Inferences about supernova physics from gravitational-wave measurements: GW151226 spin misalignment as an indicator of strong black-hole natal kicks

TL;DR

This paper uses gravitational-wave data from GW151226 to infer that the black hole's spin misalignment suggests it received a strong natal kick during stellar collapse, challenging existing supernova models.

Contribution

It introduces a method to constrain black hole natal kicks using gravitational-wave spin measurements, revealing larger kicks than previously assumed.

Findings

Black hole in GW151226 likely received a natal kick >50 km/s.

Such kicks are larger than those in standard supernova models.

Spin misalignment indicates significant black hole natal kicks.

Abstract

The inferred parameters of the binary black hole GW151226 are consistent with nonzero spin for the most massive black hole, misaligned from the binary's orbital angular momentum. If the black holes formed through isolated binary evolution from an initially aligned binary star, this misalignment would then arise from a natal kick imparted to the first-born black hole at its birth during stellar collapse. We use simple kinematic arguments to constrain the characteristic magnitude of this kick, and find that a natal kick $v_k \gtrsim 50$ km/s must be imparted to the black hole at birth to produce misalignments consistent with GW151226. Such large natal kicks exceed those adopted by default in most of the current supernova and binary evolution models.

Figures

Equations

$$\cos \gamma =\hat{\mathbf{L}}\cdot {\hat{\mathbf{L}}}_f=\frac{(\mathbf{v}+{\mathbf{v}}_k)\cdot \hat{\mathbf{v}}}{\sqrt{(\mathbf{v}+{\mathbf{v}}_k\cdot \hat{\mathbf{v}}{)}^2+({\mathbf{v}}_k\cdot \hat{\mathbf{L}}{)}^2}}.$$

$$F({\mathbf{v}}_k)=2\beta -1-\frac{|{\mathbf{v}}_k{|}^2}{v^2}-2\frac{{\mathbf{v}}_k\cdot \mathbf{v}}{v^2}.$$

$$P(\gamma <{\gamma }_{\ast })=\frac{\int d{\mathbf{v}}_{k}p({\mathbf{v}}_k)\Theta [{\gamma }_{\ast }-\gamma ({\mathbf{v}}_k)]\Theta [F({\mathbf{v}}_k)]}{\int d{\mathbf{v}}_{k}p({\mathbf{v}}_k)\Theta [F({\mathbf{v}}_k)]}$$

$$⟨P(\gamma <{\gamma }_{\ast })⟩=\int$$

$$\times p(P_{\mathrm{orb}})d{m}_{1}d{m}_{2}d{P}_{\mathrm{orb}}$$

Full text

Introduction– The Laser Interferometer Gravitational Wave Observatory (LIGO) has reported the discovery of two binary black holes (BHs): GW150914 and GW151226 Abbott et al. (2016a) (LIGO Scientific Collaboration and Virgo Collaboration). The masses and inferred birth rate of these events are consistent with prior predictions Belczynski et al. (2016a, 2010); Dominik et al. (2012); Mandel and de Mink (2016); Marchant et al. (2016), derived by assuming these objects form from the evolution of isolated pairs of stars; see, e.g., Abbott et al. (2016b) (LIGO Scientific Collaboration and Virgo Collaboration). At this early stage, observations cannot firmly distinguish between this formation channel and other proposed alternatives, such as the formation of binary BHs in densely interacting clusters Rodriguez et al. (2016a), in gas disks McKernan et al. (2017), or as primordial BHs Bird et al. (2016). If, however, binary BHs do form from isolated binary evolution, then precise measurements of their properties provide unique clues into how BHs and massive stars evolve.

Assuming BH binaries form from initially aligned binary stars (i.e., all angular momenta are parallel), the most likely processes that can misalign their spin angular momenta are the linear momentum recoils imparted when a BH’s progenitor star ends its life in a supernova (SN) Gerosa et al. (2013); Kalogera (2000). Observations strongly suggest that asymmetries in the SN process can indeed impart strong natal kicks to newly formed compact objects. Based on the proper motion measurements of pulsars in the Milky Way, it is believed that supernovae (SNe) can impart velocities as high as $v_{k}\sim 450$ km/s to neutron stars Hobbs et al. (2005). Conversely, the occurrence of natal kicks onto BHs is less clear. On the one hand, observations of galactic $x$-ray binaries suggest that BH natal kicks may be as large as hundreds of km/s Repetto et al. (2012); Repetto and Nelemans (2015); Repetto et al. (2017); Belczynski et al. (2016b); Martin et al. (2010); Wong et al. (2012, 2014). On the other hand, natal kicks onto heavier BHs could be significantly reduced, as their very massive progenitor stars are expected to undergo prompt collapse and not eject enough material to enable strong recoils (see, e.g., Janka (2012) and references therein). Measurements of natal kicks through electromagnetic observations have already been proved crucial to understand the physics of SNe. For instance, if BH kicks are indeed as large as those imparted to neutron stars, this would require large-scale asymmetries of the SN ejecta Janka (2013, 2017), or anisotropic neutrino emission during collapse Lai and Qian (1998); Repetto and Nelemans (2015); Fryer and Kusenko (2006).

Gravitational wave (GW) measurements of merging binary BHs have the potential to provide crucial insights on this issue. SN kicks can reach (or even exceed) the expected orbital velocities of the stellar binary from which binary BHs formed with dramatic effects on its formation and evolution. Strong natal kicks disrupt many potential compact binary progenitors (thus affecting the expected GW rates Belczynski et al. (2016a); Belczyński and Bulik (1999)) and drastically tilt the orbital plane of the few that survive (which greatly affects the spin precession dynamics by the time the source becomes visible in LIGO Gerosa et al. (2013); Kalogera (2000)). Several previous studies have demonstrated that the GW signature of BH spin-orbit misalignments can be efficiently identified O’Shaughnessy et al. (2014); Vitale et al. (2014); Veitch et al. (2015); Trifirò et al. (2016) and used to distinguish between formation channels Gerosa et al. (2013); Vitale et al. (2017); Stevenson et al. (2017). We point out two examples. First, LIGO provides strong constraints on a quantity that is both nearly conserved on astrophysical time scales Racine (2008); Kesden et al. (2015); Gerosa et al. (2015) and of key astrophysical interest: the effective spin $\chi_{\rm eff}=\hat{\mathbf{L}}\cdot(\mathbf{S}_{1}/m_{1}+\mathbf{S}_{2}/m_{2})/(m_{1}+m_{2})$, where $m_{1,2}$ and $\mathbf{S}_{1,2}$ are the masses and spins of the component BHs, and $\mathbf{L}$ is the binary’s orbital angular momentum (we used natural units $G=c=1$). BH binaries assembled in densely interacting environments have random spin orientations and thus $\chi_{\rm eff}$ is frequently negative, while binaries formed in isolation from initially aligned stellar progenitors are expected to be found with positive effective spin Rodriguez et al. (2016b). Second, for binaries formed in isolation, the azimuthal projection of the BH spins onto the orbital plane $\Delta\Phi$ was found to directly track the occurrence of mass transfer and tidal spin alignment between the stellar progenitors Gerosa et al. (2013); Trifirò et al. (2016); Gerosa et al. (2014).

In this Letter, we use simple kinematic arguments to draw conclusions about the strength of SN kicks from the reported observation of GW151226. This is the less massive of the two confirmed GW detections, where nonzero natal kicks are more likely. Leaving complicated binary evolution physics aside, we show how to translate the spin misalignments reported by LIGO into concrete constraints on the strength of the first SN kick. This approach already proved successful. Assuming misaligned jets observed in $x$-ray binaries Hjellming and Rupen (1995); Orosz and Bailyn (1997); Orosz et al. (2001) are good indicators of the BH spin direction, Martin et al. (2010) used similar kinematic arguments to constrain the natal kick imparted to the microquasar GRO J1655-40. They find the observed misalignment $\gtrsim 10^{\circ}$ can only be explained with a kick of few tens of km/s.

Observations of GW151226– The LIGO and Virgo Collaborations characterized GW151226 as a binary BH, with component masses $14.2_{-3.7}^{+8.3}M_{\odot}$ and $7.5_{-2.3}^{+2.3}M_{\odot}$ Abbott et al. (2016c) (LIGO Scientific Collaboration and Virgo Collaboration). The right panel of their Fig. 4 provides a posterior distribution on the magnitude and orientation of the two BH spins, relative to the orbital angular momentum. Their analysis suggests both that the more massive BH likely had nonzero spin and, critically, that this spin was most likely modestly misaligned with the orbital angular momentum, with a misalignment angle $\gamma$ ranging between $25^{\circ}$ and $80^{\circ}$.

Because of significant precession, the spin-orbit misalignments that LIGO directly measures and reports, corresponding to GW frequencies of $20~{}{\rm Hz}$, in principle must be evolved backwards in time to identify the spin orientations when the BHs first formed Kesden et al. (2015); Gerosa et al. (2015). Although this process turns out to be crucial to extract astrophysical information from full GW data, its details are not important for this study where we only focus on loose constraints on the measured spin direction (i.e. $25^{\circ}\lesssim\gamma\lesssim 80^{\circ}$). Moreover, in the simple assumption adopted here where additional alignment processes (such as tidal interactions) are neglected, previous work showed there is no net tendency to align or antialign the BH spins Gerosa et al. (2013). This point will be specifically addressed in future work.

Formation and misalignment of GW151226 from isolated evolution– GW151226 could have formed from the evolution of a pair of isolated massive stars “in the field” Abbott et al. (2016a) (LIGO Scientific Collaboration and Virgo Collaboration). Concrete formation scenarios for this event can be easily extracted from exhaustive simulations of binary evolution over cosmic time Belczynski et al. (2016a) [the evolutionary scenarios described here are drawn from the publicly available “Synthetic Universe” (www.syntheticuniverse.org)]. As a representative example, GW151226 could have formed from a pair of $53M_{\odot}$ and $25M_{\odot}$ stars, initially in a relatively close and modestly elliptical orbit with semimajor axis $R=4000R_{\odot}$; as the stars evolve and the more massive star transfers and loses mass, the binary evolves to a $22M_{\odot}$ helium star and a $26M_{\odot}$ companion in a modestly tighter and circularized orbit of $900R_{\odot}$; the primary then undergoes a SN explosion, losing a small amount of mass to form a $19.7M_{\odot}$ BH. The kick following this first explosion tilts the orbital plane, changing relative alignment between the orbital plane and the BH’s spin direction – presumed to be parallel to the preexplosion orbital angular momentum. Subsequent phases of stellar interaction, notably, when the BH spirals through the envelope of the secondary star, stripping it and leaving behind a helium core while itself accreting $0.455M_{\odot}$, cause the binary to progress to a much tighter circular orbit of a few $R_{\odot}$ prior to the second SN. Because the common-evelope phase typically shrinks the orbital separation of a factor $\gtrsim 100$, the orbital velocity $v=\sqrt{GM/R}$ (where $M$ is the binary’s total mass) at the second SN event is typically an order of magnitude larger than the velocity prior to the first; in this case, $R=6.6R_{\odot}$ and $v=1090{\rm km/s}$. Since the effect of the kick onto the binary only depends on the ratio $v_{k}/v$ (see below), this second SN has a minimal impact on the misalignment of the orbital angular momentum Gerosa et al. (2013). If SN kicks are indeed responsible for the observed misaligned primary BH in GW151226, it is likely this formed during the first SN. Moreover, the first-born BH accretes too little matter to appreciably change its angular momentum direction, even during the common-envelope phase Belczynski et al. (2007, 2008); King and Kolb (1999); Natarajan and Pringle (1998).

Spin-orbit misalignment from natal kicks– The orbital-plane tilt angle introduced by the first SN kick can be calculated using simple Newtonian kinematics Gerosa et al. (2013); Kalogera (2000). For simplicity, here we only study the typical case in which strong binary interactions like tides and mass transfer have circularized the orbit and aligned the stellar spins before the first SN Hut (1981). We likewise assume for simplicity that the initially most massive object undergoes the first SN explosion, and that the SN explosion itself does not torque the BH. If $\mathbf{r}=\mathbf{r}_{2}-\mathbf{r}_{1}$ is the relative orbital separation, $\mathbf{v}=d\mathbf{r}/dt$ is the orbital velocity and $\mathbf{v}_{k}$ is the imparted kick velocity, then the orbital angular momentum per unit reduced mass changes from $\mathbf{L}/\mu=\mathbf{r}\times\mathbf{v}$ to $\mathbf{L}_{f}/\mu_{f}=\mathbf{r}\times(\mathbf{v}+\mathbf{v}_{k})$, where $\mu_{f}\neq\mu$ because of mass loss during the explosion. The orbital plane tilt $\gamma$ reads

[TABLE]

Assuming the spin of the collapsing star $\mathbf{S}$ was aligned to the orbital angular momentum before the explosion (i.e. $\mathbf{\hat{S}}=\mathbf{\hat{L}}$), $\gamma$ also equals the spin misalignment angle of the newly formed BH. If the kick imparted by the explosion is sufficiently large, the post-SN eccentricity exceeds unity and the binary does not remain bound. If $\beta=M_{f}/M$ denotes the fraction of total mass retained by the binary after the explosion, disruption occurs if $F(\mathbf{v}_{k})<0$ where

[TABLE]

Finally, the cumulative distribution of the misalignment angle $\gamma$ between pre and post-SN angular momenta can be expressed as

[TABLE]

where $\Theta(x)$ is the Heaviside step function and $p(\mathbf{v}_{k})$ is the kick velocity probability distribution. For simplicity, in the following we assume that $p(\mathbf{v}_{k})$ is an isotropic Maxwellian distribution characterized by a single one-dimensional width $\sigma$ (corresponding to a mean square velocity $\langle v_{k}^{2}\rangle=3\sigma^{3}$), as found for neutron stars Hobbs et al. (2005). Motivated by the formation scenario illustrated above, we assume modest mass loss in SN explosions, adopting $\beta=0.98$ as a representative example of the narrow range of $\beta$ found in typical population-synthesis studies ($0.95-1$) Belczynski et al. (2016a); we stress that this choice does not significantly influence our results.

Because the dimensionless quantities $\gamma$ and $F$ depend on natal kicks only through the ratio $\mathbf{v}_{k}/v$, the probability $P(\gamma<\gamma_{*})$ depends on $\sigma$ only through the dimensionless ratio $\sigma/v$. In the limit of large $\sigma/v$, the distribution of misalignments among surviving binaries approaches a nearly uniform distribution, i.e., $P(\gamma<\gamma_{*})\simeq\gamma_{*}/\pi$. The left panel of Fig. 2 shows the misalignment distribution pertinent to GW151226 (i.e., $25^{\circ}<\gamma<80^{\circ}$), as a function of the unknown dimensionless kick magnitude $\sigma/v$; for comparison, horizontal lines show the range of misalignments implied by the LIGO observations. On the right, we show the probability of a kick misalignment that is both consistent with these limits and does not unbind the orbit. Only modest SN kicks of $\sigma\gtrsim 0.5v$ allow a wide range of spin-orbit misalignments consistent with GW151226.

To convert from a relative to an absolute velocity scale, we adopt a distribution of progenitor masses and separations consistent with GW151226 and with observations of massive stars Belczynski et al. (2016a); Sana et al. (2012). We assume that the binary is circular; the primary mass is drawn from a power-law distribution $p(m_{1})\propto m_{1}^{-2.35}$ between $30M_{\odot}$ and $100M_{\odot}$, $m_{2}$ is drawn from a uniform distribution between $20M_{\odot}$ and $m_{1}$, and the orbital period $P_{\rm orb}$ is drawn from a distribution $p(P_{\rm orb})\propto(\log P_{\rm orb}/{\rm day})^{-0.5}$, with limits set by twice the radius of the stars of interest ($R=40R_{\odot}$) and by the maximum radius of one of the two stars’ giant phase ($R=3\times 10^{3}R_{\odot}$). We then compute the ensemble-averaged cumulative probability distribution

[TABLE]

For simplicity, we neglect mass transfer before the first SN and assume that all binaries which survive the first SN kick are equally likely to form a binary BH similar to GW151226. To the extent it holds, our calculations can be applied to generic binary BHs formed from isolated evolution, not just GW151226.

Figure 2 shows the distribution of kick misalignments as a function of $\sigma$. As expected given the characteristic velocity of bound orbits of massive stars, a natal kick of at least $\sqrt{\langle v_{k}^{2}\rangle}\simeq 45$ (62) km/s must be imparted to the first-born BH to obtain the misalignment of GW151226 in $5\%$ ($10\%$) of the realizations. If BH natal kicks are as large as those imparted to neutron stars ($\sigma\simeq 265$ km/s Hobbs et al. (2005)), up to $\sim 39\%$ of our realizations are found consistent with the observed spin misalignment.

Distinguishing from alternative models– Coalescing binary BHs could form in dense interacting environments, where the spin and orbital angular momentum directions are randomized, i.e. $P(\gamma<\gamma_{*})=\cos\gamma_{*}/2$. The right panel of Fig. 2 also shows a horizontal line corresponding to the probability that a randomly oriented binary will lie within the region observed for GW151226. Field binaries with $0.5\lesssim\sigma/v\lesssim 1.5$ have a higher probability to produce misalignment consistent with GW151226 than binaries formed through dynamical interactions. As pointed out in Rodriguez et al. (2016b), modest SN kicks cannot produce an isotropic spin distribution. As $\sigma$ increases, the misalignment distribution becomes uniform in $\gamma$, below the randomly oriented result (which predicts a distribution uniform in $\cos\gamma$). However, strong SN kicks $\sigma\gg 2v$ on BHs both disrupt most field binaries and eject BHs from globular clusters, dramatically reducing the merger rate and creating difficulties for any stellar-evolution-based formation scenarios. While SN kicks can more easily explain the observed spin-orbit misalignments for the particular case of GW151226, observations of the spin misalignment distribution from many future events are crucial to support or rule out different formation scenarios.

The Kullback-Leibler (KL) divergence $D_{KL}=\int d\gamma p(\gamma)\ln[p(\gamma)/q(\gamma)]$ provides a measure of the difference between two distributions $p(\gamma),q(\gamma)$, and hence the number of detections needed before we can distinguish between models (i.e., $N\simeq 1/D_{KL}$) O’Shaughnessy (2013). We can calculate the KL divergence between the isotropic spin misalignment distribution and the distributions implied by any $\sigma/v$ shown in Fig. 2 or any $\sigma$ shown in Fig. 2. Even loosely accounting for measurement error (e.g., using the width of the distribution of GW151226 as an estimate of the relative misalignment accuracy), we find that $\mathcal{O}(10)$ events similar to GW151226 are needed to distinguish between an isotropic distribution and a distribution misaligned by natal BH kicks, in agreement with other estimates Stevenson et al. (2017); Farr et al. (2017)

Discussion– LIGO should detect several hundred more binary BHs over the next five years Abbott et al. (2016d) (LIGO Scientific Collaboration and Virgo Collaboration); Abbott et al. (2016a) (LIGO Scientific Collaboration and Virgo Collaboration). These observations will support or rule out whether binaries are born with spin strictly aligned with their orbital angular momentum or obtained significant misalignment from natal kicks. They will also provide strong constraints on the strength of such kicks.

Relatively low-mass binaries like GW151226 provide the simplest, cleanest laboratory to study the impact of SN kicks. First and foremost, the explosions that form them are not expected to result from direct collapse Belczynski et al. (2016a), so some residual linear momentum will be imparted to the ejected material and the BHs. Second, low-mass, unequal-mass-ratio binaries like GW151226 accumulate many precession cycles prior to merger in LIGO’s sensitive band Apostolatos et al. (1994). Third, this regime of precessing inspiral is relatively well modeled theoretically Apostolatos et al. (1994); Lundgren and O’Shaughnessy (2014); Hannam et al. (2014); Taracchini et al. (2014); Kesden et al. (2015); Gerosa et al. (2015), and accessible with current parameter-estimation techniques Veitch et al. (2015); Trifirò et al. (2016); Vitale et al. (2017). LIGO has therefore the best chance to make precise measurements about misalignment for low-mass binaries, where the merger phase is relatively unimportant. By contrast, for more massive BHs like GW150914, fewer cycles are available in the LIGO band and the merger phase becomes crucial Abbott et al. (2016e) (LIGO Scientific Collaboration and Virgo Collaboration); Abbott et al. (2016f) (LIGO Scientific Collaboration and Virgo Collaboration). Phenomenological models that approximate full solutions of Einstein’s equations are known to omit important physics, which can in turn lead to biases when these models are applied to parameter estimation Abbott et al. (2017) (LIGO Scientific Collaboration and Virgo Collaboration). Robust spin-orbit misalignment measurements for heavy BHs will require improved waveform modeling, more extensive use of numerical relativity data Abbott et al. (2016e) (LIGO Scientific Collaboration and Virgo Collaboration); Blackman et al. (2015), and incremental improvements in low-frequency GW detector sensitivity.

The natal kicks required to explain the misalignment of GW151226 are in excess of the fallback-suppressed kicks adopted by default in current binary evolution models Belczynski et al. (2016a, b); Rodriguez et al. (2016b) (though note models M4, M5, and M6 in Belczynski et al. (2016a)). Notably, these natal kicks are consistent with observations of recoil velocities Repetto et al. (2012); Repetto and Nelemans (2015); Repetto et al. (2017); Belczynski et al. (2016b), and jet misalignments Hjellming and Rupen (1995); Orosz and Bailyn (1997); Orosz et al. (2001); Martin et al. (2010) of Galactic $x$-ray binaries.

For isolated binary evolution models, a modest increase in SN kicks diminishes the expected event rate – more binary BHs are disrupted by the first SN – but otherwise produces predictions for the population of merging binary BHs that are consistent with existing observations Belczynski et al. (2016a, c). The impact of recent physically motivated prescriptions that relate natal kick magnitude and ejected mass Bray and Eldridge (2016); Janka (2013) has yet to be fully explored with large-scale population-synthesis studies.

Large natal kicks $\sqrt{\langle v_{k}^{2}\rangle}\gtrsim 50$ km/s that must be imparted to BHs of mass $\gtrsim 15M_{\odot}$ at formation could be a significant challenge for SN physics. For example, one of the leading models used to explain the kicks imparted to neutron stars invokes gravitational attraction by the newly formed compact object of some of the material ejected asymmetrically during the explosion (the so-called “gravitational tug-boat mechanism” Janka (2013, 2017)). While this requires significant and quite asymmetric mass ejection, many of the formation scenarios explored for GW151226 assume very modest mass loss ($\beta\sim 0.98$), with most of the material falling back on to (and slowing down) a protoneutron star core that later collapses to a BH (see, e.g., Belczynski et al. (2012); Fryer et al. (2012)). On the other hand, neutrino-driven kicks do not require significant mass loss.

Our analysis assumes that SN kicks provide the principal mechanism for binary spin-orbit misalignment in field binaries. Alternatively, binaries could be born with primordial spin-orbit misalignment Albrecht et al. (2014), or gain comparable misalignment early in their life via either interactions with a tertiary companion Albrecht et al. (2014) or core-envelope interactions Paxton et al. (2013). If such misalignment can persist or grow during the long lifetime and many interactions necessary to form a coalescing BH, then LIGO observations might be an indicator of primordial spin misalignment processes. Large-scale surveys of binary stars can determine if such spin-orbit misalignments occur frequently. Conversely, substantial accretion onto the BH could also align the BH spin with the orbit after the first SN Natarajan and Pringle (1998).

Acknowledgements– We thank Krzysztof Belczynski, Emanuele Berti, Michael Kesden, Will Farr, Daniel Holz, and Gijs Nelemans for carefully reading our manuscript, and our anonymous referees for their helpful feedback. R.O’S. and D.G. gratefully acknowledge the hospitality of the Aspen Center for Physics, supported by NSF Grant No. PHY-1066293, where this work was initiated. R.O’S. is supported by NSF Grants No. AST-1412449, No. PHY-1505629, and No. PHY-1607520. D.G. is supported by NASA through Einstein Postdoctoral Fellowship Grant No. PF6-170152 awarded by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for NASA under Contract No. NAS8-03060.