The Universal Eccentricity Distribution for Dynamical Gravitational-Wave Merger Channels

Abstract

We argue that all dynamical astrophysical black hole merger channels are expected to result in a common eccentricity distribution at gravitational wave (GW) frequencies relevant for LIGO/Virgo/KAGRA (LVK) in the high eccentricity limit. This follows from the large separation of scales between the GW regime required for creating eccentric mergers in LVK, and the underlying astrophysical formation environment. Our analytical solution shows exceptional agreement with numerical studies. This finding has important implications for both theoretical studies and ongoing searches for eccentric GW sources.

Figures (5)

• Figure 1: The Pinhole Regime.Top: For a given encounter to result in an eccentric merger, the two interacting BHs must pass each other at a relatively small pericentre distance. This restriction can be mapped to a corresponding small area at the encounter surface, which we refer to as the 'pinhole'. As the size of this area is generally much smaller than any of the underlying astrophysical scales, the distribution of BHs passing through this area is approximately random, which maps to our proposed universal eccentricity distribution, as described in Sec. \ref{['sec:universal']}. Bottom: At high eccentricity, all of the channels are expected to converge towards the same functional form, which is a result of the separation of scales. At lower eccentricity, the astrophysical environment will start to show up, however, LVK is not expected to resolve this limit yet with the current sensitivity.
• Figure 2: Analytical Eccentricity Distributions. The probability density function of the log of eccentricity, $P(\log e) \propto eP(e)$, as calculated in Eqs. \ref{['eq:P_peak']} and \ref{['eq:PT']} for the peak and orbital frequencies correspondingly (in dashed and dashed-dotted lines), compared to the asymptotic solution (solid line) and log-flat distribution (dotted line). The distributions will break below a characteristic eccentricity, $e_c$, set by the astrophysical environment, and at high eccentricity when finite effects are included (see Sec. \ref{['sec:universal']}).
• Figure 3: Astrophysical Eccentricity Distributions.Stellar cluster: Outcomes from 2.5PN binary-single interactions evaluated at $f_p = 50~\text{Hz}$ between equal mass BHs with $m=20 \ M_{\odot}$, and initial SMA $a_0 = 0.1\ \rm{AU},\ 0.01\ \rm{AU}$, are shown in solid red and blue, respectively. Open red in the Zoom-in figure shows the $a_0 = 0.1\ \rm{AU}$ case at $f_p = 20~\text{Hz}$. Triples: Outcomes from hierarchical triple evolution resulting in BBH mergers, as adopted and described in 2014ApJ...781...45A, are shown in grey for $f_p = 10~\text{Hz}$. Our derived general scaling $P(\log e) \propto e^{-12/19}$ is shown with a black dashed line. A log-flat uniform prior is shown with a black dotted line. Exceptional agreement is seen between the different channels and our predicted scaling. Note that each distribution is scaled to match at high eccentricity to ease comparison.
• Figure 4: The probability density function of eccentricity, as calculated in Appendix \ref{['appendix:Jacobian']}, at different orbital frequencies (solid blue lines), compared to log-flat distribution (dotted-dashed line), approximated solution (Eq. \ref{['eq:PT']}, dotted) and asymptotic (solid red line).
• Figure 5: Initial pericenter distance scaled by $R_s = 2GM/c^2$, $r/R_s$, as a function of eccentricity, $e_f$, derived using Eqs. \ref{['eq:r_EM']} and \ref{['eq:r_EM2']}. Results for $f_p$ and $f_T$ are shown in red and blue, respectively. For this figure we have assumed $M=40M_{\odot}$ and $f=20~\text{Hz}$.