Fast frequency-domain phenomenological modeling of eccentric aligned-spin binary black holes

TL;DR

IMRPhenomXE delivers a fast, frequency-domain IMR waveform for eccentric, non-precessing BBHs by extending the quasi-circular IMRPhenomXAS baseline and incorporating eccentric inspiral dynamics via orbit-averaged 3PN quasi-Keplerian evolution with spin effects. It uses a stationary-phase approximation on an eccentricity-expanded time-domain PN waveform (up to $^{12}$) to produce a frequency-domain $(2,2)$ mode, while merger–ringdown remains anchored to the quasicircular XAS description. Validation against 186 eccentric NR waveforms shows unfaithfulness typically below $3\%$ for $e_0\lesssim0.4$, with accuracy degrading at higher eccentricities due to expansion convergence and SPA limits; computational benchmarks demonstrate superior speed relative to other eccentric IMR models. Bayesian inference studies, including zero-noise NR injections and analyses of GW150914, GW151226, and GW190521, confirm the model’s practicality for large catalogs, yielding parameter estimates consistent with previous results and no strong evidence for eccentricity in these events. IMRPhenomXE thus offers a ready-to-use, efficient tool for eccentric BBH gravitational-wave astronomy and paves the way for extensions to higher modes and spin-precession.

Abstract

We present the IMRPhenomXE frequency-domain phenomenological waveform model for the dominant mode of inspiral-merger-ringdown non-precessing binary black holes in elliptical orbits. IMRPhenomXE extends the quasi-circular IMRPhenomXAS waveform model for the dominant $(\ell, |m|) =$ (2,2) modes to eccentric binaries. For the inspiral part, orbit-averaged equations of motion within the quasi-Keplerian parametrization up to third post-Newtonian order, including spin effects, are evolved, and the waveform modes are computed using the stationary phase approximation on eccentricity expanded expressions up to $\mathcal{O}(e^{12})$. The model assumes circularization at merger-ringdown, where it adopts the underlying quasicircular IMRPhenomXAS baseline. We show that IMRPhenomXE reduces to the accurate IMPhenomXAS model in the quasi-circular limit. Compared against 186 public numerical relativity waveforms from the Simulating eXtreme Spacetimes catalog with initial eccentricities up to $~0.8$, IMRPhenomXE provides values of unfaithfulness below $3\%$ for $72\%$ of simulations with initial eccentricities below 0.4. For larger eccentricities, the unfaithfulness degrades up to $\gtrsim 10\%$ due to the underlying small eccentricity expansions and additional modelling approximations. In terms of speed, IMRPhenomXE outperforms any of the existing inspiral-merger-ringdown eccentric waveform models. We demonstrate the efficiency, robustness, and modularity of IMRPhenomXE through injections into zero noise and parameter-estimation analyses of gravitational-wave events, showing that IMRPhenomXE is a ready-to-use waveform model for gravitational-wave astronomy in the era of rapidly growing event catalogs.

Figures (9)

• Figure 1: Mismatch distribution in the effective-spin parameter and mass ratio plane between the IMRPhenomXAS, IMRPhenomXE and IMRPhenomT models for $10^5$ quasi-circular configurations randomly sampled within the IMRPhenomXAS and IMRPhenomT validity regions: mass ratio $q\in[1,20]$, total mass $M\in[10,200]\ M_{\odot}$, dimensionless spin components $\chi_i\in[-0.99,0.99]$, using an azimuthal phase $\varphi=0^\circ$ and inclination angle $\iota=0^\circ$. Each configuration is color-coded by its maximum value of the mismatch within the total mass range. From left to right, the panels correspond to the IMRPhenomXE-IMRPhenomXAS, IMRPhenomXE-IMRPhenomT and IMRPhenomXAS-IMRPhenomT comparisons.
• Figure 2: Parameter space distribution (in initial eccentricity $e_0$, mass ratio $q$, and effective-spin parameter $\chi_{\rm eff}$) for the 186 NR simulations from the public SXS catalog used in Sec. \ref{['sec:ecc_NRcomparison']}.
• Figure 3: Mismatches of the IMRPhenomXE and SEOBNRv5E models against the 186 SXS eccentric simulations in Fig. \ref{['fig:nr_param_space']}. From left to right, IMRPhenomXE mismatches computed with (N$_{\rm harm},e^X)=\{(25,e^{12}),(13,e^{12}),(13,e^{6})\}$, respectively, where N$_{\rm harm}$ corresponds to the number of mean anomaly harmonics and $e^X$ to the highest order in the eccentricity expansions considered in the waveform. The last panel corresponds to the SEOBNRv5E model. Each curve corresponds to a NR simulation containing the $(2,|2|)$-modes, color-coded by the initial eccentricity $e^{\mathrm{NR}}_0$ extracted from the metadata. The mismatches are calculated over a total mass range of $M\in[20,200]\ M_{\odot}$.
• Figure 4: Waveform comparison between the NR waveform SXS:BBH:2522 (blue) and the best-fitting IMRPhenomXE waveforms generated with 5 (orange) and 13 (green dashed) mean anomaly harmonics (N$_{\rm harm}$). Top panels show the frequency-domain amplitude of the $h_+$ polarization, while the lower row shows the time-domain $h_+$ polarization. In each row, the right panel zooms into the merger-ringdown of the waveform. The NR waveform corresponds to an equal-mass, non-spinnning configuration with initial eccentricity $e_0=0.4$ (see Table \ref{['tab:nr']} for details).
• Figure 5: Upper plot: Mismatch of IMRPhenomXE against the NR waveform SXS:BBH:2522 as a function of the mean anomaly harmonics, N$_{\rm harm}$, for time-domain amplitudes expanded up to $\mathcal{O}(e^{12})$. The horizontal dashed line corresponds to the mismatch of IMRPhenomXE evaluated with $e=0$. Lower panel: Mismatch of IMRPhenomXE against the NR waveform SXS:BBH:2527, as a function of the mean anomaly harmonics, N$_{\rm harm}$, and eccentricity order, i.e. $\mathcal{O}(e^j)$ with $j=3,5,...,25$, in the time-domain amplitudes. Each point is color-coded by its value of mismatch. In both panels the total mass is fixed to be $20 M_\odot$.