Bayesian analysis of late-time tails in spin-aligned eccentric binary black hole mergers

TL;DR

This work analyzes late-time gravitational-wave tails from 15 spin-aligned eccentric binary black hole mergers in the extreme mass-ratio limit ($q=1000$) using high-accuracy point-particle BH perturbation theory. By solving the time-domain Teukolsky equation and decomposing the Weyl scalar $\Psi_4$ into spin-weighted modes, the authors quantify a power-law tail with exponent $p_{\mathrm tail}^{\ell m}=-(\ell+4)$, confirming theoretical predictions and revealing universality across $m$ for fixed $\ell$. They compare frequentist and Bayesian tail fits via the gwtails package, demonstrating that late-time windows yield accurate, near-asymptotic exponents, and that longer evolutions reduce biases present in earlier studies. The results enhance understanding of late-time GW relaxation, providing a robust, publicly available toolkit for tail analysis and guiding future work on tail modeling in more general (e.g., precessing) binary configurations.

Abstract

We present a comprehensive analysis of late-time tails in gravitational radiation from merging spin-aligned eccentric binary black holes, using high-accuracy point-particle black hole perturbation theory simulations. We simulate the late-time evolution of 15 binary black hole mergers with mass ratio $q = 1000$, dimensionless spins $χ= [-0.9, -0.6, 0.0, 0.6, 0.9]$ and eccentricity at the last stable orbit $e_{\rm LSO} = [0.8, 0.9, 0.95]$. We track the tail amplitudes and exponents up to a retarded time coordinate $t = 9000M$ after merger for the six spin-weighted spherical harmonic modes $(2,1)$, $(2,2)$, $(3,2)$, $(3,3)$, $(4,3)$, and $(4,4)$ employing both frequentist and Bayesian approaches. We note that the tails are increasingly pronounced for binaries with high eccentricity $e_{\rm LSO}$ and large negative spin $χ$. We find that the overall late-time exponents closely approach their predicted asymptotic values ($p=-\ell-4$ for Weyl curvature scalar $ψ_{4,\ell m}$ where $\ell$ is the spin-weighted spherical harmonic index), while estimates restricted to the latest portion of the data exactly recover them. We further verify numerically that modes with the same spherical index $\ell$ share identical tail exponents, while variations in $m$ do not affect the tail behavior. Our analysis framework is publicly available through the gwtails Python package.

Figures (10)

• Figure 1: We show the tail amplitudes (upper panel (a)) and the local tail exponents (panel (b)) for six representative modes, $[(2,1), (2,2), (3,2), (3,3), (4,3), (4,4)]$, corresponding to binaries with varying spins and eccentricities. The lines are color-coded according to the spin values. In panel (b), the expected asymptotic values are also shown as red horizontal dashed lines. We find that the tail exponents gradually approach their expected asymptotic values as the post-merger evolution progresses. More details are in Section \ref{['sec:result']}.
• Figure 2: We show the post-merger amplitudes and frequencies (inset) are shown for six representative modes, $[(2,1), (2,2), (3,2), (3,3), (4,3), (4,4)]$, corresponding to a binary characterized by $[\chi, e_{\rm LSO}] = [0.6, 0.8]$. For reference, we also indicate the best-fit exponents obtained through a Bayesian analysis performed with the gwtails package, which employs the emcee sampler. We find that the best-fit tail exponents are close to their expected asymptotic values. We find that the best-fit tail exponents are close to their expected asymptotic values. Furthermore, the post-merger frequencies $\omega_{\ell m}$ approach zero at different times for different spin-weighted spherical harmonic modes, indicating that the onset of the tail phase occurs at different times for each mode. More details are in Section \ref{['sec:result']}.
• Figure 3: We present the corner plot for the tail parameters $A_{\mathrm{tail}}^{\ell m}$, $p_{\mathrm{tail}}^{\ell m}$, and $c_{\mathrm{tail}}^{\ell m}$ obtained from Bayesian fits for four representative modes: $(2,1)$, $(2,2)$, $(3,3)$, and $(4,4)$. The results correspond to a binary characterized by $[\chi, e_{\rm LSO}, ] = [0.6, 0.8]$. The Bayesian analysis is performed using the gwtails package, which employs the emcee sampler for MCMC sampling. The analysis is performed over the time window $t = [1200, 8000]M$ for the $(2,1)$, $(2,2)$, and $(3,3)$ modes, while for the $(4,4)$ mode we restrict the fit to $t = [1200, 4000]M$ due to enhanced numerical noise at later times. The scatter points in each corner plot are color-coded according to their corresponding log-likelihood values. More details are in Section \ref{['sec:example_bbh']}.
• Figure 4: We show the local tail exponents $p(t)$ (Left) and histogram of the estimated tail exponent $p_{\rm tail}^{\ell m}$ using Bayesian sampler (Right) for six representative modes, $[(2,1),(2,2),(3,2),(3,3),(4,3),(4,4)]$, for a binary characterized by $[\chi, e_{\rm LSO}]=[0.6,0.8]$. The analysis is performed over the time window $t = [1200, 8000]M$ for the $(2,1)$, $(2,2)$, and $(3,3)$ modes, while for the $(4,4)$ mode we restrict the fit to $t = [1200, 4000]M$ due to enhanced numerical noise at later times. We find that not only are the tail exponents close to their expected asymptotic values, but modes with the same $\ell$ also yield sufficiently similar values. For $\ell = 3$, we observe a slight discrepancy between the $p_{\mathrm{tail}}^{\ell m}$ values inferred from the $(3,3)$ and $(3,2)$ modes. We attribute this difference to enhanced numerical noise present in the late-time portion of the waveform, which is also visible in the evolution of $p_{\rm local}(t)$. More details are in Section \ref{['sec:example_bbh']}.
• Figure 5: We show the $(2,2)$ mode tail amplitude along with the best-fit predictions obtained from different fit assumption and techniques for a binary with $[\chi, e_{\rm LSO}]=[0.6,0.8]$. The analysis is performed over the time window $t = [1200, 8000]M$. Furthermore, we show the residuals for each fit in the insets and report the corresponding reduced $\chi^2$ values in the legends. For reference, we also show the overall best-fit power-law color-coded to match the corresponding best-fit lines. More details are in Section \ref{['sec:example_bbh']}.