass range 10 − M ⊙ . M . − M ⊙ to thedark-matter content of the Universe. Primordial black holes (PBHs) that may have formedin the early Universe (see, e.g., [1, 2]) have long been con-sidered candidates for contributing to, if not accountingfor, the mysterious and elusive dark matter (see, e.g., [3],as well as [4] for a recent review). Constraints on thePBH contribution to the dark matter have been estab-lished by a number of different observations. Sufficientlysmall PBHs ( M . × g) would have evaporateddue to Hawking radiation [5] within less than a Hubbletime, while, for larger black-hole masses, different typesof observations have resulted in limits for different massranges (see, e.g., [6, 7] and references therein; see also[8] for a review of constraints arising from gravitational-wave observations).One compelling constraint on the PBH contributionto the dark matter in the mass range 10 − M ⊙ . M . − M ⊙ (which currently is not well-constrained byother observations; see, e.g., [4]) results from the factthat PBHs can be captured by stars, and would then ac-crete and swallow these stars (see, e.g., [1, 9]). This pro-cess is particularly efficient for capture by neutron stars(see, e.g., [6, 10]), so that the existence of neutron-starpopulations provides a limit on the density of PBHs, andhence on their contribution to the dark-matter contentof the Universe.Evidently, the above argument can provide constraintson PBHs only if the capture and subsequent accretion-driven destruction of the neutron star proceeds on time-scales shorter than the age of the oldest neutron-starpopulations; it therefore hinges on reliable time-scaleestimates for these processes. Estimates for the cap-ture of BPHs by neutron stars via gravitational focusing,followed by the dissipation of PBH kinetic energy viadynamical friction, accretion, surface and gravitationalwaves that lead to the settling down of the PBH near thecenter of the neutron star, have recently been revisitedby [10]. The subsequent accretion rate of the star by theendoparasitic black hole is often estimated analytically(see [11, 12] for numerical simulations) by adopting the spherical, steady-state, Bondi accretion formula,˙ M = 4 πλ M ρ a (1)for adiabatic flow ([13]; see also [14] for a textbook treat-ment, including its relativistic generalization). Here ρ and a are the “asymptotic” rest-mass density and thesound speed, respectively, measured at large distancesfrom the black hole, but, for small black holes, stillwell inside the neutron star’s nearly homogeneous centralcore. The mass of the gas within the accretion radius isassumed to be negligible in comparison with the blackhole mass. The parameter λ is a dimensionless “accre-tion eigenvalue”, which, for 1 ≤ Γ ≤ /
3, is a constant oforder unity.In a general relativistic formulation, the dot in (1) de-notes a derivative with respect to time as measured bya static observer far from the black hole but again wellinside the neutron star core. This time advances at a ratesomewhat shorter than that measured at infinity due tothe gravitational redshift from the core by a factor givenby the “lapse” function 0 . . α c < rest mass, which enhances the blackhole’s gravitational mass by a similar amount for strictlyadiabatic flow as long as the asymptotic internal energyof the gas is small in comparison to the rest-mass en-ergy. For accretion onto black holes, a relativistic treat-ment for Schwarzschild black holes shows that the re-quirement that the sound speed be less than the speed oflight demands that the flow pass through a critical point,yielding a unique value for λ [14]. Adopting a polytropicequation of state (EOS) P = Kρ Γ0 , (2)where K is a constant, Γ = 1 + 1 /n is the adiabaticindex and n is the polytropic index, that unique valuefor 1 ≤ Γ ≤ /
3, assuming a ≪
1, is λ = 14 (cid:18) − (cid:19) (5 − / − , (3)Typeset by REVTEXwhich is the value found in the Newtonian formulation.Note that here and throughout we adopt geometrizedunits with G = 1 = c .While the accretion timescale may be small comparedto the total capture and settling time for many blackhole masses, there is some uncertainty in determiningthe actual accretion timescale when applying the Bondiaccretion rate given by Eq. (1). We first observe that (1)depends on both ρ and a , which may vary from star tostar (even though their order of magnitude is probablysimilar for all neutron stars). More importantly, the ac-cretion eigenvalues λ are easily derived only for soft EOSswith 1 ≤ Γ ≤ /
3, as in Eq. (3). As a result, some au-thors (e.g. [10, 15]) have resorted to approximating theaccretion rate by adopting values for Γ such as Γ = 4 / &
2, forwhich (3) clearly does not apply. Finally, to make mat-ters worse, consider that the polytropic EOS (2) impliesthat the sound speed is given by a ≃ Γ Kρ Γ − (4)whenever P ≪ ρ , where ρ is the total mass-energy den-sity. This is the standard Newtonian relation. Inserting(4) into (1) results in˙ M ≃ πλM a (5 − / (Γ − . (5)We now observe that, for a →
0, (5) suggests that ˙ M becomes infinite whenever Γ > /
3, seemingly makingthis expression totally unreliable for stiff EOSs.In this short paper we clarify this issue. Building onour relativistic treatment of Bondi accretion for stiff poly-tropic EOSs (see [16]) we use our finding there that thereexists a finite, nonzero, minimum accretion rate wheneverΓ ≥ / maximum accretion time fora PBH to swallow a neutron star. This maximum accre-tion time depends only weakly on the stiffness for EOSswith Γ ≥ /
3, thereby providing a nearly universal es-timate, depending only on the initial black-hole mass,for the maximum time that a neutron star can surviveaccretion by an endoparasitic black hole.We first observe that, for Γ = 5 /
3, the accretion rate(5) becomes independent of the sound speed a . A rel-ativistic treatment shows that this value represents a minimum accretion rate , since relativistic corrections forhigher sound speeds a . M min = 4 π ¯ λ Γ / (Γ − M K / (Γ − , (6) for all 5 / ≤ Γ ≤
3. Here ¯ λ is defined as¯ λ = ¯ x (5 − / (2Γ − (cid:18) Γ − − − ¯ x III (cid:19) / (Γ − (1 + 3¯ x III ) / , (7)with ¯ x III = Γ − / − (12Γ − / / , (8)where we have adopted the notation of [16]. The exis-tence of the minimum accretion rate (6) results from thefact that, for Γ > / a , the accre-tion eigenvalues λ can be written as λ = ¯ λ a (3Γ − / (Γ − ,so that the dependence on a in (5) cancels out, therebyleaving ˙ M finite but non-zero. Note that (6) reduces to(5) for Γ = 5 /
3, in which case λ = ¯ λ = 1 / dMdt = ˙ M ≥ ˙ M min , (9)we can find the maximum time by integrating t max = Z M NS M dM ˙ M ≤ Z M NS M dM ˙ M min = κK / (Γ − Z M NS M dMM = κK / (Γ − (cid:18) M − M NS (cid:19) . (10)Here M is the initial black-hole mass, M NS is the initialneutron star mass, and we have defined κ = Γ / (Γ − π ¯ λ . (11)Assuming that M ≪ M NS we may neglect the last termin (10) and obtain t max = κ K / (Γ − M . (12)This maximum time is dominated by the initial phaseof accretion, when M and the accretion rate given byEq. 6 assume their lowest values. During this phase theaccretion proceeds in a quasistationary fashion and theneutron star maintains its initial state (e.g., Γ and K ) togood approximation.We can now determine the value of the constant K forany polytropic EOS by matching its dimensionless max-imum mass ¯ M max = K − / (2Γ − M max to the best cur-rent value value for the maximum mass of a nonrotating,isolated neutron star, M max . The quantity ¯ M max is ob-tained by setting K equal to unity and integrating theTolman-Oppenheimer-Volkoff equations (see [17, 18]) tofind the mass at the turning point along an equilibriumsequence of stars parametrized by their central density.We can therefore express K as K = (cid:18) M max ¯ M max (cid:19) − . (13) n Γ ¯ λ κ ¯ M max κ/ ¯ M n is the polytropicindex, Γ = 1 + 1 /n the adiabatic index, the coefficients ¯ λ and κ are defined in Eqs. (7) and (11), and ¯ M max = K − /n M max is the maximum dimensionless mass allowed for non-rotatingneutron stars. Note that the combination κ/ ¯ M , whichappears in the maximum accretion time (16), depends onlyweakly on Γ for stiff EOSs (with Γ &
2, say).
Inserting (13) into (12) we now obtain t max = 5 × − s κ ¯ M (cid:18) M max M ⊙ (cid:19) (cid:18) M ⊙ M (cid:19) , (14)where we have used that, in geometrized units, 1 M ⊙ ≃ × − s.We list values for the polytropic parameters appear-ing in (14) in Table I. Note that, for moderately stiffEOSs with n . . &
2, say, the factor κ/ ¯ M varies only very moderately, with values between 3 and4. Adopting the maximum of these values, κ ¯ M (cid:12)(cid:12)(cid:12)(cid:12) max ≃ , (15)we can rewrite (12) as t max ≃ × s (cid:18) M max M ⊙ (cid:19) (cid:18) − M ⊙ M (cid:19) . (16)We next observe that, according to Fig. 4 of [16], ac-cretion rates for Γ > / M max = 2 . M ⊙ . Thismass is consistent with the likely range of values esti-mated by several groups (see, e.g., [19–22]) using datafrom the binary neutron star merger event GW170817 de-tected by LIGO/Virgo in gravitational waves [23], as wellas from the counterpart gamma-ray burst GRB17081Aand kilonova AT 2017gfo. It is also consistent with the measurements of [24], which revealed the highest pulsarmass to date. We then may rewrite (16) as t max ≃ × s (cid:18) − M ⊙ M (cid:19) ≃ × s (cid:18) g M (cid:19) , (17)resulting in values that are remarkably close to thoseadopted by, for example, [10] (see their Eq. 39). Thisclose agreement, though reassuring, is rather coinciden-tal, as our value results from a detailed treatment ofBondi accretion for stiff EOSs in full general relativity.Regardless, results quite similar to (17) have been in-voked by previous investigators (see, e.g., [6, 10] and ref-erences therein) to constrain PBHs in the mass range10 − M ⊙ . M . − M ⊙ as dark-matter candidates.Corresponding to Eqs. (14) and (17), the minimumaccretion rate given by Eq. (6) can be evaluated to yieldinitially˙ M min = 2 × M ⊙ s ¯ M κ (cid:18) M ⊙ M max (cid:19) (cid:18) M M ⊙ (cid:19) . × − M ⊙ yr (cid:18) M − M ⊙ (cid:19) , (18)where, in the last step, we have again assumed (15) and M max = 2 . M ⊙ .To summarize, we provide estimates for the accretiontimes of black holes residing in neutron stars. While ourresults are very similar to previously adopted values, theyare based on a rigorous, relativistic treatment of sphericalBondi accretion for stiff EOSs. We also demonstrate thatthere exists a maximum accretion time , depending chieflyon the initial black hole mass alone, by which time theblack hole will have consumed the entire neutron star.We further argue that actual accretion times will notdiffer much from the maximum accretion time, so thatthe latter provides an approximate universal value forthe lifetime of a neutron star with a small black holeresiding at its center.While our discussion here centers on PBHs and theircontribution to the dark matter, the same estimates ap-ply to some alternative scenarios as well. Specifically,the neutron star could also capture other dark matterparticles which, under sufficiently favorable conditions,could form a high density object that collapses to forma small black hole in the neutron star interior and ulti-mately consume the entire star. Invoking observations ofneutron star populations, several authors have used thesearguments to derive constraints and bounds on dark mat-ter particles (see, e.g., [25–29]).Our arguments build on a number of assumptions, ofcourse. We assume that the accretion process is domi-nated by spherical, steady-state, adiabatic Bondi accre-tion, and we ignore effects of rotation (which seems justi-fied according to the findings of [11]; but see also [9, 15]).We also ignore radiation and radiation pressure, althoughthe trapping radius for photons [30] is likely very largeand radiation may be totally ineffective in holding backthe hyper-Eddington accretion that can arise here (see,e.g., [31]). Radiation processes, including the possiblerole of neutrinos, as well as the role of conduction, needto be probed further and the effects of fermion degen-eracy must be taken into account. The possible effectsof magnetic fields also have been ignored and should beinvestigated. We implicitly assume that the weak depen-dence of the accretion rates and times on the polytropicindex of the EOS indicate a similarly weak dependence onthe detailed properties of realistic nuclear EOSs. Therecould, in principle, be phase transitions and other effectsthat might occur in the accretion flow, but we note thatthe Bondi critical radius occurs rather close to the blackhole for typical cases (see Fig 2 in [16]). This meansthat the density and temperature in the accreting gas donot increase substantially above their core values beforethe gas is captured. In spite of these considerations, webelieve that our result provides an interesting limit onthe timescale for the demise of a neutron star by an en-doparasitic black hole, not least because it provides morerigorous justification for a key assumption used in con-straining the contribution of PBHs to the dark mattercontent of the Universe.It is a pleasure to thank Gordon Baym and ChloeRichards for interesting discussions. This work was sup-ported in part by National Science Foundation (NSF)grant PHY-2010394 to Bowdoin College, and NSFgrants PHY-1662211 and PHY-2006066 and NationalAeronautics and Space Administration (NASA) grant80NSSC17K0070 to the University of Illinois at Urbana-Champaign. [1] S. Hawking, Gravitationally collapsed objects of very lowmass, Mon. Not. R. Astron. Soc. , 75 (1971).[2] B. J. Carr and S. W. Hawking, Black holes in the earlyUniverse, Mon. Not. R. Astron. Soc. , 399 (1974).[3] G. F. Chapline, Cosmological effects of primordial blackholes, Nature (London) , 251 (1975).[4] B. Carr and F. K¨uhnel, Primordial Black Holesas Dark Matter: Recent Developments, An-nual Review of Nuclear and Particle Science , 10.1146/annurev-nucl-050520-125911 (2020),arXiv:2006.02838 [astro-ph.CO].[5] S. W. Hawking, Black hole explosions?,Nature (London) , 30 (1974).[6] F. Capela, M. Pshirkov, and P. Tinyakov,Constraints on primordial black holes asdark matter candidates from capture by neu-tron stars, Phys. Rev. D , 123524 (2013),arXiv:1301.4984 [astro-ph.CO].[7] F. K¨uhnel and K. Freese, Constraints onprimordial black holes with extended massfunctions, Phys. Rev. D , 083508 (2017),arXiv:1701.07223 [astro-ph.CO].[8] M. Sasaki, T. Suyama, T. Tanaka, and S. Yokoyama, Primordial blackholes—perspectives in gravitational wave astronomy,Classical and Quantum Gravity , 063001 (2018),arXiv:1801.05235 [astro-ph.CO].[9] D. Markovic, Evolution of a primordialblack hole inside a rotating solar-type star,Mon. Not. R. Astron. Soc. , 25 (1995).[10] Y. G´enolini, P. D. Serpico, and P. Tinyakov,Revisiting primordial black hole capture intoneutron stars, Phys. Rev. D , 083004 (2020),arXiv:2006.16975 [astro-ph.HE].[11] W. E. East and L. Lehner, Fate of a neutronstar with an endoparasitic black hole and implica-tions for dark matter, Phys. Rev. D , 124026 (2019),arXiv:1909.07968 [gr-qc].[12] C. B. Richards, T. W. Baumgarte, and S. L. Shapiro,Accretion onto small black holes at the centers of neutronstars (2021), in preparation.[13] H. Bondi, On spherically symmetrical accretion,Mon. Not. R. Astron. Soc. , 195 (1952).[14] S. L. Shapiro and S. A. Teukolsky, Black holes, whitedwarfs, and neutron stars : the physics of compact objects (Wiley Interscience, 1983).[15] C. Kouvaris and P. Tinyakov, Growth ofblack holes in the interior of rotating neu-tron stars, Phys. Rev. D , 043512 (2014),arXiv:1312.3764 [astro-ph.SR].[16] C. B. Richards, T. W. Baumgarte, and S. L. Shapiro,Relativistic Bondi accretion for stiff equations of state,arXiv:2101.08797 [astro-ph.HE] (2021).[17] J. R. Oppenheimer and G. M. Volkoff, On Massive Neu-tron Cores, Physical Review , 374 (1939).[18] G. M. Volkoff, On the Equilibrium of Massive Spheres,Physical Review , 413 (1939).[19] B. Margalit and B. D. Metzger, Con-straining the Maximum Mass of NeutronStars from Multi-messenger Observations ofGW170817, Astrophys. J. Lett. , L19 (2017),arXiv:1710.05938 [astro-ph.HE].[20] M. Shibata, S. Fujibayashi, K. Hotokezaka, K. Ki-uchi, K. Kyutoku, Y. Sekiguchi, and M. Tanaka,Modeling GW170817 based on numerical relativityand its implications, Phys. Rev. D , 123012 (2017),arXiv:1710.07579 [astro-ph.HE].[21] L. Rezzolla, E. R. Most, and L. R. Weih, UsingGravitational-wave Observations and Quasi-universalRelations to Constrain the Maximum Mass of Neu-tron Stars, Astrophys. J. Lett. , L25 (2018),arXiv:1711.00314 [astro-ph.HE].[22] M. Ruiz, S. L. Shapiro, and A. Tsokaros,GW170817, general relativistic magnetohydrody-namic simulations, and the neutron star max-imum mass, Phys. Rev. D , 021501 (2018),arXiv:1711.00473 [astro-ph.HE].[23] B. P. Abbott, LIGO Scientific Collaboration, andVirgo Collaboration, GW170817: Observationof Gravitational Waves from a Binary NeutronStar Inspiral, Phys. Rev. Lett. , 161101 (2017),arXiv:1710.05832 [gr-qc].[24] H. T. Cromartie, E. Fonseca, S. M. Ransom, P. B. De-morest, Z. Arzoumanian, H. Blumer, P. R. Brook, M. E.DeCesar, T. Dolch, J. A. Ellis, R. D. Ferdman, E. C.Ferrara, N. Garver-Daniels, P. A. Gentile, M. L. Jones,M. T. Lam, D. R. Lorimer, R. S. Lynch, M. A. McLaugh- lin, C. Ng, D. J. Nice, T. T. Pennucci, R. Spiewak, I. H.Stairs, K. Stovall, J. K. Swiggum, and W. W. Zhu, Rela-tivistic Shapiro delay measurements of an extremely mas-sive millisecond pulsar, Nature Astronomy , 72 (2020),arXiv:1904.06759 [astro-ph.HE].[25] I. Goldman and S. Nussinov, Weakly inter-acting massive particles and neutron stars,Phys. Rev. D , 3221 (1989).[26] A. de Lavallaz and M. Fairbairn, Neutron stars asdark matter probes, Phys. Rev. D , 123521 (2010),arXiv:1004.0629 [astro-ph.GA].[27] J. Bramante and T. Linden, Detecting DarkMatter with Imploding Pulsars in the Galac-tic Center, Phys. Rev. Lett. , 191301 (2014), arXiv:1405.1031 [astro-ph.HE].[28] J. Bramante and F. Elahi, Higgs portals topulsar collapse, Phys. Rev. D , 115001 (2015),arXiv:1504.04019 [hep-ph].[29] J. Bramante, T. Linden, and Y.-D. Tsai, Search-ing for dark matter with neutron star mergersand quiet kilonovae, Phys. Rev. D , 055016 (2018),arXiv:1706.00001 [hep-ph].[30] M. J. Rees, Accretion and the quasar phenomenon,Physica Scripta , 193 (1978).[31] K. Inayoshi, Z. Haiman, and J. P. Ostriker, Hyper-Eddington accretion flows on to massive blackholes, Mon. Not. R. Astron. Soc.459