Spins of primordial binary black holes before coalescence
PPrepared for submission to JCAP
Spins of primordial binary black holesbefore coalescence
K.A. Postnov, a , b , c , N.A. Mitichkin b a Sternberg Astronomical Institute, M.V. Lomonosov Moscow State University,13, Universitetskij pr., 119234, Moscow, Russia b Faculty of Physics, M.V. Lomonosov Moscow State University,Leninskie Gory, 1, 119991, Moscow, Russia c Department of Physics, Novosibirsk State University,Pirogova 2, 630090, Novosibirsk, RussiaE-mail: [email protected], [email protected]
Abstract.
Primordial stellar-mass black holes, which may contribute to dark matter and tothe observed LIGO binary black hole coalescences, are expected to be born with very lowspins. Here we show that accretion mass gain by the components of a primordial black holebinary from the surrounding matter could lead to noticeable spins of the components priorto the coalescence provided high initial orbital eccentricities. Corresponding author. a r X i v : . [ a s t r o - ph . H E ] J un ontents The discovery of coalescing binary black holes (BHs) heralded the advent of gravitationalwave (GW) astronomy [1]. Presently, the GWTC-1 catalog of binary coalescences detected byLIGO/Virgo GW interferometers includes 10 BH+BH binaries and one NS+NS (GW170817)binary [2]. A statistical analysis of properties of coalescing binary BHs [3] suggests that thespin distribution of BHs prior to coalescence favors low spins of the components .The origin of the observed BH binaries is not fully clear. While the evolution of massivebinary systems [5–7] is able to reproduce the observed masses and effective spins of theLIGO BH+BH sources [8–11], the alternative (or additional) mechanisms of the binary BHformation is not yet excluded. These channels include, in particular, the dynamical formationof close binary BH in dense stellar clusters [12, 13] or coalescences of primordial black holebinaries which can constitute substantial fraction of dark matter [14–20]. Primordial BHsmay form clusters (see [21] for a review) facilitating the formation of binary BHs.In this note, we focus on the last possibility in order to understand whether primordialbinary BHs formed in the early Universe can have noticeable spins before the coalescence.Originally, the spins of primordial BHs should be close to zero (at a percent level at most,see e.g. recent studies [22, 23]). However, when in a binary system, accretion of matterwill inevitably bring angular momentum, and the components of a binary BH should acquirespins.The spin of a BH with mass M and angular momentum J is characterized by thedimensionless parameter a = J /( GM / c ) , where G and c are the Newtonian gravity constantand speed of light, respectively. Below we will use geometrical units G = c = . We willmeasure masses in solar mass units, M (cid:12) = × g, m = M / M (cid:12) , so that the length unit is [ cm ] = / × − m , the time unit is [ s ] = × m , etc.It is easy to estimate the final spin of an initially Schwarzschild BH. Assuming that noaccreted mass ∆ M = M f − M is radiated away, the BH spin after acquiring mass ∆ M reads[24]: a ∗ = (cid:114) (cid:18) M M f (cid:19) − (cid:115) (cid:18) M M f (cid:19) − . (1.1) An independent analysis of LIGO O1 data discovered one more possible BH+BH binary, GW151216,which may have rapidly spinning aligned components, but with a low astrophysical probability ∼ . [4]. – 1 –his formula is valid insofar as M f / M < √ . For a larger final BH mass, a ∗ = a ∗ max = (more precisely, a ∗ (cid:39) . , if one takes into account photon drag from accretion-generatedradiation, [25]). If ∆ M (cid:28) M , the acquired BH spin is a ∗ (cid:39) /√ ( ∆ M / M ) .It is easy to estimate the accretion mass gain for a single BH. Suppose it is immersed in amedium with sound velocity c s . Typically, in the interstellar medium c s ∼ √ T (cid:39) − (cid:112) T / or less (here T is the temperature of the medium). Assuming a Bondi-Hoyle-Lyttleton accre-tion onto the BH, we find ∆ M / M ≈ π ρ m /( v + c s ) / × t , (1.2)where ρ is the density of the medium, v is the proper velocity of the BH relative to themedium, t is the duration of the accretion. For example, for the typical ISM density ρ ∼ − g cm − = ( / ) × − m − and a maximum possible Hubble time t = t H = × s = × m , by neglecting the BH motion, from Eq. (1.2) we obtain ∆ M / M (cid:39) . − m (cid:28) and the final spin a ∗ (cid:39) . ∆ M / M (cid:39) . m . For a − M (cid:12) BH this would give a noticeablevalue but it is hard to measure the mass and spin of a single BH.The situation is less certain for the initially non-rotating components of a binary BHthat is able to coalesce over the Hubble time. Below we calculate the accretion mass gain bythe components of such a binary and show that the acquired spins can be interesting only ifthe initial orbital eccentricity of the binary is sufficiently large.
Consider a binary system consisting of two point-like masses m , m = m / q ( q is the binarymass ratio). The total mass is M = m + m = m ( + q ) , the orbital period T is found fromthe 3d Kepler’s law π / T = M / a , where a is the orbital semi-major axis. Let us start with the simplest case of a circular orbit. For typical BH+BH binares with m ∼ − , orbital velocities even at the maximum initial separations allowing for thecoalescence over the Hubble time are much larger than the ISM sound velocity, so we willneglect c s in the Bondi-Hoyle-Lyttleton formula. For the i -th component ( i = , , j = − i )moving with the velocity υ i , the mass accretion rate (see Section 3 for the discussion of thenumerical coefficient) is (cid:219) M i = π ρ m υ i = π ρ m i a / M / m j , (2.1)where we have used the expression for the Keplerian orbital velocity of the i -th component υ i = (cid:113) m j / aM . The binary loses the energy and angular momentum due to emission ofgravitational waves, and during the time before the coalescence the mass gain by the -stcomponent (for definiteness) will to good accuracy read ∆ M = t ∫ (cid:219) M dt = ∫ a dM dt dtda da , (2.2)where the initial orbital separation a of the binary is uniquely determined from the GW-driven coalescence time t = a Mm m (2.3)– 2 – M /M sun -9 -8 -7 -6 -5 -4 l og ( M / M ) Circular orbits q = 6q = 3q = 1q = 1/3q = 1/6 M/M ~ M
Figure 1 . The fractional accretion mass gain by a coalescing binary BH system in a circular orbitover the Hubble time in a cold medium with density 1 cm − . and dt / da is found from the quadrupole GW formula for a circular binary system: dt = − a m m M da . (2.4)After taking the integral in Eq. (2.2) and substituting a through t from Eq. (2.3), we arriveat: ∆ M M (cid:12)(cid:12)(cid:12)(cid:12) = π ρ M / a / m = (cid:18) (cid:19) / π ρ t / H m / q / ( + q ) / . (2.5)The plot of ∆ M / M as a function of m is shown in Fig. 1 for different mass ratios q = m / m for the fiducial ISM density 1 g cm − . Clearly, the effect increases both with m and q but even for large q > (i.e., when we consider the mass gain by the heaviestbinary component) is desperately small to enable astrophysically interesting BH spins, evenfor larger densities. The case of initially elliptical orbits is more interesting. Elliptical orbits of binary BHs arepossible in both the dynamical channel of binary BH formation in dense stellar clusters andfor primordial BHs.Consider a Keplerian binary in an elliptical orbit with eccentricity e . The mass accretedover one orbital revolution with period T reads: δ M = T ∫ (cid:219) M dt = π ∫ (cid:219) M (cid:18) dtd θ (cid:19) d θ = π ρ q ( + q )( − e ) a I ( e ) (2.6)– 3 –here I ( e ) = π ∫ (cid:18) ( + e cos θ ) ( + e cos θ + e ) / (cid:19) − d θ . (2.7)Here we have used the expressions for the orbital velocity υ i ( θ ) = (cid:112) M / a ( − e )( + e cos θ + e ) / ( m j / M ) , the orbital angular momentum conservation r d θ / dt = (cid:112) Ma ( − e ) and r = a ( − e )/( + e cos θ ) for the Keplerian motion.The mass accretion rate averaged over one orbital period T is (cid:104) (cid:219) M (cid:105) = δ M T , T = π (cid:115) a M ( + q ) . (2.8)In a way similar to the circular case, we find the accretion mass gain by the 1-st componentof a binary BH with initial orbital eccentricity e coalescing over the Hubble time: ∆ M ( e ) = ∫ e (cid:104) (cid:219) M (cid:105) (cid:18) dtde (cid:19) de , (2.9)where de / dt reads [26] dedt = − m m Me a ( − e ) / (cid:18) + e (cid:19) . (2.10)For the coalescing binary, the expression a ( e ) reads [26] a ( e ) = C e / ( − e ) (cid:18) + e (cid:19) / , (2.11)where the constant C is determined by substituting a ( e ) into the formula for the binarycoalescence time t in the case of elliptical orbit [26]: t = a ( e ) Mm m ( − e ) e / (cid:18) + e (cid:19) − / I ( e ) , I ( e ) = e ∫ (cid:18) + e (cid:19) / e / ( − e ) / de . (2.12)Substituting Eq. (2.8) into Eq. (2.9) with an account of Eq. (2.11) and Eq. (2.12), wefinally obtain the accretion mass gain for the elliptical orbit: ∆ M M (cid:12)(cid:12)(cid:12)(cid:12) e = (cid:18) I ( e ) (cid:19) / ρ t / H m / q / ( + q ) / e ∫ I ( e ) e / (cid:18) + e (cid:19) / de . (2.13)It can be written in the form ∆ M M (cid:12)(cid:12)(cid:12)(cid:12) e = ∆ M M (cid:12)(cid:12)(cid:12)(cid:12) · K ( e ) , (2.14)where the enhancement factor K ( e ) reads: K ( e ) = π (cid:18) I ( e ) (cid:19) / e ∫ I ( e ) e / (cid:18) + e (cid:19) / de . (2.15)– 4 – e l og ( K ( e )) K(e) = M e / M -log(1-e ) l og ( K ( e )) Figure 2 . Left: The enhancement factor K ( e ) of the fractional mass accretion gain in elliptical orbitby a component of a coalescing binary BH relative to the circular case as a function of the orbiteccentricity e . Right: log K ( e ) - (− log ( − e ) ) plot manifestly showing the asymtptotic power-lawbehaviour at large e (cid:39) , K ( e ) ∼ ( − e ) − . . Clearly, the enhancement factor with respect to the circular orbit is a function of the initialorbital eccentricity only, and is shown in Fig. 2, left panel. In the e → limit, I ( e ) ∼ π and I ( e ) ∼ ( / ) e / , and K e → . In the more interesting limit of large eccentricities e (cid:39) , we find from numerical integration I ( e ) ∼ ( − e ) − . . Therefore, in this limit K ( e ) ∼ ( − e ) / × ( − e ) − . = ( − e ) − . . This power-law asymptotic is clearly seenon the plot log K ( e ) − log 1 /( − e ) shown in Fig. 2, right panel. Therefore, in the limit ofhigh initial orbital eccentricities, we find approximately ∆ M M (cid:12)(cid:12)(cid:12)(cid:12) e ≈ − (cid:16) ρ − g cm − (cid:17) (cid:16) M M (cid:12) (cid:17) / q / ( + q ) / (cid:18) . − e (cid:19) . ≈ − (cid:16) ρ − g cm − (cid:17) (cid:16) M M (cid:12) (cid:17) / q ( + q ) (cid:18) . − e (cid:19) . . (2.16)In the last equality, we have introduced the chirp mass of the binary system M ≡ ( M M ) / / M / = M ( q ( + q )) − / that is directly read off the chirp GW signal from binary coalescences. Notethat Eq. (2.16) does not violate the isolated BH mass gain estimate, ∆ M / M (cid:39) . × − m for the orbital eccentricities − e > . ( m / ) . [ q / ( + q ) / ] / . , i.e. e < . for m = and q = . This limit, however, is weaker than imposed by the validity of negligiblesound velocity in the Bondi-Hoyle-Lyttleton formula (see the next Section).It is seen that for eccentric orbits with e (cid:38) . (i.e. /( − e ) (cid:38) ) ) this factor canbring the mass accretion gain into astrophysically interesting region for BH spin, especiallyfor more massive component of a binary with large mass ratio q > . As an example,in Fig. 3 we show the fractional accretion mass gain by a BH with mass m = as afunction of the initial orbital eccentricity e for different binary mass ratios q . Formally,for the assumed ISM density, a noticeable spin of the primary BH component before thecoalescence, a ∗ ∼ . ( ∆ M / M ) , could be achieved only for very eccentric orbits with e (cid:39) .The effect is stronger for more massive BHs and higher surrounding densities.– 5 – e -8 -7 -6 -5 -4 -3 -2 -1 l og ( M / M ) Elliptical orbits (M =30M sun ) q = 6q = 3q = 1q = 1/3q = 1/6 Figure 3 . The fractional accretion mass gain by a M (cid:12) black hole in an elliptical binary as a functionof the initial orbital eccentricity e . The cold medium density is 1 cm − . In our analysis, we have neglected the sound velocity c s in the Bondi-Hoyle-Lyttleton formulaEq. (1.2). This needs to be justified in the case of strongly eccentric orbits because theaccretion rate is determined by the maximal of the orbital velocity and the sound velocity c s . The orbital velocity of star M at apastron is v a ( M ) = (cid:115) M ( − e ) a ( − e ) M M = (cid:115) M ( − e ) a ( + e ) q ( + q ) . (3.1)Then the condition v a ( M ) > c s can be written as − e > c s ( + e ) q ( + q ) (cid:18) aM (cid:19) . (3.2)Clearly, if the initial orbital eccentricity e satisfies this inequality, it will hold always truein the subsequent binary evolution due to GW losses. Making use of Eq. (2.12) to express a ( e )/ M , in the limit e → of interest here we find − e > c s q / ( + q ) / (cid:18) t H m (cid:19) / (cid:18) ( − e ) (cid:19) / (cid:18) (cid:19) / [ I ( e → )] − / Noticing that I ( e → ) ≈ (cid:18) (cid:19) / ( − e ) − / , plugging t H / m for the fiducial m = ,after making arrangements, we arrive at the inequality − e , max ( m ) > . (cid:16) c s − (cid:17) / (cid:16) m (cid:17) − / ( q ( + q )) / (3.3)– 6 – -1 @ e ff Figure 4 . Maximum possible effective spin of a coalescing BH with chirp mass M = M (cid:12) that hadaccreted matter from cold ISM ( c s = − ) with particle number density 1 cm − . that restricts the applicability of our approximation. Proceeding exactly in the same way asfor m , we obtain the restriction for the initial orbital eccentricity for m : − e , max ( m ) > . (cid:16) c s − (cid:17) / (cid:16) m (cid:17) − / q − / ( + q ) / . (3.4)This limit for the initial binary eccentricity, e , max < − . for the fiducial parameters,leaves quite a room for a significant enhancement factor K ( e ) . To see this, let us estimate themaximum possible effective spin of a coalescing binary BH, χ eff = ( m a ∗ + m a ∗ )/ M , whichcan be inferred from GW observations [2]. Note that in our setup fully aligned BH spinsare expected. As a ∗ i ∼ ( ∆ M / M ) i , we need to calculate also the mass gain by the secondarycomponent, ( ∆ M / M ) . This is obviously done by substituting M → M in Eq. (2.16), i.e.– 7 –imply changing q → / q : ∆ M M ≈ − (cid:18) ρ − g cm − (cid:19) (cid:18) M M (cid:12) (cid:19) / q − ( + q ) (cid:32) . − e (cid:33) . . (3.5)If there would be no initial eccentricity restrictions (e.g., in the limit of a cold mediumwith very low sound velocities c s ), the effective spin of the coalescing BH binary with M = M (cid:12) would be χ eff = q + q a ∗ + + q a ∗ ≈ . × − (cid:16) ρ − g cm − (cid:17) (cid:16) M M (cid:12) (cid:17) / (cid:18) . − e (cid:19) . ( + q )( q + q − ) > . × − (cid:16) ρ − g cm − (cid:17) (cid:16) M M (cid:12) (cid:17) / (cid:18) . − e (cid:19) . (3.6)for any mass ratio q because the function f ( q ) = ( + q )( q + q − ) reaches the minimum f ( q min ) = at q min = . However, taking into account the initial eccentricity limits, Eq. (3.3)and Eq. (3.4), due to finite sound velocity of the medium, we find χ eff < χ eff , max = q + q a ∗ ( e , max ( m )) + + q a ∗ ( e , max ( m )) ∝ M . c − . s Ψ ( q ) , (3.7)where Ψ ( q ) is a symmetric function of the mass ratio with maximum at q = that canbe readily calculated by substituting the factors − e , max ( m , ) for a [Eq. (3.3)] and a [Eq. (3.4)], respectively, into Eq. (3.6). The plot of χ eff , max for the fiducial values c s = − , ρ = − g cm − and M = M (cid:12) is shown in Fig. 4. Roughly, we can take χ eff , max (cid:39) . ( ρ / − g cm − )(M/ M (cid:12) ) . ( c s / − ) − . for any mass ratio . < q < . This esti-mate shows that coalescing primordial binary BHs can acquire measurable values of χ eff ∼ afew percents due to the accretion mass gain in the galactic ISM. Here we presented the results of calculation of the mass gain by components of a BH+BHbinary system, which can coalesce over the Hubble time, due to the Bondi-Hoyle-Lyttletonaccretion from a relatively cold ( c s ∼ a few km s − ) surrounding medium. The angularmomentum by the accreted material can spin up an the initially Schwarzschild BH up tonoticeable values if the initial binary orbit had a high eccentricity e ∼ . Such eccentricitiesare in principle possible in primordial BH binaries that can be formed in the early Universeand coalesce at the present time.In our calculations we have assumed the simplest formula for the accretion rate onto abinary components, which is, of course, a rough estimate. For example, recent 3D simula-tions of Bondi-Hoyle accretion [27] suggest an orbital-averaged reduction of the Bondi-Hoyleaccretion efficiency by a factor of ∼ / in circular binaries. However, for our purposes thisreduction is not very important in view of much more uncertain density of matter surroundingthe coalescing binary. This density can be an order of magnitude higher or smaller dependingon the location of the binary in a galaxy or in the galactic halo. Moreover, primordial binaryBHs are thought to have high velocity dispersion ∼
300 km s − , which drastically reduces theefficiency of matter accretion. Still, some BH binaries could have rather small velocities and– 8 –an be found inside the galactic ISM. Therefore, in principle, the components of such BHbinaries can acquire noticeable aligned spins prior to the coalescence.The accretion-gained spins should be higher in more massive binaries ( a ∗ ∼ ∆ M / M ∼ m / , see Fig. 1). Interestingly, the most massive LIGO binary BH, GW170729 [2] and(not very reliable) recently reported BH binary GW151216 [4] show appreciable and likelyaligned spins of the components. Of course, presently it is difficult to separate different for-mation channels of the observed coalescing binary BHs, and increased statistics of binary BHcoalsecences in the forthcoming O3 LIGO/Virgo run could help disentangling various scenar-ios of binary BH formation and evolution. However, we stress that even primordial binaryBHs could have appreciable aligned spins before the coalescence due to matter accretion ingalaxies. Acknowledgments
We thank the anonymous referee for constructive criticism and Prof. A.D. Dolgov for en-couraging discussions. The work of KAP is supported by RSF grant 19-42-02004. NAMacknowledges support by the Program of development of M.V. Lomonosov Moscow StateUniversity (Leading Scientific School ’Physics of stars, relativistic objects and galaxies’).
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