Bayesian inference from gravitational waves in fast-rotating, core-collapse supernovae

Core-collapse supernovae (CCSNe) are prime candidates for gravitational-wave detectors. The analysis of their complex waveforms can potentially provide information on the physical processes operating during the collapse of the iron cores of massive stars. In this work we analyze the early-bounce rapidly rotating CCSN signals reported in the waveform catalog of Richers et al. 2017. This catalog comprises over 1800 axisymmetric simulations extending up to about 10 ms of postbounce evolution. It was previously established that for a large range of progenitors, the amplitude of the bounce signal, $D·\mathrm{\Delta }h$, is proportional to the ratio of rotational-kinetic energy to potential energy, $T/|W|$, and the peak frequency, $f_{\mathrm{peak}}$, is proportional to the square root of the central rest-mass density, $\sqrt{{\rho }_{\mathrm{c}}}$. In this work, we exploit these relations to suggest that it could be possible to use such waveforms to infer protoneutron star properties from a future gravitational wave observation, but only if the distance and inclination are well known and the rotation rate is sufficiently low. Our approach relies on the ability to describe a subset of the waveforms in the early postbounce phase in a simple form—a master waveform template—depending only on two parameters, $D·\mathrm{\Delta }h$ and $f_{\mathrm{peak}}$. We use this template to perform a Bayesian inference analysis of waveform injections in Gaussian colored noise for a network of three gravitational wave detectors formed by Advanced LIGO and Advanced Virgo. We show that, for a Galactic event ($D\sim 10\mathrm{kpc}$), it is possible to recover the peak frequency and amplitude with an accuracy better than 10% for $\sim 80\%$ and $\sim 60\%$ of the signals, respectively, given known distance and inclination angle. However, inference on waveforms from outside the Richers catalog is not reliable, indicating a need for carefully verified waveforms of the first 10 ms after bounce of rapidly rotating supernovae of different progenitors with agreement between different codes.

Figure 1

Waveform amplitude, $D·\mathrm{\Delta }h$, vs peak frequency, $f_{\mathrm{peak}}$, for all models in the waveform catalog of [17]. Red (blue) markers indicate models included (excluded) in our analysis. The two kinds of models occupy regions of the parameter space that are mostly disjoint.

Figure 2

Strain as a function of time (left panel) and normalized strain as a function of the normalized time (right panel) for five illustrative examples. These waveforms have been selected from simulations that are as different as possible from each other to display the diversity of waveforms. The right panel shows the universality of the shape of the normalized waveform. The simulation model names have the form AXwY_Z, indicating X the degree of differential rotation, Y the value of ${\mathrm{\Omega }}_0$ in $\mathrm{rad}/\mathrm{s}$, and Z the EOS used (see more details in [17]).

Figure 3

Master template. Left panel shows $H_{\mathrm{T}}(\widehat{t})$ for the master template (red line) together with the 402 waveforms used for its construction. Right panel shows the same master template with the $2\text{−}\sigma$ ($\sim 95\%$) confidence interval computed using $\mathrm{\Sigma }(\widehat{t})$ (dashed lines).

Figure 4

Logarithm of the Bayes factor as a function of the distance of the injected signal. Both signal (blue and green markers) and noise (red dots) injections are plotted. The violet solid line (only visible in the bottom panel since it has negative values) and violet area display the mean value for noise injections and the 99% confidence interval (see main text for details on the noise model). The gray dashed line in the top plot shows the $1/D^2$ dependence. The lower panel shows a detailed view of the region with low values of the Bayes factor in linear scale.

Figure 5

Logarithm of the Bayes factor as a function of the network SNR for different types of injections. The dashed gray line shows a quadratic dependence with the network SNR. The violet area corresponds to values of ${\log }_{10}\mathcal{B}<0.5$, which approximately corresponds to the 99% confidence interval of the noise injections.

Figure 6

Corner plot of the posterior probability distribution for a master template injection. The network SNR of the signal is 27.5 and ${\log }_{10}\mathcal{B}=157$. Contours show $1\text{−}\sigma$, $2\text{−}\sigma$ and $3\text{−}\sigma$ levels. Dashed blue lines show the $1\text{−}\sigma$ confidence intervals, and orange lines correspond to the injected values. The correlation between $\psi$ and $D\mathrm{\Delta }h·{\sin }^2\theta$ is a result of the particular sky location and detector orientation.

Figure 7

Bayesian inference for master template injections. Upper panels: Median of the inferred posteriors for $f_{\mathrm{peak}}$ (left panel), $D·\mathrm{\Delta }h·{\sin }^2\theta$ (middle panel), and $\psi$ (right panel) as a function of the true value of the injected waveform. Colors indicate the logarithm of the Bayes factor. Values of ${\log }_{10}\mathcal{B}>100$ are indicated in red and ${\log }_{10}\mathcal{B}<0.5$ in gray. The blue solid line in the diagonal of all plots corresponds to equal true and inferred values. Lower panels: error in the inferred values in term of standard deviations. Horizontal dashed lines indicate the $2\text{−}\sigma$ interval.

Figure 8

P-P plot for master template injections: Observed cumulative probability distribution of the deviation of the inferred values from the injected values vs the expected cumulative distribution when comparing the posterior distribution of each injection with the corresponding injected value. For this analysis we consider a set on injections in which the parameter distribution of the injections matches the priors. Dashed black line indicates equal probability.

Figure 9

Corner plot of the posterior probability distribution for an example injection of the Richers et al. catalog. The network SNR of the signal is 57 and ${\log }_{10}\mathcal{B}=463$. Blue contours show the posterior distribution computed by the Bayesian analysis and black contours, the corresponding distribution after incorporating the error in the template. Contours show $1\text{−}\sigma$, $2\text{−}\sigma$, and $3\text{−}\sigma$ levels. Dashed black lines display the $1\text{−}\sigma$ confidence intervals, and orange lines correspond to the injected values.

Figure 10

Bayesian inference of $f_{\mathrm{peak}}$, $D·\mathrm{\Delta }h·{\sin }^2\theta$ and $\psi$ as a function of the true value of the injected waveform, when using injections from the Richers et al. catalog. See Fig. 7 for details.

Figure 11

Normalized Fourier transform of the bounce signal (between 5 ms before to 6 ms after bounce) as a function of the frequency normalized to the peak frequency, for all the waveforms used from the Richers et al. catalog (right) and for all the waveforms from Table 1 (left). The latter show in general a richer structure with multiple peaks compared to the former. Highlighted in red the waveform A634w3.00_SFHo_ecapture_0.1, mentioned in the text, which also has a secondary peak.

Figure 12

Bayesian inference of $f_{\mathrm{peak}}$, $D·\mathrm{\Delta }h·{\sin }^2\theta$ and $\psi$ as a function of the true value of the injected waveform, when using injections from Table 1. The description of the plots is as in Fig. 7.

Figure 13

Left panels: relative error in the inference of $f_{\mathrm{peak}}$ (top), $D·\mathrm{\Delta }h·{\sin }^2\theta$ (middle), and $\psi$ (bottom) as a function of the network SNR for both master template injections (green symbols) and injections from the Richers et al. catalog [17] (blue symbols). Dashed vertical lines mark the approximate detection threshold of $\mathrm{SNR}=20$. Right panels: fraction of injections at a given distance with a relative error smaller than a certain threshold (10%, 1% and 0.1%) for the three parameters studied.

Figure 14

Reconstructed (blue) and injected (orange) signal for the same case shown in Fig. 9. Purple region is the $2\text{−}\sigma$ confidence interval estimated with the error of the master templates. Cyan curves correspond to waveforms generated with 100 random samples of the posterior distribution (including template errors).

Figure 15

Bayesian inference of ${\rho }_{\mathrm{c}}$ and $T/|W|{\sin }^2\theta$ as a function of the true value of the injected waveform, when using injections from the Richers et al. catalog [17]. See Fig. 7 for details.

Figure 16

Upper panels show the parameters ${\mathrm{\Theta }}_{\mathrm{int},\mathrm{real}}$ for each template (real value) as a function of the value of the maximum likelihood of ${\mathrm{\Theta }}_{\mathrm{int},\mathrm{MT}}$ (MT value) for this template, for the parameters $f_{\mathrm{peak}}$ (left panel), and $D·\mathrm{\Delta }h·{\sin }^2\theta$ (right panel). Dashed lines indicate equal value of the MT and real values (identity model) and the blue line, the result of a linear regression (fit model). Lower panels show the corresponding relative errors considering the identity and fit models. Dashed lines show $2\text{−}\sigma$ confidence interval for the fit model.

Figure 17

Upper panels show the values of the maximum likelihood of ${\mathrm{\Theta }}_{\mathrm{int},\mathrm{MT}}$ (MT value) for a template with parameters ${\mathrm{\Theta }}_{\mathrm{int},\mathrm{phys}}$ (physical value), for the parameters ${\rho }_c$ (left panel), and $T/|W|{\sin }^2\theta$ (right panel). The blue line shows the result of a linear regression. Lower panels show the corresponding relative errors with respect to the fit. Dashed lines show the $2\text{−}\sigma$ confidence interval for the fit. The visible groups of points correspond to models with the same initial rotation rate but different rotation law.