Sparse Representations of Gravitational Waves from Precessing Compact Binaries

Many relevant applications in gravitational wave physics share a significant common problem: the seven-dimensional parameter space of gravitational waveforms from precessing compact binary inspirals and coalescences is large enough to prohibit covering the space of waveforms with sufficient density. We find that by using the reduced basis method together with a parametrization of waveforms based on their phase and precession, we can construct ultracompact yet high-accuracy representations of this large space. As a demonstration, we show that less than 100 judiciously chosen precessing inspiral waveforms are needed for 200 cycles, mass ratios from 1 to 10, and spin magnitudes $\le 0.9$. In fact, using only the first 10 reduced basis waveforms yields a maximum mismatch of 0.016 over the whole range of considered parameters. We test whether the parameters selected from the inspiral regime result in an accurate reduced basis when including merger and ringdown; we find that this is indeed the case in the context of a nonprecessing effective-one-body model. This evidence suggests that as few as $\sim 100$ numerical simulations of binary black hole coalescences may accurately represent the seven-dimensional parameter space of precession waveforms for the considered ranges.

Figure 1

Left: maximum $\phi$-domain ($\phi \mathit{D}$) projection error (red) from (9) for $7\mathrm{D}$ post-Newtonian precession waveforms versus basis size. The contributions from the time function (dotted), waveform in the minimally rotating frame (dashed), and quaternion (solid) are also shown. Right: maximum time-domain, inertial frame mismatches from (10) for ${10}^7$ randomly selected waveforms ($+$) using the first 10, 20, and 50 RB waveforms. Also plotted are $\phi$-domain projection errors for non-precessing PN waveforms (dashed) and the time-domain (TD) projection errors from using the latter parameter values selected by the greedy algorithm to represent EOB waveforms (solid black), which additionally include merger and ringdown phases.

Figure 2

Mass ratios (top), $x\text{−}y$ components of both spins (second row), projection of inertial frame spins onto initial orbital angular momentum unit vector (bottom left), and both spin magnitudes (bottom right) selected by our greedy algorithm.