Revisiting the Coprecessing Frame in the Presence of Orbital Eccentricity

Abstract

Accurate inclusion of both spin precession and orbital eccentricity effects in gravitational waveform models represents a key hurdle in our ability to fully characterize the properties of compact binaries. Virtually all efforts to model precession rely on a coprecessing frame transformation, a time-dependent spatial rotation that tracks the dominant emission direction and simplifies the waveform morphology. We assess the utility of the coprecessing frame transformation to separate out the effect of the precession of the orbital plane from the waveform in the presence of non-negligible orbital eccentricity. We rely on 20 numerical relativity simulations, which include the complete physical effects of spin precession and eccentricity in the strong-field, and compare waveforms in both the inertial and coprecessing frames. Comparing against the eccentric, spin-aligned model SEOBNRv5EHM, we find that while the waveform mismatches decrease in the coprecessing frame, they remain above the level required for accurate waveform modeling, $\sim$ 0.01 or higher for large inclinations. Further improvements, e.g., modeling mode asymmetries as already pursued for quasicircular binaries, will likely prove essential. We also find that by removing the dominant amplitude and phase modulations from the waveform, the coprecessing frame facilitates surrogate modeling, achieving lower errors at a fixed number of basis elements compared to the inertial frame. Our results demonstrate both the utility and the limitations of the coprecessing frame as a cornerstone in waveform modeling for eccentric and precessing binaries.

Figures (6)

• Figure 1: Effect of the coprecessing frame on waveform modes, the $(2,2)$ mode (left column) and $(2,1)$ mode (right column), for four different binary configurations: eccentric and precessing (top row), precessing and quasicircular (second row), non-precessing and eccentric (third row), and quasicircular and non-precessing (bottom row). In the non-precessing cases, the coprecessing frame leaves the waveform modes identical, but for the top two panels it simplifies the precession modulations and restores the mode hierarchy. The binary parameters are that of the SXS waveform SXS:BBH:4286, with the full NR waveform shown in the top row, while the other three rows use waveforms with the same parameters but removing the in-plane spins or eccentricity as appropriate, generated using either SEOBNRv5PHMRamos-Buades:2023ehm or SEOBNRv5EHMGamboa:2024hli. In the aligned spin cases we also plot the waveform mode envelopes, to show the effect of the eccentricity on the waveform. Waveforms are plotted as a function of time in units of the total mass $M$.
• Figure 2: Mismatch $\mathcal{M}\mathcal{M}$ between the NR simulations from Tab. \ref{['tab:NRSimsParams']} and the best-fitting SEOBNRv5EHM waveform, for $\iota=0$ (left), $\iota=\pi/4$ (middle), and $\iota=\pi/2$ (right). Results are plotted as a function of the NR eccentricity $e_{\text{gw}}$ and coloured by the value of the precessing spin $\chi_p$ at the simulation reference time. Mismatches between SEOBNRv5EHM and NR in the inertial frame are shown by circles, while the equivalent mismatches for the same NR simulations transformed into the coprecessing frame are shown with grey triangles; mismatches in the two frames for the same simulation are joined by a grey dotted line. The red box marks simulation SXS:BBH:3714, examined further in Fig. \ref{['fig:3714strain']}. The eccentricity $e_{\text{gw}}$ is computed from the NR waveforms, as detailed in Sec. \ref{['subsec:NRSims']}, and it is the same in the two frames.
• Figure 3: NR and best-fit SEOBNRv5EHM waveforms (plus polarization) as a function of time in the inertial (top panel) and coprecessing (bottom panel) frames for simulation SXS:BBH:3714 and an inclination of $\iota=\pi/4$. This case is marked with a red box in Fig. \ref{['fig:strainmismatches']} and demonstrates one of the largest mismatch reductions among all cases considered.
• Figure 4: Mismatches $\mathcal{M}\mathcal{M}$ in both frames (main panel) and mismatch difference between the inertial and coprecessing frame (top panel) as a function of $\chi_{\text{p}}$ (at the simulation reference time) and for $\iota=\pi/4$. These are the same data as the middle panel of Fig. \ref{['fig:strainmismatches']}, now plotted as a function of the precessing spin.
• Figure 5: Eccentricities of the NR simulation ($e_{\text{NR}}$) and the best-fitting SEOBNRv5EHM waveform ($e_{\text{SEOBNRv5EHM}}$), in both the inertial (coloured circles) and coprecessing (grey triangles) frames for $\iota=\pi/4$. The circles are coloured by the value of the mismatch difference between the inertial and coprecessing frames. The red box marks simulation SXS:BBH:3714, examined further in Fig. \ref{['fig:3714strain']}, where the eccentricity remains similar despite a large mismatch improvement. The blue box marks a case (SXS:BBH:3711), where the eccentricity becomes more different than the NR one after the coprecessing transformation.