Formation Redshift of the Massive Black Holes Detected by LIGO
FFormation Redshift of the Massive Black Holes Detected by LIGO
Razieh Emami ∗ and Abraham Loeb † Institute for Theory and Computation, Harvard University, 60 Garden Street, Cambridge, MA 02138, USA
We compare the event rate density detected by LIGO to the comoving number density of isolated stellarprogenitors and find a range for their formation redshift. Our limit depends on the threshold mass for making theblack holes (BHs) but only weakly on the metallicity of their progenitor. If 10% of all BHs are in coalescingbinaries, then enough progenitors have formed by 2 < z f < The detection of the gravitational waves (GWs) by the LaserInterferometer Gravitational Wave Observatory (LIGO) [1–6],from the merger of binary black holes (BBHs) ushered a newera of multi-messenger astronomy. In this
Letter we focus onthe events with BH masses above 20 M (cid:12) which are accordingto most recent data from the LIGO / Virgo Collaboration (LVC)[7, 8] are about 70 % of the detected events.The LIGO results prompted two theoretical challenges. Oneinvolves the large masses of the detected BHs and the secondinvolves their assembly into binaries that coalesce within theage of the universe. There are three main channels proposedto address these issues.. “
Dynamical formation ” requires adense star cluster. In this channel, BHs are formed throughthe evolution of massive stars and segregate to the cluster corewhere they pair as BBHs [9–11]. “
Classical isolated binaryevolution ” leads to the formation of a BBH through a commonenvelope ejection of an expanded envelope beyond its RocheLobe [12–16]. In “ chemically homogeneous evolution ”, mix-ing plays an important role in spreading the helium producedat the core throughout the envelope and causing an almosthomogeneous evolution of the progenitor stars to BHs [17, 18].Alternative formation channels may also be possible [19, 20].Supposing that BBHs are the remnants of gravitationallycollapsed progenitors and ignoring the possibility of primordialBHs [21], it is important to establish the connection betweentheir mass and their progenitor mass at zero age on the mainsequence. Following the process of gravitational collapse, theresulting BH mass depends not only on the progenitor massbut also on other parameters, including the metallicity, rotationand magnetic field. Therefore, instead of a one-to-one matchbetween the progenitor and the BH mass, we consider a rangeof progenitor masses for each BH mass [22–24] as follows: • ≤ (cid:12) : Collapse to neutron stars or light BHs. Sinceour discussion focuses on massive BHs, we ignore thisrange. • −
80 M (cid:12) : Collapse to a BH. The mass of remnantdepends on the metallicity of star. Metal poor stars witha main sequence mass above 50 M (cid:12) , could lead to aBH with mass about 20 M (cid:12) . On the other hand, singlestars with very low metalicity, Z (cid:46) − Z (cid:12) , barely losemass and so the remnant mass is close to their originalmass [25]. Likewise in the chemically homogeneously ∗ razieh.emami [email protected] † [email protected] evolving stars in very metal poor environment the entirestar can be turned to BHs with large masses [26]. • − (cid:12) : Pulsational pair-instability supernovae(PPSN). The mass loss during the pulsation depends onmetallicity but allows BH remnants with a mass above30 M (cid:12) as mentioned in [27]. • ≥ (cid:12) : Stars with a mass between 150 − M (cid:12) yield pair-instability supernovae (PISN) with no remnant.Stars with masses above 260 M (cid:12) collapse to heavy BHs.Since LIGO did not observe BH masses above 40 M (cid:12) [28], we neglect all progenitor masses above 150 M (cid:12) inour analysis.Hereafter, we consider stars with main sequence masses in therange of 50 − M (cid:12) as progenitors of BHs with mass (cid:38) M (cid:12) .We use the most recent data from LIGO Virgo Collaboration(LVC) for the merger rate density of the BBH [7, 8] given by R L = . + . − . Gpc − yr − . This range is associated with theflat merger rate, constant in time. In addition, the LVC foundan upper mass for each of BHs in BBH as M max = . + . − . M (cid:12) ,with a lower mass limit M Lmin = . M (cid:12) . Here, we wish tofind the merger rate density for the most massive BHs with m ≥ . M (cid:12) , hereafter R . We compare this rate with thecosmological star formation density (SFD) and find a rangefor the formation redshift happens when these two numbersmatch.The results depends on various parameters, includingthe BH formation e ffi ciency from massive star progenitorsas well as the fraction of BHs that reside in binaries whichcoalesce within the age of the universe.We start by computing the merger rate density for the pop-ulations of BHs with masses in the range M min ≤ m ≤ M max .As mentioned above, for massive BHs we may adopt M min = M (cid:12) . We notice that this is di ff erent than the LIGO lowerlimit M Lmin = M (cid:12) . Here, a super script L refers to the LIGOchoice.More precisely, we wish to find which fraction of the globalevent rate density detected by LIGO, hereafter R L ,[7, 8], orig-inates from BBHs above some mass range. For this purpose,we use the standard expression for R L [3, 28], R L ≡ Λ L (cid:104) VT (cid:105)| L , (1)where Λ L denotes the expected number of the BBHs, and (cid:104) VT (cid:105)| L refers to the population-averaged spacetime volume a r X i v : . [ a s t r o - ph . H E ] F e b [3, 28], (cid:104) VT (cid:105)| L = (cid:90) M max M min dm (cid:90) m M min dm VT ( m , m ) p pop ( m , m ) , (2)with the outer integral taken over M min ≤ m ≤ M max . Here-after we use the above values for M min and M max . VT ( m , m ) represents the spacetime volume in which LIGOcan detect the binaries, based on the search time and detectorsensitivity. It has been shown [28] that VT ∝ m k with k ∼ . p pop ( m , m ) denotes the mass distribution of BBHs[7, 8]. We focus on the following power-low choice of themass distribution, as adopted by the LVC. This enables us touse the most recent observational results and convert them toour mass limit. p pop ( m , m ) ∝ m − α m − M Lmin , α = . + . − . . (3)In the following, we use the above form of p pop ( m , m ) with M Lmin = M (cid:12) and we only allow the normalization vary fromthe original mass distribution. Using the new form of the massdistribution, we compute (cid:104) VT ( m , m ) (cid:105) and corresponding R .In addition, we use the most recent power-low index, α , from[7, 8].Therefore new LIGO event rate is given by, R = Λ (cid:104) VT (cid:105)| = N (cid:32) (cid:104) VT (cid:105)| L (cid:104) VT (cid:105)| (cid:33) R L , (4)where N refers to the ratio between the expected LIGO ratefor BH masses above some threshold and the total expectedrates. In our analysis, we adopt the latest results from [7, 8]which imply that about 70% of the events are associated with m ≥ M (cid:12) . . This yields N = / R = (cid:32) (cid:33) (cid:82) M max M Lmin m . − α dm (cid:82) M max M Lmin m . − α dm (cid:82) M max M min m . − α (cid:18) m − M min m − M Lmin (cid:19)(cid:82) M max M min m − α (cid:18) m − M min m − M Lmin (cid:19) − R L . = (cid:16) . + . − . (cid:17) Gpc − yr − . (5)Next, we compute the mass density, ρ , of LIGO BBHsprogenitors with masses above 20 M (cid:12) . We assume that eachcomponent of the binary originated from the collapse of a starwith a zero age main sequence mass above a threshold mass,hereafter M (cid:63), min , which we take to be in the range 50 M (cid:12) ≤ M (cid:63), min ≤ M (cid:12) . Furthermore since the BH mass is only afraction of the main sequence star, we use a simple mappingbetween the stars on the progenitor star mass and the BH mass.Throughout our analysis, we use the mapping of Ref. [13]in their figure 5 for two di ff erent metallicities Z = . Z (cid:12) and Z = Z (cid:12) . Each binary system requires two stellarprogenitors. For the initial mass function of progenitor stars,we adopt the Kroupa form [29], Φ ( M (cid:63) ). We integrate R ( t ) overcosmic time and take into account the lower mass limit for the progenitor mass, M (cid:63), min . The required comoving mass densityof progenitor stars is therefore, ρ ( M (cid:63), min ) = (cid:32)(cid:90) t H R ( t ) dt (cid:33) × (cid:82) M (cid:63), min M BH ( M (cid:63) ) Φ ( M (cid:63) ) dM (cid:63) (cid:82) M (cid:63), min Φ ( M (cid:63) ) dM (cid:63) (cid:39) R t H × (cid:18) (cid:82) M (cid:63), min M BH ( M (cid:63) ) Φ ( M (cid:63) ) dM (cid:63) (cid:82) M (cid:63), min Φ ( M (cid:63) ) dM (cid:63) (cid:19) . (6)Here we use one of the most recent models used by the LVCwith a constant merger rate in time [7, 8]. It is satisfactoryto generalize this study to the more complicated case with apower-low redshift dependence as well as a time delay in theBBH formation, which goes beyond the limited scope of thispaper. Thus we use the average value of R ( t ) over the age ofthe universe t H = . × yr to be close to the estimatedvalue by LIGO at z (cid:39) .
18 [4].Next, we compute the star formation density ρ (cid:63) for somefraction of the stars above a threshold, 50 M (cid:12) ≤ M (cid:63), min ≤ M (cid:12) . For this purpose, we adopt the star formation ratedensity (SFRD) as presented in [30] for z ≤ z ≥
8. The uncertainty in the SFRD at z (cid:38) ff ect on our results.We integrate the SFRD over cosmic time to get the globalstar formation density, as inferred from the observed UV lumi-nosity density in the universe as a function of redshift. Sincethe UV emission is dominated by massive stars, we do notexpect our results to be very sensitive to the assumed formof Φ ( M (cid:63) ). As noted above, we need to make sure that theremnants of the stars are within our desired mass range forLIGO’s BBHs. This can be done by multiplying the ρ (cid:63) withthe factor, f min ( M (cid:63), min ) ≡ (cid:32)(cid:90) M (cid:63), min M (cid:63) Φ ( M (cid:63) ) dM (cid:63) (cid:33) (cid:30) (cid:32)(cid:90) M (cid:63) Φ ( M (cid:63) ) dM (cid:63) (cid:33) , (7)where we consider 50 M (cid:12) ≤ M (cid:63), min ≤ M (cid:12) .So far we have assumed that all of stars with mass in therange 50 − M (cid:12) end up in a BBH. In reality, only a fractionof them collapse to BH in the desired mass range above 20 M (cid:12) .This fraction depends on various parameters, such as the metal-licity, magnetic field and rotation. In addition, not all of thegenerated BHs would end up in su ffi ciently tight BBHs thatcoalesce within t H . We combine both of these factors througha parameter, (cid:15) bin . In principle, (cid:15) bin could be time dependent,but for simplicity we take it to be a constant. Therefore, ourderived limits on (cid:15) bin should be taken as the constrains on theaverage of (cid:15) bin accounting for all astrophysical channels.Combining the di ff erent factors mentioned above, the result-ing progenitor mass density is, ρ (cid:63) ( z , M (cid:63), min ) = f min ( M (cid:63), min ) (cid:15) bin (cid:90) t ( z )0 ˙ ρ (cid:63) dt . (8)Figure 1 shows ρ (cid:63) as a function of the formation redshift, here-after z f . On the left panel, we adopt M (cid:63), min = M (cid:12) andwe show the star density for (cid:15) bin = . , . , .
1. Increasing z f * ( M / p c ) ( M , min = 100 M ) LIGO , Z = 10% Z bin = 0.01 bin = 0.1 LIGO , Z = 0.5% Z bin = 0.05 z f * ( M / p c ) ( Z = 10% Z ) LIGOM , min = 100 M , bin = 0.05 M , min = 80 M , bin = 0.05 M , min = 120 M , bin = 0.05 FIG. 1. Comparison between the LIGO event density(shaded band), and the comoving star density ρ (cid:63) . On the left panel, we choose M (cid:63), min = M (cid:12) and show the star density for (cid:15) bin = . , . , .
1, for two choices of metalliciy, Z = . Z (cid:12) and Z = Z (cid:12) . On the rightpanel, we show the case with M (cid:63), min = , , M (cid:12) and with (cid:15) bin = .
05, and Z = Z (cid:12) . (cid:15) bin pushes us to higher redshift and so enhances the forma-tion redshift. This makes sense as for higher e ffi ciencies weincrease the percentage of the star density for the BBH. Wehave also plotted the LIGO region for two di ff erent metalicites, Z = . Z (cid:12) and Z = Z (cid:12) . Interestingly, the metallicitydoes not a ff ect the observational limit significantly implyingthat our results for the isolated stars are not strongly modeldependent. This yields us 0 . < z f < .
13 for the above rangeof parameters. In the right panel, we present the results for M (cid:63), min = , , M (cid:12) . Here we have adopted (cid:15) bin = . Z = Z (cid:12) . This showsthat increasing M (cid:63), min decreases ρ (cid:63) and also pushes towardslower redshifts. This makes sense since increasing M (cid:63), min we decreases f min in Eq. (8). This yields a redshift range3 . < z f < .
94 for the above range of M (cid:63), min , including LIGOerror bars.Figure 2 presents the required progenitor mass fraction incoalescing binaries log (cid:15) bin , as a function of formation red-shift, z f , for sourcing the LIGO event mass density for BHswith masses above 20 M (cid:12) . Here we present the results for M (cid:63), min = , , M (cid:12) . Again, increasing M (cid:63), min pushestoward lower redshifts.In conclusion, mapping the LIGO merger density to the starformation density we have found a range for the formationredshift of the LIGO progenitors. Our limit depends slightlyon the threshold mass for the BHs progenitors. Assuming thatall of the missing stars yield in the coalescing binaries requirestheir formation redshift to be 4 . < z f <
14. It is howevermore realistic to assume that only a fraction of the stars appearas binaries. If we assume that 10% of the massive stars are inthe form of the BBHs, we get 2 < z f <
9. Finally for the casewith 1% of stars in the for of binaries, the formation redshiftis 0 < z f <
8. A late time suppression of (cid:15) bin could resultfrom the increase of metallicity in newly formed stars at low redshifts.While in this work we have considered a one-to-one map- z f l o g b i n M , min = 80 MM , min = 100 MM , min = 120 M FIG. 2. LIGO constraints on the progenitor mass fraction in coalescingBBH, log (cid:15) bin , as a function of formation redshift, z f , for M (cid:63), min = , , M (cid:12) . ping between the progenitor stars and the observed binaries, itwould be interesting to generalize the current analysis to thecase with non-zero delay time for the binary formation. Weleave this investigation to a future paper.We thank Daniel D’Orazio and John Forbes for helpfulcomments. We also very grateful to two anonymous refereesfor their constructive suggestions. R.E. acknowledges sup-port by the Institute for Theory and Computation at Harvard-Smithsonian Center for Astrophysics. This work was alsosupported in part by the Black Hole Initiative at Harvard Uni-versity which is funded by a JTF grant. [1] B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations],Phys. Rev. X , no. 4, 041015 (2016) Erratum: [Phys. Rev. X ,no. 3, 039903 (2018)] [arXiv:1606.04856 [gr-qc]].[2] B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations],Phys. Rev. D , no. 12, 122003 (2016) [arXiv:1602.03839[gr-qc]].[3] B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations],Astrophys. J. , no. 1, L1 (2016) [arXiv:1602.03842 [astro-ph.HE]].[4] B. P. Abbott et al. [LIGO Scientific and VIRGO Collaborations],Phys. Rev. Lett. , no. 22, 221101 (2017) Erratum: [Phys. Rev.Lett. , no. 12, 129901 (2018)] [arXiv:1706.01812 [gr-qc]].[5] B. . P. .Abbott et al. [LIGO Scientific and Virgo Collaborations],Astrophys. J. , no. 2, L35 (2017) [arXiv:1711.05578 [astro-ph.HE]].[6] B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations],Phys. Rev. Lett. , no. 14, 141101 (2017) [arXiv:1709.09660[gr-qc]].[7] B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations],arXiv:1811.12940 [astro-ph.HE].[8] B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations],arXiv:1811.12907 [astro-ph.HE].[9] C. L. Rodriguez, M. Morscher, B. Pattabiraman, S. Chatterjee,C. J. Haster and F. A. Rasio, Phys. Rev. Lett. , no. 5, 051101(2015) Erratum: [Phys. Rev. Lett. , no. 2, 029901 (2016)][arXiv:1505.00792 [astro-ph.HE]].[10] C. L. Rodriguez, S. Chatterjee and F. A. Rasio, Phys. Rev. D ,no. 8, 084029 (2016) [arXiv:1602.02444 [astro-ph.HE]].[11] D. Park, C. Kim, H. M. Lee, Y. B. Bae and K. Belczyn-ski, Mon. Not. Roy. Astron. Soc. , no. 4, 4665 (2017)[arXiv:1703.01568 [astro-ph.HE]].[12] K. Belczynski, A. Buonanno, M. Cantiello, C. L. Fryer,D. E. Holz, I. Mandel, M. C. Miller and M. Walczak, Astrophys.J. , no. 2, 120 (2014) [arXiv:1403.0677 [astro-ph.HE]].[13] K. Belczynski, D. E. Holz, T. Bulik and R. O’Shaughnessy,Nature , 512 (2016) [arXiv:1602.04531 [astro-ph.HE]].[14] S. E. Woosley, Astrophys. J. , no. 1, L10 (2016)[arXiv:1603.00511 [astro-ph.HE]].[15] C. L. Rodriguez and A. Loeb, Astrophys. J. , no. 1, L5 (2018)[arXiv:1809.01152 [astro-ph.HE]]. [16] N. Choksi, M. Volonteri, M. Colpi, O. Y. Gnedin and H. Li,[arXiv:1809.01164 [astro-ph.GA]].[17] S. E. de Mink, M. Cantiello, N. Langer and O. R. Pols, AIPConf. Proc. , 291 (2010).[18] S. E. de Mink and I. Mandel, Mon. Not. Roy. Astron. Soc. ,no. 4, 3545 (2016) [arXiv:1603.02291 [astro-ph.HE]].[19] A. Loeb, Astrophys. J. , no. 2, L21 (2016) [arXiv:1602.04735[astro-ph.HE]].[20] D. J. D’Orazio and A. Loeb, Phys. Rev. D , no. 8, 083008(2018) [arXiv:1706.04211 [astro-ph.HE]].[21] S. Bird, I. Cholis, J. B. Muoz, Y. Ali-Hamoud,M. Kamionkowski, E. D. Kovetz, A. Raccanelli and A. G. Riess,Phys. Rev. Lett. , no. 20, 201301 (2016) [arXiv:1603.00464[astro-ph.CO]].[22] S. E. Woosley and A. Heger, Astrophys. Space Sci. Libr. ,199 (2015) [arXiv:1406.5657 [astro-ph.SR]].[23] T. Sukhbold, S. Woosley and A. Heger, Astrophys. J. , no. 2,93 (2018) [arXiv:1710.03243 [astro-ph.HE]].[24] A. Heger, C. L. Fryer, S. E. Woosley, N. Langer and D. H. Hart-mann, Astrophys. J. , 288 (2003) [astro-ph / , no.4, 4739 (2017) doi:10.1093 / mnras / stx1576 [arXiv:1706.06109[astro-ph.SR]].[26] P. Marchant, N. Langer, P. Podsiadlowski, T. M. Taurisand T. J. Moriya, Astron. Astrophys. , A50 (2016)doi:10.1051 / / , no. 2, 244 (2017)doi:10.3847 / / / /
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