Reduced basis representations of multi-mode black hole ringdown gravitational waves

Abstract

We construct compact and high accuracy Reduced Basis (RB) representations of single and multiple quasinormal modes (QNMs). The RB method determines a hierarchical and relatively small set of the most relevant waveforms. We find that the exponential convergence of the method allows for a dramatic compression of template banks used for ringdown searches. Compressing a catalog with a minimal match MM = 0.99, we find that the selected RB waveforms are able to represent any QNM, including those not in the original bank, with extremely high accuracy, typically less than 10 −13 . We then extend our studies to two-mode QNMs. Inclusion of a second mode is expected to help with detection, and might make it possible to infer details of the progenitor of the final black hole. We find that the number of RB waveforms needed to represent any two-mode ringdown waveform with the above high accuracy is smaller than the number of metric-based, one-mode templates with MM = 0.99. For unconstrained two-modes, which would allow for consistency tests of General Relativity, our high accuracy RB has around 10 4 fewer waveforms than the number of metric-based templates for MM = 0.99. The number of RB elements grows only linearly with the number of multipole modes versus exponentially with the standard approach, resulting in very compact representations even for many multiple modes. The results of this paper open the possibility of searches of multi-mode ringdown gravitational waves.

Fast and accurate inference on gravitational waves from precessing compact binaries

“…A variety of ROM-type techniques have recently appeared in the GW literature [10,[12][13][14][15][16][17]. We shall use a combination of the reduced-basis method and the empirical interpolation method, whose favorable computational efficiency, ease-of-parallelization and numerical stability make them attractive candidates for tackling precessing waveform systems and other challenging models.…”

Fast Prediction and Evaluation of Gravitational Waveforms Using Surrogate Models

“…Recent work [39][40][41][42] has shown that for fixed but arbitrary physical and parameter ranges, a small number of basis functions is indeed sufficient to accurately represent any waveform of the same physical model and with an exponentially decaying greedy error (9). Such observations are expected for functions with smooth parameter dependence, as is the case with gravitational waveforms.…”

“…whereas waveforms from T (even if not in the training set) continue to be well approximated by the RB if the training set is dense enough [39][40][41][42]. Since the waveform space is numerically finite dimensional [39], one can verify sufficiently dense training sets through convergence as M gets larger or by checking how well the basis represents randomly selected waveforms (see Appendix A).…”

“…For an underlying model that requires prohibitively expensive numerical solutions, one may use a simpler model to propose a training set building strategy. If there is sufficient similarity amongst the members of the original set, then m ≪ M. This is found to be the case for gravitational waveforms [39,41,42]. Let ϵ be a user-specified tolerance whose role is to guarantee that the approximation error for waveforms in the training set, which we will call the greedy error σ m , is bounded by ϵ,…”

“…Work over the last few years has shown that gravitational waveforms exhibit redundancy in the parameter space [37][38][39][40][41][42], suggesting that the amount of information necessary to represent a fiducial waveform model is smaller than might be anticipated. This reduction can be captured accurately using only a remarkably few number m of representative waveforms.…”