Black hole spins in coalescing binary black holes
MMNRAS , 1–17 (2018) Preprint 5 December 2018 Compiled using MNRAS L A TEX style file v3.0
Black hole spins in coalescing binary black holes
K.A. Postnov , (cid:63) , A.G. Kuranov , Sternberg Astronomical Institute, Moscow M.V. Lomonosov State University, Universitetskij pr., 13, Moscow 119234, Russia Kazan Federal University, Kremlevskaya 18, 420008 Kazan, Russia Russian Foreign Trade Academy, 4a Pudovkin str., 119285 Moscow, Russia
Received ... Accepted ...
ABSTRACT
The possible formation mechanisms of massive close binary black holes (BHs) that canmerge in the Hubble time to produce powerful gravitational wave bursts detected during ad-vanced LIGO O1 and O2 science runs include the evolution from field low-metallicity massivebinaries, the dynamical formation in dense stellar clusters and primordial BHs. Di ff erent for-mation channels produce di ff erent source distributions of total masses M tot and e ff ective spins χ e ff of coalescing binary BHs. Using a modified bse code, we carry out extensive popula-tion synthesis calculations of the expected e ff ective spin and total mass distributions from thestandard field massive binary formation channel for di ff erent metallicities of BH progenitors(from zero-metal Population III stars up to solar metal abundance), di ff erent initial rotationsof the binary components, stellar wind mass loss prescription, di ff erent BH formation modelsand a range of common envelope e ffi ciencies. The stellar rotation is treated in two-zone (core-envelope) approximation using the e ff ective core-envelope coupling time and with an accountof the tidal synchronization of stellar envelope rotation during the binary system evolution.The results of our simulations, convolved with the metallicity-dependent star-formation his-tory, show that the total masses and e ff ective spins of the merging binary black holes detectedduring LIGO O1-O2 runs but the heaviest one (GW170729) can be simultaneously repro-duced by the adopted BH formation models. Noticeable e ff ective spin of GW170729 requiresadditional fallback from the rotating stellar envelope. Key words: stars: black holes, binaries: close, gravitational waves
The discovery of the first gravitational wave (GW) sourceGW150914 from coalescing binary black hole (BH) system (Ab-bott et al. 2016b) not only heralded the beginning of gravita-tional wave astronomy era, but also stimulated a wealth of workson fundamental physical and astrophysical aspects of the forma-tion and evolution of binary BHs. The LIGO binary BH detec-tions GW150914 (Abbott et al. 2016b), GW151226 (Abbott et al.2016c), GW170104 (Abbott et al. 2017a), GW170608 (Abbottet al. 2017c) and recently announced additional binary BH coales-cences (LIGO / Virgo Scientific Collaboration 2018), as well as thefirst LIGO / VIRGO BH binary merging event GW170814 (Abbottet al. 2017b) enables BH masses and spins before the merging, theluminosity distance to the sources and the binary BH merging ratein the Universe to be estimated (Abbott et al. 2016a). Astrophysicalimplications of these measurements were discussed, e.g., in Abbottet al. (2016e,d). This discovery was long awaited and anticipatedfrom the standard scenario of evolution of massive stars (see e.g. (cid:63)
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Mandel & Farmer (2018) for recent and Grishchuk et al. (2001) forearly references).The formation of double BHs from field stars is based on theevolution of single massive stars (Woosley et al. 2002) and mas-sive binary evolution scenario first proposed by van den Heuvel &Heise (1972) and independently by Tutukov & Yungelson (1973).To produce a massive BH with M (cid:38) − M (cid:12) in the end of evolu-tion, the progenitor star should have a large mass and low mass-lossrate at evolutionary stages preceding the core collapse. The mass-loss rate is strongly dependent on the metallicity, which plays thekey role in determining the final mass of the stellar remnant (seee.g. Belczynski et al. 2010; Spera et al. 2015).In addition to the metallicity that a ff ects the intrinsic evolutionof the binary components, the most important uncertainty in the bi-nary evolution is the e ffi ciency of the common envelope (CE) stagewhich is required to form a compact double BH binary capable ofmerging within the Hubble time. In a dedicated study (Kruckowet al. 2016), high CE e ffi ciencies ( α CE <
1) were found to be re-quired for the possible formation of binary BH systems with pa-rameters similar to GW150914 and GW151226 through the CEchannel. The common envelope e ffi ciency remains a highly debat-able issue. Recent model hydro simulations (Ohlmann et al. 2016; c (cid:13) a r X i v : . [ a s t r o - ph . H E ] D ec Postnov & Kuranov
Ricker et al. 2018) failed to produce a high CE e ffi ciency in bothlow-massive and massive binary stars, while successful CE calcu-lations were reported by other groups (see, e.g., Nandez & Ivanova(2016)). It is not excluded that the so-called stable ’isotropic re-emission’ mass transfer mode can be realized in high-mass X-ray binaries with massive BHs, thus helping to avoid the mergingof the binary system components in the common envelope (vanden Heuvel et al. 2017). This stable mass transfer mode can ex-plain the surprising stability of kinematic characteristics observedin the galactic microquasar SS433 (Cherepashchuk et al. 2018). Ofcourse, much more empirical constraints on and hydro simulationsof the common evolution formation and properties are required.To avoid the ill-understood common envelope stage, severalalternative scenarios of the binary BH formation from massivestars were proposed. For example, in short-period massive binarysystems chemically homogeneous evolution due to rotational mix-ing can be realized. The stars remain compact until the core col-lapse, and close binary BH system is formed without common en-velope stage (Mandel & de Mink 2016; de Mink & Mandel 2016;Marchant et al. 2016). In this scenario, a pair of nearly equal mas-sive BHs can be formed with the merging rate comparable to theempirically inferred one from the first LIGO observations. This sce-nario, however, can be challenged by recent observations of slowrotation of WR stars in LMC (Vink & Harries 2017).Another possible way to form massive binary BH system isthrough dynamical interactions in a dense stellar systems (e.g.,globular clusters). This scenario was earlier considered by Sigurds-son & Hernquist (1993). In the core of a dense globular clusters,stellar-mass BH form multiple systems, and BH binaries are dy-namically ejected from the cluster. This mechanism can be e ffi -cient in producing 30 + M (cid:12) merging binary BHs (Rodriguez et al.2016b; Askar et al. 2017), and binary BH formed in this way canprovide a substantial fraction of all binary BH mergings in the localUniverse (Rodriguez et al. 2016a; Rodriguez & Loeb 2018).Finally, there can be more exotic channels of binary BH for-mation. For example, primordial black holes (PBHs) formed in theearly Universe can form pairs which could be e ffi cient sources ofgravitational waves (Nakamura et al. 1997). After the discovery ofGW150914, the interest to binary PBHs has renewed (Bird et al.2016). Stellar-mass PBHs can form a substantial fraction of darkmatter in the Universe (Carr et al. 2016). The PBHs formed at theradiation-dominated stage can form pairs like GW150914 with themerging rate compatible with empirical LIGO results, being onlysome fraction of all dark matter (Eroshenko 2016; Sasaki et al.2016). A di ff erent class of PBHs with a universal log-normal massspectrum produced in the frame of a modified A ffl eck-Dine super-symmetric baryogenesis mechanism (Dolgov & Silk 1993; Dolgovet al. 2009) were also shown to be able to match the observed prop-erties of GW150914 Blinnikov et al. (2016) without violating theexisting constraints on stellar-mass PBH as dark matter (or at leastits significant fraction).The aim of this paper is to calculate the total mass and ef-fective spin distributions of coalescing binary black holes in theastrophysical scenario of BH-BH formation from initially massivebinary stars. To do this, we use the population synthesis methodbased on the open bse (Binary Stellar Evolution) code elaboratedin Hurley et al. (2000, 2002). The code was modified to take intoaccount the BH formation from massive low-metallicity and zero-metallicity stars and was supplemented with a treatment of the stel-lar core rotation during the evolution of massive stars in binaries.In Section 2, we discuss the low e ff ective spins inferred from GWobservations of coalescing binary BHs. In Section 3, we describe modifications of the BSE code and the model assumptions used inthe calculations. In Section 4, we describe in more detail the spinevolution of the binary components. Section 5 presents the resultsof our simulations, and Section 6 discusses and summarizes themain results. Examples of di ff erent types of calculated evolution-ary tracks for several initial metallicites are given in the Appendix. / VIRGOBINARY BH MERGINGS
In General Relativity, a BH is fully characterized by its mass M BH and dimensionless angular momentum a = J / M (in geometri-cal units G = c =
1) (the possible BH electric charge is neg-ligible in real astrophysical conditions). The LIGO observationsenable measurements of both masses of the coalescing BH com-ponents, M and M , the total mass of the system, M tot , and thechirp mass that determines the strength of the gravitational wavesignal M = ( M M ) / / M / . From the analysis of waveformsat the inspiral stage, individual BH spins before the merging arepoorly constrained, but their mass-weighted total angular momen-tum parallel to the orbital angular momentum, χ e ff , can be estimatedwith acceptable accuracy (Abbott et al. 2016a). This parameter is χ e ff = ( M a cos θ + M a cos θ ) / M tot , where θ i is the angle be-tween the angular momentum of the i-th BH and orbital angularmomentum of the binary system. The current LIGO / VIRGO de-tections suggest that observed merging events are consistent withinmeasurements errors with χ e ff (cid:39)
0. The O2 LIGO event GW170104may also have even slightly negative e ff ective spin χ e ff = − . + . − . (Abbott et al. 2017a), suggesting that with a probability of around0.8 the spin of one of the BHs prior to merging is directed by the an-gle more than 90 degrees relative to the orbital angular momentumof the binary system.Here we should note that some uncertainties in the GW dataanalysis are still not excluded. For example, recently, the possibil-ity of decreasing the BH masses by a factor of three compared tothose as inferred from the GW signal analysis due to strong gravi-tational lensing was discussed by Broadhurst et al. (2018). Also, anindependent analysis of the reported GW signal from GW150914with an account of the waveform degeneracy from coalescing bi-nary BHs (Creswell et al. 2018) allows a possible increase in bothmasses and spins of the BH companions. Also, the LIGO analy-sis assumes random spin distributions of the components. Relaxingthis prior could change the estimates (see, e.g., the discussion inBelczynski et al. (2017a)). Clearly, only future more precise obser-vations can solve these open issues, and therefore below we willadopt the observational parameters of coalescing binary BHs as re-ported in the original LIGO / Virgo papers.Several explanations have been proposed to the low e ff ectivespins of the observed binary BH coalescences. For example, therecan be a degeneracy between the eccentricity and spin correctionsto the binary inspiral waveforms (Huerta et al. 2018). It is also pos-sible that the BH spins can eventually lie in the binary orbital planedue to he dynamical evolution in triple systems, even for high initialBH spins coaligned with the orbital angular momentum (Antoniniet al. 2018).The low e ff ective BH spins as inferred from GW observa-tions can have important evolutionary implications (see e.g. Kush-nir et al. (2016); Hotokezaka & Piran (2017); Belczynski et al.(2017a); Wysocki et al. (2018)). They suggest a slow rotation ofthe BH progenitors, which by itself strongly constrains, for exam-ple, chemically homogeneous pathways mentioned above in which MNRAS000
0. The O2 LIGO event GW170104may also have even slightly negative e ff ective spin χ e ff = − . + . − . (Abbott et al. 2017a), suggesting that with a probability of around0.8 the spin of one of the BHs prior to merging is directed by the an-gle more than 90 degrees relative to the orbital angular momentumof the binary system.Here we should note that some uncertainties in the GW dataanalysis are still not excluded. For example, recently, the possibil-ity of decreasing the BH masses by a factor of three compared tothose as inferred from the GW signal analysis due to strong gravi-tational lensing was discussed by Broadhurst et al. (2018). Also, anindependent analysis of the reported GW signal from GW150914with an account of the waveform degeneracy from coalescing bi-nary BHs (Creswell et al. 2018) allows a possible increase in bothmasses and spins of the BH companions. Also, the LIGO analy-sis assumes random spin distributions of the components. Relaxingthis prior could change the estimates (see, e.g., the discussion inBelczynski et al. (2017a)). Clearly, only future more precise obser-vations can solve these open issues, and therefore below we willadopt the observational parameters of coalescing binary BHs as re-ported in the original LIGO / Virgo papers.Several explanations have been proposed to the low e ff ectivespins of the observed binary BH coalescences. For example, therecan be a degeneracy between the eccentricity and spin correctionsto the binary inspiral waveforms (Huerta et al. 2018). It is also pos-sible that the BH spins can eventually lie in the binary orbital planedue to he dynamical evolution in triple systems, even for high initialBH spins coaligned with the orbital angular momentum (Antoniniet al. 2018).The low e ff ective BH spins as inferred from GW observa-tions can have important evolutionary implications (see e.g. Kush-nir et al. (2016); Hotokezaka & Piran (2017); Belczynski et al.(2017a); Wysocki et al. (2018)). They suggest a slow rotation ofthe BH progenitors, which by itself strongly constrains, for exam-ple, chemically homogeneous pathways mentioned above in which MNRAS000 , 1–17 (2018) lack hole spins in binary BH M L 2
MBH M Z A M S
Z = 0 . 0 0 2 Z = 0 . 0 2 Z = 0 . 0 M L 1 Z = 0 . 0 0 0 2
Figure 1.
BH remnant mass in the delayed core collapse mechanism (Fryeret al. 2012) from stars of di ff erent metallicity for two stellar wind mass-loss models ML1 (Giacobbo et al. (2018), the solid lines) and ML2 (Vinket al. (2001), the dashed lines). For zero- and low-metallicity stars, the BHmass drop at around 60 M (cid:12) is due to taking into account PPISN. For solarmetal abundance (the bottom solid curve), the curves for Ml1 and ML2wind mass-loss models are virtually indistinguishable. the tidally induced rotation of the close binary components playsthe key role. Massive stars are often observed to be rapid rotators(Maeder 2009). No significant angular momentum loss is expectedduring evolution of single stars with a low mass-loss rate by stellarwind and at the pre-collapse stage as required to produce massiveBHs (Spera et al. 2015). Note that low e ff ective spin values canimply either small intrinsic BH spins a ∼
0, or unusual orienta-tions of BH spins with respect to the orbital angular momentum atthe inspiral stage. The unusual spin orientations can, for example,be obtained in the dynamical formation scenario (Rodriguez et al.2016a), where the BH spins are not expected to be correlated withthe orbital angular momentum, or can result from natal BH kicks.The BH spin misalignment is also expected in merging BH bina-ries produced by Lidov-Kozai oscillations in triple stellar systems(Liu & Lai 2018). In the PBH scenario, BH spins must be intrin-sically small as there are no vorticity in primordial cosmologicalperturbations.Therefore, the mass-spin distribution of BHs can serve asa sensitive tool to discriminate between di ff erent astrophysicalformation channels of coalescing massive binary BHs (Talbot &Thrane 2017; Ng et al. 2018; Piran & Hotokezaka 2018). The open-access bse code has been widely used by many au-thors to make independent population synthesis calculations. Themost recent modification was reported by (Giacobbo et al. 2018).We added the code with the treatment of the evolution of zero-metallicity (primordial Population III) stars parametrized as in Kin-ugawa et al. (2014) and with the treatment of the rotation of thecore of a star in a binary system using the e ff ective core-envelopecoupling time Postnov et al. (2016), as described below. • The initial parameters of binaries with non-zero metal abun- dances are: the primary mass is distributed according to the Salpeterlaw, dN / dM ∝ M − . (0 . M (cid:12) (cid:54) M (cid:54) M (cid:12) ), the binary massratio q = M / M (cid:54) dN / dq = const , the binary orbital separation are distributed follow-ing Sana et al. (2012), and a flat distribution for orbital eccentrici-ties in the range [0,1] . For zero-metallicity stars, we adopt di ff erentdistributions as discussed in (Belczynski et al. 2017b) (model FS1from Table 2 in that paper). • Stellar wind mass loss is recognized to be one of the mostimportant parameters that determines the mass and rotation of thestellar remnant. In the present calculations, we used the metallicity-dependent stellar wind mass loss from O-B stars with radiationpressure corrections decreasing the stellar wind power as describedin (Giacobbo et al. 2018) (model ML1 below), or without them(model ML2, Vink et al. (2001)). Note that more massive BH rem-nants are produced in the ML1 stellar wind model than in the ML2case (see Fig. 1). We assume no wind mass loss in zero-metal Pop-ulation III stars. • As the BH formation is not yet fully understood, to determinethe mass of the BH remnant we have considered two cases:(i) The BH remnant with a mass equal to that of the pre-collapseC-O core of the BH progenitor (as calculated in the bse code), M BH = . M CO ; the total mass of the coalescing binary BH in thiscase is then (with an account of the 10% gravitational mass defect) M tot = . M CO , + M CO , ).(ii) The BH remnant with a mass M BH as calculated by Fryeret al. (2012) (the delayed model in that paper) and parametrized inthe Appendix of (Giacobbo et al. 2018). In this case, the mass of aBH is defined as: M BH = . M Fe + ∆ M ) (1)where the mass of the proto-compact object (in fact, the iron stellarcore) depends on the mass of the C-O core M CO : M Fe = . M (cid:12) if M core / M (cid:12) < . . M (cid:12) if 3 . (cid:54) M CO / M (cid:12) < . . M (cid:12) if 6 . (cid:54) M CO / M (cid:12) < . . M (cid:12) if 11 . (cid:54) M CO / M (cid:12) . (2)The additional matter falling on the collapsing iron core from theouter stellar envelope is: ∆ M = . M (cid:12) if M CO / M (cid:12) < . . M CO − . M (cid:12) if 2 . (cid:54) M CO / M (cid:12) < . α D M CO + β D )( M fin − M Fe ) if 3 . (cid:54) M CO / M (cid:12) < . M fin − M Fe ) if 11 . (cid:54) M CO / M (cid:12) , (3)where M fin is the total mass of the pre-collapse star (i.e., the coreplus envelope), and α D ≡ . − . M fin − M Fe ; β D ≡ − α D . (4)The total mass of the coalescing binary BH is then M tot = M BH , + M BH , . The pulsation pair instability expected in the helium coresof very massive stars 32 M (cid:12) (cid:46) M He (cid:46) M (cid:12) (Woosley 2017) is as-sumed to prevent the formation of BH remnants with masses above ≈ M (cid:12) . The form of the initial binary eccentricity distribution is found to insignif-icantly a ff ect final results.MNRAS , 1–17 (2018) Postnov & Kuranov
Z = 0 . 0 aBH M C O
Z = 0 . 0 0 0 2 Z = 0 . 0 0 2 Z = 0 . 0 2
Z = 0 . 0 0 2 Z = 0 . 0 Z = 0 . 0 2 aBH M C O
Z = 0 . 0 0 0 2
Figure 2.
Left: Spins of single black holes formed from the collapsing cores of massive rotating stars of di ff erent metallicity in the BH formation scenariowith M BH = . M CO , where M CO is the mass of the C-O core, as a function of the C-O core mass M CO . Right: Spins of single black holes assuming the stellarenvelope fallback ( M BH is calculated as in the delayed model by Fryer et al. 2012). The spin of the BH remnant is calculated using Eq. (7). • The BH remnant masses resulted from the delay core collapsesSN mechanism for single stars with di ff erent metallicity are pre-sented in Fig. 1 as a function of the mass of the progenitor zero-agemain-sequence (ZAMS) star for two adopted models of the stellarwind mass loss ML1 and ML2. The drop in the BH masses seenat around 60 M (cid:12) is due to the pulsation pair-instability. The pair-instability supernovae (PISNe) and pulsation pair-instability super-novae (PPISNe) are treated using the formalism described in (Gia-cobbo et al. 2018). • To calculate the spin of the BH remnant, we assume the angu-lar momentum conservation of the collapsing C-O core. Therefore,in the first BH formation scenario ( M BH = . M CO ), the BH spin isdetermined by the angular momentum of the collapsing C-O coreonly: J BH = J CO . In the second case (the delayed core collapsemechanism), the BH angular momentum increases due to the fall-back of matter from the rotating outer envelope onto the C-O core: J BH = J CO + ∆ J fb . Here ∆ J fb = ∆ M fb j fb , (5)where ∆ M fb is the fallback mass, ∆ M fb = max { M BH − . M CO , } (6)which is the di ff erence between the mass of the final BH and themass of the collapsed C-O core. Below we shall refer to the BHformation models (i) and (ii) as the case without and with fallbackfrom the envelope, respectively.In the case (ii), we assume j fb = δ GM BH / c to be the mean spe-cific angular momentum of the matter falling onto BH from therotating shell around the collapsing core. We set the dimension-less factor δ =
2, which is a compromise between the specific an-gular momentum of particles at the innermost stable orbit in theSchwarzschild metric ( δ = √
3) and in the extreme Kerr metric( δ = / √ a = min (cid:32) , ( J CO + min { ∆ J fb , J env } ) cGM (cid:33) (7) where G , c are the Newtonian gravitational constant and the speedof light, respectively, J CO is the angular momentum of the C-Ocore which is assumed to be conserved during the gravitational col-lapse, J env is the angular momentum of the stellar envelope prior tothe core collapse. The results of more detailed calculations can befound in the recent paper by the Geneva group (Qin et al. 2018). • In both BH formation scenarios (i) and (ii) we assume a natalBH kick ( V kick ): V kick = M fin − M BH M fin − M Fe A kick , (8)Here the amplitude A kick is a Maxwellian distribution with a 1D-rms value 265 km s − . • The common envelope phase is α -parametrized: ∆ E env = α CE ∆ E orb , where ∆ E orb is the orbital energy of the binary lostin the common envelope stage, ∆ E env is the binding energyof the envelope (Webbink 1984; Iben & Tutukov 1984). Toavoid the λ -description of the envelope binding energy, ∆ E env = GM env M core / ( λ R ), we have directly calculated ∆ E env using theopen-access code described in Loveridge et al. (2011). As the e ff ective spin χ e ff of a coalescing BH-BH binary dependson the value and orientation of BH spins, we should specify how tocalculate BH spins and their orientation relative to the binary orbitalangular momentum. Here the following processes have been takeninto account.The value of a BH remnant spin a depends on the rotationalevolution of the stellar core, which is ill-understood and stronglymodel-dependent. For massive binaries, one possible approach isto match theoretical predictions of the core rotation with observedperiod distribution of young neutron stars observed as radio pulsars(Postnov et al. 2016). Initially, a star is assumed to rotate rigidly,but after the main sequence the stellar structure can be separated intwo parts – the core and the envelope, with some e ff ective couplingbetween these two parts. In the present calculations, the separation MNRAS000
3) and in the extreme Kerr metric( δ = / √ a = min (cid:32) , ( J CO + min { ∆ J fb , J env } ) cGM (cid:33) (7) where G , c are the Newtonian gravitational constant and the speedof light, respectively, J CO is the angular momentum of the C-Ocore which is assumed to be conserved during the gravitational col-lapse, J env is the angular momentum of the stellar envelope prior tothe core collapse. The results of more detailed calculations can befound in the recent paper by the Geneva group (Qin et al. 2018). • In both BH formation scenarios (i) and (ii) we assume a natalBH kick ( V kick ): V kick = M fin − M BH M fin − M Fe A kick , (8)Here the amplitude A kick is a Maxwellian distribution with a 1D-rms value 265 km s − . • The common envelope phase is α -parametrized: ∆ E env = α CE ∆ E orb , where ∆ E orb is the orbital energy of the binary lostin the common envelope stage, ∆ E env is the binding energyof the envelope (Webbink 1984; Iben & Tutukov 1984). Toavoid the λ -description of the envelope binding energy, ∆ E env = GM env M core / ( λ R ), we have directly calculated ∆ E env using theopen-access code described in Loveridge et al. (2011). As the e ff ective spin χ e ff of a coalescing BH-BH binary dependson the value and orientation of BH spins, we should specify how tocalculate BH spins and their orientation relative to the binary orbitalangular momentum. Here the following processes have been takeninto account.The value of a BH remnant spin a depends on the rotationalevolution of the stellar core, which is ill-understood and stronglymodel-dependent. For massive binaries, one possible approach isto match theoretical predictions of the core rotation with observedperiod distribution of young neutron stars observed as radio pulsars(Postnov et al. 2016). Initially, a star is assumed to rotate rigidly,but after the main sequence the stellar structure can be separated intwo parts – the core and the envelope, with some e ff ective couplingbetween these two parts. In the present calculations, the separation MNRAS000 , 1–17 (2018) lack hole spins in binary BH of the star into the ’core’ and the ’envelope’ is done according tothe scheme used in Hurley et al. (2000).The coupling between the core and envelope rotation canbe mediated by magnetic dynamo (Spruit 2002), internal gravitywaves (Fuller et al. 2015), etc. In this approximation, the time evo-lution of the angular momentum of the stellar core reads d J c dt = − I c I e I c + I e Ω c − Ω e τ c , (9)where I c and I e is the core and envelope moment of inertia, respec-tively, calculated as in the bse code and Ω c and Ω e are their angularvelocity vectors, which can be misaligned in due course of the evo-lution (see below). Long τ c correspond to the case of an almostindependent rotational evolution of the stellar core and the enve-lope, while short τ c describes the opposite case of a very strongcore-envelope rotational coupling. For the initially rigidly rotatingsingle stars with Ω c = Ω e this equation implies a slowing downof the core during the evolution because of the envelope radius in-crease and stellar wind mass loss. In the case of binary stars, thetidal e ff ects can change Ω e di ff erently, and the evolution of Ω c be-comes more complicated (see below).The validity of such an approach was checked by direct MESAcalculations of the rotational evolution of a 15 M (cid:12) star (Postnovet al. 2016). It was found that the observed period distribution ofyoung pulsars can be reproduced if the e ff ective coupling time be-tween the core and envelope of a massive star is τ c = × years(see Fig. 1 in Postnov et al. (2016)). Below we shall assume thatthis parametrization of the core-envelope angular momentum cou-pling is also applicable to the evolution of very massive stars leav-ing behind BH remnants. In our calculations, we varied the valueof τ c from 10 years (very strong coupling) to 10 years (very weakcoupling). The initial rotational velocity of the binary components was cho-sen according to the empirical relation between the mass of main-sequence stars M and their equatorial velocities (as used, e.g., inthe bse code (Hurley et al. 2002)) v = M . + M . km s − (10)(here M is in solar units). The main-sequence stars were assumedto be initially uniformly rotating. This assumption has some sup-port from Kepler asteroseismology (Moravveji 2017).To check the e ff ect of the initial rotational velocity, we per-formed calculations for (a) initially non-rotating stars, v rot =
0, (b)stars rotating according to Eq. (10), v rot = v , and (c) stars rotatingwith v rot = min(4 v , v crit ), where v crit = √ (2 / GM / R is the lim-iting equatorial (break-up) velocity for a rigidly rotating star withmass M and polar radius R . / misalignment The initial spins of the components of a binary system are likely tobe coaligned with the orbital angular momentum ˆL . This assump-tion is supported by recent observations of coaligned spins of starsin old stellar clusters (Corsaro et al. 2017). However, due to vio-lent turbulence in proto-stellar clouds and possible dynamical inter-actions, spins of the binary component (especially for su ffi cientlylarge orbital separations) can be initially misaligned. The latter pos-sibility is supported by observations of misaligned protostellar and protoplanetary discs in binary systems (see, e.g., observations ofHK Tau Jensen & Akeson (2014), IRS 43 (Brinch et al. 2016)),which can be explained by the binary formation in the turbulentfragmentation process (e.g. O ff ner et al. (2016)). Therefore, in ourcalculations we will consider two extreme cases: (i) the initial spinsof the binary components aligned with the orbital angular momen-tum and (ii) totally independent (random) initial spin orientation ofthe binary components.In the course of the binary evolution, the spin-orbit misalign-ment can be also produced by an additional kick velocity duringthe BH formation (Postnov & Prokhorov 1999; Kalogera 2000;Grishchuk et al. 2001). The possibility of BH generic kicks isactively debated in the literature; see, e.g., the recent discussionof potential constraining BH natal kicks from GW observationsin (O’Shaughnessy et al. 2017; Zevin et al. 2017; Wysocki et al.2018). In our calculations, we adopted the fallback-dependent BHkicks described by Eq. (8). During the evolution of a binary system, we assume that the ro-tation of the stellar envelope gets tidally synchronized with theorbital motion with the characteristic synchronization time t sync ,and the processes of tidal synchronization and orbital circulariza-tion are treated as in the bse code (see Hurley et al. (2002), Eqs.(11), (25), (26), (35)). Due to a possible misalignment of the spinvectors of the stars with the binary orbital angular momentum ˆL as discussed above, we separately treated the change of the coreand the envelope spin components parallel and perpendicular to ˆL , J c , e = J || ( c , e ) + J ⊥ ( c , e ) . On evolutionary stages prior to the compactremnant formation, for each binary component we assumed thatdue to the tidal interactions the stellar envelope spin components J e ( || , ⊥ ) evolve with the characteristic time t sync : d J e dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) tid = I e d Ω e dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) tid , (11)where the parallel and perpendicular to the orbital angular momen-tum components of the envelope angular velocity change as˙ Ω tide , || = − Ω e , || − Ω orb t sync (12)and ˙ Ω tide , ⊥ = − Ω e , ⊥ t sync , (13)respectively. Here we have assumed that the tidal interactions tendto exponentially synchronize the envelope’s parallel rotation withthe orbital motion (Eq. (12)), and to decrease the perpendicularcomponent of the envelope’s rotation (Eq. (13)) on the same timescale t sync . Clearly, under this assumption the spin-orbit alignmenttime is di ff erent from t sync . This time is also model-dependent. Inour calculations, we used Eqs. (25) and (26) from Hurley et al.(2002) for the tidal circularization and synchronization time, re-spectively. As the tidal interaction can be not as e ff ective as adoptedin that paper (see, e.g., Claret (2007)), we repeated calculationswith the circularization and synchronization times multiplied byfactor 100. No significant di ff erence in the final results were foundbecause of very e ffi cient tidal e ff ects at the stage of the Roche lobeoverflow even with the increased characteristic times.The spin of the stellar envelope, J e , also changes due to thecore-envelope interaction with the characteristic time τ c (see above, MNRAS , 1–17 (2018)
Postnov & Kuranov - 0 . 40 . 00 . 40 . 8 c eff , Z = 2e-2 - 0 . 40 . 00 . 40 . 8 c eff , Z = 2e-3 - 0 . 40 . 00 . 40 . 8 a C E = 4 . 0 a C E = 1 . 0 a C E = 0 . 5 a C E = 0 . 1 c eff , Z = 2e-4 c eff M t o t = 0 . 9 ( M C O 1 + M
C O 2 ) M L 1 | | - 0 . 40 . 00 . 40 . 8 c eff , Z = 2e-2 - 0 . 40 . 00 . 40 . 8 c eff , Z = 2e-3 - 0 . 40 . 00 . 40 . 8 a C E = 4 . 0 a C E = 1 . 0 a C E = 0 . 5 a C E = 0 . 1 c eff , Z = 2e-4 c eff M t o t = M B H 1 + M
B H 2
M L 1 | |
Figure 3.
The normalized total mass – e ff ective spin M tot − χ e ff distribution for the coalescing black hole binaries for di ff erent stellar metallicities (1st-3drow), the assumed metallicity-dependent star-formation rate history (Eldridge et al. (2019), 4th row), and di ff erent CE e ffi ciencies (1st-4th column). The lesse ff ective stellar wind mass loss with radiation pressure corrections (model ML1, Giacobbo et al. (2018)) is assumed. The e ff ective core-envelope coupling timeis τ c = × years. The initial binary component spins are coaligned with the orbital angular momentum. The natal BH kick is given by Eq. (8). Open circleswith error bars show the observed BH-BH systems from LIGO GWTC-1 catalogue (LIGO / Virgo Scientific Collaboration 2018). Upper panel: BH formationmodel without the fallback from the stellar envelope, M tot = . M CO , + M CO , ). Lower panel: BH formation model with the fallback from the envelope, M tot = M BH , + M BH , . MNRAS000
The normalized total mass – e ff ective spin M tot − χ e ff distribution for the coalescing black hole binaries for di ff erent stellar metallicities (1st-3drow), the assumed metallicity-dependent star-formation rate history (Eldridge et al. (2019), 4th row), and di ff erent CE e ffi ciencies (1st-4th column). The lesse ff ective stellar wind mass loss with radiation pressure corrections (model ML1, Giacobbo et al. (2018)) is assumed. The e ff ective core-envelope coupling timeis τ c = × years. The initial binary component spins are coaligned with the orbital angular momentum. The natal BH kick is given by Eq. (8). Open circleswith error bars show the observed BH-BH systems from LIGO GWTC-1 catalogue (LIGO / Virgo Scientific Collaboration 2018). Upper panel: BH formationmodel without the fallback from the stellar envelope, M tot = . M CO , + M CO , ). Lower panel: BH formation model with the fallback from the envelope, M tot = M BH , + M BH , . MNRAS000 , 1–17 (2018) lack hole spins in binary BH - 0 . 40 . 00 . 40 . 8 c eff , Z = 2e-2 - 0 . 40 . 00 . 40 . 8 c eff , Z = 2e-3 - 0 . 40 . 00 . 40 . 8 a C E = 4 . 0 a C E = 1 . 0 a C E = 0 . 5 a C E = 0 . 1 c eff , Z = 2e-4 c eff M t o t = 0 . 9 ( M C O 1 + M
C O 2 ) M L 1 - 0 . 40 . 00 . 40 . 8 c eff , Z = 2e-2 - 0 . 40 . 00 . 40 . 8 c eff , Z = 2e-3 - 0 . 40 . 00 . 40 . 8 a C E = 4 . 0 a C E = 1 . 0 a C E = 0 . 5 a C E = 0 . 1 c eff , Z = 2e-4 c eff M t o t = M B H 1 + M
B H 2
M L 1
Figure 4.
The same as in Fig. 3 for the initial binary component spins randomly oriented relative to the orbital angular momentum.MNRAS , 1–17 (2018)
Postnov & Kuranov - 0 . 40 . 00 . 40 . 8 c eff , Z = 2e-2 - 0 . 40 . 00 . 40 . 8 c eff , Z = 2e-3 - 0 . 40 . 00 . 40 . 8 a C E = 4 . 0 a C E = 1 . 0 a C E = 0 . 5 a C E = 0 . 1 c eff , Z = 2e-4 c eff M t o t = 0 . 9 ( M C O 1 + M
C O 2 ) M L 2 - 0 . 40 . 00 . 40 . 8 c eff , Z = 2e-2 - 0 . 40 . 00 . 40 . 8 c eff , Z = 2e-3 - 0 . 40 . 00 . 40 . 8 a C E = 4 . 0 a C E = 1 . 0 a C E = 0 . 5 a C E = 0 . 1 c eff , Z = 2e-4 c eff M t o t = M B H 1 + M
B H 2
M L 2
Figure 5.
The same as in Fig. 4 for the more e ff ective stellar wind mass loss model ML2 (Vink et al. 2001).MNRAS000
The same as in Fig. 4 for the more e ff ective stellar wind mass loss model ML2 (Vink et al. 2001).MNRAS000 , 1–17 (2018) lack hole spins in binary BH - 0 . 40 . 00 . 40 . 8 II random - 0 . 40 . 00 . 40 . 8 a C E = 4 . 0 a C E = 1 . 0 a C E = 0 . 5 a C E = 0 . 1 II random c eff c eff M t o t = 0 . 9 ( M C O 1 + M
C O 2 ) M t o t = M B H 1 + M
B H 2
Figure 6.
The same as in Fig. 3 for zero-metal Pop III stars (Kinugawa et al. 2014) assuming no mass-loss. Upper and bottom panels show the results forthe BH formation without and with fallback from the envelope, respectively, for the initially coaligned (upper rows) and randomly misaligned (bottom rows)binary component spins.
Section 4.1), the mass loss (mass gain) due to the stellar wind lossesand the mass exchange between the components: d J e dt = − d J c dt + d J e dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) tid + d J e dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˙ M (14)The spin evolution of the binary components described above wasadded to the updated bse population synthesis code. With the additions to the bse code as described above, a populationsynthesis of typically 1,000,000 binaries per run has been carriedout for di ff erent parameters of binary evolution (the common enve-lope stage e ffi ciency α CE , stellar metallicities, stellar wind model)and assuming coaligned or misaligned initial spins of the binarycomponents.To compare the results of simulations with the BH masses ande ff ective spins as inferred from the gravitational-wave observations,we need to take into account the time delays from the formationand coalescence of a given binary system and the history of thestar-formation rate in the Universe as a function of time. We usedthe method described in detail, e.g., in Dominik et al. (2015); Bel-czynski et al. (2016c), with the star-formation dependence on themetallicity from Eldridge et al. (2019). The fractional mass den-sity of star formation at and below metallicity mass fraction of Z atgiven redshift z is factorized as Ψ (cid:32) z , ZZ (cid:12) (cid:33) = ψ ( z ) Φ ( Z / Z (cid:12) ) (15) where the cumulative metallicity distribution is Φ ( Z / Z (cid:12) ) = ˆ Γ [0 . , ( Z / Z (cid:12) ) . z ] Γ (0 . , (16)and ˆ Γ and Γ are the incomplete and complete Gamma functions,respectively. The ψ ( z ) is star-formation rate density as a function ofredshift z : ψ ( z ) = .
015 (1 + z ) . + ((1 + z ) / . . M (cid:12) yr − Mpc − . (17)The time-redshift relation is calculated using the standard Λ CDMcosmological model with H =
70 km s − Mpc − , Ω Λ = . Ω m = .
3. We have carried out simulations of stellar populationswith the metallicity ranged from Z mi = − to Z max = × − binned in 10 intervals in each decade so that the probability tofind a system with given Z i at redshift z is P ( Z i ) = ( Φ ( Z i + ∆ Z ) − Φ ( Z i )) / ( Φ ( Z max ) − Φ ( Z min )[ ∆ Z / ( Z max − Z min )]. The calculated delay-time distribution of coalescing binary BHs produced by systemswith given metallicity is convolved with the adopted star-formationrate redshift and metallicity distribution. We have used 100 ml yrstime intervals for the redshift convolution.The horizon distance for aLIGO detector for a binarysystem with the chirp mass M (cid:99) is taken to be D h (cid:39) M c / . M (cid:12) ) / . Corrections to the non-spherical responsefunction of the detector are neglected. While the detector responsefunction a ff ects the detector horizon for a given M c , its e ff ect isfound to be subdominant compared to the convolution with the star-formation rate history. Besides, for coalescing BH binaries with no-ticeable (and possibly misaligned) spins, there are uncertainties in MNRAS , 1–17 (2018) Postnov & Kuranov waveforms caused by the BH spin values and misalignment whichrequire dedicated studies (cf. the waveform e ff ects for non-spinningbinaries shown in Figs.3-4 in Dominik et al. (2015)). Our main re-sults are presented in Figs. 3-8. ff ect of the fallback from the envelope during BHformation The possible fallback from the rotating stellar envelope onto a BHformed from the collapsing stellar core is found to mostly a ff ectthe distribution of coalescing binary black holes on the total mass –e ff ective spin M tot − χ e ff plane (see Figs. 3-5). In these Figures, theobserved BH-BH binaries from LIGO / Virgo GWTC-1 catalogue(LIGO / Virgo Scientific Collaboration 2018) are shown by open cir-cles with error bars in the order of decreasing total mass M tot for theguidance.The upper panel of Fig. 3 shows the results of calculations fordi ff erent common envelope e ffi ciencies α CE (1st-4th columns for α CE = . , . , ,
4, respectively ), the initial metallicity of the bi-nary components (1st-3d row for Z = . , . , . ff erent metal-licites with the adopted star-formation rate history in the Universe(Eldridge et al. 2019) (4th row). The results are shown for the stel-lar wind mass loss with radiation pressure corrections (model ML1,Giacobbo et al. (2018)) and the BH formation model without fall-back from the outer envelope, M tot = . M CO , + M CO , ). The initialspins of the binary components are coaligned with the orbital angu-lar momentum, so that the negative e ff ective spins of the coalescingBH systems arise solely due to the natal BH kicks (see Eq. (8)).The bottom panel of Fig. 3 shows the results obtained underthe same assumptions as those presented in the upper panel butfor the BH formation model with fallback from the outer envelope, M tot = M BH , + M BH , . Particular types of evolutionary tracks lead-ing to merging binary BH are summarized in the Appendix.The comparison of the 4th rows in the upper and lower panelsof Fig. 3 suggests that within measurement errors, the observedBH-BH systems but the most massive (GW150914,GW170729)can be reproduced in the standard scenario for the adoptedmetallicity-dependent star-formation rate history (Eldridge et al.2019) if no significant fallback from the rotating envelope is as-sumed. Adding the envelope fallback (the lower panel) leads to theappearance of rapidly rotating BH components (cf. Fig. 2, rightpanel) and, consequently, of the coalescing BH binaries with highe ff ective spins χ e ff (e.g. GW170729). The systems with higher M tot are also obtained in the case with the envelope fallback.Fig. 4 shows the results of calculations assuming random ini-tial orientations of the binary component spins. Generally, the re-sults are similar to those shown in Fig. 3, with a smoother distri-bution of χ e ff and higher probability to have a negative e ff ectivespin prior to the coalescence, χ e ff <
0, for some combination of theparameters. Like in the case of the coaligned initial spins, the ad-dition of the envelope mass and angular momentum to the nascentBH (the bottom panel of Fig. 4) gives rise to coalescing binary BHswith high e ff ective spins (like GW170729).For comparison, we have run calculations by assuming themore e ff ective stellar wind mass-loss from massive stars (modelML2, Vink et al. (2001)). The results remain qualitatively the sameas in calculations with less e ff ective stellar wind mass-loss modelML1 shown in Fig. 4 and are presented in Fig. 5 for randomly di-rected initial binary spins. As expected, somewhat lighter BH rem-nants are produced for a more e ff ective wind mass-loss rate.For completeness, in Fig. 6 we present the result of calcula- tions of the evolution of the zero-metallicity Pop III stars using themodel by Kinugawa et al. (2014) after a delta-like SFR burst. Thetotal masses and e ff ective spins of binary BHs produced from thesestars can spread over a wide range covering the observed binary BHparameters. However, the evolution of these objects is less certainand model-dependent, and the coalescing binary BHs from Pop IIIstars are recognized to be subdominant to produce the observed bi-nary BH coalescences (e.g., Hartwig et al. (2016); Dvorkin et al.(2016); Belczynski et al. (2017b)). The normalized distributions of BH spins a and spin misalign-ments in coalescing binary BHs, in terms of cos θ relative to theorbital plane, for binaries convolved with the adopted metallicity-dependent star-formation history (Eldridge et al. 2019), the com-mon envelope e ffi ciency α CE =
1, the ML1 stellar wind mass lossare presented in Figs. 7 and 8 for the initially coaligned and ran-domly misaligned spins of the binary components, respectively.The BH formation model without and with the envelope fallbackare used to calculate the upper and bottom rows, respectively.While in the both case the spin and mass distributions of thecomponents ( dN / dM BH1 , , Figs. 3, 4) look similar, for the initiallyaligned spins (Fig. 7) the BH spins before the coalescence arealigned in most cases (cos θ , (cid:39) θ =
1. Interestingly, inthe initially misaligned spin case, there is a non-zero probability forthe BH components to have counter-aligned spins. Note that such apossibility is still not excluded by observations of GW150914 dueto GW waveform degeneracy (Creswell et al. 2018).Right bottom panels in Figs. 7 and 8 show the mass-weightedspin projections of BH components onto the orbital angular mo-mentum, ( M , / M tot ) a , cos θ , , before the coalescence determin-ing the e ff ective binary spin χ e ff . It is seen that in both BH formationmodels (without or with fallback from the envelope) the second BHmostly contributes to χ e ff , especially in the case of the BH forma-tion with fallback (bottom rows in these Figures).Note that a natal BH kick alone is able to produce the BHspin-orbit misalignment. Indication to the possible non-zero natalBH kick was suggested by a careful statistical analysis of the BHspin misalignments in LIGO binary BHs carried out by Wysockiet al. (2018). ff ect of the initial stellar rotation and core-envelopecoupling The influence of di ff erent assumptions about the initial rotationvelocities of the binary components is shown in Fig. 9. This Fig-ure presents the calculated distribution M tot − χ e ff for binaries con-volved with the adopted metallicity-dependent star-formation rate(Eldridge et al. 2019), di ff erent α CE , the core-envelope rotationalcoupling time τ c = × yrs and the ML1 stellar wind mass-loss MNRAS000
1. Interestingly, inthe initially misaligned spin case, there is a non-zero probability forthe BH components to have counter-aligned spins. Note that such apossibility is still not excluded by observations of GW150914 dueto GW waveform degeneracy (Creswell et al. 2018).Right bottom panels in Figs. 7 and 8 show the mass-weightedspin projections of BH components onto the orbital angular mo-mentum, ( M , / M tot ) a , cos θ , , before the coalescence determin-ing the e ff ective binary spin χ e ff . It is seen that in both BH formationmodels (without or with fallback from the envelope) the second BHmostly contributes to χ e ff , especially in the case of the BH forma-tion with fallback (bottom rows in these Figures).Note that a natal BH kick alone is able to produce the BHspin-orbit misalignment. Indication to the possible non-zero natalBH kick was suggested by a careful statistical analysis of the BHspin misalignments in LIGO binary BHs carried out by Wysockiet al. (2018). ff ect of the initial stellar rotation and core-envelopecoupling The influence of di ff erent assumptions about the initial rotationvelocities of the binary components is shown in Fig. 9. This Fig-ure presents the calculated distribution M tot − χ e ff for binaries con-volved with the adopted metallicity-dependent star-formation rate(Eldridge et al. 2019), di ff erent α CE , the core-envelope rotationalcoupling time τ c = × yrs and the ML1 stellar wind mass-loss MNRAS000 , 1–17 (2018) lack hole spins in binary BH c o s q a1 d N / d a a1 dN / d (cos q c o s q M L 1 I I , M to t = 0 . 9 ( M
C O 1 + M
C O 2 ) dN / d c c c o s q a2 a2 d N / d a dN / d (cos q c o s q dN / d c c c o s q a1 d N / d a a1 dN / d (cos q c o s q M L 1 I I , M to t = M
B H 1 + M
B H 2 dN / d c c c o s q a2 a2 d N / d a dN / d (cos q c o s q dN / d c c Figure 7.
Normalized distribution of spins a i , misalignement angles cos θ i and mass-weighted spin projections on the orbital angular momentum( M i / M tot ) a i cos θ i of components of coalescing binary BHs. Shown is the case of ML1 stellar-wind mass loss, parallel initial binary component spins and α CE =
1. Panels in the upper and bottom rows correspond to BH formation model without and with envelope fallback, respectively (cf. 4th row-3d column inthe upper and bottom panel of Fig. 3, respectively). Left: BH from the primary component. Right: BH from the secondary component. model. The initial rotation of the binary components were calcu-lated for equatorial velocities ranging from 0 to maximum possiblerotation v crit corresponding to the limiting break-up equatorial ve-locity of a rigidly rotating star.The e ff ect of the di ff erent core-envelope time τ c is shown forthe same fiducial parameters but for the fixed initial velocity lawEq. (10) in Fig. 10. Upper and bottom panels of both Figures cor-respond to the BH formation model without fallback and with fall-back from the rotating envelope, respectively.Figs. 9 and 10 are almost identical (which is evident for themiddle rows of each panels of the Figures that were calculated forthe same parameters), but for plots with v = v of the components and core-envelope couplinge ffi ciency parametrized by the time τ c . At first glance, this lookssomewhat unexpected, but the analysis of individual evolutionarytracks in both cases suggests that it is a very e ff ective tidal syn-chronization of the stellar envelopes at stages when the star fills itsRoche lobe that determines the spin of the stellar C-O core (whichwe assume to collapse into BH on left panels). To see this, consider the upper panels of these figures (BHformation models without envelope fallback). The evolution of theangular momentum of the BH remnant is determined by two terms:the initial angular momentum of the C-O core and the change due tothe core-envelope coupling. Increasing the initial star rotation fromzero to maximum value (from top to bottom rows on these Figures)changes, correspondingly, the rotation of the C-O core for a givencoupling time (Fig. 9) thus widening the final χ e ff distribution. Onthe other hand, at the given initial rotation of the star, the increasein the core-envelope coupling time (from an almost rigid couplingat low τ c to independent rotation of the core and the envelope atlarge τ c , from top to bottom rows in Fig. 10) decreases the angularmomentum removal from the core, which also widens the resulting χ e ff distribution. We remind that in our model the change in the coreangular momentum is determined by the angular velocity di ff erencebetween the stellar C-O core and the outer envelope.In the case of BH formation with fallback (bottom panels ofFigs. 9 and 10), the final BH spin is mostly determined by the an-gular momentum transferred by accretion from the envelope, andthe evolution of the angular velocity of the envelope depends on MNRAS , 1–17 (2018) Postnov & Kuranov c o s q a1 d N / d a a1 dN / d (cos q c o s q M L 1 , M to t = 0 . 9 ( M
C O 1 + M
C O 2 ) dN / d c c c o s q a2 a2 d N / d a dN / d (cos q c o s q dN / d c c c o s q a1 d N / d a a1 dN / d (cos q c o s q M L 1 , M to t = M
B H 1 + M
B H 2 dN / d c c c o s q a2 a2 d N / d a dN / d (cos q c o s q dN / d c c Figure 8.
The same as in Fig. 7 for randomly misaligned initial binary component spins (cf. 4th row-3d column in the upper and bottom panel of Fig. 4,respectively). the tidal synchronization at the Roche-lobe filling stages. The ef-fect of a short tidal synchronization is clearly visible for BH spinscalculated for less e ff ective common envelopes with α CE = In the present paper, we have calculated the possible distributionof the total masses, M tot , and e ff ective spins, χ e ff , of coalescing bi-nary black holes formed through the standard astrophysical chan-nel during evolution of massive binary stars of di ff erent metallicity.We have used a modified version of the open-access bse populationsynthesis code (Hurley et al. 2000, 2002), to which we added thedescription of evolution of zero-metal Population III stars (Kinu-gawa et al. 2014) and the treatment of the stellar core rotation intwo-zone approximation as described in (Postnov et al. 2016). Tocompare the results of calculations with the observed distributionsof total masses M tot and e ff ective spins χ e ff , we have convloved theresults of calculations for di ff erent metallicites with the metallicity-dependent star-formation rate history presented in Eldridge et al. (2019) (see Eq. (15) and Eq. (17)). For completeness, we have cal-culated the evolution of zero-metallicity Population III stars (Kin-ugawa et al. 2014) (Fig. 6).Our calculations suggest that the e ff ective spin χ e ff of binaryBH produced from massive binary star evolution (the standard as-trophysical formation scenario for coalescing binary BHs) can bedistributed in a wide range (Figs. 4-5). The assumed BH formationmodel – either without fallback from the outer rotating envelopeof the collapsing star, when the total mass of the BH binary is de-termined by the mass of the stellar C-O core prior to the collapse, M tot = . M CO , + M CO , ), or with an account of the fallback fromthe outer envelope with M tot = M BH , + M BH , , where BH massesare calculated using the model of (Fryer et al. 2012) (Fig. 1) – isfound to mostly a ff ect the final e ff ective spin of the formed BH bi-nary (cf. last rows in the upper and bottom panels of Figs. 3, 4, 5, 6.The increase in the BH spin during the BH formation with fallbackhas been independently confirmed by model calculations (Schrøderet al. 2018).The second important assumption of our model calculationsis the initial alignment of spins of the binary components. Initiallymisaligned binary components even with an account of tidal in-teraction during evolution give rise to misaligned BH spins and in MNRAS000
The same as in Fig. 7 for randomly misaligned initial binary component spins (cf. 4th row-3d column in the upper and bottom panel of Fig. 4,respectively). the tidal synchronization at the Roche-lobe filling stages. The ef-fect of a short tidal synchronization is clearly visible for BH spinscalculated for less e ff ective common envelopes with α CE = In the present paper, we have calculated the possible distributionof the total masses, M tot , and e ff ective spins, χ e ff , of coalescing bi-nary black holes formed through the standard astrophysical chan-nel during evolution of massive binary stars of di ff erent metallicity.We have used a modified version of the open-access bse populationsynthesis code (Hurley et al. 2000, 2002), to which we added thedescription of evolution of zero-metal Population III stars (Kinu-gawa et al. 2014) and the treatment of the stellar core rotation intwo-zone approximation as described in (Postnov et al. 2016). Tocompare the results of calculations with the observed distributionsof total masses M tot and e ff ective spins χ e ff , we have convloved theresults of calculations for di ff erent metallicites with the metallicity-dependent star-formation rate history presented in Eldridge et al. (2019) (see Eq. (15) and Eq. (17)). For completeness, we have cal-culated the evolution of zero-metallicity Population III stars (Kin-ugawa et al. 2014) (Fig. 6).Our calculations suggest that the e ff ective spin χ e ff of binaryBH produced from massive binary star evolution (the standard as-trophysical formation scenario for coalescing binary BHs) can bedistributed in a wide range (Figs. 4-5). The assumed BH formationmodel – either without fallback from the outer rotating envelopeof the collapsing star, when the total mass of the BH binary is de-termined by the mass of the stellar C-O core prior to the collapse, M tot = . M CO , + M CO , ), or with an account of the fallback fromthe outer envelope with M tot = M BH , + M BH , , where BH massesare calculated using the model of (Fryer et al. 2012) (Fig. 1) – isfound to mostly a ff ect the final e ff ective spin of the formed BH bi-nary (cf. last rows in the upper and bottom panels of Figs. 3, 4, 5, 6.The increase in the BH spin during the BH formation with fallbackhas been independently confirmed by model calculations (Schrøderet al. 2018).The second important assumption of our model calculationsis the initial alignment of spins of the binary components. Initiallymisaligned binary components even with an account of tidal in-teraction during evolution give rise to misaligned BH spins and in MNRAS000 , 1–17 (2018) lack hole spins in binary BH - 0 . 40 . 00 . 40 . 8 Vrot = 0 c eff - 0 . 40 . 00 . 40 . 8 a C E = 0 . 1
Vrot = V0 c eff a C E = 0 . 5 a C E = 1 . 0 a C E = 4 . 0
Vrot = min (4V0,Vcrit) c eff M t o t = 0 . 9 ( M C O 1 + M
C O 2 ) M L 1 - 0 . 40 . 00 . 40 . 8
Vrot = 0 c eff - 0 . 40 . 00 . 40 . 8 a C E = 0 . 1
Vrot = V0 c eff a C E = 0 . 5 a C E = 1 . 0 a C E = 4 . 0
M L 1
Vrot = min (4V0,Vcrit) c eff M t o t = M B H 1 + M
B H 2
Figure 9. M tot − χ e ff distribution for di ff erent initial rotation velocities of the binary components. Results for the metallicity-dependent star-formation history(Eldridge et al. 2019), α CE = . τ c = × yrs. Top: the BH formation model without fallback from the outer envelope, M tot = . M CO , + M CO , ).Bottom: the BH formation model including the fallback from the outer envelope, M tot = M BH , + M BH , .MNRAS , 1–17 (2018) Postnov & Kuranov - 0 . 40 . 00 . 40 . 8 t c = 104 yrs c eff - 0 . 40 . 00 . 40 . 8 a C E = 0 . 1 t c = 5 105 yrs c eff a C E = 0 . 5 a C E = 1 . 0 a C E = 4 . 0 t c = 107 yrs c eff M t o t = 0 . 9 ( M C O 1 + M
C O 2 ) M L 1 - 0 . 40 . 00 . 40 . 8 t c = 104 yrs c eff - 0 . 40 . 00 . 40 . 8 a C E = 0 . 1 t c = 5 105 yrs c eff a C E = 0 . 5 a C E = 1 . 0 a C E = 4 . 0
M L 1 t c = 107 yrs c eff M t o t = M B H 1 + M
B H 2
Figure 10. M tot − χ e ff distribution for di ff erent core-envelope rotational coupling time τ c . Results for the metallicity-dependent star-formation history (Eldridgeet al. 2019), α CE = . v ( Eq. (10)). Top: the BH formation model without fallback from the outer envelope, M tot = . M CO , + M CO , ). Bottom: the BHformation model including the fallback from the outer envelope, M tot = M BH , + M BH , . MNRAS000
Figure 10. M tot − χ e ff distribution for di ff erent core-envelope rotational coupling time τ c . Results for the metallicity-dependent star-formation history (Eldridgeet al. 2019), α CE = . v ( Eq. (10)). Top: the BH formation model without fallback from the outer envelope, M tot = . M CO , + M CO , ). Bottom: the BHformation model including the fallback from the outer envelope, M tot = M BH , + M BH , . MNRAS000 , 1–17 (2018) lack hole spins in binary BH some cases to negative e ff ective spin parameter χ e ff of the coalesc-ing binary BHs. Some BH spin misalignment can also be producedfor initially coaligned spins due to possible natal BH kicks.Other uncertainties, including the common envelope e ffi -ciency parameter α CE and the stellar wind mass-loss model for mas-sive stars, initial rotational velocities of the binary components ande ff ective core-envelope rotational coupling time, which we variedin the present calculations, have less strong e ff ect on the results.It is important to note that the spin of the secondary BH mostlycontributes to the e ff ective spin χ e ff (see Figs. 7, 8, the right col-umn). This conclusion is independently confirmed by the recentcalculations by the Geneva group (Qin et al. 2018).The inspection of the calculated model BH-BH binaries in Fig.4 suggests that the observed location of detected LIGO sources onthe M tot − χ e ff plane but the heaviest one, GW170729, can fall si-multaneously within the calculated range of total masses and ef-fective spins. To see this more clearly, we plot the seemingly mostlikely models (4th row -3d column in the upper and bottom panel oftheis figure) as 1-d distributions separately in Fig. 11. It is seen thatfor the BH formation from the CO-core without addition from thesurrounding envelope of the collapsing star, the total mass rangedoes not cover the heaviest source, GW170729 (left panel). Theallowance for additional fallback from the stellar envelope couldreproduce the correct mass range but results in a much wider rangeof the e ff ective spin χ e ff of the coalescing binary BHs (right panel).Apparently, a more refined treatment of BH formation is required toreproduce simultaneously masses and spins of all observed so farLIGO BH-BH binaries, or a mixture of their formation channelsshould be involved.Clearly, the calculation of the e ff ective spins of coalescing bi-nary BHs is subject to many uncertainties, which we tried to takeinto account in the present study. These include: (1) the initial stel-lar rotation, (2) the treatment of the angular momentum transportin the star before the collapse, (3) the description of the mass loss,(4) the calculation of BH spin during the collapse of a rotatingstar (see Eq. (7)). The assumption of the angular momentum con-servation of the collapsing core appears to be safe, although thefraction of matter and angular momentum during the possible fall-back is less reliable. However, connection of some long gamma-ray bursts with supernovae (Hjorth & Bloom 2012) supports thecollapsar model (Woosley & Bloom 2006) involving rapidly rotat-ing BHs from core collapses of massive stars and accompanied bymass ejection. Presently, e ff orts are being made to search for pos-sible massive progenitors of type Ic supernovae (see, e.g., Van Dyket al. (2018) and references therein), and failed supernovae (Adamset al. 2017), but the results are not fully conclusive.In our calculations we have also taken into account a possi-ble natal kick during the BH formation. This assumption remainsmodel-dependent, but can be used to produce BH spin misalign-ments in the frame of the standard production channel of coalescingbinary BHs from massive binary stars (e.g. Wysocki et al. 2018, andreferences therein). Note here that the BH kick law we used in ourpopulation synthesis calculations, Eq. (8), is only one among manypossible, which is di ffi cult to specify at present. Moreover, allow-ing for o ff -center random kicks during the compact object forma-tion could change the angular momentum as well (see e.g. Spruit& Phinney 1998; Postnov & Prokhorov 1998, for the case of theneutron star rotation). Clearly, for the BH case this issue remainsopen and requires further studies.Our calculations suggest (see Fig. 9 and 10) that there is adegeneracy between the initial rotation velocity of the binary com-ponents and the core-envelope coupling e ffi ciency: the evolution Table 1.
The number of merging binary BHs per unit mass for di ff erent stel-lar metallicities, the common envelope e ffi ciencies and stellar wind models Z α CE X ML [ M − (cid:12) ] X ML [ M − (cid:12) ]0 0.1 2.2E-06 2.2E-060 0.5 2.2E-06 2.2E-060 1.0 6.7E-06 6.7E-060 4.0 4.8E-04 4.8E-040.0002 0.1 2.5E-06 8.6E-070.0002 0.5 7.9E-06 4.1E-060.0002 1.0 2.5E-05 2.6E-050.0002 4.0 1.4E-04 1.1E-040.002 0.1 2.4E-05 1.1E-050.002 0.5 2.8E-05 1.1E-050.002 1.0 1.6E-05 5.9E-060.002 4.0 6.3E-05 5.1E-050.02 0.1 1.3E-07 1.3E-070.02 0.5 1.5E-07 1.5E-070.02 1.0 2.6E-07 2.6E-070.02 4.0 1.5E-06 1.5E-06 of initially more rapidly rotating components and the evolution ofmildly rotating (or even initially non-rotating) stars with less strongcore-envelope coupling produce similar final e ff ective BH spin dis-tributions. We also find that the fallback of matter from rotatingenvelope during BH formation always leads to higher e ff ective BHspins, almost independently of the initial rotational velocity of thecomponents v and the core-envelope coupling time τ c .To facilitate the comparison with other recent population syn-thesis studies (e.g. Giacobbo et al. 2018), we also computed therelative number of merging binary BHs per unit mass defined as X = N DBH / Σ M i , where Σ M i is the total initial mass of stars cal-culated in each run with adopted distributions of masses and massratios (see Section 3). The results are listed in Table 1 for di ff erentchemical compositions and parameters α CE . Generally, our resultsagree with those calculations (cf. Table 3 in Giacobbo et al. (2018))because we have used very similar assumptions on the binary starevolution and BH formation but the description of the common en-velope stage (those authors fixed both α CE and λ parameters, whilewe have explicitly calculated the binding energy of the stellar en-velope as described in Loveridge et al. 2011). The results are pre-sented for two stellar wind mass loss models ML1 and ML2, whichare identical for zero-metallicity stars (no wind mass loss was as-sumed) and are almost indistinguishable for solar metallicity stars.Of course, the predictive power of multi-parametric popula-tion synthesis calculations should not be overestimated. In addi-tion to the distribution of the total mass M tot and e ff ective spin χ e ff we consider in the present paper (with the reservations discussedabove in Section 2), the occurrence rate of double BH mergingscan also be used to constrain their evolutionary formation channels(see, e.g., Dominik et al. (2013); Belczynski et al. (2016a); Dvorkinet al. (2016); Belczynski et al. (2017a); Rodriguez & Loeb (2018),among others).Presently, there are di ff erent viable pathways of producingmassive binary BHs that merge in the Hubble time. They can beformed from low-metallicity massive field stars, primordial Pop-ulation III remnants, can be a result of dynamical evolution indense stellar clusters or even primordial black holes. It is not ex-cluded that all channels contribute to the observed binary BH pop-ulation. For example, the discovery of very massive ( M > M (cid:12) )Schwarzschild BHs would be di ffi cult to reconcile with the stan-dard massive binary evolution (Belczynski et al. 2016b), but canbe naturally explained by primordial black holes (Blinnikov et al.2016). MNRAS , 1–17 (2018) Postnov & Kuranov
M L 1 c eff a C E = 1 . 0 d N / d c e f f dN / dMtot M t o t = 0 . 9 ( M C O 1 + M
C O 2 ) M L 1 a C E = 1 . 0 c eff d N / d c e f f dN / dMtot M t o t = M B H 1 + M
B H 2
Figure 11.
Normalized distribution of total mass – e ff ective spin M tot − χ e ff of coalescing binary BH components. Shown is the case of ML1 stellar-windmass loss, randomly misaligned initial binary component spins and α CE =
1. Left: BH formation model without fallback from the stellar envelope, M tot = . M CO , + M CO , ) (cf. 4th row-3d column in the upper panel of Fig. 4). Right: BH formation model with fallback from the envelope, M tot = M BH , + M BH , (cf. 4th row-3d column in the bottom panel of Fig. 4). With the current LIGO sensitivity, the detection horizon of bi-nary BH with masses around 30 M (cid:12) reaches ∼
700 Mpc (ignoringpossible strong gravitational lensing, see Broadhurst et al. (2018)).So far the statistics of binary BH merging rate as a function ofBH mass as inferred from reported LIGO events is consistent witha power-law dependence, dR / dM ∼ M − . (Hotokezaka & Piran2017), which does not contradict the general power-law behaviorof the stellar mass function. Our calculations presented in this pa-per confirm that presently the formation of LIGO coalescing binaryblack holes can be explained in the frame of the standard astrophys-ical formation scenario, but the discovery of a very massive BH-BHbinary with large and possible negative e ff ective spin may requireadditional formation channels to these extreme objects.Clearly, the increased statistics of BH masses and spins in-ferred from GW observations of binary BH mergings will be help-ful to distinguish between the possible binary BH populationswhich can be formed at di ff erent stages of the evolution of the Uni-verse. ACKNOWLEDGEMENTS
The authors thank the anonymous referee for constructive criticismand useful notes. KP acknowledges the support from RSF grant 16-12-10519. AK acknowledges support from RSF grant 14-12-00146(modification of the bse code) and the M.V. Lomonosov MoscowState University Program of Development (Scientific School SAIMSU).
REFERENCES
Abbott B. P., et al., 2016a, Physical Review X, 6, 041015Abbott B. P., et al., 2016b, Physical Review Letters, 116, 061102Abbott B. P., et al., 2016c, Physical Review Letters, 116, 241103Abbott B. P., et al., 2016d, ApJL, 818, L22Abbott B. P., et al., 2016e, ApJL, 833, L1Abbott B. P., et al., 2017a, Phys. Rev. Lett., 118, 221101Abbott B. P., et al., 2017b, Physical Review Letters, 119, 141101 Abbott B. P., et al., 2017c, ApJL, 851, L35Adams S. M., Kochanek C. S., Gerke J. R., Stanek K. Z., Dai X., 2017,MNRAS, 468, 4968Antonini F., Rodriguez C. L., Petrovich C., Fischer C. L., 2018, MNRAS,480, L58Askar A., Szkudlarek M., Gondek-Rosi´nska D., Giersz M., Bulik T., 2017,MNRAS, 464, L36Belczynski K., Bulik T., Fryer C. L., Ruiter A., Valsecchi F., Vink J. S.,Hurley J. R., 2010, ApJ, 714, 1217Belczynski K., Holz D. E., Bulik T., O’Shaughnessy R., 2016a, Nature, 534,512Belczynski K., et al., 2016b, A&A, 594, A97Belczynski K., Repetto S., Holz D. E., O’Shaughnessy R., Bulik T., BertiE., Fryer C., Dominik M., 2016c, ApJ, 819, 108Belczynski K., et al., 2017a, preprint, ( arXiv:1706.07053 )Belczynski K., Ryu T., Perna R., Berti E., Tanaka T. L., Bulik T., 2017b,MNRAS, 471, 4702Bird S., Cholis I., Mu˜noz J. B., Ali-Ha¨ımoud Y., Kamionkowski M., KovetzE. D., Raccanelli A., Riess A. G., 2016, Physical Review Letters, 116,201301Blinnikov S., Dolgov A., Porayko N. K., Postnov K., 2016, J. CosmologyAstropart. Phys., 11, 036Brinch C., Jørgensen J. K., Hogerheijde M. R., Nelson R. P., Gressel O.,2016, ApJL, 830, L16Broadhurst T., Diego J. M., Smoot III G., 2018, preprint,( arXiv:1802.05273 )Carr B., K¨uhnel F., Sandstad M., 2016, Phys. Rev. D, 94, 083504Cherepashchuk A. M., Postnov K. A., Belinski A. A., 2018, MNRAS, 479,4844Claret A., 2007, A&A, 467, 1389Corsaro E., et al., 2017, Nature Astronomy, 1, 0064Creswell J., Liu H., Jackson A. D., von Hausegger S., Naselsky P., 2018, J.Cosmology and Astro-Particle Physics, 2018, 007Dolgov A., Silk J., 1993, Phys. Rev. D, 47, 4244Dolgov A. D., Kawasaki M., Kevlishvili N., 2009, Nuclear Physics B, 807,229Dominik M., Belczynski K., Fryer C., Holz D. E., Berti E., Bulik T., MandelI., O’Shaughnessy R., 2013, ApJ, 779, 72Dominik M., et al., 2015, ApJ, 806, 263Dvorkin I., Vangioni E., Silk J., Uzan J.-P., Olive K. A., 2016, MNRAS,461, 3877 MNRAS000
Abbott B. P., et al., 2016a, Physical Review X, 6, 041015Abbott B. P., et al., 2016b, Physical Review Letters, 116, 061102Abbott B. P., et al., 2016c, Physical Review Letters, 116, 241103Abbott B. P., et al., 2016d, ApJL, 818, L22Abbott B. P., et al., 2016e, ApJL, 833, L1Abbott B. P., et al., 2017a, Phys. Rev. Lett., 118, 221101Abbott B. P., et al., 2017b, Physical Review Letters, 119, 141101 Abbott B. P., et al., 2017c, ApJL, 851, L35Adams S. M., Kochanek C. S., Gerke J. R., Stanek K. Z., Dai X., 2017,MNRAS, 468, 4968Antonini F., Rodriguez C. L., Petrovich C., Fischer C. L., 2018, MNRAS,480, L58Askar A., Szkudlarek M., Gondek-Rosi´nska D., Giersz M., Bulik T., 2017,MNRAS, 464, L36Belczynski K., Bulik T., Fryer C. L., Ruiter A., Valsecchi F., Vink J. S.,Hurley J. R., 2010, ApJ, 714, 1217Belczynski K., Holz D. E., Bulik T., O’Shaughnessy R., 2016a, Nature, 534,512Belczynski K., et al., 2016b, A&A, 594, A97Belczynski K., Repetto S., Holz D. E., O’Shaughnessy R., Bulik T., BertiE., Fryer C., Dominik M., 2016c, ApJ, 819, 108Belczynski K., et al., 2017a, preprint, ( arXiv:1706.07053 )Belczynski K., Ryu T., Perna R., Berti E., Tanaka T. L., Bulik T., 2017b,MNRAS, 471, 4702Bird S., Cholis I., Mu˜noz J. B., Ali-Ha¨ımoud Y., Kamionkowski M., KovetzE. D., Raccanelli A., Riess A. G., 2016, Physical Review Letters, 116,201301Blinnikov S., Dolgov A., Porayko N. K., Postnov K., 2016, J. CosmologyAstropart. Phys., 11, 036Brinch C., Jørgensen J. K., Hogerheijde M. R., Nelson R. P., Gressel O.,2016, ApJL, 830, L16Broadhurst T., Diego J. M., Smoot III G., 2018, preprint,( arXiv:1802.05273 )Carr B., K¨uhnel F., Sandstad M., 2016, Phys. Rev. D, 94, 083504Cherepashchuk A. M., Postnov K. A., Belinski A. A., 2018, MNRAS, 479,4844Claret A., 2007, A&A, 467, 1389Corsaro E., et al., 2017, Nature Astronomy, 1, 0064Creswell J., Liu H., Jackson A. D., von Hausegger S., Naselsky P., 2018, J.Cosmology and Astro-Particle Physics, 2018, 007Dolgov A., Silk J., 1993, Phys. Rev. D, 47, 4244Dolgov A. D., Kawasaki M., Kevlishvili N., 2009, Nuclear Physics B, 807,229Dominik M., Belczynski K., Fryer C., Holz D. E., Berti E., Bulik T., MandelI., O’Shaughnessy R., 2013, ApJ, 779, 72Dominik M., et al., 2015, ApJ, 806, 263Dvorkin I., Vangioni E., Silk J., Uzan J.-P., Olive K. A., 2016, MNRAS,461, 3877 MNRAS000 , 1–17 (2018) lack hole spins in binary BH Eldridge J. J., Stanway E. R., Tang P. N., 2019, MNRAS, 482, 870Eroshenko Y. N., 2016, preprint, ( arXiv:1604.04932 )Fryer C. L., Belczynski K., Wiktorowicz G., Dominik M., Kalogera V., HolzD. E., 2012, ApJ, 749, 91Fuller J., Cantiello M., Lecoanet D., Quataert E., 2015, ApJ, 810, 101Giacobbo N., Mapelli M., Spera M., 2018, MNRAS, 474, 2959Grishchuk L. P., Lipunov V. M., Postnov K. A., Prokhorov M. E.,Sathyaprakash B. S., 2001, Physics Uspekhi, 44, R01Hartwig T., Volonteri M., Bromm V., Klessen R. S., Barausse E., Magg M.,Stacy A., 2016, MNRAS, 460, L74Hjorth J., Bloom J. S., 2012, The Gamma-Ray Burst - Supernova Connec-tion. pp 169–190Hotokezaka K., Piran T., 2017, ApJ, 842, 111Huerta E. A., et al., 2018, Phys. Rev. D, 97, 024031Hurley J. R., Pols O. R., Tout C. A., 2000, MNRAS, 315, 543Hurley J. R., Tout C. A., Pols O. R., 2002, MNRAS, 329, 897Iben Jr. I., Tutukov A. V., 1984, ApJ Suppl. Ser., 54, 335Jensen E. L. N., Akeson R., 2014, Nature, 511, 567Kalogera V., 2000, ApJ, 541, 319Kinugawa T., Inayoshi K., Hotokezaka K., Nakauchi D., Nakamura T.,2014, MNRAS, 442, 2963Kruckow M. U., Tauris T. M., Langer N., Sz´ecsi D., Marchant P., Podsiad-lowski P., 2016, A&A, 596, A58Kushnir D., Zaldarriaga M., Kollmeier J. A., Waldman R., 2016, MNRAS,462, 844LIGO / Virgo Scientific Collaboration 2018, preprint ( arXiv:1811.12907 )Liu B., Lai D., 2018, ApJ, 863, 68Loveridge A. J., van der Sluys M. V., Kalogera V., 2011, ApJ, 743, 49Maeder A., 2009, Physics, Formation and Evolution of Rotating Stars,doi:10.1007 / arXiv:1806.05820 )Mandel I., de Mink S. E., 2016, MNRAS, 458, 2634Marchant P., Langer N., Podsiadlowski P., Tauris T. M., Moriya T. J., 2016,A&A, 588, A50Moravveji E., 2017, in European Physical Journal Web of Conferences. p.02004 ( arXiv:1612.03092 ), doi:10.1051 / epjconf / ff ner S. S. R., Dunham M. M., Lee K. I., Arce H. G., Fielding D. B., 2016,ApJL, 827, L11Ohlmann S. T., R¨opke F. K., Pakmor R., Springel V., 2016, ApJL, 816, L9Piran T., Hotokezaka K., 2018, preprint, ( arXiv:1807.01336 )Postnov K. A., Prokhorov M. E., 1998, Astronomy Letters, 24, 568Postnov K. A., Prokhorov M. E., 1999, ArXiv Astrophysics e-prints (astro-ph / arXiv:1811.03656 )Rodriguez C. L., Loeb A., 2018, ApJ, 866, L5Rodriguez C. L., Chatterjee S., Rasio F. A., 2016a, Phys. Rev. D, 93, 084029Rodriguez C. L., Haster C.-J., Chatterjee S., Kalogera V., Rasio F. A.,2016b, ApJL, 824, L8Sana H., et al., 2012, Science, 337, 444Sasaki M., Suyama T., Tanaka T., Yokoyama S., 2016, Physical ReviewLetters, 117, 061101Schrøder S. L., Batta A., Ramirez-Ruiz E., 2018, ApJ, 862, L3Sigurdsson S., Hernquist L., 1993, Nature, 364, 423Spera M., Mapelli M., Bressan A., 2015, MNRAS, 451, 4086Spruit H. C., 2002, A&A, 381, 923Spruit H., Phinney E. S., 1998, Nature, 393, 139Talbot C., Thrane E., 2017, Phys. Rev. D, 96, 023012 Tutukov A., Yungelson L., 1973, Nauchnye Informatsii, 27, 70Van Dyk S. D., et al., 2018, ApJ, 860, 90Vink J. S., Harries T. J., 2017, A&A, 603, A120Vink J. S., de Koter A., Lamers H. J. G. L. M., 2001, A&A, 369, 574Webbink R. F., 1984, ApJ, 277, 355Woosley S. E., 2017, ApJ, 836, 244Woosley S. E., Bloom J. S., 2006, ARAA, 44, 507Woosley S. E., Heger A., Weaver T. A., 2002, Reviews of Modern Physics,74, 1015Wysocki D., Gerosa D., O’Shaughnessy R., Belczynski K., Gladysz W.,Berti E., Kesden M., Holz D. E., 2018, Phys. Rev. D, 97, 043014Zevin M., Pankow C., Rodriguez C. L., Sampson L., Chase E., Kalogera V.,Rasio F. A., 2017, ApJ, 846, 82de Mink S. E., Mandel I., 2016, MNRAS, 460, 3545van den Heuvel E. P. J., Heise J., 1972, Nature Physical Science, 239, 67van den Heuvel E. P. J., Portegies Zwart S. F., de Mink S. E., 2017, MNRAS,471, 4256 APPENDIX A: SUMMARY OF DIFFERENT TYPES OFEVOLUTIONARY TRACKS
Here we present in the table form a summary of di ff erent evolu-tionary tracks leading to the formation of coalescing binary BHs(Table A1). Columns: Z – stellar metallicity; α CE – common en-velope e ffi ciency; CE1(2) – stages of the components (primary –star 1, secondary – star 2) at the beginning of the first (second, ifhappens) CE stage (MS – main sequence, HeMS – helium mainsequence; HeHG – helim star Hertzspring gap; CHeB – core he-lium burning, RSG – red super giant, BH – black hole); NN DBH –fraction of this type of tracks among all DBH binaries for this Z and α CE ; X [ M − (cid:12) ] – the fraction of coalescing binary BH per totalmass of calculated binaries; (cid:104) M tot (cid:105) – mean total mass of the coa-lescing BH binaries; (cid:104) χ e ff (cid:105) , (cid:104) a (cid:105) , (cid:104) a (cid:105) – mean e ff ective spin of theBH binary and individual BH spins at the merging, respectively, (cid:104) cos θ , (cid:105) – mean misalgnment angle cosines. (cid:104) M tot (cid:105) , (cid:104) χ e ff (cid:105) , (cid:104) a , (cid:105) , (cid:104) cos θ , (cid:105) are shown for two BH formation models: with fallbackfrom the envelope when M tot = M BH1 + M BH2 , and without fall-back, M tot = . M CO1 + M CO2 ).The tracks were obtained in population synthesis runs of1,000,000 binaries with given metallicity Z and common envelopee ffi ciency α CE , with fixed initial parameter distributions (see Sec-tion 3) and other evolutionary parameters described in the maintext: ML1 stellar wind mass loss model, the core-envelope cou-pling time τ c = × yrs, the initial rotation of the componentswith velocity v (Eq. (10)), and random initial spin misalgnment ofthe binary components. This paper has been typeset from a TEX / L A TEX file prepared by the author.MNRAS , 1–17 (2018) Postnov & Kuranov
Table A1.
Merging double BHs in population synthesis run of 1,000,000 binaries with given Z and α CE for two di ff erent BH formation models (see text) CE1 CE2 M tot = M BH1 + M BH2 M tot = . M CO1 + M CO2 )Z α CE star 1 star 2 star 1 star 2 NN DBH X [ M − (cid:12) ] (cid:104) M tot (cid:105) (cid:104) χ e ff (cid:105) (cid:104) a (cid:105) (cid:104) a (cid:105) (cid:104) cos θ (cid:105) (cid:104) cos θ (cid:105) (cid:104) M tot (cid:105) (cid:104) χ e ff (cid:105) (cid:104) a (cid:105) (cid:104) a (cid:105) (cid:104) cos θ (cid:105) (cid:104) cos θ (cid:105) MNRAS000