Consistency of spin effects between numerical relativity and perturbation theory for inspiraling comparable-mass black hole binaries

Abstract

Numerical relativity (NR) provides the most accurate waveforms for comparable-mass binary black holes but becomes prohibitively expensive for increasingly asymmetric mass ratios. Point-particle black hole perturbation theory (ppBHPT), which expands the Einstein equations in the small-mass-ratio limit, offers a computationally efficient alternative but is expected to break down in the comparable-mass regime because it neglects nonlinear effects. Nonetheless, several recent studies have shown that ppBHPT can model non-spinning binaries with high accuracy when supplemented by simple calibrations or a first post-adiabatic (PA) correction. Here we assess the applicability of ppBHPT to quasi-circular binaries with a spinning primary by comparing waveform amplitudes, orbital frequencies, and orbital phases. We find that spin effects in ppBHPT waveforms (without additional spin information beyond adiabatic order) are in surprisingly close agreement with the corresponding NR calculation (outperforming some post-Newtonian models) over the last $\approx 20$ orbital cycles. This suggests that, after incorporating higher-order corrections into ppBHPT waveforms in the non-spinning limit -- via second-order self-force results or semi-analytical fits -- only modest spin-dependent adjustments may be required to achieve NR-faithful ppBHPT waveforms. We also show that combining non-spinning NR information with adiabatic ppBHPT can provide a reasonably accurate inspiral waveform for spins $χ\lesssim 0.5$ mass ratios $q \gtrsim 5$.

Figures (8)

• Figure 1: Ratios of waveform amplitudes and frequencies defined in Eq. \ref{['eq:sef']} for two representative binaries: $[q,\chi]=[8,-0.5]$ (upper panel) and $[q,\chi]=[6,0.4]$ (lower panel). The quantities shown are $R_{2,2}^{\mathrm{amp}}$, $R_{3,3}^{\mathrm{amp}}$, $R_{4,4}^{\mathrm{amp}}$, and $R_{2,2}^{\mathrm{freq}}$. Solid curves denote NR results from SXS simulations, while dashed curves give the corresponding adiabatic ppBHPT predictions. The adiabatic ppBHPT ratios closely track the NR results across the entire inspiral portion of the available waveforms. During the inspiral, the ppBHPT ratios remain in close agreement with NR, with deviations becoming more pronounced once the ppBHPT systems enter the geodesic plunge (gray shaded regions). The NR waveforms exhibit additional oscillations not present in ppBHPT, the origin of which is not yet understood (possibly related to residual eccentricity). The combinations of spinning and non-spinning SXS NR simulation IDs used to compute these ratios are indicated by color-coded labels.
• Figure 2: Ratios of $(2,2)$ waveform mode amplitudes defined in Eq. \ref{['eq:sef']} for three representative binaries with mass ratios $q = [5, 10, 15]$ and fixed spin $\chi=0.5$. Solid curves denote NR results from SXS simulations, while dashed lines indicate the corresponding ppBHPT results. During the inspiral, the ppBHPT ratios remain in close agreement with NR, with deviations becoming more pronounced once the ppBHPT systems enter the geodesic plunge (gray shaded regions). As in Fig. \ref{['fig:R_nr_bhpt_example']}, the NR waveforms display additional oscillations absent in ppBHPT. We also mention the corresponding combination of spinning and non-spinning SXS NR simulation ID, as color-coded text, used to compute these quantities.
• Figure 3: Dephasing between NR and ppBHPT-based orbital phases is shown for the NR simulation SXS:BBH:2185 with $[q,\chi]=[6,0.4]$. The NR-ppBHPT dephasing (solid blue) increases significantly during the inspiral. In contrast, combining post-adiabatic phase information from the non-spinning case with the adiabatic ppBHPT spin phase reduces the dephasing by 1 to 2 orders of magnitude (solid green), demonstrating the effectiveness of spin information at adiabatic order. The residual $\delta \phi_{\rm orb}^{\rm PA,spin}$, defined in Eq. \ref{['eq:pa_est']}, captures the remaining phase error from neglecting higher-order spin-dependent PA terms. Horizontal lines provide reference thresholds of $\pi/4$ (dashed red) and $\pi/16$ (dotted black). The shaded gray region indicates the system's transition from inspiral to geodesic plunge. Additionally, we show the dephasing between the two highest-resolution NR simulations as a benchmark (solid purple line).
• Figure 4: We show the estimated phase correction that arises from neglecting higher-order spin-dependent PA terms, defined in Eq.(\ref{['eq:pa_est']}), for all five BBHs considered in Figure \ref{['fig:R_nr_bhpt_example']} and \ref{['fig:q_5_10_15']}. Horizontal lines provide reference thresholds of $\pi/4$ (dashed red) and $\pi/16$ (dotted black). Shaded gray region indicates the time span from the end of the adiabatic inspiral through the transition and into the geodesic plunge in the ppBHPT waveforms. Additionally, we show the average dephasing between the two highest-resolution NR simulations for the five BBH systems (cyan-shaded region) as a benchmark.
• Figure 5: We show the percentage error in the $(2,2)$ mode amplitude and frequency spin-enhancement ratios from Eq. \ref{['eq:sef']} -- comparing spinning and non-spinning waveforms -- computed using NR and ppBHPT for a total of 43 NR simulations obtained from the SXS catalog (upper panel). The corresponding estimated PA phase corrections associated with spin effects are shown in the lower panel. Additionally, we show the average dephasing between the two highest-resolution NR simulations for each mass ratio (cyan-shaded region) as a benchmark.