Equation of state effects on gravitational waves from rotating core collapse

Gravitational waves (GWs) generated by axisymmetric rotating collapse, bounce, and early postbounce phases of a galactic core-collapse supernova are detectable by current-generation gravitational wave observatories. Since these GWs are emitted from the quadrupole-deformed nuclear-density core, they may encode information on the uncertain nuclear equation of state (EOS). We examine the effects of the nuclear EOS on GWs from rotating core collapse and carry out 1824 axisymmetric general-relativistic hydrodynamic simulations that cover a parameter space of 98 different rotation profiles and 18 different EOS. We show that the bounce GW signal is largely independent of the EOS and sensitive primarily to the ratio of rotational to gravitational energy, $T/|W|$, and at high rotation rates, to the degree of differential rotation. The GW frequency ($f_{\text{peak}}\sim 600–1000\mathrm{Hz}$) of postbounce core oscillations shows stronger EOS dependence that can be parametrized by the core’s EOS-dependent dynamical frequency $\sqrt{G{\overline{\rho }}_c}$. We find that the ratio of the peak frequency to the dynamical frequency $f_{\text{peak}}/\sqrt{G\overline{{\rho }_c}}$ follows a universal trend that is obeyed by all EOS and rotation profiles and that indicates that the nature of the core oscillations changes when the rotation rate exceeds the dynamical frequency. We find that differences in the treatments of low-density nonuniform nuclear matter, of the transition from nonuniform to uniform nuclear matter, and in the description of nuclear matter up to around twice saturation density can mildly affect the GW signal. More exotic, higher-density physics is not probed by GWs from rotating core collapse. We furthermore test the sensitivity of the GW signal to variations in the treatment of nuclear electron capture during collapse. We find that approximations and uncertainties in electron capture rates can lead to variations in the GW signal that are of comparable magnitude to those due to different nuclear EOS. This emphasizes the need for reliable experimental and/or theoretical nuclear electron capture rates and for self-consistent multidimensional neutrino radiation-hydrodynamic simulations of rotating core collapse.

Figure 1

EOS Constraints from experiment and NS mass measurements. The maximum cold neutron star gravitational mass $M_{\max }$, the incompressibility $K$, symmetry energy $J$, and the derivative of the symmetry energy $L$ are plotted. For $M_{\max }$, the bottom of the plot is 0, the minimum line is at $1.97M_{\odot }$, and the maximum line is not used. The other constraints are normalized so the listed minima and maxima lie on the minimum and maximum lines. EOS that are within all of these simple constraints are colored, and the color code is consistent throughout the paper. Note that there are additional constraints on the NS mass-radius relationship, which we show in Fig. 2, and joint constraints on $J$ and $L$ [26] that we do not show.

Figure 2

Neutron star mass-radius relations. The relationship between the gravitational mass and radius of a cold neutron star is plotted for each EOS. The EOS employed in this study cover a wide swath of parameter space. EOS that lie within the constraints depicted in Fig. 1 are colored, and the color code is consistent throughout the paper. We show the $2\sigma$ mass-radius constraints from “model A” of [27] as a shaded region between two dashed lines. These constraints were obtained from a Bayesian analysis of observations of type-I x-ray bursts in combination with theoretical constraints on nuclear matter. The EOS that agree best with these constraints are SFHo, SFHx, and LS220.

Figure 3

$Y_e(\rho )$ deleptonization profiles. For each EOS, radial profiles of the electron fraction $Y_e$ as a function of density $\rho$ are taken from spherically symmetric GR1D radiation-hydrodynamics simulations using two-moment neutrino transport at the point in time when the central $Y_e$ is smallest (roughly at core bounce) and are plotted here. We manually extend the curves out to high densities with a constant $Y_e$ to ensure that simulations never encounter a density outside the range provided in these curves. In the two-dimensional simulations, $Y_e$ is determined by the density and one of these curves until core bounce.

Figure 4

EOS variability in waveforms. The time-domain waveforms (left panel) and Fourier transforms scaled by $\sqrt{f}$ (right panel) of signals from all 18 EOS for the $\mathrm{A}=634\mathrm{km}$, $\mathrm{\Omega }=5.0\mathrm{rad}{\mathrm{s}}^{-1}$ rotation profile (moderately rapidly rotating, $T/|W|=0.069–0.074$ at core bounce, depending on the EOS) are plotted assuming a distance of 10 kpc and optimal orientation, along with the Advanced LIGO [30, 110], VIRGO [32], and KAGRA in the zero detuning VRSE configuration [31, 111] design sensitivity curves. $t_b$ is the time of core bounce; $t_{\mathrm{be}}$ is the end of the bounce signal and the beginning of the postbounce signal. We use data only until $t_{\mathrm{be}}+6\mathrm{ms}$ to exclude the GW signal from prompt convection from our analysis. The differences in postbounce oscillation rates can be seen both in phase decoherence of the waveform and the peak location of the Fourier transform. The colored curves correspond to EOS that satisfy the constraints depicted in Fig. 1.

Figure 5

Velocity field. We plot the entropy, density, and velocity for the ${\mathrm{\Omega }}_0=4.0\mathrm{rad}{\mathrm{s}}^{-1}$ (left) and ${\mathrm{\Omega }}_0=8.0\mathrm{rad}{\mathrm{s}}^{-1}$ (right) simulations with $A=634\mathrm{km}$ at 4.5 ms after core bounce. The color map shows entropy. Blue regions belong to the unshocked inner core. The density contours show densities of $10^{\{13.5,13.75,14.0,14.25\}}\mathrm{g}{\mathrm{cm}}^{-3}$ from outer to inner. The vectors represent only the poloidal velocity (i.e. the rotational velocity is ignored) and are colored for visibility. At low rotation rates (left) the flow in the inner core is largely quadrupolar. At high rotation rates (right), rotation significantly deforms the inner core and couples $\ell =2$, $m=0$ quadrupole oscillations to other modes.

Figure 6

Bounce signal amplitude. We plot the difference between the maximum and minimum strain $\mathrm{\Delta }{h}_{+}$ before $t_{\mathrm{be}}$ assuming $D=10\mathrm{kpc}$ and optimal source-detector orientation as a function of the ratio of rotational to gravitational energy $T/|W|$ of the inner core at bounce. Each two-dimensional simulation is a single point and the SFHo simulations with the same differential rotation parameter $A$ are connected to guide the eye. $A1-A5$ corresponds to $A=300$, 467, 634, 1268, 10000 km, respectively. Simulations with all EOS and values of $A$ behave similarly for $T/|W|\lesssim 0.06$, but branch out when rotation becomes dynamically important. We plot a dashed line representing the expected perturbative behavior with $T/|W|$, using representative values of $M=0.6M_{\odot }$ and $R=65\mathrm{km}$. All 1704 collapsing simulations are included in this figure.

Figure 7

Peak frequency determination. The full GW spectrum for the $A3=634\mathrm{km}$, $\mathrm{\Omega }=5.0\mathrm{rad}{\mathrm{s}}^{-1}$ SFHo simulation is plotted in black. To prevent convection contributions from entering into the analysis, we cut the GW signal at 6 ms after the end of core bounce ($t_{\mathrm{be}}+6\mathrm{ms}$, blue line). The green line is the spectrum for the time series through the end of core bounce. To remove the bounce signal from the spectrum, we subtract the green line from the blue line to get the red line. The maximum of the red line within the depicted window of 600–1075 Hz determines the peak frequency $f_{\text{peak}}$.

Figure 8

Peak frequencies. Top: The peak frequencies of GWs emitted by postbounce PNS oscillations in all 1704 collapsing simulations are plotted as a function of the ratio of rotational to gravitational energy $T/|W|$ of the inner core at bounce. Red lines connect SFHo simulations with the same differential rotation parameter $A$. There is a large spread in the peak frequencies due to both the EOS and differential rotation. Bottom: We can remove most of the effects of the different EOS by normalizing the peak frequency by the dynamical frequency $\sqrt{G{\overline{\rho }}_c}$ and multiply by $2\pi$ to make it an angular frequency. However, significant differences due to differing amounts of differential rotation remain for rapidly spinning models. The transition from slow to Rapid Rotation Regimes occurs at $T/|W|\approx 0.06$ and it becomes difficult for our analysis scripts to find the $\ell =2$ $f$-mode peak at $T/|W|\gtrsim 0.17$. Each panel contains 1704 data points, and there are 1487 good points with $T/|W|<0.17$.

Figure 9

Universal relation: All differential rotation parameters and EOS result in simulations that obey the same relationship between the normalized peak frequency and the normalized maximum rotation rate ${\mathrm{\Omega }}_{\max }$. The kink in the plot where ${\mathrm{\Omega }}_{\max }=\sqrt{G{\overline{\rho }}_c}$ corresponds to $T/|W|\approx 0.06$. The dashed line is described by $2\pi f_{\text{peak}}/\sqrt{G{\overline{\rho }}_c}=0.5(1+{\mathrm{\Omega }}_{\max }/\sqrt{G{\overline{\rho }}_c})$. This figure includes all 1487 simulations with $T/|W|<0.17$.

Figure 10

Demystifying the universal relation. To better understand the relation in Fig. 9, we plot the peak frequency $f_{\text{peak}}$ and the maximum rotation rate ${\mathrm{\Omega }}_{\max }$ separately, each as a function of the dynamical frequency. The dramatic kink in Fig. 9 is due to a sharp change in the behavior of $f_{\text{peak}}$ once $2\pi f_{\text{peak}}>\sqrt{G{\overline{\rho }}_c}$. An approximate nuclear saturation density of ${\rho }_{\mathrm{nuc}}=2.7\times 10^{14}\mathrm{g}{\mathrm{cm}}^{-3}$ is plotted as well for reference. The top panel contains the 1487 simulations with $T/|W|<0.17$, while the bottom panel contains all 1704 collapsing simulations to show the decrease in ${\mathrm{\Omega }}_{\max }$ at extreme rotation rates.

Figure 11

Rotation changes oscillation mode character. In the top panel, we plot the GW signals for 20 cores collapsed with $A3=634\mathrm{km}$ and the SFHo EOS, color-coded according to their initial central rotation rate. The center and bottom panel show the radial velocity at 5 km from the origin along the equatorial and polar axis, respectively. We indicate core bounce with a vertical dashed line. In the Rapid Rotation Regime (starting at around the transition from red to green color), postbounce GW frequency and velocity oscillation frequency increase visibly. In the same regime, the oscillation mode structure changes. The polar velocities continue to increase, while the oscillations are damped along the equator.

Figure 12

Correlation coefficients. We calculate linear correlation coefficients between several parameters and observables in our collapsing simulations. The cell color represents the number within the cell, with positive correlations being red and negative correlations blue. Bottom left: Correlation coefficients for 874 simulations with ${\mathrm{\Omega }}_{\max }<\sqrt{G{\overline{\rho }}_c}$ (i.e. slowly rotating). Top right: Correlation coefficients for 613 simulations with ${\mathrm{\Omega }}_{\max }>\sqrt{G{\overline{\rho }}_c}$ (i.e. rapidly rotating) and $T/|W|<0.17$. $M_{\mathrm{IC},\mathrm{b}}$ is the mass of the inner core, defined by the region in sonic contact with the center, at core bounce. $j_{\mathrm{IC}}$ is the angular momentum of the inner core at bounce. $T/|W|$ is the inner core’s ratio of rotational kinetic to gravitational potential energy at core bounce. ${\mathrm{\Omega }}_{\max }$ is the maximum rotation rate obtained at any time in the simulation outside of $R=5\mathrm{km}$ and ${\tilde{\mathrm{\Omega }}}_{\max }={\mathrm{\Omega }}_{\max }/\sqrt{G{\overline{\rho }}_c}$. $f_{\text{peak}}$ is the peak frequency of GWs from postbounce PNS oscillations, and ${\tilde{f}}_{\text{peak}}=f_{\text{peak}}/\sqrt{G{\overline{\rho }}_c}$. ${\mathrm{\Omega }}_0$ is the precollapse maximum rotation rate and $A$ is the precollapse differential rotation parameter. $Y_{e,c}$ is the central electron fraction at core bounce. The incompressibility $K$, symmetry energy $J$, density derivative of the symmetry energy $L$, radius of a $1.4M_{\odot }$ star $R_{1.4}$, and $M_{\max }$ are properties of the EOS described in Sec. 2.

Figure 13

Signal to noise ratios. The SNR for all 1704 collapsing simulations that result in collapse and core bounce, assuming the rotation axis is perpendicular to the line of sight, the aLIGO interferometer is optimally oriented and at design sensitivity in the high-power zero-detuning configuration, and the source is 10 kpc away. A SNR of $\gtrsim 10$ is considered detectable. The colors correspond to the EOS in Fig. 1. A line is drawn through all the SFHo simulations to guide the eye. Each of the five branches corresponds to a different value of the differential rotation parameter $A$, where $A=300\mathrm{km}$ is the longest branch and $A=10000\mathrm{km}$ is the shortest.

Figure 14

GW differences due to the EOS. The GW mismatch [see Eq. (17)] is integrated between SFHo and each of the other EOS for the same rotation parameters ($A$,${\mathrm{\Omega }}_0$) for all 1704 collapsing simulations. Note that $T/|W|$ at bounce is slightly different between simulations with the same initial rotation parameters due to EOS effects. Only data through 6 ms after the end of the bounce signal are used to avoid contributions from prompt convection. Differences between EOS decrease with faster rotation as the bounce signal becomes stronger and rotational effects become more important. The HShen and HShenH EOS (not identified by color and shown in gray) have the consistently largest mismatches with SFHo in the Slow and Rapid Rotation Regimes. Mismatch calculations at $T/|W|\lesssim 0.02$ are unreliable due to a very weak GW signal. In the Extreme Rotation Regime, some EOS develop larger mismatches with SFHo. This occurs because simulations with these EOS transition to a centrifugal bounce at subnuclear density at lower rotation rates than SFHo. The resulting qualitative and quantitative change in the waveforms leads to larger mismatches.

Figure 15

$Y_e(\rho )$ profiles from variations in electron capture treatment. We plot our fiducial $Y_e(\rho )$ profile for the SFHo EOS along with $Y_e(\rho )$ profiles obtained with the approach of Sullivan et al. [73] for the SFHo EOS using detailed tabulated nuclear electron capture rates (SFHo_ecap1.0) and also rates multiplied by 0.1 (SFHo_ecap0.1) and 10 (SFHo_ecap10.0) as a proxy for systematic uncertainties in the actual rates. Note that these $Y_e(\rho )$ profiles differ substantially from our fiducial profile, leading to different inner core masses and GW signals.

Figure 16

Changes in GW observables with variations in electron capture rates. We show results for $\mathrm{\Delta }{h}_{+}$ (at 10 kpc, top panel) and $f_{\text{peak}}$ for SFHo EOS simulations with $A3=634\mathrm{km}$ with our fiducial $Y_e(\rho )$ profile and with the new $Y_e(\rho )$ profiles from simulations with detailed tabulated nuclear electron capture rates (cf. Fig. 15). Differences in electron capture treatment and uncertainties in capture rates lead to differences in the key GW observables that are as large as those induced by switching EOS.

Figure 17

Discerning the EOS. We plot the GW peak frequency against the bounce signal amplitude for each of our 1704 collapsing cores. Data from the $A1=300\mathrm{km}$ simulations are connected with lines to guide the eye. We predict a region of parameter space where we can reasonably expect rapidly rotating core-collapse GW bounce and early postbounce signals to lie given uncertainties in the nuclear EOS. For signals with $\mathrm{\Delta }{h}_{+}\lesssim 15\times 10^{-21}$ (at 10 kpc), we may be able to distinguish the EOS from GW signals if the distance and orientation can be accurately determined. Peak frequencies at the slowest rotation rates (corresponding to $\mathrm{\Delta }{h}_{+}\lesssim 2\times 10^{-21}$ in the figure) are unreliable due to extremely weak GW signals.

Figure 18

Test $Y_e(\rho )$ profiles. We plot the different possibilities for deleptonization functions one might input into the two-dimensional GRHD simulations. The solid red line is the $Y_e(\rho )$ directly taken from the radial profile at the moment when the central $Y_e$ is lowest. The solid black line is also directly taken from the radial data of a GR1D simulation using “shellular” rotation with $A=634\mathrm{km}$, ${\mathrm{\Omega }}_0=5.0\mathrm{rad}{\mathrm{s}}^{-1}$. The dot-dashed line is a fit to the nonrotating $Y_e(\rho )$ using the same parameters as [99] in addition to a high-density slope. The dashed line is the G15 fit from [99]. The dotted line is a record of the central $Y_{e,c}({\rho }_c)$ throughout nonrotating collapse, appended to the $Y_e(\rho )$ profile at $t=0$.