Interpolation in waveform space: Enhancing the accuracy of gravitational waveform families using numerical relativity

Matched filtering for the identification of compact object mergers in gravitational wave antenna data involves the comparison of the data stream to a bank of template gravitational waveforms. Typically the template bank is constructed from phenomenological waveform models, since these can be evaluated for an arbitrary choice of physical parameters. Recently it has been proposed that singular value decomposition (SVD) can be used to reduce the number of templates required for detection. As we show here, another benefit of SVD is its removal of biases from the phenomenological templates along with a corresponding improvement in their ability to represent waveform signals obtained from numerical relativity (NR) simulations. Using these ideas, we present a method that calibrates a reduced SVD basis of phenomenological waveforms against NR waveforms in order to construct a new waveform approximant with improved accuracy and faithfulness compared to the original phenomenological model. The new waveform family is given numerically through the interpolation of the projection coefficients of NR waveforms expanded onto the reduced basis and provides a generalized scheme for enhancing phenomenological models.

Figure 1

The dashed line in the top panel traces the mismatch between the equal-mass, zero-spin NR and PhenomB waveforms of the same total mass $M$. The mismatch is reduced (solid line) by searching to find the mass $M^{\prime }$ for which $\mathcal{M}[{\mathbf{h}}_{\mathrm{NR}}(M),{\mathbf{h}}_{\mathrm{PB}}(M^{\prime })]$ is a minimum. We generally find that $M^{\prime }<M$, as shown in the bottom panel where the solid line traces the mass bias $(M-M^{\prime })/M$. For comparison, the dot-dashed curve in the bottom panel traces the mass spacing $(M_k-M_{k-1})/M_k$ for a template bank of PhenomB waveforms satisfying $\mathcal{M}[{\mathbf{h}}_{\mathrm{PB}}(M_{k-1}),{\mathbf{h}}_{\mathrm{PB}}(M_k)]=8^{-2}/2$.

Figure 2

Representation mismatch for the reduced SVD basis with expected tolerance $⟨\mathcal{M}⟩={10}^{-6}$ (traced by dotted line). Open squares (open circles) show the representation mismatch for PhenomB (NR) waveforms evaluated at the PhenomB template locations, while the thin solid line (thin dashed line) traces the representation mismatch of PhenomB (NR) waveforms evaluated between templates. The NR waveforms cannot be represented as well as their PhenomB counterparts, although their total match is improved over using PhenomB waveforms alone. This is evidenced by the thick solid line tracing the mass-optimized mismatch between NR and PhenomB waveforms (i.e., the solid line in the top panel of Fig. 1).

Figure 3

Convergence of representation mismatch with tightening tolerance $⟨\mathcal{M}⟩$. As a function of $⟨\mathcal{M}⟩$, we plot the averages of the four data sets shown in Fig. 2. The representation mismatch of the PhenomB waveforms for masses in the template bank used to construct the SVD basis decays roughly with $⟨\mathcal{M}⟩$. The representation mismatch of PhenomB waveforms for masses between the masses in the template bank reaches a plateau of $\sim {10}^{-5}$ [the precise value depends on ${\mathcal{O}}^{\prime }$, cf. Eq. (5)]. The representation mismatch of NR waveforms is yet larger with a plateau of $\sim {10}^{-3}$ that flattens for larger $⟨\mathcal{M}⟩$ (this flattening depends only mildly on ${\mathcal{O}}^{\prime }$).

Figure 4

Plotted are the real parts of the complex coefficients obtained from projecting NR waveforms onto the PhenomB SVD basis waveforms ${\mathbf{u}}_1$ (top panel), ${\mathbf{u}}_{50}$ (third panel), and ${\mathbf{u}}_{123}$ (fifth panel). The $x$ axis has been constructed according to Eq. (13) and is ideal for performing a Chebyshev interpolation. Open circles show the projection coefficients sampled at the collocation points in Eq. (15) for an $n=175$ Chebyshev interpolation; solid lines trace the resultant interpolation for each basis mode. Interpolation is performed separately on the real and imaginary parts of ${\mu }_k(x)$ and combined afterward to obtain the complex function ${\mu }_k^{\prime }(x)$. The small panels below each large panel show the absolute value of the interpolation residual $|r_k|$ defined in Eq. (16).

Figure 5

The solid line traces the interpolation error $R_k^2$ in Eq. (27) maximized over mass for each mode $k$. This is used in Eq. (28) to define an upper bound to the interpolation error arising in Eq. (26). The cumulative sum in the latter expression is traced by the dashed curve and shows that the interpolation error is dominated by the lowest-order basis waveforms.

Figure 6

Open circles show the total mismatch of our new waveform family obtained by interpolating the coefficients of NR waveforms projected onto a reduced SVD basis of PhenomB waveforms. To highlight the error introduced by interpolation, the solid curve traces only the mismatch of NR waveforms represented by the reduced basis (i.e., the dashed curve in Fig. 2). The total interpolation mismatch is lower than the dotted line tracing the mass-optimized mismatch between NR and PhenomB waveforms (i.e., the solid line in the top panel of Fig. 1) and demonstrates the ability of SVD to boost the accuracy of the PhenomB family.

Figure 7

Left panels: Real components of the smoothed PhenomB projection coefficients ${\tilde{\mu }}_3(M,q)$ (top) and ${\tilde{\mu }}_{16}(M,q)$ (bottom) for mass ratios $1\le q\le 6$ and total masses $50{\mathrm{M}}_{\odot }\le M\le 70{\mathrm{M}}_{\odot }$. Middle panels: Real components of the smoothed NR projection coefficients ${\tilde{\mu }}_3(M,q)$ (top) and ${\tilde{\mu }}_{16}(M,q)$ (bottom) for mass ratios $q=\{1,2,3,4,6\}$ and total mass $50{\mathrm{M}}_{\odot }\le M\le 70{\mathrm{M}}_{\odot }$. Right panels: Real components of the smoothed PhenomB projection coefficients ${\tilde{\mu }}_3(M,q)$ (top) and ${\tilde{\mu }}_{16}(M,q)$ (bottom) coarsened to the same set of mass ratios as the middle panels. For both the middle and rightmost panels, an artificial row of white space has been plotted for $q=5$ in order to ease comparison with the leftmost panels.