Bayesian reconstruction of gravitational wave burst signals from simulations of rotating stellar core collapse and bounce

Presented in this paper is a technique that we propose for extracting the physical parameters of a rotating stellar core collapse from the observation of the associated gravitational wave signal from the collapse and core bounce. Data from interferometric gravitational wave detectors can be used to provide information on the mass of the progenitor model, precollapse rotation, and the nuclear equation of state. We use waveform libraries provided by the latest numerical simulations of rotating stellar core collapse models in general relativity, and from them create an orthogonal set of eigenvectors using principal component analysis. Bayesian inference techniques are then used to reconstruct the associated gravitational wave signal that is assumed to be detected by an interferometric detector. Posterior probability distribution functions are derived for the amplitudes of the principal component analysis eigenvectors, and the pulse arrival time. We show how the reconstructed signal and the principal component analysis eigenvector amplitude estimates may provide information on the physical parameters associated with the core collapse event.

Figure 1

Sample waveforms of the core-bounce signal for three models with varying initial rotation states while the mass of the progenitor model and EoS are fixed. Note the relatively small signal peak at the time of core bounce and the significant late-time contribution from postbounce convection for the slowly rotating model s20a1o05_shen, and the overall lower signal frequency for the rotation-dominated and centrifugally bouncing model s20a3o15_shen. The three signals presented here adequately cover the waveform morphology of our signal catalog.

Figure 2

The top three principal components (PCs) derived from the catalog of waveforms described in Sec. 3.

Figure 3

Achievable match $\mu$ and scaling factor ${\gamma }^2$ for given numbers $k$ of PCs across the whole catalog used here. A particular number $k$ of PC basis vectors yields a certain match $\mu$ for each of the 128 waveforms in the catalog. Shown here is the distribution of matches across the catalog for increasing values of $k$, as characterized by mean, median, and some more quantiles.

Figure 4

Marginal posterior probability distributions of the parameters of example signal $A$ with SNR $\rho =10$, indicating the values and uncertainties of the model parameters as inferred from the data. Only the first four coefficients ${\beta }_1$ to ${\beta }_4$ corresponding to the top four principal components are shown here; there are $k=10$ coefficients in total. The dashed line in the top right plot shows the prior probability distribution in comparison to the (very similar) posterior.

Figure 5

Reconstruction of the example signal $A$ with SNR $\rho =10$ using $k=10$ PCs.

Figure 6

A repeated analysis and reconstruction of the same example signal $B$ with SNR $\rho =10$ using different numbers of PCs ($k=10$ and $k=20$). Note that a larger number of basis vectors in the model does not necessarily improve signal recovery.

Figure 7

Reconstruction of the same example signal $C$ at different overall amplitudes (and, with that, SNRs $\rho =10$ and $\rho =20$), both times using $k=10$ PCs.

Figure 8

The resulting first three PC coefficients ${\beta }_1$, ${\beta }_2$, and ${\beta }_3$ when matching each of the 128 waveforms in the catalog against the set of PCs (no noise involved). The waveforms in the catalog all correspond to a signal from a fixed distance (10 kpc). Note that the coefficients only occupy a very restricted region in parameter space. The diamonds show the posterior mean values of the PC coefficients for the three examples (labeled $A$, $B$, and $C$) discussed in the text. For comparison they are scaled down (by their known distance) to the same magnitude.

Figure 9

Comparison of the distributions of all PC coefficients ${\beta }_i$ to the best-matched sets of coefficients found in the original catalog. Posterior means and standard deviations (error bars) are shown in black, the (scaled) sets of the top five best-matching coefficients are shown in different shades of gray.