Inferring Eccentricity of Binary Black Holes from Spin-Orbit Misalignment

TL;DR

The paper tackles the difficulty of directly measuring BBH orbital eccentricity by proposing an indirect method: the natal kick from the second supernova imprints a spin-orbit misalignment that encodes the formation eccentricity. By relating the tilt angle $\theta$ to the kick via $v_k/v_{\text{orb}} = \tan\theta$ and deriving $e_f = \frac{1}{\cos^2\theta} \cdot \frac{m_1+M_2}{m_1+m_2} - 1$ (with $e_{f,\min}=\tan^2\theta$), the authors translate GW-measured spin tilts into $e_f$ constraints. Applying the framework to GW190412 and GW241011 under isolated binary evolution yields meaningful, though tilt-dependent, constraints on formation eccentricity and natal kick magnitudes, with GW241011 offering notably tighter constraints. The study also discusses future prospects: higher-SNR detectors and multiband observations with LISA can further sharpen $e_f$ measurements and, by recovering formation redshift $z_f$ and birth separation $a_f$, illuminate BBH birth environments and core-collapse physics.

Abstract

Orbital eccentricity remains one of the least accessible parameters in observations of binary black hole (BBH) systems, largely erased by gravitational radiation long before detection. We introduce a new method to recover this lost parameter by using a more accessible and routinely measurable quantity: spin-orbit misalignment. In isolated binary evolution, a natal kick from the second supernova both tilts the orbital plane and injects orbital eccentricity, forging a direct and quantifiable connection between spin-tilt and post-supernova eccentricity. By measuring this spin-tilt using gravitational waves, we can not only constrain the natal kick, but we can also reconstruct the binary's formation eccentricity. We apply this method to GW190412 and GW241011, assuming an isolated formation channel, and show how their eccentricity at formation can be constrained even in the absence of direct eccentricity measurements. As more advanced detectors come online, improved signal-to-noise ratios will tighten spin-tilt constraints, allowing more precise and reliable estimates of BBH formation eccentricity. Combining this method with multiband observations from LISA and next-generation (XG) detectors will allow us to recover not only eccentricity but also the binary's orbital separation and redshift at formation, offering a clearer picture of the birth environments of BBH systems and processes that drive their merger.

Figures (4)

• Figure 1: Inferred posterior distribution of the formation eccentricity ($e_f$) of GW190412 (orange) and GW241011 (green), shown as a function of the $m_2/M_2$ (representing mass loss during the second supernova). The violin plots show the probability density of $e_f$ for $m_2/M_2 = 0.25$, $0.5$, $0.75$, and $1.0$ (no mass loss). The continuous shaded regions represent the $90\%$ confidence intervals for $e_f$ across all $m_2/M_2$. The dashed lines, corresponding to the right y-axis, plot the survival probability, $p(e_f < 1)$.
• Figure 2: Posterior distribution of the inferred natal kick velocity ($v_k$) imparted to the secondary black hole in GW190412 (orange) and GW241011 (green). Left panel: $v_k$ as a function of the fractional mass ratio $m_2/M_2$, assuming a fixed formation redshift of $z_f = 3$. Violin plots show the posterior density of $v_k$ at representative values of $m_2/M_2 = 0.25$, $0.5$, $0.75$, and $1$. Right panel: $v_k$ as a function of the formation redshift $z_f$, assuming $m_2 = M_2$. Violin plots show the posterior density at $z_f = 1/3$, $1$, $3$, and $10$. The continuous shaded regions represent $90\%$ confidence intervals for $v_k$ for full range of $m_2/M_2$ and $z_f$.
• Figure 3: Projected constraints on the BBH formation eccentricity ($e_f$) as a function of improved detector sensitivity. We model a GW190412-like binary with a spin-tilt of $30^\circ$ and assumes no mass loss, for which the theoretical eccentricity is $e_f = 1/3$ (vertical dashed line). Each curve represents the posterior probability density function (PDF) for $e_f$ at a different signal-to-noise ratio (SNR), indicated by the SNR gain $\rho/\rho_0$ with respect to GW190412. As the SNR increases, the constraint on the spin-tilt tightens, resulting in a progressively narrower and more sharply peaked PDF for the eccentricity.
• Figure 4: Posterior distributions for the inferred formation parameters of a BBH system, assuming multiband GW observations for two scenarios for a BBH with masses $m_{1} = m_{2} = 30 M_{\odot}$ merging at $z_{m} = 0.1$: one with formation eccentricity $e_{f} = 0.5$ and formation orbital separation $a_{f} = 10 R_\odot$ (orange), and another with $e_{f} = 0.8$ and $a_{f} = 20 R_\odot$ (teal). The violin plots show the probability distributions for the natal kick velocity ($v_{k}$), the orbital separation at formation ($a_{f}$), and the formation redshift ($z_{f}$). The dashed vertical lines represent the true values for each parameter.