Unveiling the nature of gravitational-wave emission in core-collapse supernovae with perturbative analysis

Gravitational waves (GWs) can provide crucial information about the central engines of core-collapse supernovae (CCSNe). In order to unveil the nature of GW emission in CCSNe, we apply perturbative analyses with the same underlying equations as simulations to diagnose oscillations of the proto-neutron star (PNS) during $\sim 1\mathrm{s}$ postbounce. In the pseudo-Newtonian case, we find that radial profiles of GW emission match well between the perturbative analysis with $l=2$ and simulations inside the PNS at any frequency and time. This confirms that the GW emission of CCSNe arises from the global PNS oscillations in the perturbative regime. Based on this, we solve for the discrete eigenmodes with a free PNS surface and tentatively identify a set of $g$ modes and the $f$ mode contributing to the peak GW emission. We also offer a possible explanation for the power gap in the GW spectrum found in simulations that lies at the frequency with vanishing cumulative emission of the PNS. Our results enhance the predictive power of perturbative analyses in the GW signals of CCSNe.

Figure 1

Gravitational-wave spectrogram in the logarithmic scale for the model m0.75. The power spectral density is normalized to the maximum value after 0.1 s postbounce. The red shaded regions indicate the resimulated interval with a high cadence of data recording ($2\times {10}^{-5}\mathrm{s}$). These regions are centered at 0.4, 0.8, 0.9, and 1.0 s postbounce with an interval of 40 ms.

Figure 2

The 2D color map plots the radial profile of gravitational-wave power spectral density in the logarithmic scale for the model m0.75 in the 40 ms interval centered at 0.4 s postbounce. The vertical dotted line denotes the proto-neutron star (PNS) surface with $\rho ={10}^{11}{\mathrm{g}\mathrm{cm}}^{-3}$ and the horizontal black dashed lines denote the frequencies (725, 1050, and 1500 Hz) chosen for matching the perturbative functions. The lines plotted in the left panel show the overall gravitational-wave spectrum for the regions inside 100 km (solid) and the PNS (dashed).

Figure 3

Deviation from the hydrostatic condition ${\partial }_{r}P=-\rho {\partial }_r\mathrm{\Phi }$ at three postbounce times, 0.4, 0.8, and 1.0 s in the model m0.75. The black stars mark the proto-neutron star surface defined as where the spherically averaged density equals ${10}^{11}\mathrm{g}{\mathrm{cm}}^{-3}$.

Figure 4

Examples of matching radial profiles of gravitational-wave power spectral density between the simulation (black solid lines, also see Fig. 2) and perturbative analysis (red dashed lines) for the model m0.75 in the 0.04 s interval centered at 0.4 s postbounce. The frequencies in the simulation are 725, 1050, and 1500 Hz from left to right. Note that we have searched for the best-fit perturbative function in a $\pm 100\mathrm{Hz}$ window centered at the simulated frequency. The blue dotted lines show the perturbative function with the same frequency as the simulation when the best-fit frequency differs from that. Note that we multiply the perturbative function with a frequency-dependent constant to match the simulated signals.

Figure 5

Left panel: gravitational-wave (GW) spectrogram seen in Fig. 1 with the frequencies of perturbative eigenmodes overplotted as digits denoting the corresponding numbers of radial nodes. Right panel: same as the left panel, but with the red dashed line denoting the quadratic fitting curve of the GW peak frequency as a function of postbounce time. The digits denote the frequencies of perturbative eigenmodes closest to the red line.

Figure 6

Left panel: total power ($P(f)/C(f)$ in Eq. (15) of the perturbative mode functions inside the proto-neutron star (PNS) as a function of frequency at 0.4, 0.8, and 1.0 s postbounce. The PNS surface is defined as the locus of ${10}^{11}{\mathrm{g}\mathrm{cm}}^{-3}$. The vertical dotted lines denote the local minimum around the gap frequency in the simulation ($\sim 1250\mathrm{Hz}$). Right panel: same as Fig. 1, but with the gap frequencies overplotted as white open circles. The red dashed line denotes the frequency for the local minimum power inside PNS as the vertical dotted lines in the left panel.

Figure 7

Comparison of gap frequencies in the simulation (open circles) and perturbative analysis (solid lines) for all the supernova models considered in this paper. The gap frequencies from the perturbative analysis are shifted lower by 30 Hz to better match those from the simulations.

Figure 8

Same as Fig. 4, but for the 0.04 s intervals centered at 0.8, 0.9, and 1.0 s postbounce in the model m0.75 from top to bottom. The blue dotted lines show the perturbative function with the same frequency as the simulation when the best-fit frequency differs from that.

Figure 9

Gravitational-wave spectrograms as Fig. 1 but for the models m0.55 (left panel) and m0.95 (right panel). They are overplotted with a red line denoting the quadratic fitting curve of the peak frequency as a function of time. The frequencies of perturbative eigenmodes are marked with digits that denote the corresponding numbers of radial nodes.