Broadening of universal relations at the birth and death of a neutron star

Certain relations among neutron-star observables that are insensitive to the underlying nuclear matter equation of state are known to exist. Such universal relations have been shown to be valid for cold and stationary neutron stars. Here, we study these relations in more dynamic scenarios: protoneutron stars and hypermassive neutron stars, allowing us to investigate the time evolution of these relations from the birth to the death of a neutron star. First, we study protoneutron stars. We use an effective equation of state, extracted from three-dimensional core-collapse supernova simulations, to obtain the structure of spherically symmetric protoneutron stars. We then consider nonradial oscillations to compute their $f$-mode frequency ($f$), as well as slow rotation and small tidal deformation, to compute their moment of inertia ($I$), spin-induced quadrupole moment ($Q$), and Love number. We find that well-established universal relations for cold neutron stars involving these observables (namely, the $I$-Love-$Q$ and $f$-Love relations) are approximately valid for protoneutron stars, with a deviation below $\approx 10\%$ for a postbounce time above $\approx 0.5\mathrm{s}$, considering eight different supernova progenitors and one equation of state (SFHo). Next, we study hypermassive neutron stars. The bulk of a neutron star is defined as the region enclosed by the isodensity surface that corresponds to the maximum compactness ($C$) inside the star. We obtain a new universal relation between the $f$-mode frequency and the compactness of cold and nonrotating neutron stars, using bulk quantities. We show that this relation has an equation-of-state-variation of $\approx 3\%$, considering a set of ten equations of state. Bulk quantities of postmerger remnants can be obtained from numerical-relativity simulations. Using results from binary neutron star merger simulations, we study the evolution of hypermassive neutron stars on the $f\text{−}C$ plane, considering two different mass ratios and one equation of state (SFHo). We find that the relation between the peak frequency of the gravitational-wave signal and the compactness from these hypermassive neutron stars deviates from the universal $f\text{−}C$ relation by $\approx 70\%–80\%$, when the peak frequency is taken directly as a proxy for the $f$-mode. As our results are limited to a single equation of state, the deviations we obtain when universal relations for cold neutron stars are used for protoneutron stars or hypermassive neutron stars can be conservatively considered as a lower limit. Finally, we discuss the reliability of the universal relations in the context of future observations of gravitational waves from remnants of core-collapse supernovae or binary neutron star mergers.

Figure 1

Relation between pressure $p$ and total mass density $\rho$ for various postbounce times $t_{\mathrm{pb}}$ for the PNS generated by the $9M_{\odot }$ progenitor (see Table 1), where the SFHo EOS [94] was used. For reference, we include the zero-temperature EOS.

Figure 2

Left: time evolution of the circumferential radius $R$ and the gravitational mass $M$ for PNSs generated by eight different progenitors (see Table 1). The radius is determined by the density cutoff ${\rho }_{\mathrm{B}}^{\mathrm{srf}}={10}^{10}\mathrm{g}/{\mathrm{cm}}^3$ (see main text). As expected, the mass increases and the radius decreases over time (asymptotically approaching cold NS values), owing to matter accretion and neutrino emission, respectively. Right: time evolution of the oscillation frequency $f_f$ and the damping time ${\tau }_f$ of the $f$-mode for PNSs generated by eight different progenitors (see Table 1). The frequencies are lower and the damping times are (much) longer, compared to the ones for cold NSs ($f_f^{\mathrm{NS}}\approx 1.5–3.0\mathrm{kHz}$ and ${\tau }_f^{\mathrm{NS}}\approx 0.1–0.5\mathrm{s}$), due to the thermal effects on the effective EOS.

Figure 3

Upper left: relation between the $f$-mode frequency $f_f$ and the average density $\sqrt{M/R^3}$ for PNSs from eight progenitors. This relation is universal against the progenitor mass and the deviation is $|1-f_{f,\mathrm{fit}}/f_f|\lesssim 2\%$. The fitting coefficients are shown in Table 2. For reference, we include the fit from Rodriguez et al. [66] (see main text), that we denote as $f_{f,\mathrm{R}}$, and we see that $|1-f_{f,\mathrm{fit}}/f_{f,\mathrm{R}}|\lesssim 21\%$. The vertical dashed line indicates the starting point ($\approx 0.005M_{\odot }^{1/2}/{\mathrm{km}}^{3/2}$) for the validity of the Cowling classification for the $f$-mode in [66]. Upper right: relation between the dimensionless $f$-mode frequency ${\widehat{f}}_f$ (see main text) and the postbounce time $t_{\mathrm{pb}}$ for PNSs from eight progenitors. This relation is not as tight as the relation between $f_f$ and $\sqrt{M/R^3}$ since the deviation is $|1-{\widehat{f}}_{f,\mathrm{fit}}/{\widehat{f}}_{f,\mathrm{num}}|\lesssim 12\%$. The fitting coefficients are shown in Table 2. For reference, we include the fit from Afle et al. [91] (see main text), that we denote as ${\widehat{f}}_{f,\mathrm{A}}$, and we see that $|1-{\widehat{f}}_{f,\mathrm{fit}}/{\widehat{f}}_{f,\mathrm{A}}|\lesssim 43\%$. The vertical dashed line indicates the starting point ($\approx 0.2\mathrm{s}$) for the $f$-mode in the GW spectrogram, as considered by [91]. Lower left: time evolution of the dominant frequency extracted from the GW spectrograms, $f_{\text{peak}}$ (see Appendix pp1), and the $f$-mode frequency computed from perturbation theory, $f_{\text{pert}}$ (see Sec. 2b1), for PNSs from eight progenitors. We fit the relations to a second-order fit and the relative difference between $f_{\text{pert}}$ and $f_{\text{peak}}$ is $\lesssim 43\%$, which agrees with the deviation in the relation between ${\widehat{f}}_f$ and $t_{\mathrm{pb}}$ (not being the case for the relation between $f_f$ and $\sqrt{M/R^3}$, see main text). Lower right: relative difference between $f_{\text{pert},\mathrm{fit}}$ ($f_{\text{peak},\mathrm{fit}}$) and $f_{\text{pert}}$ ($f_{\text{peak}}$). The progenitor-mass-variation is $\lesssim 9\%$.

Figure 4

Left: time evolution of $I$-Love-$Q$ and $f$-Love relations for PNSs generated by eight different progenitors (see Table 1). We consider $\overline{\mathrm{\Lambda }}$ as the independent variable and show its correlation with $\overline{I}$ and $\overline{Q}$, and with Re(${\overline{\omega }}_f$) and Im(${\overline{\omega }}_f$). For reference, we include the relations for cold NSs, considering ten zero-temperature EOSs (see main text). We note that, as the postbounce time $t_{\mathrm{pb}}$ increases (from high $\overline{\mathrm{\Lambda }}$ to low $\overline{\mathrm{\Lambda }}$), the PNS relations approach the NS relations. Right: time evolution of the relative difference $\mathrm{\Delta }y=|1-y_{\mathrm{PNS}}/y_{\mathrm{NS}}|$ between PNS observables $y_{\mathrm{PNS}}$ and the NS observables $y_{\mathrm{NS}}$, where $y=y(x)$, for $x=\overline{\mathrm{\Lambda }}$ and $y\in \{\overline{I},\overline{Q},\mathrm{Re}({\overline{\omega }}_f),\mathrm{Im}({\overline{\omega }}_f)\}$. We note that $\mathrm{\Delta }y$ decreases as $t_{\mathrm{pb}}$ increases. In particular, $\mathrm{\Delta }y\lesssim 10\%$ for $t_{\mathrm{pb}}\gtrsim 0.5\mathrm{s}$ (see horizontal and vertical dashed lines).

Figure 5

Relation between the compactness ${\tilde{C}}_{\mathrm{V}}=M_{\mathrm{B}}/R_{\mathrm{V}}$ and baryonic mass density ${\rho }_{\mathrm{B}}$ for $0.5M_{\odot }$, $1.0M_{\odot }$, $1.5M_{\odot }$, and $2.0M_{\odot }$ NSs described by the zero-temperature SFHo EOS. We also show the relation between the compactness of the bulk ${\tilde{C}}_{\mathrm{V}}^{\mathrm{blk}}=M_{\mathrm{B}}^{\mathrm{blk}}/R_{\mathrm{V}}^{\mathrm{blk}}$ and the surface density of the bulk ${\rho }_{\mathrm{B}}^{\mathrm{blk}}$ for the same EOS.

Figure 6

Relation between the dimensionless $f$-mode frequency of the bulk $\mathrm{Re}({\tilde{\omega }}_f^{\mathrm{blk}})=\mathrm{Re}(M_{\mathrm{B}}^{\mathrm{blk}}{\omega }_f^{\mathrm{blk}})$ and compactness of the bulk ${\tilde{C}}_{\mathrm{V}}^{\mathrm{blk}}=M_{\mathrm{B}}^{\mathrm{blk}}/R_{\mathrm{V}}^{\mathrm{blk}}$, considering ten zero-temperature EOSs (see main text). The “spiraling” dashed curves show the instantaneous GW frequency of the simulated HMNSs on the $\mathrm{Re}({\tilde{\omega }}_f^{\mathrm{blk}})-{\tilde{C}}_{\mathrm{V}}^{\mathrm{blk}}$ plane and the “plus” markers are the normalized peak frequencies, for two mass ratios $q$ (see Table 3). The relative difference $|1-y_{\mathrm{sim}}/y_{\text{SFHo}}|$ between the normalized peak frequency from the simulation $y_{\mathrm{sim}}({\tilde{C}}_{\mathrm{V}}^{\mathrm{blk}})$ and the frequency for the cold and nonrotating NS described by the SFHo EOS $y_{\text{SFHo}}({\tilde{C}}_{\mathrm{V}}^{\mathrm{blk}})$, for the same compactness ${\tilde{C}}_{\mathrm{V}}^{\mathrm{blk}}$, is $\approx 73\%$ for $q=0.9$ and $\approx 78\%$ for $q=1.0$, where $y=\mathrm{Re}({\tilde{\omega }}_f^{\mathrm{blk}})$. The vertical dashed line indicates the maximum compactness of the bulk for the SFHo EOS, beyond which the simulated HMNS collapses to a black hole.

Figure 7

Left: time evolution of circumferential radius $R$, gravitational mass $M$, $f$-mode oscillation frequency $f_f$, and $f$-mode damping time ${\tau }_f$ for three different density cutoffs: ${\rho }_{\mathrm{B}}^{\mathrm{srf}}={10}^{10}\mathrm{g}/{\mathrm{cm}}^3$, ${\rho }_{\mathrm{B}}^{\mathrm{srf}}={10}^{11}\mathrm{g}/{\mathrm{cm}}^3$, and ${\rho }_{\mathrm{B}}^{\mathrm{srf}}={10}^{12}\mathrm{g}/{\mathrm{cm}}^3$, considering the PNS generated by the $9M_{\odot }$ progenitor. The differences are more significant at early $t_{\mathrm{pb}}$ and decrease as $t_{\mathrm{pb}}$ increases. Right: time evolution of relative difference $\mathrm{\Delta }y=|1-y_{\mathrm{PNS}}/y_{\mathrm{NS}}|$ between PNS observables $y_{\mathrm{PNS}}$ and NS observables $y_{\mathrm{NS}}$ for three different density cutoffs, where $y\in \{\overline{I},\overline{Q},\mathrm{Re}({\overline{\omega }}_f),\mathrm{Im}({\overline{\omega }}_f)\}$. In general, $\mathrm{\Delta }y$ decreases as ${\rho }_{\mathrm{B}}^{\mathrm{srf}}$ increases. We obtained similar results for the other progenitors.

Figure 8

Left: radial profiles for the gravitational potential $\mathrm{\Phi }(r)$, gravitational mass $m(r)$, total mass density $\rho (r)$, and pressure $p(r)$ for our TOV solutions and the simulation data. The profiles are for the PNSs generated by the $9M_{\odot }$ progenitor at $t_{\mathrm{pb}}=0.1\mathrm{s}$ and $t_{\mathrm{pb}}=1.0\mathrm{s}$. We note that the $\mathrm{\Phi }(r)$ profiles for our TOV solutions are deeper, slightly modifying the $\rho (r)$ and $p(r)$ profiles, and thus affecting the $m(r)$ profile. Right: relative differences between PNS observables from TOV solutions $y_{\mathrm{TOV}}$ and simulation data $y_{\mathrm{sim}}$, where $y\in \{R,M,\overline{\rho }=\sqrt{M/R^3},\overline{C}=M/R\}$, and for $t_{\mathrm{pb}}\gtrsim 0.2\mathrm{s}$. In general, $y_{\mathrm{TOV}}<y_{\mathrm{sim}}$, considering the eight different progenitors (see Table 1).

Figure 9

GW spectrogram for the $25M_{\odot }$ progenitor. The $f$-mode frequency ($f_{\text{pert}}$, computed using perturbation theory) is lower than the simulation frequency ($f_{\text{peak}}$, obtained from the short-time Fourier transform), and we attribute this difference to the approximations used in the simulations in [90] (see main text).

Figure 10

Left: $M(t_{\mathrm{pb}})$ vs $R(t_{\mathrm{pb}})$ relation for the PNS generated by the $9M_{\odot }$ progenitor. The black points show the stars that we are actually considering in this work, since their central density is the same as the one in the simulation data. We extend these configurations to lower central densities, and the colorful curves show the slope of the mass-radius relation for each time slice. We note that, for $t_{\mathrm{pb}}\gtrsim 0.5\mathrm{s}$, the black points are in stable branches (i.e., the mass is increasing and the radius is decreasing). Right: time evolution of the fundamental mode squared $F_0^2$ for the radial oscillations of PNSs generated by eight different progenitors (see Table 1). We note that, for the $9M_{\odot }$ progenitor: (i) if $\mathrm{\Gamma }={\mathrm{\Gamma }}_0$, $F_0^2>0$ for $t_{\mathrm{pb}}\gtrsim 0.5\mathrm{s}$, which agrees with the qualitative analysis using the turning point principle; (ii) if $\mathrm{\Gamma }={\mathrm{\Gamma }}_1$, $F_0^2>0$ for all $t_{\mathrm{pb}}$, thus breaking the validity of the maximum-mass criterion (see main text). We obtained similar results for the other progenitors.

Figure 11

Time evolution of the $f$-mode frequency (upper panel) and damping time (lower panel) for the $9M_{\odot }$ progenitor, computed with perturbation theory, considering $\mathrm{\Gamma }={\mathrm{\Gamma }}_0$ and $\mathrm{\Gamma }={\mathrm{\Gamma }}_1$ (see main text). For comparison, we also show the GW frequency obtained from the spectrogram, with a short-time Fourier transform (see Figs. 3 and 9). We obtained similar results for the other progenitors.

Figure 12

Left: time evolution of the oscillation frequency $f_w$ and the damping time ${\tau }_w$ of the $w$-mode of PNSs generated by eight different progenitors (see Table 1). Similarly to the $f$-mode case (see Fig. 2), the frequencies are lower while the damping times are (slightly) longer than those for cold NSs ($f_w^{\mathrm{NS}}\approx 5–10\mathrm{kHz}$ and ${\tau }_w^{\mathrm{NS}}\approx 0.01–0.05\mathrm{ms}$). Right: time evolution of the $w$-Love relation. We consider $\overline{\mathrm{\Lambda }}$ as the independent variable and show its correlation with Re(${\overline{\omega }}_w$) and Im(${\overline{\omega }}_w$). For reference, we include the relations for cold NSs, considering ten zero-temperature EOSs (as in Fig. 4). Similarly to the case for $I$-Love-$Q$ or $f$-Love, the PNS relations approach the cold NS ones as the postbounce time $t_{\mathrm{pb}}$ increases (from high $\overline{\mathrm{\Lambda }}$ to low $\overline{\mathrm{\Lambda }}$).

Figure 13

Upper: NS and bulk $f\text{−}C$ relations for different definitions (see main text). The bulk relations are below the NS ones (except for the imaginary part, when the compactness is close to its maximum). The maximum compactness (see vertical dashed lines) for the tilde-defined (defined in terms of the baryonic mass) quantities is $\approx 0.43$, and for the bar-defined (defined in terms of the gravitational mass) quantities is $\approx 0.35$. The relative difference $|1-y_{\mathrm{fit}}/y|$ between $y_{\mathrm{fit}}$ and $y$ is $\lesssim 4\%$ for the real part relations and $\lesssim 2\%$ for the imaginary part relations (see the horizontal dashed lines), where $y$ is a dependent variable. Lower panels: $f\text{−}C$ relations corresponding to the different definitions (see main text), for the SFHo EOS. For this compactness range (that is similar to the one in Fig. 6), the relations are very similar, although the maximum range of their validity is significantly different (see vertical dashed lines).