Eccentric binary black holes: A new framework for numerical relativity waveform surrogates

TL;DR

This work introduces a radial-phase reparameterization to decompose eccentric binary black hole waveforms, enabling high-accuracy surrogate modelling of inspiral, merger, and ringdown. By expressing the inspiral in the radial phase $\zeta$ and solving quasi-Keplerian PN equations to obtain $\zeta(t)$, the authors achieve highly compressible data pieces $A_{22}(\zeta)$ and $t(\zeta)$ that are modelled with Gaussian process regression, while a short-time merger-ringdown surrogate covers the final evolution in time. The resulting NRSurE$_{q4\text{NoSpin}}_{22}$ surrogate reproduces NR waveforms with maximum mismatches $\sim 5\times10^{-4}$ and median $\sim 2\times10^{-5}$ for non-spinning, eccentric binaries in the $(2,2)$ mode, enabling efficient parameter estimation for eccentric sources and advancing tests of general relativity. The method paves the way for longer, more accurate eccentric waveform models and can be extended to broader parameter spaces, including potential precession in future work.

Abstract

Mounting evidence indicates that some of the gravitational wave signals observed by the LIGO/Virgo/KAGRA observatories might arise from eccentric compact object binaries, increasing the urgency for accurate waveform models for such systems. While for non-eccentric binaries, surrogate models are efficient and accurate, the additional features due to eccentricity have posed a challenge. In this letter, we present a novel method for decomposing eccentric numerical relativity waveforms which makes them amenable to surrogate modelling techniques. We parameterize the inspiral in the radial phase domain, factoring out eccentricity-induced dephasing and thus enhancing compressibility and accuracy. This is combined with a second surrogate for the merger-ringdown in the time-domain and a novel technique to take advantage of the approximate periodicity with radial oscillations during the inspiral. We apply this procedure to the $(2,2)$ mode for non-spinning black hole binaries, and demonstrate that the resulting surrogate, NRSurE_q4NoSpin_22, is able to faithfully reproduce the underlying numerical relativity waveforms, with maximum mismatches of $5\times10^{-4}$ and median mismatches of $2\times10^{-5}$. This technique paves the way for high-accuracy parameter estimation with eccentric models, a key ingredient for astrophysical inference and tests of general relativity.

Figures (5)

• Figure 1: Incompressibility of data pieces. We show the amplitude of the $(2,2)$ mode $A_{22}(t)$ for sequences of waveforms generated using the surrogate model. From top to bottom, we first vary all 3 parameters (eccentricity $e$, mean anomaly $\ell$, and mass ratio $q$) together, then vary $e$, $\ell$, and $q$ individually while keeping the other parameters constant. Each sequence of waveforms is offset vertically and horizontally for clarity.
• Figure 2: Example inspiral waveform decomposition. Plotted are the $A_{22}(\zeta)$ and $t(\zeta)$ data pieces used for construction of the inspiral surrogate, for the same parameters as the top family in Fig. \ref{['fig:problem']}. Varying radial frequencies are captured by $t(\zeta)$, while relative phasing is handled by how the inspiral is stitched to merger. Here $t_0$ ($\zeta_0$) corresponds to the time (relativistic anomaly) at the final periastron passages considered for the inspiral surrogate. The inset of the bottom panel shows the difference $t(\zeta)-t^{\rm QC}(\zeta)$, where the time $t^{\rm QC}(\zeta)$ for a quasi-circular inspiral is obtained by evaluating the surrogate for the same $(q,\ell_{-1200M})$, but with $e_{-3000M}$ set to zero.
• Figure 3: Demonstration of mean-anomaly periodicity in the inspiral surrogate.Top: Illustration of how one could assign $\ell_{\rm gw}$ at $t=-1200M$ for three different reference periastron locations using one waveform. Bottom Left: The corresponding portion of waveform that would be used for each case, in time domain. For visibility, we vertically offset the curves from each other and plot only the final 4 radial periods that would be used. Bottom Right: The corresponding data pieces that are ultimately added to the surrogate training set, ensuring periodicity is captured for $\ell_{\rm gw}\in[0,2\pi]$.
• Figure 4: Demonstration of surrogate using new waveform decomposition.Left: Worst validation case waveform, with mismatch of $5\times10^{-4}$. The dashed orange line is the surrogate evaluation, while the solid blue line is the corresponding NR waveform. The bottom panel shows the absolute waveform error $\Delta h_{22} = |h_{22}^{\rm Sur} - h_{22}^{\rm NR}|$. Right: Comparison of surrogate mismatches with NR mismatches. The solid orange lines correspond to the leave-eight-out cross validation mismatches for the surrogate, while the blue dashed curves correspond to the mismatches between the two highest resolutions of each NR waveform. Also plotted are mismatches for a surrogate constructed using modern methods for quasi-circular systems. Vertical lines indicate medians.
• Figure 5: Example evaluations of the inspiral, merger and full IMR surrogate.Top: The real part of $h_{22}$ after evaluating the surrogate, performing the time shift and rotating the waveform. Middle: The same for the merger surrogate evaluation. Bottom: The full IMR waveform after stitching both component surrogates together.