Chase Orbits, not Time: A Scalable Paradigm for Long-Duration Eccentric Gravitational-Wave Surrogates

Abstract

Surrogate modeling of eccentric binary black hole waveforms has remained challenging. The complicated morphology of these waveforms due to the eccentric orbital timescale variations makes it difficult to construct accurate and efficient surrogate models, especially for waveforms long enough to cover the sensitivity band of the current ground-based gravitational wave detectors. We present a novel and scalable surrogate building technique which makes surrogate modeling of long-duration eccentric binary black hole waveforms both feasible and highly efficient. The technique aims to simplify the harmonic content of the intermediate eccentric surrogate data pieces by modeling them in terms of an angular orbital element called the mean anomaly, instead of time. We show that this novel parameterization yields an order of magnitude fewer surrogate basis functions than using the contemporary parameterization in terms of time. We show that variations in surrogate data-pieces across parameter space become much more regular when expressed in terms of the instantaneous waveform eccentricity and mean anomaly, greatly easing their parameter-space fitting. The methods presented in this work make it feasible to build long-duration eccentric surrogates for the current as well as future third-generation gravitational wave detectors.

Figures (7)

• Figure 1: Amplitude $A_{22}$, eccentric residual amplitude $\Delta A$, residual phase $\Delta \phi_1$, and detrended residual phase $\Delta \phi$ (c.f. Eq. \ref{['eqn: amp-phase-decomp']}--\ref{['eqn: residual-circular-phase']}) for a few representative $20,000M$ long eccentric InspiralESIGMA waveforms. Also shown are the amplitudes for the corresponding quasi-circular systems as solid black lines.
• Figure 2: Number of basis functions required to achieve a greedy error threshold Field:2013cfa of $10^{-5}$ for representing the training spaces of the eccentric residual amplitude $\Delta A$ and phase $\Delta \phi$ for different surrogate lengths for time-parameterized (dashed orange) and mean anomaly parameterized (solid blue) surrogate methodologies. The mean anomaly parameterization achieves the same accuracy with an order of magnitude fewer basis functions. The minimum binary mass for which each surrogate can be evaluated from $15$Hz across its parameter space is indicated at the top, with additional metrics listed in Table \ref{['tab: surrogate-metrics']}.
• Figure 3: Variation of time/mean-anomaly period of oscillations $P_t(\Delta A)$/$P_l(\Delta A)$ and $P_t(\Delta \phi)$/$P_l(\Delta \phi)$ in $\Delta A$ and $\Delta \phi$ respectively as a function of time for a representative sample of 5 binaries. Their time period of oscillations secularly decreases due to the gradual inspiral of the binary (orange solid and light blue dashed curves), which is the typical chirp of a GW signal. This secular decrease is also encoded in the rate of change of mean anomaly angle of the system (pink dash-dot curve), as evidenced by the mutual overlap of these three sets of curves. Therefore, the oscillation period of $\Delta A$ and $\Delta \phi$ in terms of the mean anomaly for any binary system becomes constant (solid lines; they are vertically offset for visual clarity). In this manner, the chirping behavior of the $\Delta A$ and $\Delta \phi$ oscillations can be factored away into the non-oscillatory time evolution of mean anomaly, simplifying their harmonic content significantly.
• Figure 4: Mismatches of the $2.77 \times 10^6M$ long mean anomaly parameterized surrogate (c.f. Table \ref{['tab: surrogate-metrics']}) computed against $10,000$InspiralESIGMA waveforms randomly sampled across the surrogate parameter space for binaries of different masses. The mismatches are computed over the full surrogate length, assuming zero-detuning high-power noise power spectral density for the advanced LIGO detector LIGO-T0900288-v3LIGO-T070247-v1. The dashed lines show the median mismatch values.
• Figure 5: Waveform evaluation time of the base model InspiralESIGMA (orange distribution), and the corresponding speedup (blue distribution) and the median evaluation time (blue dots) of its $2.77 \times 10^6M$ long mean anomaly parameterized surrogate for different binary masses. All the waveforms are generated from $15$Hz at a sampling rate of $4096$Hz at 1000 points randomly drawn across the parameter space of the surrogate for each binary mass. The lowest total mass shown corresponds to the smallest value for which the surrogate can be evaluated across its entire parameter space from $15$Hz. Markers also indicate the respective median evaluation times/speedup. The study was performed on an AMD EPYC 7352 processor operating at 2.3 GHz.