Binary Black Hole Spins: Model Selection with GWTC-3

Abstract

The origin of the spins of stellar-mass black holes is still controversial, and angular momentum transport inside massive stars is one of the main sources of uncertainty. Here, we apply hierarchical Bayesian inference to derive constraints on spin models from the 59 most confident binary black hole merger events in the third gravitational-wave transient catalogue (GWTC-3). We consider up to five parameters: chirp mass, mass ratio, redshift, effective spin, and precessing spin. For the model selection, we use a set of binary population synthesis simulations spanning drastically different assumptions for black hole spins and natal kicks. In particular, our spin models range from the maximal to minimal efficiency of angular momentum transport in stars. We find that if we include the precessing spin parameter into our analysis, models predicting only vanishingly small spins are in tension with GWTC-3 data. On the other hand, models in which most spins are vanishingly small but that also include a subpopulation of tidally spun-up black holes are a good match to the data. Our results show that the precessing spin parameter has a crucial impact on model selection.

1. Introduction

The third observing run (O3) of the Advanced LIGO [1] and Virgo [2] detectors has brought the number of compact binary merger observations up to 90 events with a probability of astrophysical origin $>0.5$ [3,4,5,6]. The intrinsic distribution of primary black hole (BH) masses inferred by the LIGO–Virgo–KAGRA collaboration (hereafter, LVK) shows several substructures, including a main peak at ≈10 M${}_{\odot }$, a secondary peak at ≈30–40 M${}_{\odot }$, and a long tail extending up to ∼80 M${}_{\odot }$, e.g., [7]. The inferred distribution of mass ratios has a strong preference for equal-mass systems, but several BBHs are confidently unequal-mass systems (e.g., GW190412 [8]; GW190517 [4]). The analysis of LVK data safely excludes that all BHs are maximally spinning [3,9,10]. Typical spin magnitudes in BBHs are small, with ∼50% of BHs having $\chi \lesssim 0.3$, e.g., [4,11], although not all BHs in the LVK sample have zero spin [12,13]. For example, GW151226 [14] and GW190517 [7] confidently possess spin. LVK data also support some mild evidence for spin–orbit misalignment, e.g., [4,7,15,16,17,18,19,20].

These results provide crucial insights to understand BBH formation and evolution, e.g., [10,11,12,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46]. Moreover, the mass and spin of BHs carry the memory of their progenitor stars and therefore are a key to unravel the details of massive star evolution and collapse, e.g., [47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67]. In particular, the spin magnitude of a stellar-origin BH should retain the imprint of the spin of the core of its progenitor star, e.g., [56,57,68,69,70,71,72].

Several models have been proposed to infer the spin magnitude of a BH from that of the progenitor star. The main open question concerns the efficiency of angular momentum transport within a star, e.g., [73,74,75]. If angular momentum is efficiently transferred from the core to the outer layers, mass loss by stellar winds can dissipate most of it, leading to a low-spinning stellar core and then to a low-spinning BH. If instead the core retains most of its initial angular momentum until the final collapse, the BH will be fast spinning.

In the shellular model [76,77,78,79], angular momentum is mainly transported by meridional currents and shear instabilities, leading to relatively inefficient spin dissipation. In contrast, according to the Tayler–Spruit dynamo mechanism [80], the differential rotation induces the formation of an unstable magnetic field configuration, leading to an efficient transport of angular momentum via magnetic torques. Building upon the Tayler–Spruit mechanism, Fuller et al. 2019 [70] derived a new model with an even more efficient angular momentum dissipation, predicting that the core of a single massive star might end its life with almost no rotation.

Electromagnetic observations yield controversial results. Asteroseismology favours slowly rotating cores in the late evolutionary stages, but the vast majority of stars with an asteroseismic estimate of the spin are low-mass stars [81,82,83]. The continuum-fitting derived spins of BHs in high-mass X-ray binaries are extremely high, e.g., [84,85,86], but such measurements might be affected by substantial observational biases, e.g., [84]. Finally, BH spins inferred from quasi-periodic oscillations yield notably smaller values than continuum fitting. For example, the estimate of the dimensionless spin of the BH in GRO J1655–40 is $\chi =0.7\pm 0.1$ and $0.290\pm 0.003$ from continuum fitting [87] and quasi-periodic oscillations [88], respectively.

In a binary system, the evolution of the spin is further affected by tidal forces and accretion, which tend to spin up a massive star, whereas nonconservative mass transfer and common-envelope ejection enhance mass loss, leading to more efficient spin dissipation [68,89,90,91]. For example, the model by Bavera et al. (2020, [56]) shows that the second-born BH can be highly spinning if its progenitor was tidally spun up when it was a Wolf–Rayet (WR) star orbiting about the first-born BH.

Furthermore, the orientation of the BH spin with respect to the orbital angular momentum of the binary system encodes information about binary evolution processes. In a tight binary system, tides and mass transfer tend to align the stellar spins with the orbital angular momentum ([92], but see Stegmann et al. 2021 [93] for a possible spin-flip process induced by mass transfer). If the binary system is in the field, the supernova kick is the main mechanism that can misalign the spin of a compact object with respect to the orbital angular momentum, by tilting the orbital plane, e.g., [94]. Finally, the spins of BHs in dynamically formed binary systems are expected to be isotropically distributed, because close encounters in a dense stellar cluster reset any previous signature of alignment, e.g., [23,41].

Here, we perform a model-selection hierarchical Bayesian analysis on confident LVK BBHs ($p_{\mathrm{astro}}>0.9$ and $\mathrm{FAR}<0.25{\mathrm{yr}}^{-1}$). We consider models of field BBHs for three of the most used angular-momentum transport models: (i) the shellular model as implemented in the Geneva stellar evolution code [77], (ii) the Tayler–Spruit dynamo model as implemented in the mesa code [74], and (iii) the model by Fuller et al. (2019, [70]). Hereafter, we refer to these three models simply as GENEVA (G), MESA (M), and FULLER (F) models [57]. For each of these models, we consider an additional variation accounting for the WR star tidal spin-up mechanism described by Bavera et al. (2020, [56]). Also, we account for spin tilts induced by core-collapse supernova explosions.

We selected these cases because they encompass the differences in current astrophysical models, ranging from very effective (F) to highly inefficient angular momentum transport (G). We compare them agnostically, as toy models, without focusing on their astrophysical features.

This paper is organized as follows. Section 2 presents our population-synthesis models. Section 3 describes the hierarchical Bayesian framework we used and discusses the LVK events used in our study. We lay down the results in Section 4 and summarize our conclusions in Section 5.

2. Models

2.1. MOBSE and Natal Kicks

We simulated our binary systems with the code mobse [95,96]. mobse is a custom and upgraded version of bse [97,98], in which we introduced metallicity-dependent stellar winds for OB [99], WR [49], and luminous blue variable stars [100]. mobse includes a formalism for electron-capture [101], core-collapse [102], and (pulsational) pair-instability supernovae [103]. Here, we adopt the rapid core-collapse supernova prescription, which enforces a gap between the maximum mass of neutron stars and the minimum mass of BHs (2–5 M${}_{\odot }$) [104,105]. Appendix A shows our results for the delayed core-collapse supernova model [102], which instead predicts a smooth transition between the maximum mass of a neutron star and the minimum BH mass.

Considering that there are still large uncertainties about natal kicks, e.g., [102,106,107], we decided to compare the following three models because they encompass the main uncertainties and are simple enough (1–3 free parameters) to allow us to easily interpret the results:

• A unified kick model, in which both neutron stars and BHs receive a kick $v_{\mathrm{kick}}\propto m_{\mathrm{ej}}/m_{\mathrm{rem}}$, where $m_{\mathrm{ej}}$ is the mass of the ejecta and $m_{\mathrm{rem}}$ the mass of the compact remnant [108]. This model naturally produces low kicks for electron-capture, stripped, and ultrastripped supernovae [109,110]. This model is similar to the one presented by Bray and Eldridge [106] and Bray and Eldridge [107]. It is also analogous to the model by Fryer et al. [102] but with a relevant difference: GM20 normalizes the kick by the BH mass, while Fryer et al. [102] normalize it by the total final mass of the star. Hereafter, we call this model GM20.
• A model in which compact-object kicks are drawn from a Maxwellian curve with one-dimensional root-mean-square $\sigma =265$ km s${}^{-1}$, consistent with observations of galactic pulsars [111]. This can be considered as an upper limit for BH natal kicks, because we assume that the natal kick distribution is the same for neutron stars and BHs, regardless of the larger BH mass. Hereafter, we name this model $\sigma 265$.
• A model in which compact-object kicks are drawn from a Maxwellian curve with $\sigma =150$ km s${}^{-1}$. This value of $\sigma$ is more similar to what is suggested from indirect measurements of galactic BH kicks [e.g., [112,113]. Hereafter, we refer to this model as $\sigma 150$.

We show the three models in Figure 1. The peak at zero kick in the PDF for model GM20 corresponds to BHs born from a direct collapse (i.e., with no ejected mass $m_{\mathrm{ej}}=0$).

For more details about mobse, see [100]. mobse is an open-source code and can be downloaded from https://gitlab.com/micmap/mobse_open (accessed on 4 September 2023).

2.2. Spin Magnitude

We implemented four models for the spin magnitude in mobse, the first three from Belczynski et al. (2020, [57]) and the fourth from Bouffanais et al. (2019, [33]). Given the large uncertainties on angular momentum transport, we do not claim that these four models are a complete description of the underlying physics: our models must be regarded as toy models, which encompass current uncertainties on BH spin magnitudes.

2.2.1. Geneva (G) Model

In the Geneva (hereafter, G) model, the dimensionless natal spin magnitude of a BH ($\chi$) can be approximated as:

$$\chi =\left\{\begin{array}{cc}0.85\hfill & M_{\mathrm{CO}}\le m_1\hfill \\ a{M}_{\mathrm{CO}}+b\hfill & m_1<M_{\mathrm{CO}}<m_2\hfill \\ a_{\mathrm{low}}\hfill & M_{\mathrm{CO}},\ge m_2\hfill \end{array}\right.$$

(1)

where $a=-0.088$ for all models, $M_{\mathrm{CO}}$ is the final carbon–oxygen mass of the progenitor star, while the values of b, $m_1$, $m_2$, and $a_{\mathrm{low}}$ depend on metallicity, as indicated in

Table 1. Parameters adopted in model G. See Equation (1) for details.

b	${\mathit{m}}_1\left({\mathbf{M}}_{\odot }\right)$	${\mathit{m}}_2\left({\mathbf{M}}_{\odot }\right)$	${\mathit{a}}_{\mathbf{low}}$	Z
2.258	16.0	24.2	0.13	$\ge 0.010$
3.578	31.0	37.8	0.25	$[0.004,0.010)$
2.434	18.0	27.7	0.0	$[0.0012,0.004)$
3.666	32.0	38.8	0.25	$<0.0012$

. This model comes from a fit [57] to the Geneva evolutionary tracks described by Ekström et al. [77], in which angular momentum transport is relatively inefficient.

2.2.2. MESA (M) Model

In the M model, we use the fits done by Belczynski et al. (2020, [57]) to a set of stellar tracks run with the mesa code. mesa models the transport of angular momentum according to the Tayler–Spruit magnetic dynamo (Spruit et al. [80], see also Cantiello et al. [74]). This yields a dimensionless natal BH spin

$$\chi =\left\{\begin{array}{cc}{a}_1{M}_{\mathrm{CO}}+b_1\hfill & \mathrm{if}{M}_{\mathrm{CO}}\le m_1\hfill \\ a_2{M}_{\mathrm{CO}}+b_2\hfill & \mathrm{if}{M}_{\mathrm{CO}}>m_1,\hfill \end{array}\right.$$

(2)

where $a_1$, $b_1$, and $m_1$ are given in

Table 2. Parameters adopted in model M. See Equation (2) for details.

${\mathit{a}}_1$	${\mathit{b}}_1$	${\mathit{a}}_2$	${\mathit{b}}_2$	${\mathit{m}}_1\left({\mathbf{M}}_{\odot }\right)$	Z
$-0.0016$	0.115	-	-	∞	$\ge 0.010$
$-0.0006$	0.105	-	-	∞	$[0.004,0.010)$
$0.0076$	0.050	$-0.0019$	0.165	$12.09$	$[0.0012,0.004)$
$-0.0010$	0.125	-	-	∞	$\le 0.0012$

.

2.2.3. Fuller (F) Model

Fuller et al. (2019, [70]) predict that angular momentum transport can be even more efficient than the one predicted by the Tayler–Spruit dynamo. Belczynski et al. (2020, [57]) summarize the results of the model by Fuller et al. (2019, [70]) simply as $\chi =0.01$ for all single stars and metallicities.

2.2.4. Maxwellian Model (Max)

Finally, we also introduce a toy model in which we represent the spin of a BH as a random number drawn from a Maxwellian curve with one-dimensional root-mean-square $\sigma {}_{\chi }=0.1$ and truncated to ${\chi }_{\max }=1.0$. This model [33] is a good match to the distribution arising from LVK data, e.g., [3,4,7]. Hereafter, we indicate this Maxwellian toy model as Max, for brevity.

2.3. Tidal Spin-Up

The progenitor star of the second-born BH can be substantially spun up by tidal interactions, as we expect from the fundamentals of tidal evolution theory [97,114]. In the scenario explored by Bavera et al. (2020, [56]), a common-envelope or an efficient stable mass transfer episode can lead to the formation of a BH–WR binary system, in which the WR star is the result of mass stripping. The orbital period of this BH–WR binary system can be sufficiently short to lead to efficient tidal synchronisation and spin–orbit coupling. The WR star is then efficiently spun up. If the WR star then collapses to a BH directly, the final spin of the BH will retain the imprint of the final WR spin.

Bavera et al. (2021, [115]) derive a fitting formula to describe the spin-up of the WR star and the final spin of the second-born BH:

$$\chi =\left\{\begin{array}{cc}{\alpha }_{\mathrm{WR}}{log}_{10}^2(P/[\mathrm{d}\mathrm{a}\mathrm{y}\left]\right)+{\beta }_{\mathrm{WR}}{log}_{10}(P/\mathrm{d}\mathrm{a}\mathrm{y})\hfill & \mathrm{if}P\le 1\mathrm{d}\hfill \\ 0\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(3)

where P is the orbital period of the BH–WR system, ${\alpha }_{\mathrm{WR}}=f\left(M_{\mathrm{WR}},c_1^{\alpha },c_2^{\alpha },c_3^{\alpha }\right)$ and ${\beta }_{\mathrm{WR}}=f\left(M_{WR},c_1^{\beta },c_2^{\beta },c_3^{\beta }\right)$. In this definition,

$$f\left(M_{\mathrm{WR}},c_1,c_2,c_3\right)=\frac{-c_1}{c_2+exp\left(-c_3{M}_{\mathrm{WR}}/\left[{\mathrm{M}}_{\odot }\right]\right)},$$

(4)

where $M_{\mathrm{WR}}$ is the mass of the WR star, while the coefficients $c_1$, $c_2$, and $c_3$ are determined through nonlinear least-square minimization and can be found in [115]. These fitting formulas were derived assuming the delayed model for a core-collapse supernova [102]. In Appendix A, we compare the results of the delayed and rapid supernova model and find negligible differences. Another important assumption of the model is that the common envelope ejection efficiency is $\alpha =1$ (which is the same as we assume here). Different values for the common-envelope efficiency parameters and the natal kicks affect the results. Bavera et al. (2021, [115]) discuss these uncertainties in detail. In mobse, we can use these fits for the spin of the second-born BH, while still adopting one of the models presented in the previous subsections (G, M, F, and Max) for the first-born BH.

2.4. Spin Orientation

We assume that natal kicks are the only source of misalignment between the orbital angular momentum vector of the binary system and the direction of BH spins [23,92]. Furthermore, we conservatively assume that accretion onto the first-born BH cannot change the direction of its spin [116]. For simplicity, we also neglect the spin-flip process [93]. Under such assumptions, we can derive the angle between the direction of the spins of the two compact objects and that of the orbital angular momentum of the binary system as [21,23]

$$cos\delta =cos\left(\nu {}_1\right)cos\left(\nu {}_2\right)+sin\left(\nu {}_1\right)sin\left(\nu {}_2\right)cos\left(\varphi \right),$$

(5)

where $\nu {}_i$ is the angle between the new (${\overrightarrow{L}}_{\mathrm{new}}$) and the old (${\overrightarrow{L}}_{\mathrm{old}}$) orbital angular momentum after a supernova ($i=1,2$ corresponding to the first and second supernova), so that $cos\left(\nu \right)={\overrightarrow{L}}_{\mathrm{new}}·{\overrightarrow{L}}_{\mathrm{old}}/\left(L_{\mathrm{new}}{L}_{\mathrm{old}}\right)$, while $\varphi$ is the phase of the projection of the orbital angular momentum into the orbital plane.

2.5. Setup of mobse Runs

Hereafter, we consider eight possible models for the spins (see also Table 3):

• The first four models (hereafter, G, M, F, and Max) adopt the Geneva, Mesa, Fuller, and Maxwellian models for both the first- and second-born BHs;
• The other four models (hereafter, G_B21, M_B21, F_B21, and Max_B21) adopt the fits by Bavera et al. (2021, [115]) for the second-born BH and the Geneva, Mesa, Fuller, and Maxwellian models for the first-born BH.

Table 3. Description of the runs performed for this work. ${}^a$ Model for the spin magnitude (Section 2.2). ${}^b$ Correction of the spin magnitude accounting for tidal spin-up, as described in B21 (Section 2.3). ${}^c$ Model for the natal kick (Section 2.1).

Model Name	Spin Magnitude ${}^{\mathit{a}}$	B21 ${}^{\mathit{b}}$	Kick Model ${}^{\mathit{c}}$
G	Geneva (G)	no	GM20, $\sigma 265$, $\sigma 150$
G_B21	Geneva (G)	yes	GM20, $\sigma 265$, $\sigma 150$
M	MESA (M)	no	GM20, $\sigma 265$, $\sigma 150$
M_B21	MESA (M)	yes	GM20, $\sigma 265$, $\sigma 150$
F	Fuller (F)	no	GM20, $\sigma 265$, $\sigma 150$
F_B21	Fuller (F)	yes	GM20, $\sigma 265$, $\sigma 150$
Max	Maxwellian (Max)	no	GM20, $\sigma 265$, $\sigma 150$
Max_B21	Maxwellian (Max)	yes	GM20, $\sigma 265$, $\sigma 150$

Table 3. Description of the runs performed for this work. ${}^a$ Model for the spin magnitude (Section 2.2). ${}^b$ Correction of the spin magnitude accounting for tidal spin-up, as described in B21 (Section 2.3). ${}^c$ Model for the natal kick (Section 2.1).

Model Name	Spin Magnitude ${}^{\mathit{a}}$	B21 ${}^{\mathit{b}}$	Kick Model ${}^{\mathit{c}}$
G	Geneva (G)	no	GM20, $\sigma 265$, $\sigma 150$
G_B21	Geneva (G)	yes	GM20, $\sigma 265$, $\sigma 150$
M	MESA (M)	no	GM20, $\sigma 265$, $\sigma 150$
M_B21	MESA (M)	yes	GM20, $\sigma 265$, $\sigma 150$
F	Fuller (F)	no	GM20, $\sigma 265$, $\sigma 150$
F_B21	Fuller (F)	yes	GM20, $\sigma 265$, $\sigma 150$
Max	Maxwellian (Max)	no	GM20, $\sigma 265$, $\sigma 150$
Max_B21	Maxwellian (Max)	yes	GM20, $\sigma 265$, $\sigma 150$

The model by Bavera et al. [115] was derived assuming a very efficient angular momentum transport and simulating the binary evolution with mesa [117]. Hence, the fitting formula by Bavera et al. [115] can be naturally associated with models F and M, which are based on an efficient angular momentum transport. Here, we also coupled it with models Max (toy model) and G (inefficient angular momentum transport), because our purpose was to encompass all possible uncertainties and compare our models agnostically.

For each of the aforementioned eight spin models, we considered three different kick models: the GM20, $\sigma 265$, and $\sigma 150$ models discussed in Section 2.1.

Finally, for each of these 24 models, we considered 12 metallicities ($Z=0.0002$, 0.0004, 0.0008, 0.0012, 0.0016, 0.002, 0.004, 0.006, 0.008, 0.012, 0.016, and 0.02). For each metallicity, we ran ${10}^7$ ($2\times {10}^7$) binary systems if $Z\le 0.002$ ($Z\ge 0.004$). Hence, for each model we ran $1.8\times {10}^8$ binary systems, for a total of $4.32\times {10}^9$ binary systems encompassing the eight models.

We sampled the initial conditions for each binary system as follows. We randomly drew the zero-age main sequence mass of the primary stars from a Kroupa [118] initial mass function in the range $5-150$ M${}_{\odot }$. The initial orbital parameters (semimajor axis, orbital eccentricity, and mass ratio) of binary stars were randomly drawn as already described in [119]. In particular, we derived the mass ratios $q\equiv m_2/m_1$ (with $m_2\le m_1$) as $\mathcal{F}\left(q\right)\propto q^{-0.1}$ with $q\in [0.1,1]$, the orbital period P from $\mathcal{F}(\Pi )\propto -0.55$ with $\Pi ={log}_{10}(P/\mathrm{d})\in [0.15,5.5]$, and the eccentricity e from $\mathcal{F}\left(e\right)\propto e^{-0.42}$ with $0\le e\le 0.9$. These distributions resulted from fitting the observational data of massive binary systems in nearby young clusters [120].

As to the main binary evolution parameters, here, we used $\alpha =1$ for the common envelope, while the parameter $\lambda$ depended on the stellar structure as described in [121]. The other binary evolution parameters were set up as described in [119].

Each of the resulting eight mobse catalogues consisted of a number of BBH mergers ranging from $1.47\times {10}^5$ for the high-kick case ($\sigma 265$) to $1.05\times {10}^6$ for the lowest-kick model (GM20). These numbers are sufficient to claim that differences in the kick (Figure 1) and mass (Figure 2) distributions are robust to stochastic fluctuations, as already discussed by Iorio et al. [122].

2.6. Merger Rate Density

We estimated the evolution of BBH mergers with a redshift by using our semianalytic code Cosmo$\mathcal{R}$ate [119,123]. With Cosmo$\mathcal{R}$ate, we convolved our mobse catalogues (Section 2.5) with an observation-based metallicity-dependent star formation rate (SFR) density evolution of the Universe, SFRD$(z,Z)$, in order to estimate the merger rate density of BBHs as

$${\mathcal{R}}_{\mathrm{BBH}}\left(z\right)={\int }_{z_{\max }}^z\left[{\int }_{Z_{\min }}^{Z_{\max }}\mathrm{SFRD}(z^{\prime },Z)\mathcal{F}(z^{\prime },z,Z)\mathrm{d}Z\right]\frac{\mathrm{d}t\left(z^{\prime }\right)}{\mathrm{d}{z}^{\prime }}\mathrm{d}{z}^{\prime },$$

(6)

where

$$\frac{\mathrm{d}t\left(z^{\prime }\right)}{\mathrm{d}{z}^{\prime }}={\left[H_0(1+z^{\prime })\right]}^{-1}{[{(1+z^{\prime })}^3{\mathrm{\Omega }}_M+{\mathrm{\Omega }}_{\Lambda }]}^{-1/2}.$$

(7)

In the above equation, $H_0$ is the Hubble constant, ${\mathrm{\Omega }}_M$ and ${\mathrm{\Omega }}_{\Lambda }$ are the matter and energy density, respectively. We adopted the values in [124]. The term $\mathcal{F}(z^{\prime },z,Z)$ is given by:

$$\mathcal{F}(z^{\prime },z,Z)=\frac{1}{{\mathcal{M}}_{\mathrm{TOT}}\left(Z\right)}\frac{\mathrm{d}\mathcal{N}(z^{\prime },z,Z)}{\mathrm{d}t\left(z\right)},$$

(8)

where ${\mathcal{M}}_{\mathrm{TOT}}\left(Z\right)$ is the total simulated initial stellar mass, and $\mathrm{d}\mathcal{N}(z^{\prime },z,Z)/\mathrm{d}t\left(z\right)$ is the rate of BBHs forming from stars with initial metallicity Z at redshift $z^{\prime }$ and merging at z, extracted from our mobse catalogues. In Cosmo$\mathcal{R}$ate, $\mathrm{SFRD}(z,Z)$ is given by

$$\mathrm{SFRD}(z^{\prime },Z)=\psi \left(z^{\prime }\right)p(z^{\prime },Z),$$

(9)

where $\psi \left(z^{\prime }\right)$ is the cosmic SFR density at formation redshift $z^{\prime }$, and $p(z^{\prime },Z)$ is the log-normal distribution of metallicities Z at fixed formation redshift $z^{\prime }$, with average $\mu \left(z^{\prime }\right)$ and spread ${\sigma }_Z$. Here, we took both $\psi \left(z\right)$ and $\mu \left(z\right)$ from Madau and Fragos (2017, [125]). Finally, we assumed a metallicity spread ${\sigma }_Z=0.3$.

2.7. Hyperparametric Model Description

For each of our models (Table 3), described by their hyperparameters $\lambda$, we predicted the distributions of BBH mergers

$$\frac{\mathrm{d}N}{\mathrm{d}\theta }\left(\lambda \right)=N_{\lambda }p\left(\theta \right|\lambda ),$$

(10)

where $\theta$ are the merger parameters, and $N_{\lambda }$ is the total number of mergers predicted by the model. Assuming an instrumental horizon redshift $z_{\max }=1.5$, $N_{\lambda }$ can be calculated as

$$N_{\lambda }={\int }_0^{z_{\max }}\mathcal{R}\left(z\right)\frac{\mathrm{d}{V}_{\mathrm{c}}}{\mathrm{d}z}\frac{T_{\mathrm{obs}}}{(1+z)}\mathrm{d}z,$$

(11)

where $\frac{\mathrm{d}{V}_{\mathrm{c}}}{\mathrm{d}z}$ is the comoving volume and $T_{\mathrm{obs}}$ the observation duration.

To model the population of merging BBHs, we chose five observable parameters $\theta =\{{\mathcal{M}}_{\mathrm{c}},q,z,{\chi }_{\mathrm{eff}},{\chi }_{\mathrm{p}}\}$, where ${\mathcal{M}}_{\mathrm{c}}={\left(m_1{m}_2\right)}^{3/5}/{(m_1+m_2)}^{1/5}$ is the chirp mass in the source frame with $m_1$ ($m_2$) the masses of the primary (secondary) BH of the binary, $q=m_2/m_1$, and z is the redshift of the merger. In addition, we used two spin parameters: the effective spin (${\chi }_{\mathrm{eff}}$) and the precessing spin (${\chi }_{\mathrm{p}}$). The effective spin ${\chi }_{\mathrm{eff}}$ is the mass-weighted projection of the two individual BH spins on the binary orbital angular momentum $\overrightarrow{L}$

$${\chi }_{\mathrm{eff}}=\frac{({\overrightarrow{\chi }}_1+q{\overrightarrow{\chi }}_2)}{1+q}·\frac{\overrightarrow{L}}{L},$$

(12)

where ${\overrightarrow{\chi }}_{1,2}={\overrightarrow{s}}_{1,2}c/\left(G{m}_{1,2}^2\right)$ is the dimensionless spin parameter of the two BHs. The precessing spin ${\chi }_{\mathrm{p}}$ is defined as

$${\chi }_{\mathrm{p}}=\max \left({\chi }_{1,\perp },A{\chi }_{2,\perp }\right),$$

(13)

where ${\chi }_{1,\perp }$ (${\chi }_{2,\perp }$) is the spin component of the primary (secondary) BH perpendicular to the orbital angular momentum vector $\overrightarrow{L}$, and $A=\left(4q+3\right)q/\left(4+3q\right)$.

To compute the distributions $p\left(\theta \right|\lambda )$, we constructed a catalogue of ${10}^6$ sources for all possible combinations of hyperparameters $\lambda$, using the merger rate density and the metallicity given by Cosmo$\mathcal{R}$ate. From these catalogues, we derived continuous estimations of $p\left(\theta \right|\lambda )$ by making use of a Gaussian kernel density estimation assuming a bandwidth of 0.15.

3. Hierarchical Bayesian Inference

The likelihood associated to an astrophysical model, given a dataset $\mathcal{H}={\left\{h^k\right\}}_{k=1}^{N_{\mathrm{obs}}}$ of $N_{\mathrm{obs}}$ GW observations, can be written as, e.g., [33,34,35,126,127,128]:

$$\mathcal{L}\left(\lambda \right|\mathcal{H})=\prod _{k=1}^{N_{\mathrm{obs}}}\frac{{\mathcal{I}}^k}{\beta \left(\lambda \right)},$$

(14)

where $N_{\mathrm{obs}}$ is the number of events observed by the LVK, with an ensemble of parameters $\theta$, ${\mathcal{I}}^k$ is the match of the $k^{th}$ event with the model $\lambda$ and $\beta \left(\lambda \right)$ the detection efficiency associated to the model. The detection efficiency of a model $\lambda$ is defined as the ratio of the number of eventually detected mergers ${\mu }_{\lambda }$ over all the mergers predicted by this model $N_{\lambda }$ and can be expressed as:

$$\beta \left(\lambda \right)=\frac{{\mu }_{\lambda }}{N_{\lambda }}={\int }_{\theta }p\left(\theta \right|\lambda )p_{\det }\left(\theta \right)\mathrm{d}\theta ,$$

(15)

where $p_{\det }\left(\theta \right)$ is the detection probability for a set of parameters $\theta$.

Finally, ${\mathcal{I}}^k$ is the integral of an event’s log-likelihood derived from posteriors and priors from data samples released by the LVK collaboration. It is approximated with a Monte Carlo approach as:

$${\mathcal{I}}^k=\int {\mathcal{L}}^k\left(h^k\right|\theta )p\left(\theta \right|\lambda )\mathrm{d}\theta \approx \frac{1}{N_s^k}\sum _{i=1}^{N_s^k}\frac{p\left({\theta }_i^k\right|\lambda )}{{\pi }^k\left({\theta }_i^k\right)},$$

(16)

where ${\theta }_i^k$ is the $i^{\mathrm{th}}$ posterior sample of the $k^{\mathrm{th}}$ detection and $N_s^k$ is the total number of posterior samples for the $k^{\mathrm{th}}$ detection. To compute the prior term in the denominator, we also used a Gaussian kernel density estimation.

The standard model comparison used to compute the Bayes factors B between two models ${\lambda }_i$ and ${\lambda }_j$ is defined by the ratio of posteriors:

$$B=\frac{p\left({\lambda }_i\right|\mathcal{H})}{p\left({\lambda }_j\right|\mathcal{H})}.$$

(17)

In practice, we assumed the same prior for all models in order to avoid any preference for any models. Therefore, the Bayes factor expression simplified as:

$$B=\frac{\mathcal{L}\left({\lambda }_i\right|\mathcal{H})}{\mathcal{L}\left({\lambda }_j\right|\mathcal{H})}.$$

(18)

We adopted the formalism described in Equations (14)–(18) to perform a hierarchical Bayesian inference to compare the astrophysical models presented in Section 2 with the third gravitational-wave transient catalogue (GWTC-3, [6,7]). GWTC-3 contains 90 event candidates with a probability of astrophysical origin $p_{\mathrm{astro}}>0.5$. From GWTC-3, we extracted 59 confident detections of BBHs with a false alarm rate $\mathrm{FAR}<0.25$ yr${}^{-1}$. In this subsample, we did not include binary neutron stars and neutron star–BH systems, and we also excluded the other BBH candidates with a higher FAR. Our chosen FAR threshold ensured a sufficiently pure sample for our analysis [7]. A list of the events used in this study is available in Appendix C. For the observable parameters $\theta$, we used the choice described in Section 2.7, namely, $\theta =\{{\mathcal{M}}_{\mathrm{c}},q,z,{\chi }_{\mathrm{eff}},{\chi }_{\mathrm{p}}\}$.

4. Results

4.1. Masses

The primary BH mass (Figure 2) and mass ratio distributions (Figure 3) do not depend on the spin model, by construction. Therefore, we only show different natal kicks models in these figures. Models $\sigma 150$ and $\sigma 265$ show a similar distribution of primary masses with two peaks of similar importance, one at $m_1\approx 8$ M${}_{\odot }$ and the other (broader) peak at $m_1\approx 18$ M${}_{\odot }$. In contrast, model GM20 has a dominant peak at $m_1\approx 8$ M${}_{\odot }$. The main reason for this difference is that the natal kick is independent of both BH mass and ejected mass in both models $\sigma 150$ and $\sigma 265$ [122]. Hence, binaries hosting low-mass BHs break up more easily during a supernova explosion in models $\sigma 150$ and $\sigma 265$ compared to model GM20. In contrast, most BHs receive low natal kicks in model GM20, and their binaries do not break.

Figure 2 and Figure 3 also compare the distribution of our models with the distributions inferred by LVK detections adopting the Power Law + Peak parametric model [7]. The models present a significant excess in the range $m_1\approx 15-25$ M${}_{\odot }$ to the data. Finally, the peak at $m_1\approx 9$ M${}_{\odot }$ in the data approximately matches the peak at $m1\approx 8$ M${}_{\odot }$ in the models. The main features of our population synthesis models (in particular, the peaks at $m_1\approx 8-10$ M${}_{\odot }$ and $m_1\approx 15-20$ M${}_{\odot }$) are also common to other population-synthesis models, e.g., [57,63], and mostly stem from the core-collapse SN prescriptions by [102]. The features of these models fall outside the 90% credible region for the primary BH mass with respect to the Power Law + Peak model shown in Figure 2. The difference is particularly evident for $m_1\approx 15-20$ M${}_{\odot }$ and for the high-mass tail. Alternative core-collapse SN models, e.g., [59,61,103,122,129,130], produce different features and deserve further investigation. Furthermore, here, we did not consider the dynamical formation channels in star clusters and AGN discs, which can significantly affect the mass distribution and add several degrees of freedom to this issue [23,43,131,132,133].

4.2. Spin Parameters

Figure 4 shows the distribution of spin parameters ${\chi }_{\mathrm{p}}$ and ${\chi }_{\mathrm{eff}}$ for all of our models. By construction, large spins are much more common in models G and G_B21, while models F and F_B21 have a strong predominance of vanishingly small spins. Models M, M_B21, Max, and Max_B21 are intermediate between the other two extreme models. Including or not the correction by B21 has a negligible impact on the distribution of ${\chi }_{\mathrm{p}}$ and ${\chi }_{\mathrm{eff}}$ for models G, because of the predominance of large spin magnitudes. In contrast, introducing the spin-up correction by B21 has a key impact on models F, because it is the only way to account for mild to large spins in these models. The correction by B21 is important also for models M and Max, being responsible for the large-spin wings.

Finally, our model with slow kicks (GM20) results in a distribution of ${\chi }_{\mathrm{p}}$ that is more peaked at zero (for models G, M, and Max) with respect to the other two kick models ($\sigma 150$ and $\sigma 265$). In fact, the supernova kicks in model GM20 are not large enough to appreciably misalign BH spins (see Figure 1). A similar effect is visible in the distribution of ${\chi }_{\mathrm{eff}}$: model $\sigma 265$ produces a distribution of ${\chi }_{\mathrm{eff}}$ that is less asymmetric about the zero with respect to models $\sigma 150$ and especially GM20.

4.3. Model Selection

Figure 5 and

Table 4. Log-likelihood $\mathcal{L}$ (Equation (18)) estimated with five merger parameters $\theta =\left\{{\mathcal{M}}_{\mathrm{c}},z,{\chi }_{\mathrm{eff}},q,{\chi }_{\mathrm{p}}\right\}$.

Model Name	GM20	$\mathit{\sigma }$150	$\mathit{\sigma }$265
G	−1	149	145
G_B21	−12	150	141
M	0	162	171
M_B21	36	232	232
F	−∞	−∞	−∞
F_B21	88	250	242
Max	92	255	254
Max_B21	106	257	250

report the values of the log-likelihood $log\mathcal{L}$ defined in Equation (14). The purpose of the log-likelihood values is just to compare models to each other: they do not tell us if a model is a good match to the data in the absolute sense. Therefore, we can quantify the difference between two models A and B by computing the average absolute difference in percentage

$$\mathrm{\Delta }\log \mathcal{L}\left(\mathrm{A},\mathrm{B}\right)={〈\frac{2\left|\log {\mathcal{L}}^{\mathrm{A}}-\log {\mathcal{L}}^{\mathrm{B}}\right|}{\log {\mathcal{L}}^{\mathrm{A}}+\log {\mathcal{L}}^{\mathrm{B}}}〉}_{var},$$

(19)

on the non-A, B variation $var$ ($var$ would be a kick (spin) if A and B are spin (kick) models). For example to compare the two models G and G_B21, A and B become G_B21 and G, and $var=\{\mathrm{GM}20,\sigma 150,\sigma 265\}$.

The tidal spin-up mechanism (B21) affects the spin of a small part of the population of each model (Figure 4). However, it improves the likelihood of the F and M models significantly (e.g., $\mathrm{\Delta }\log \mathcal{L}(\mathrm{M}\_\mathrm{B}21,\mathrm{M})=89\%$, Table 4). This improvement of the log-likelihood can be explained by the presence of higher values of ${\chi }_{\mathrm{p}}$ and ${\chi }_{\mathrm{eff}}$ in the distribution of populations M_B21 and F_B21 compared to M and F (Figure 4).

Model F yields $\mathcal{L}\left(\mathrm{F}\right)=-\infty$, because the LVK data have support for non vanishingly small spins, i.e., outside the values permitted by model F ($|{\chi }_{\mathrm{eff}}|>0.05$). However, it is sufficient to inject a tiny subpopulation of spinning BHs, by switching on the B21 correction, and the F model becomes one of the best considered models. In fact, the F_B21 model only includes 0.4% of BHs with $\chi >0.01$ and achieves $log\mathcal{L}>200$ (for kick models $\sigma 150$ and $\sigma 265$).

The G and G_B21 spin models exhibit lower log-likelihood values than the others for all kick models: $\log \mathcal{L}⩽150$ for $\sigma 150$ and $\sigma 265$, and $\log \mathcal{L}<0$ for GM20. This happens because the LVK data have little support for extreme values ${\chi }_{\mathrm{eff}}<-0.5$ and ${\chi }_{\mathrm{eff}}>0.5$ (Figure 4).

The kick models $\sigma 150$ and $\sigma 265$ show similar results ($\mathrm{\Delta }\log \mathcal{L}\left(\sigma 150,\sigma 265\right)<3\%)$ for every spin assumptions. Also, for all spin assumptions, the GM20 kick model scores a significantly lower likelihood than the other models $\sigma 150$ and $\sigma 265$ with $\mathrm{\Delta }\log \mathcal{L}\left(\sigma 150,\mathrm{GM}20\right)\sim \mathrm{\Delta }\log \mathcal{L}\left(\sigma 265,GM20\right)\sim$150%. This result can be explained by the high peak of model GM20 at low chirp masses (${\mathcal{M}}_{\mathrm{c}}\sim 8$ M${}_{\odot }$, see Section 4.1 and Figure 2) and by the low value of $\chi {}_{\mathrm{p}}$ compared to the other kick models (Figure 4).

Models Max and Max_B21 are possibly the best match to the data, but this is not surprising, because they were built as a toy model to visually match the data. Among the astrophysically motivated models (i.e., after excluding the Max model), M, M_B21, and F_B21 (with kick models $\sigma 150$ and $\sigma 265$) are the most favoured by the data. This might be interpreted as a support for the Tayler–Spruit instability mechanism (adopted in models M) and for the tidal spin-up model by B21.

4.4. Importance of $\chi {}_{\mathrm{p}}$

The $\chi {}_{\mathrm{p}}$ parameter encodes information on the spin component in the orbital plane. Its impact on gravitational-wave signals is much lower than that of ${\chi }_{\mathrm{eff}}$, and therefore, its measurement is less precise. To understand the impact of ${\chi }_{\mathrm{p}}$ on our results, we reran the analysis without this parameter. The results are shown in

Table 5. Log-likelihood $\mathcal{L}$ (Equation (18)) estimated with four merger parameters $\theta =\left\{{\mathcal{M}}_{\mathrm{c}},z,{\chi }_{\mathrm{eff}},q\right\}$. Here, we ignore ${\chi }_{\mathrm{p}}$.

Model Name	GM20	$\mathit{\sigma }$150	$\mathit{\sigma }$265
G	35	146	147
G_B21	47	149	154
M	141	192	190
M_B21	130	199	180
F	85	146	138
F_B21	185	207	180
Max	161	208	155
Max_B21	160	206	200

and in Figure 5 with empty markers. Figure 5 shows that if we do not include ${\chi }_{\mathrm{p}}$, the models M and M_B21 have almost the same log-likelihood, and even the F model yields a positive log-likelihood. Furthermore, the analysis without ${\chi }_{\mathrm{p}}$ results in significantly larger values of $\mathcal{L}$ for the kick model GM20. Our results demonstrate that the measured ${\chi }_{\mathrm{p}}$ of GWTC-3 BBHs carries substantial information, despite the large uncertainties.

5. Discussion

The spin magnitude of BHs is largely uncertain, mostly because we do not fully understand angular momentum transport in massive stars. In order to encompass the main uncertainties, we took a number of toy models for the BH spin, implemented them into our population-synthesis code mobse, and compared them against GWTC-3 data within a hierarchical Bayesian framework.

The data did not support models in which the entire BH population had vanishingly small spins (model F). This result was mainly driven by the ${\chi }_{\mathrm{p}}$ parameter. This is in agreement with, e.g., the complementary analysis presented in [20]. They employed a variety of complementary methods to measure the distribution of spin magnitudes and orientations of BBH mergers and concluded that the existence of a subpopulation of BHs with vanishing spins was not required by the current data. Ref. [20] found that the fraction of nonspinning BHs could comprise up to ∼60–70% of the total population. In our F_B21 models, we had ∼99.6% of BHs with $\chi <0.01$.

Other authors [134,135,136,137] recently claimed the existence of a subpopulation of zero-spin BHs. From our analysis, we cannot exclude the existence of such subpopulation, as the F model with B21 correction (F_B21) still represented a good match of the data. We found that models with large spins (G, G_B21) were less favoured by the data, but they were still acceptable if we allowed for large kicks.

Overall, we found a preference for large natal kicks. This result goes into the same direction as the work by Callister et al. 2021, [18]. Actually, this preference for large natal kicks is degenerate with the adopted formation channel. Had we included the dynamical formation channel in dense star clusters, we would have added a subpopulation of isotropically oriented spins (see, e.g., Figure 8 of Mapelli et al. [42]). In a forthcoming study, we will extend our analysis to a multichannel analysis. While it is unlikely that BBH mergers only originate from one single channel, adding more formation channels to a hierarchical Bayesian analysis dramatically increases the number of parameters, making it more difficult to reject some portions of the parameter space. Finally, some of our results depend on the usage of the mobse population-synthesis models. This mostly affects BBH masses, which depend on the assumptions on stellar tracks, binary evolution (e.g., the parameter of common-envelope efficiency), core-collapse supernovae (see, e.g., the comparison between rapid and delayed supernova model in Appendix A), and natal kicks. We refer to Iorio et al. [122] for a detailed comparison of various population-synthesis models. On the other hand, our main result, i.e., the poor performance of model F when ${\chi }_{\mathrm{p}}$ was included in our hierarchical analysis, is robust with respect to such model assumptions. In fact, the estimated log-likelihood of model F ($\mathcal{L}=-\infty$, Table 4) did not change when we considered different kick models, even if these significantly affected the mass distribution. Our result confirms that not all LVK BBHs have vanishingly small spins.

6. Summary

The origin of BH spins is still controversial, and angular momentum transport inside massive stars is one of the main sources of uncertainty. Here, we applied hierarchical Bayesian inference to derive constraints on spin models from the 59 most confident BBH merger events in GWTC-3. We considered five parameters: chirp mass, mass ratio, redshift, effective spin, and precessing spin.

For the model selection, we used a set of binary population-synthesis simulations spanning different assumptions for BH spins and natal kicks. In particular, our spin models accounted for relatively inefficient (G), efficient (Max and M), and very efficient angular-momentum transport (F). A higher efficiency of angular momentum transport was associated with lower BH spins. In particular, model F predicted vanishingly small spins for the entire BH population. For each of our models, we also included the possibility that some BHs were tidally spun up (B21). These assumptions should be regarded as toy models, encompassing the main uncertainties on BH spin magnitude.

We also considered three different natal kick models: according to models $\sigma 265$ and $\sigma 150$, we randomly drew the kicks from a Maxwellian curve with $\sigma =265$ and 150 km s${}^{-1}$, respectively; in the third model (G20), we also derived the kicks from a Maxwellian curve with $\sigma =265$ km s${}^{-1}$, but the kick magnitude was then modulated by the ratio between the mass of the ejecta and the mass of the BH.

We summarize our main results as follows.

• Data from GWTC-3 do not support models in which the entire BH population has vanishingly small spins (model F).
• In contrast, models in which most spins are vanishingly small but that also include a subpopulation of tidally spun-up BHs (model F_B21) are a good match to the data.
• The models in which angular momentum transport is relatively inefficient (G and G_21) yield log-likelihood values that are lower than models with efficient angular momentum transport (M, M_B21, Max, and Max_B21).
• Models with large BH kicks ($\sigma 150$ and $\sigma 265$ ) are favoured by our analysis with respect to low-kick models (G20).
• Our results show that the precessing spin parameter ${\chi }_{\mathrm{p}}$ plays a crucial role in constraining the spin distribution of BBH mergers.