Statistical gravitational waveform models: What to simulate next?

Models of gravitational waveforms play a critical role in detecting and characterizing the gravitational waves (GWs) from compact binary coalescences. Waveforms from numerical relativity (NR), while highly accurate, are too computationally expensive to produce to be directly used with Bayesian parameter estimation tools like Markov-chain-Monte-Carlo and nested sampling. We propose a Gaussian process regression (GPR) method to generate reduced-order-model waveforms based only on existing accurate (e.g. NR) simulations. Using a training set of simulated waveforms, our GPR approach produces interpolated waveforms along with uncertainties across the parameter space. As a proof of concept, we use a training set of IMRPhenomD waveforms to build a GPR model in the 2-d parameter space of mass ratio $q$ and equal-and-aligned spin ${\ensuremath{\chi}}_{1}={\ensuremath{\chi}}_{2}$. Using a regular, equally-spaced grid of 120 IMRPhenomD training waveforms in $q\ensuremath{\in}[1,3]$ and ${\ensuremath{\chi}}_{1}\ensuremath{\in}[\ensuremath{-}0.5,0.5]$, the GPR mean approximates IMRPhenomD in this space to mismatches below $4.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$. Our approach could in principle use training waveforms directly from numerical relativity. Beyond interpolation of waveforms, we also present a greedy algorithm that utilizes the errors provided by our GPR model to optimize the placement of future simulations. In a fiducial test case we find that using the greedy algorithm to iteratively add simulations achieves GPR errors that are $\ensuremath{\sim}1$ order of magnitude lower than the errors from using Latin-hypercube or square training grids.

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