Likelihood smoothing using gravitational wave surrogate models

Likelihood surfaces in the parameter space of gravitational wave signals can contain many secondary maxima, which can prevent search algorithms from finding the global peak and correctly mapping the distribution. Traditional schemes to mitigate this problem maintain the number of secondary maxima and thus retain the possibility that the global maximum will remain undiscovered. By contrast, the recently proposed technique of likelihood transform can modify the structure of the likelihood surface to reduce its complexity. We present a practical method to carry out a likelihood transform using a Gaussian smoothing kernel, utilizing gravitational wave surrogate models to perform the smoothing operation analytically. We demonstrate the approach with Newtonian and post-Newtonian waveform models for an inspiralling circular compact binary.

Figure 1

Gravitational waveform from a compact binary with chirp mass $\mathcal{M}=2.2M_{\odot }$ in the final 0.01 s before coalescence. Inset is the same waveform for a period of 0.4 s before coalescence. The amplitude is set by the requirement that $(h|h)=1$.

Figure 2

The representation error of our RB for the 0PN waveform family as a function of the number of elements in the basis.

Figure 3

The phase of the gravitational waveform, evaluated at the first empirical node, as a function of chirp mass. The dots show the values of the phase evaluated at the greedy points selected while constructing the RB. The line is a fit to the data, which in this case is exact.

Figure 4

The smoothed waveform (26), computed for our injection parameters with a chirp mass $\mathcal{M}=2.2M_{\odot }$. Thicker lines correspond to larger values of the smoothing width $\sigma$. The $\sigma =0$ waveform is identical to that in Fig. 1.

Figure 5

The smoothed SNR (18), computed for our injection parameters with a chirp mass ${\mathcal{M}}_{*}=2.2M_{\odot }$, as a function of smoothing width.

Figure 6

The smoothed SNRs (18), computed across the allowed range of chirp masses, for a selection of smoothing widths. The inset plot shows the behavior around the injection value ${\mathcal{M}}_{*}=2.2M_{\odot }$. Thicker lines correspond to larger values of $\sigma$.

Figure 7

The likelihood function computed using the smoothed waveform model for different values of the smoothing width. Thicker lines correspond to larger values of $\sigma$.

Figure 8

Distribution of final chirp masses after a short MCMC, using likelihood functions with different amounts of smoothing. The narrowness of the peaks is indicative that the chains have located local maxima. The vertical grey dashed line is positioned at the true value ${\mathcal{M}}_{*}=2.2M_{\odot }$.

Figure 9

The final chirp mass, as a function of the seed mass, for MCMCs using different smoothed likelihood functions. The dashed sloped line is ${\mathcal{M}}_{\text{final}}={\mathcal{M}}_{\text{seed}}$, indicating chains that did not move far from their starting point. The dotted line ${\mathcal{M}}_{\text{final}}={\mathcal{M}}_{*}$ denotes the true value.

Figure 10

Distribution of final chirp masses for chains at different temperatures (dashed lines), compared to the smoothed chain (solid line).

Figure 11

PN gravitational waveform from a compact binary with chirp mass $\mathcal{M}=2.2M_{\odot }$ and symmetric mass ratio $\nu =0.18$ in the final 0.01 s before coalescence. Inset is the same waveform for a period of 0.1 s before coalescence. The amplitude is set by the requirement that $(h|h)=1$.

Figure 12

The representation error of the RB for the PN waveform family as a function of the number of elements in the basis.

Figure 13

The smoothed PN waveform, computed for our injection parameters with a chirp mass $\mathcal{M}=2.2M_{\odot }$ and symmetric mass ratio $\nu =0.18$. Thicker lines correspond to larger values of the smoothing width $\sigma$. The $\sigma =0$ waveform is identical to that in Fig. 11.

Figure 14

The smoothed SNR for the PN waveform family, computed for our injection parameters with a chirp mass ${\mathcal{M}}_{*}=2.2M_{\odot }$ and symmetric mass ratio ${\nu }_{*}=0.18$, as a function of smoothing width.

Figure 15

The smoothed log-likelihoods for our PN waveform family using a selection of smoothing widths. The black ellipses are centered on the injection values and are sized according to the smoothing width.