Gravitational Waves from a Core $g$ Mode in Supernovae as Probes of the High-Density Equation of State

Using relativistic supernova simulations of massive progenitor stars with a quark-hadron equation of state (EOS) and a purely hadronic EOS, we identify a distinctive feature in the gravitational-wave signal that originates from a buoyancy-driven mode ($g$ mode) below the proto-neutron star convection zone. The mode frequency lies in the range $200\lesssim f\lesssim 800\mathrm{Hz}$ and decreases with time. As the mode lives in the core of the proto-neutron star, its frequency and power are highly sensitive to the EOS, in particular the sound speed around twice saturation density.

Figure 1

GW spectrograms for z85 (top) and z35 (bottom) using the CMF EOS (left) and SFHx EOS (right). The same logarithmic color scale for the amplitude $|h^{+}|$ is used for all models. Models z85:CMF, z85:SFHx, and z35:CMF exhibit a distinct second frequency band from the ${{}^{2}g}_1$ mode, which branches off the dominant band after a few hundred milliseconds.

Figure 2

First and second panels: amplitude $\tilde{q}(r,f)$ of the Fourier transform of the quadrupolar perturbation $q(r,t)$ as a function of radius and frequency $f$ for the time intervals $\mathrm{\Delta }{t}_1$ and $\mathrm{\Delta }{t}_2$ around 0.4 s and 0.55 s (indicated in red shades in the third panel) for model z35:CMF. Third panel: color contour plot of the quark fraction $n_q/n_{\text{B}}$ as a function of time and radius, combined with isocontours for the adiabatic index (solid lines in different shades of gray) and a thick black line indicating the radius corresponding to nuclear saturation density. Fourth panel: Brunt-Väisälä frequency ${\omega }_{\mathrm{BV}}^2$ (solid lines) and spherically averaged specific entropy per baryon $s$, (dashed) at five different times, which are indicated on the time axis of the third panel as vertical lines of the respective color. The last entropy profile (orange) shows a blip at a radius of less than 2 km, which is due to a convective plume that penetrates the PNS core when the buoyancy barrier at the inner edge of the PNS convection zone is eroded (${\omega }_{\mathrm{BV}}^2\approx 0$).

Figure 3

Color contour plots of the squared sound speed $c_s^2$ as a function of density $\rho$ and temperature $T$ for an electron fraction of $Y_e=0.25$ for models z85 (top) and z35 (bottom) with the CMF EOS (left), and the SFHx EOS (right). Solid and dashed curves show ${\omega }_{\mathrm{BV}}^2$ and (spherically averaged) temperature profiles $T(\rho )$ at various times (indicated by line color). The density where ${\omega }_{\mathrm{BV}}^2$ peaks is indicated by a dot on the curves. The white lines indicate five constant radii; the radius (in km) is indicated on top. Note a temperature blip at high densities due to the overshooting of plumes into the PNS at late times (blue curves) in the CMF models.