Inferring physical properties of stellar collapse by third-generation gravitational-wave detectors

Galactic core-collapse supernovae are among the possible sources of gravitational waves. We investigate the ability of gravitational-wave observatories to extract the properties of the collapsing progenitor from the gravitational waves radiated. We use simulations of supernovae that explore a variety of progenitor core rotation rates and nuclear equations of state and examine the ability of current and future observatories to determine these properties using gravitational-wave parameter estimation. We use principal component analysis of the simulation catalog to determine the dominant features of the waveforms and create a map between the measured properties of the waveform and the physical properties of the progenitor star. We use Bayesian parameter inference and the parameter map to calculate posterior probabilities for the physical properties given a gravitational-wave observation. We demonstrate our method on a random sample of the waveform catalog that was excluded from construction of the principal component analysis and estimate the ratio of the progenitor’s core rotational kinetic energy to potential energy ($\beta$) and the post bounce oscillation frequency. For a supernovae at the distance of the galactic center (8.1 kpc) with $\beta =0.02$ our method can estimate $\beta$ with a 90% credible interval of 0.004 for Advanced LIGO, improving to 0.0008 for Cosmic Explorer, the proposed third-generation detector. We demonstrate that if the core is rotating sufficiently rapidly for a signal source within the Milky Way observed by Cosmic Explorer, our method can also extract the post bounce oscillation frequency of the protoneutron star to a precision of within 5 Hz (90% credible interval) allowing us to constrain the nuclear equation of state. For a supernovae at the distance of the Magellanic Clouds (48.5 kpc) Cosmic Explorer’s ability to measure these parameters decreases slightly to 0.003 for rotation and 11 Hz for the postbounce oscillation frequency (90% credible interval). Sources in Magellanic Clouds with $\beta <0.02$ will be too distant for Advanced LIGO to measure these properties.

Figure 1

Gravitational wave strain assuming the distance to the progenitor of 10 kpc as function of time for bounce and postbounce oscillation phases of a core-collapse process. The waveforms are zero buffered to make them 1 second long, and the time of bounce is aligned at 0.5 seconds for all the waveforms. The top panel shows the waveforms for the SFHx equation of state with varying rotation rates between $\beta =0.02$ and $\beta =0.06$. The strain amplitude at the bounce increases with increasing $\beta$, while the postbounce oscillation frequency remains almost the same for all the waveforms corresponding to a given equation of state. Bottom panel shows the waveforms for $\mathrm{\Omega }=2.50\mathrm{rad}/\sec$ and $A=467\mathrm{km}$ for the equations of state listed in Table 1. The bounce amplitude remains almost the same for the waveforms with the same core rotation rate, while the postbounce oscillation frequency varies for different equations of state.

Figure 2

Frequency of postbounce oscillations is plotted on the vertical axis against $\beta$ of the waveforms on the horizontal axis. The crosses represent the waveforms that are used to build the principal component basis. This also includes the green crosses, showing the waveforms that are affected the most by only considering 15 principal components and not more. The simulations that use the SFHx equation of state are shown in brown crosses. The $f_{\mathrm{peak}}$ value for a given equation of state is independent of $\beta$ for $0.02\le \beta \le 0.06$. The dashed lines represent the average $f_{\mathrm{peak}}$ values of the waveforms of a given equation of state in this range, also given in Table 1. The orange dots represent the parameter values of the waveforms that are used as astrophysical signals in this study.

Figure 3

The plot shows how well can a given number of principal components (plotted on the horizontal axis) reconstruct the original waveform. We quantify this by computing the overlap between the original waveform and the reconstructed waveform, and show it on the vertical axis. Each of the waveforms is represented by a gray line, and the mean overlap of all the waveforms as a function of number of basis vectors used for construction is represented by the red line.

Figure 4

The coefficients of the first four principal components as a function of $\beta$. The coefficient of the first principle component, ${\alpha }_1$ (shown in blue) is most strongly correlated with $\beta$, exhibiting a roughly linear relation. The correlation between the other three coefficients and $\beta$ can be seen to be weaker. The values of the coefficients spread as $\beta$ increases because of different equations of state used in simulation of the waveforms.

Figure 5

For each waveform in the catalog, a principal component basis set is constructed using all remaining waveforms. Using this basis set, $\beta ({\alpha }_1,\dots ,{\alpha }_k)$ maps are constructed using interpolation with the first $k=2,\dots ,8$ model parameters, and $\beta$ of the excluded waveform is estimated using these maps. Least square fit for a straight line is used while using just the first model parameter to construct the map $\beta ({\alpha }_1)$. The median error in reconstructing $\beta$ through various maps and the respective failure rate in interpolation are plotted on the vertical axes. Using more number of model parameters reduces the error in interpolation, however increases the number of times the interpolation fails.

Figure 6

The ${\alpha }_2$ (vertical axis) vs ${\alpha }_1$ (horizontal axis) parameter plane for the waveforms in the catalog. The colorbar shows the $\beta$ corresponding to each of the waveforms. The two dimensional convex hull of the all the points is shown by the dashed black line. Interpolation fails for a point outside the convex hull. We can construct a three-dimensional convex hull if we also incorporate ${\alpha }_3$. We constrain our MCMC samples to be within the three dimensional convex hull.

Figure 7

Predicted noise power spectral densities for Advanced LIGO, CE1, and CE2 detectors.

Figure 8

The vertical axis shows the signal-to-noise ratios of waveforms used as astrophysical signals. The horizontal axis shows the $\beta$ of the core progenitor at bounce. These sources are assumed to be at distances of 8.1 kpc, 23 kpc, 40.5 kpc, 48.5 kpc, 115 kpc, and 242 kpc and the signals are observed in the CE1, CE2, and Advanced LIGO (ALIGO) gravitational wave detectors. We ignore the waveforms with signal-to-noise ratios below 8 (shown as purple dots) and do not perform parameter estimation on them.

Figure 9

The 90% credible interval width of the posteriors obtained for $\beta$ as a function of the $\beta$ of the injected signal waveform. We note that the signals observed in Cosmic Explorer 1 (blue) and Cosmic Explorer 2 (orange) are measured an order of magnitude more precisely than the signals in Advanced LIGO (shown in green). On an average, the 90% credible interval width for signals observed in Cosmic Explorer 1 is 1.5 times that of the signals observed in Cosmic Explorer 2.

Figure 10

The ${\alpha }_1$ and ${\alpha }_2$ posteriors obtained for the signal with $\beta =0.0299$ at 23 kpc observed in Cosmic Explorer 1 (shown in blue) and Cosmic Explorer 2 (shown in orange). The $({\alpha }_1,{\alpha }_2)$ point corresponding to the injected signal (shown as the red star) is within the 90% contour region of both posteriors. The 90% contour region for the posterior of signal observed in Cosmic Explorer 2 is smaller than that of Cosmic Explorer 1 because the signal has higher signal-to-noise ratio in the former. However, when these posteriors are transformed into the posteriors of $\beta$, the error in median values of $\beta$ is larger for Cosmic Explorer 2 than Cosmic Explorer 1.