Probing mass-radius relation of protoneutron stars from gravitational-wave asteroseismology

The gravitational-wave (GW) asteroseismology is a powerful technique for extracting interior information of compact objects. In this work, we focus on spacetime modes, the so-called $w$ modes, of GWs emitted from a proto-neutron star (PNS) in the postbounce phase of core-collapse supernovae. Using results from recent three-dimensional supernova models, we study how to infer the properties of the PNS based on a quasi-normal mode analysis in the context of the GW asteroseismology. We find that the $w_1$-mode frequency multiplied by the PNS radius is expressed as a linear function with respect to the ratio of the PNS mass to the PNS radius. This relation is insensitive to the nuclear equation of state (EOS) employed in this work. Combining with another universal relation of the $f$-mode oscillations, we point out that the time dependent mass-radius relation of the PNS can be obtained by observing both the $f$- and $w_1$-mode GWs simultaneously. Our results suggest that the simultaneous detection of the two modes could provide a new probe into finite-temperature nuclear EOS that predominantly determines the PNS evolution.

Figure 1

(Spherically-averaged) radial profiles of the rest-mass density at 48, 148, and 248 ms after core bounce. The left and right panel corresponds to SFHx and TM1, respectively.

Figure 2

Time evolution of the PNS radius (left panel) and its gravitational mass (right panel) as a function of the postbounce time. The circles and diamonds corresponds to SFHx and TM1, respectively. The surface of the PNS is defined at a fiducial rest-mass density of ${\rho }_s={10}^{10}\mathrm{g}{\mathrm{cm}}^{-3}$.

Figure 3

Left: Same as Fig. 2, but for the time evolution of the stellar compactness after bounce. Right: Sequences of the masses and radii of PNSs for SFHx and TM1. Note that the points at the left (smaller PNS radius) correspond to late postbounce phase, whereas the points at the right correspond to early phase (larger PNS radius).

Figure 4

Frequency and damping rate of the axial spacetime modes for PNSs. The left and right panels correspond to the results for SFHx and TM1 EOSs, respectively, where the circles and diamonds are shown for the PNS models at 108 and 248 ms after core bounce. The open and solid marks correspond to the $w_{\mathrm{II}}$ and “ordinary” $w$ modes.

Figure 5

Evolutions of frequency $(f_{w_1})$ and damping time $({\tau }_{w_1})$ for the $w_1$ mode. The circles and diamonds correspond to SFHx and TM1, respectively.

Figure 6

The $w_1$-mode frequencies multiplied by the normalized radius $f_{w_1}(R/10\mathrm{km})$ are shown as a function of normalized compactness $(M_{\mathrm{PNS}}/1.4M_{\odot })(R_{\mathrm{PNS}}/10\mathrm{km}{)}^{-1}$, where the circles and diamonds denote the results for SFHx and TM1 EOSs, respectively. The dotted line is a fitting formula given by Eq. (10).

Figure 7

Evolutions of $f$ and $p_1$ modes in GWs from PNSs after core bounce are shown in the left panes. The solid and open marks correspond to the $f$ and $p_1$ modes, while the circles and diamonds are, respectively, the results for SFHx and TM1. The middle and right panels shows respectively the frequencies of the $f$ and $p_1$ modes as a function of average density of PNSs. The solid line denotes the linear fitting given by Eqs. (11) and (12).

Figure 8

The effective amplitude of $w_1$ modes in gravitational waves radiated from the PNSs with SFHx EOS are shown together with the sensitivity curves of KAGRA, advanced LIGO (aLIGO), Einstein Telescope (ET), and Cosmic Explorer (CE). The circles, squares, diamonds, triangles, and upside-down triangles correspond to the results with $E_{\mathrm{T}}^{(w_1)}={10}^{-4}{M}_{\odot }$, ${10}^{-5}{M}_{\odot }$, ${10}^{-6}{M}_{\odot }$, ${10}^{-7}{M}_{\odot }$, and ${10}^{-8}{M}_{\odot }$, respectively.

Figure 9

The effective (solid) and actual (cross) gravitational mass as a function of the isotropic radius $\widehat{r}$ at 48, 148, and 248 msec after core bounce. The left and right panels correspond to the results with SFHx and TM1 EOSs, respectively.

Figure 10

Typical frequency and damping rate of spacetime modes for a cold neutron star. This is a result for the neutron star with $1.5M_{\odot }$, which is constructed with the Shen EOS. The open and solid circles correspond to the $w_{\mathrm{II}}$ and “ordinary” $w$-modes, respectively.