Biases in estimates of black hole kicks from the spin distribution of binary black holes

A population of more than 50 binary black hole mergers has now been observed by the LIGO (Aasi et al., 2015) and Virgo (Acernese et al., 2015) gravitational-wave observatories, as published in a series of catalogues (Abbott et al., 2019, 2021d, 2021a; Abbott et al., 2021b).

The dominant formation scenario for binary black holes remains uncertain. One proposed scenario is the isolated evolution of a binary of two massive stars (Belczynski et al., 2016; Stevenson et al., 2017a). In this channel, two massive stars live their lives together, which interact through tides, phases of mass transfer and common envelope evolution that shrink the orbit of the binary, finally forming a binary black hole that can merge due to the emission of gravitational waves within the age of the Universe. Binary black holes formed through isolated binary evolution are expected to have their spins roughly aligned with the binary orbital angular momentum (e.g., Rodriguez et al., 2016; Stevenson et al., 2017b; Gerosa et al., 2018). Alternate formation channels such as the dynamical formation of binary black holes in dense stellar environments such as young star clusters or globular clusters have also been proposed (Sigurdsson & Hernquist, 1993; Rodriguez et al., 2019; Di Carlo et al., 2020); see Mandel & Farmer (2022) for a recent overview of binary black hole formation scenarios.

Several stages of massive binary evolution are poorly constrained. For example, we know from radio observations of isolated Galactic pulsars that neutron stars typically have large space velocities of a few hundred km/s (Lyne & Lorimer, 1994; Hobbs et al., 2005) that are thought to be imparted at birth in a supernova (Janka, 2017). However, it is still unclear what kicks black holes receive at formation (if any). Observations of the kinematics of Galactic X-ray binaries containing black holes have identified that some black holes do receive kicks comparable to neutron stars (Brandt et al., 1995; Mirabel et al., 2001), whilst other systems appear to have much smaller velocities, and some may have received no kick at all (e.g., Mirabel & Rodrigues, 2003). Fitting the distribution of observed black hole kicks typically suggests that black holes should receive kicks comparable to, or smaller in magnitude than neutron stars (Repetto et al., 2012; Repetto & Nelemans, 2015; Mandel, 2016; Repetto et al., 2017; Atri et al., 2019). The reduction may be due to mass falling back onto the proto-compact object during black hole formation (e.g., Fryer et al., 2012; Janka, 2013; Chan et al., 2018). One should also keep in mind that these samples are likely strongly biased by both observational and evolutionary selection effects, and thus they may not represent the true distribution of black hole natal kicks. The typical magnitude of black hole kicks is an important factor in determining the merger rate of binary black holes (e.g., Wysocki et al., 2018), as well as the typical angles by which the spins of the black holes are misaligned from the orbital angular momentum (Rodriguez et al., 2016).

Recently, Callister et al. (2021) analysed the distribution of spins of merging binary black holes from Abbott et al. (2021d) with a phenomenological model motivated by the isolated binary evolution formation channel and claimed that large kicks (greater than 260 km/s at 99% confidence) are required in order to reconcile the predictions of the model with the observations¹¹1O’Shaughnessy et al. (2017) also inferred that a kick greater than 50 km/s was required to explain the spin of GW151226 (Abbott et al., 2016). Such large kicks are at odds with other estimates of black hole kick magnitudes, leading Callister et al. (2021) to suggest that other channels may contribute to the formation of binary black holes, such as dynamical formation in dense star clusters (e.g., Rodriguez et al., 2019; Di Carlo et al., 2020).

In this Letter we show that making similar assumptions to Callister et al. (2021) but accounting for the population of wide merging binaries where tidal synchronisation is ineffective yields a bimodal spin magnitude distribution for the second born black hole in merging binary black holes, that agrees well with the observed distribution. We show that there is currently no evidence for large ($\sim 1000$  km/s) black hole natal kicks, and that typical kicks of order $\sim 100$ km/s (consistent with those inferred from Galactic X-ray binaries, as discussed above) are sufficient to produce the observed spin-orbit misalignments. We therefore argue that (apart from some exceptional events) the majority of the binary black hole population is consistent with forming through isolated binary evolution.

We begin by constructing a simple model that matches the assumptions made by Callister et al. (2021) as closely as possible. We briefly describe the assumptions of this model in Section 2.1. We then describe in detail the assumptions made in determining the spin of the secondary black hole in Section 2.2.

We show the distribution for the spin magnitude of the secondary black hole, $\chi_{2}$ predicted by our model in Figure 1. We see that in this model most second born black holes are also slowly rotating with $\chi_{2}\sim 0$, but there is a tail of systems with tight enough orbits that tides can efficiently spin up the helium star, resulting in a second peak around $\chi_{2}\sim 1$ (see also Bavera et al., 2020). It is clear by eye that a Gaussian distribution is a poor description of the spin magnitude distribution predicted by the tidal synchronisation model, regardless of the parameters chosen for the Gaussian.

Comparisons to gravitational-wave observations are often made using the well-measured effective spin parameter (e.g., Farr et al., 2017; Abbott et al., 2021d; Callister et al., 2021)

where $q=m_{2}/m_{1}<1$ is the binary mass ratio, $\chi_{i}$ are the dimensionless spin magnitudes of the two black holes and $\cos\theta_{i}$ are the spin tilt angles between the spins of the black holes and the orbital angular momentum. Since we assume that black holes form with no associated kick, we also assume that the spin of the secondary is perfectly aligned with the orbital angular momentum, $\cos\theta_{2}=\cos\theta_{1}=1$. Small kicks would result in only minor misalignments (Rodriguez et al., 2016; Stevenson et al., 2017b; Gerosa et al., 2018). We discuss the implications for the magnitude of black hole kicks in Section 4.

We show the distribution of effective spins $\chi_{\mathrm{eff}}$ under these assumptions in Figure 2. The distribution is strongly peaked at $\chi_{\mathrm{eff}}=0$ ($\sim 80$% of the population has $\chi_{\mathrm{eff}}<0.05$) with a tail of around 20% of the population having positive $\chi_{\mathrm{eff}}$, up to a maximum of around $\chi_{\mathrm{eff}}=0.5$, with no support for negative $\chi_{\mathrm{eff}}$ in agreement with observations (Roulet et al., 2021; Galaudage et al., 2021). For comparison, we also show the distribution of $\chi_{\mathrm{eff}}$ when using assumptions that more closely match the findings of Callister et al. (2021); in particular, we assume that the spin of the second born black hole ($\chi_{2}$) is drawn from the Gaussian distribution shown in Figure 1, and we assume that black hole kicks are drawn from a Maxwellian distribution with $\sigma=1000$ km/s (see Section 4 for details). This broadly reproduces the approximately symmetric $\chi_{\mathrm{eff}}$ distribution found by Callister et al. (2021), as shown in their Figure 3.

Whilst we do not explicitly compute the likelihood of the data under the model discussed here, Galaudage et al. (2021) found a $\chi_{\mathrm{eff}}$ distribution with a similar shape (shown in the bottom panel of Figure 2) to that found in our modified version of the Callister et al. (2021) model. The analysis of Galaudage et al. (2021) provides a substantially better match (as quantified by the Bayesian evidence) to the gravitational-wave observations than the Default spin model (Talbot & Thrane, 2017; Wysocki et al., 2019) used by Abbott et al. (2021e, c). We expect that a similar preference would be found compared to the model of Callister et al. (2021).

In the previous section we have shown that the phenomenological model for binary black hole formation from Callister et al. (2021) can be easily modified to more closely match the predictions of detailed binary evolution models (see lower panel of Figure 2). This modified model produces a distribution for the effective spin parameter $\chi_{\mathrm{eff}}$ that reproduces some of the features of the Galaudage et al. (2021) fit to current gravitational-wave observations. Some possible reasons for remaining differences are discussed in Section 5. We therefore argue that a model of binary evolution in which: (a) the progenitor of the second formed black hole in tight binaries are spun up through tides; (b) black holes formed in wide binaries are not spun up and have $\chi_{2}\sim 0$, and (c) black holes are formed with no associated kicks, naturally leads to an effective spin distribution that is compatible with gravitational-wave observations, and cannot currently be ruled out based solely on the effective spin distribution.

However, even though Galaudage et al. (2021) find no evidence for binary black holes with negative $\chi_{\mathrm{eff}}$, they still find some evidence for misaligned (precessing) black hole spins, with typical values for spin tilts of around $\cos\theta_{2}\sim 0.7$ (see Figure 3). If we assume that black hole kicks are the only mechanism capable of explaining this misalignment, then this implies that black holes receive kicks at birth. We therefore allow for black hole kicks drawn from a Maxwellian with $\sigma=100$, $300$ and $1000$ km/s, motivated by observations of Galactic X-ray binaries (Atri et al., 2019, e.g., and the results of Callister et al. (2021); see also the discussion of other observational evidence for black hole kicks in Callister et al., 2021). We assume isotropically distributed kick directions, and compute the post-supernova orbital characteristics in the standard way (Kalogera, 1996; Callister et al., 2021). As stated above, we assume that the spins are aligned prior to the second supernova, such that the spins of both objects are misaligned by the same amount ($\cos\theta_{1}=\cos\theta_{2}$). We show the distribution of misalignment angles $\cos\theta_{2}$ in Figure 3. We find that kick distributions with $\sigma=100$–$300$ km/s produce qualitatively similar distributions of black hole spin-orbit misalignments to those inferred by Galaudage et al. (2021), although the agreement is clearly not perfect. We have not tried adjusting other parameters in the Callister et al. (2021) model in order to improve the agreement; we suggest some avenues for future exploration in Section 5. Whilst most binaries still have their spins preferentially aligned with the orbital angular momentum ($\cos\theta_{2}\sim 1$), there is a tail of systems with misalignment angles $\cos\theta_{2}\lesssim 0.7.$ In this case, a small fraction of systems have $\cos\theta_{2}<0$ and $\chi_{\mathrm{eff}}<0$. For larger values of $\sigma$ (for example, $\sigma=1000$ km/s) we find that a large fraction of binaries that remain bound have $\cos\theta_{2}<0$, in contrast to the findings of Galaudage et al. (2021). While the tail of the $\cos\theta_{2}$ distribution is sensitive to the distribution of black hole kicks, we find that the $\chi_{\mathrm{eff}}$ distribution is not.

A recent study by Callister et al. (2021) suggested that black holes must receive unexpectedly large kicks at birth (greater than 260 km/s at 99% confidence) to reconcile the observed spin distribution of merging binary black holes with the observed sample, assuming that all binaries were formed through the classic isolated binary evolution channel (Belczynski et al., 2016; Stevenson et al., 2017a). This is much higher than the typical kicks that black holes are expected to receive (Atri et al., 2019), leading Callister et al. (2021) to consider alternate explanations.

We have argued that this may result arise due to an unfounded assumption made by Callister et al. (2021). In particular, Callister et al. (2021) assume that the second formed black hole in a binary black hole system can be tidally spun up, regardless of the orbital separation. Their best fit effective spin distribution has a Gaussian-like shape, peaked at slightly positive $\chi_{\mathrm{eff}}$, with a tail of negative $\chi_{\mathrm{eff}}$ systems, similar to the results obtained by Abbott et al. (2021e).

In binaries with orbital periods greater than about 1 day, tidal synchronisation is inefficient and slowly rotating black holes are formed (Qin et al., 2018; Bavera et al., 2020, 2021a). We have modified the model from Callister et al. (2021) to account for this, using fits to detailed binary evolution models from Bavera et al. (2021a). We show that this leads to a bimodal spin magnitude distribution where most binary black holes will have $\chi_{2}\sim 0$ (see Figure 1). It is unsurprising that the Gaussian spin model used by Callister et al. (2021) failed to capture such a distribution, and is likely the cause of their error. The bimodal spin magnitude distribution also results in a more complicated effective spin distribution (see Figure 2) than found by Callister et al. (2021). Galaudage et al. (2021) have recently shown that a similar effective spin distribution (as shown in Figure 2) provides a much better description (as quantified through the Bayesian evidence) of the observed binary black hole population than the model used by Abbott et al. (2021e), and that there is no evidence for binaries with negative $\chi_{\mathrm{eff}}$ (see also Roulet et al., 2021). Thus the evidence for large black hole kicks claimed by Callister et al. (2021) may simply be the result of a model misspecification.

We otherwise closely followed the assumptions made by Callister et al. (2021) regarding the properties of BH + He star binaries, and many of the caveats pertaining to that work also apply here. For example, the distribution of orbital periods of BH + He star binaries can be determined by factors such as whether mass transfer is predominantly stable (typically predicting wider binaries) or unstable, leading to common envelope evolution (Neijssel et al., 2019; Bavera et al., 2020, 2021b; Olejak et al., 2021; Gallegos-Garcia et al., 2021). As a demonstration of this we show the distribution of $\chi_{\mathrm{eff}}$ for the stable mass transfer and common envelope channels from the population synthesis models of Bavera et al. (2021b) in the lower panel of Figure 2.

We used the fits to detailed binary evolution models from Bavera et al. (2021a) to determine the spin of the secondary black hole. However, these models have their own uncertainties, such as the assumptions made about the initial rotation rates (prior to tidal spin up) of the helium stars (Qin et al., 2018; Bavera et al., 2021a; Renzo & Gotberg, 2021), and their mass-loss rates (Detmers et al., 2008; Qin et al., 2018).

We assumed that the distribution of black hole kicks is Maxwellian, following Callister et al. (2021). However, this should be seen as a starting point, as there is no strong theoretical or observational motivation for a distribution of this form. By the same token, a more complicated (e.g., bimodal) distribution is not justified by observations (Atri et al., 2019). Hence, we opted to use a Maxwellian distribution primarily for reasons of simplicity and consistency. Future analyses may also wish to incorporate the dependence of black hole kicks on the properties of the collapsing star (such as the expected anti-correlation with mass; Mandel & Müller, 2020, e.g.,), although again, this may not be justified observationally (Atri et al., 2019). If one were to include this theoretical expectation, one runs the risk of making the results strongly dependent on the model assumptions.

A logical next step to the analysis presented here would be to modify the phenomenological model from Callister et al. (2021) to account for the possibility of a bimodal spin magnitude distribution. There are several ways one might go about this. One could, for example, choose a similar model set up to that described by Callister et al. (2021), but with two arbitrarily defined subpopulations with separately modelled spin magnitude distributions; those of ‘close’ binaries in which tidal spin up is efficient, and ‘wide’ binaries where it is not. Both our analysis and previous work suggest that an orbital period of 1 day may be an appropriate dividing line between these subpopulations (Qin et al., 2018; Bavera et al., 2021a). Alternatively, one could choose to use the physical relation between orbital separations and spin magnitudes (as encoded in the Bavera et al., 2021a fitting formulae) and instead fit the distribution of orbital periods. Regardless, one could then use such a model to infer the typical magnitude of black hole kicks with a similar method to Callister et al. (2021).

In principle phenomenological models offer a method of examining the data with a model less dependent on specific choices made in modelling a particular astrophysical formation scenario. However, we have shown here that they are not without issue. While population synthesis models naturally capture correlations between various quantities (such as a possible anti-correlation between black hole mass and kick mentioned above, Mandel & Müller, 2020, e.g.,), such correlations are usually neglected in phenomenological models such as the one discussed here. Another correlation that may be important to account for is that between the orbital separation and the masses in BH + He star binaries depending on whether they form through stable or unstable mass transfer (Bavera et al., 2021b; Zevin et al., 2021; van Son et al., 2021). We have shown one example of how accounting for these correlations in a phenomenological model can influence the conclusions one draws from such a model.

In addition, this phenomenological model is broadly inspired by a single possible formation channel. Population synthesis models predict that even under the umbrella of isolated binary evolution, there are multiple formation pathways for binary black holes (e.g., Gerosa et al., 2018), such as through double core common-envelope events (Neijssel et al., 2019; Olejak & Belczynski, 2021), through close binaries experiencing chemically homogeneous evolution (Mandel & de Mink, 2016; Marchant et al., 2016) or through close binaries that experience mass transfer whilst the the primary is still on the main sequence, leading to rapidly rotating black holes (Neijssel et al., 2021). Each of these formation channels makes different predictions about the masses and orbital periods of BH + He star binaries. Even different models of the same formation channel may not fully agree on predictions for the properties of BH + He star binaries due to the many uncertainties in population synthesis models (e.g., Gallegos-Garcia et al., 2021). Future phenomenological models will need to account for these pathways in order to obtain unbiased inferences.

There is currently no evidence for black holes receiving large natal kicks ($\sim 1000$ km/s) in the spin distribution of merging binary black holes observed through gravitational-waves. The population is consistent with most binary black holes having their spins (approximately) aligned with the binary orbital angular momentum (cf. Figure 3). Given that some exceptional events such as GW190521 (Abbott et al., 2020a, c) and GW190814 (Abbott et al., 2020b) have properties that are difficult to explain through isolated binary evolution, there may be contributions to the binary black hole population from other formation channels (Bavera et al., 2021b; Zevin et al., 2021). For the remainder of the population, we have shown that at least the overall spin distribution is explainable through isolated binary evolution. There is preliminary evidence that black holes receive kicks of order $\sim 100$ km/s. In the future, it may be possible to use properties of the merging binary black hole population, such as the overall merger rate, along with the mass and spin distributions to constrain the magnitude of black hole kicks (e.g., Stevenson et al., 2015; Zevin et al., 2017; Barrett et al., 2018; Wong & Gerosa, 2019).

The results from Callister et al. (2021) are publicly available at this url. The results from Galaudage et al. (2021) are publicly available at this url. The results from Bavera et al. (2021a) are publicly available at this url. The results from Bavera et al. (2021b), as released by Zevin et al. (2021), are publicly available at this url. All data produced in this work is available from the author upon request.