Gravitational wave burst signal from core collapse of rotating stars

We present results from detailed general relativistic simulations of stellar core collapse to a proto-neutron star, using two different microphysical nonzero-temperature nuclear equations of state as well as an approximate description of deleptonization during the collapse phase. Investigating a wide variety of rotation rates and profiles as well as masses of the progenitor stars and both equations of state, we confirm in this very general setup the recent finding that a generic gravitational wave burst signal is associated with core bounce, already known as type I in the literature. The previously suggested type II (or “multiple-bounce”) waveform morphology does not occur. Despite this reduction to a single waveform type, we demonstrate that it is still possible to constrain the progenitor and postbounce rotation based on a combination of the maximum signal amplitude and the peak frequency of the emitted gravitational wave burst. Our models include to sufficient accuracy the currently known necessary physics for the collapse and bounce phase of core-collapse supernovae, yielding accurate and reliable gravitational wave signal templates for gravitational wave data analysis. In addition, we assess the possibility of nonaxisymmetric instabilities in rotating nascent proto-neutron stars. We find strong evidence that in an iron core-collapse event the postbounce core cannot reach sufficiently rapid rotation to become subject to a classical bar-mode instability. However, many of our postbounce core models exhibit sufficiently rapid and differential rotation to become subject to the recently discovered dynamical instability at low rotation rates.

Figure 1

Electron fraction ${\overline{Y}}_e$ obtained from detailed spherically symmetric calculations with Boltzmann neutrino transport versus the maximum density ${\rho }_{\max }$ in the collapsing core. The EoS is encoded in dark hues for the Shen EoS and light hues for the LS EoS with the basis color specifying the progenitor mass.

Figure 2

Time evolution of the gravitational wave amplitude $h$ for representative models with different precollapse rotation profiles using the Shen EoS (red lines) or LS EoS (blue lines). Models with slow and almost uniform precollapse rotation (e.g., s11A1O07) develop considerable prompt postbounce convection visible as a dominating lower-frequency contribution in the waveform, while the waveform for both models with moderate rotation (e.g., s11A3O13, s15A2O05, s20A2O09, s40A1O07, or e15a) and rapidly rotating models, which undergo a centrifugal bounce (e.g., s40A3O13 or e20b), exhibit an essentially regular ringdown. Time is normalized to the time of bounce $t_{\mathrm{b}}$.

Figure 3

Collapse dynamics of all investigated models in the parameter space of precollapse rotational configuration (specified by the precollapse angular velocity ${\Omega }_{\mathrm{c},\mathrm{i}}$ at the center and the precollapse differential rotation length scale $A$), progenitor mass $M_{\mathrm{prog}}$, and EoS. Models marked by unfilled/filled circles undergo a pressure-dominated bounce with/without significant early postbounce convection, while models marked with crosses show a single centrifugal bounce. The EoS is encoded as in Fig. 1, while small/medium/large symbols represent the precollapse rotation parameter A1/A2/A3. For better visibility, the symbols for the same $M_{\mathrm{prog}}$ but different EoS are spread a bit in the vertical direction. Note also that in this and the following plots that encode the precollapse rotational configuration in the form of the parameter ${\Omega }_{\mathrm{i}}$, we refrain from including models e15a, e15b, e20a, and e20b, as these have a precollapse rotation profile that is not given by the analytic rotation law (18).

Figure 4

Mass $M_{\mathrm{ic},\mathrm{b}}$ of the inner core at the time of bounce for all models versus the precollapse initial central angular velocity ${\Omega }_{\mathrm{c},\mathrm{i}}$. The progenitor model, the EoS, the initial rotation parameter $A$, and the collapse dynamics are encoded as in Fig. 3.

Figure 5

Boundary between pressure-dominated and centrifugal bounce in the ${\gamma }_{\mathrm{eff}}-{\beta }_{\mathrm{i}}$ plane for s20 progenitor models using the hybrid EoS in Newtonian gravity (dashed lines) and general relativity (solid lines). The curved dotted lines show the Newtonian results shifted by $-\Delta {\gamma }_{\mathrm{gr}}=0.015$. The transition points for models using the microphysical EoS without and with deleptonization, again for Newtonian gravity (circles) and general relativity (bullets), lie in the shaded areas around ${\gamma }_{\mathrm{eos},\mathrm{Shen}}\simeq 1.32$ and ${\gamma }_{\mathrm{eff},\mathrm{Shen}}\simeq 1.29$, respectively, for the Shen EoS (top panel) and around ${\gamma }_{\mathrm{eos},\mathrm{LS}}\simeq 1.3225$ and ${\gamma }_{\mathrm{eff},\mathrm{LS}}\simeq 1.285$, respectively, for the LS EoS (bottom panel).

Figure 6

Adiabatic index ${\gamma }_{\mathrm{eos}}$ of the Shen EoS (red line) and LS EoS (blue line) versus the maximum density ${\rho }_{\max }$ in the collapsing core for model s20A2O09. Although ${\rho }_{\max }$, which for this model is located in the center of the core, does not follow a trajectory of constant specific entropy, $s$ is still approximately conserved in the prebounce phase. Inset: Magnified view of ${\gamma }_{\mathrm{eos}}$ in the dynamically most relevant density range between ${10}^{12}$ and ${10}^{14}\mathrm{g}{\mathrm{cm}}^{-3}$. The average value of ${\gamma }_{\mathrm{eos}}$ in this density regime is roughly 1.32 for both EoSs.

Figure 7

Boundary between pressure-dominated and centrifugal bounce in the ${\gamma }_{\mathrm{eff}}-{\Omega }_{\mathrm{c},\mathrm{i}}$ plane for models of all progenitors using the hybrid EoS in general relativity. The transition points for models using the microphysical EoS with deleptonization (bullets) lie in the shaded area around ${\gamma }_{\mathrm{eff}}\simeq 1.28$. Except for the rotation profile A2 of the s15 progenitor, the locations of these points are identical for the two EoSs. Note that for the A1 profiles of any progenitor (and the A2 profile for the s11 progenitor) we do not observe a centrifugal bounce for any value of ${\Omega }_{\mathrm{c},\mathrm{i}}$. In contrast to Fig. 5, we use here the ${\Omega }_{\mathrm{c},\mathrm{i}}$ instead of ${\beta }_{\mathrm{i}}$ as parameter for the precollapse rotational configuration (see the discussion in Sec. 2e).

Figure 8

Maximum density ${\rho }_{\max ,\mathrm{b}}$ in the star at the time of bounce for all models versus the precollapse central angular velocity ${\Omega }_{\mathrm{c},\mathrm{i}}$. For moderately fast or rapidly rotating models, which do not exceed nuclear density at bounce, ${\rho }_{\max ,\mathrm{b}}$ is almost identical for the Shen EoS (dark hues) and the LS EoS (light hues), while for slowly rotating models the difference reaches $\Delta {\rho }_{\max ,\mathrm{b}}\simeq {10}^{14}\mathrm{g}{\mathrm{cm}}^{-3}$ in the nonrotating limit. The progenitor mass, the EoS, the precollapse rotation parameter $A$, and the collapse dynamics are encoded as in Fig. 3.

Figure 9

Relative change $\Delta {\rho }_{\max ,\mathrm{b},\mathrm{rel}}$ of the maximum density at bounce when changing from the Shen EoS to the LS EoS for all models versus the precollapse central angular velocity ${\Omega }_{\mathrm{c},\mathrm{i}}$. The progenitor mass and the precollapse differential rotation parameter $A$ are encoded as in Fig. 3, while the collapse dynamics are not specified.

Figure 10

Time evolution of the maximum density ${\rho }_{\max }$ for representative models with different precollapse rotation profiles using the Shen EoS (red lines) or LS EoS (blue lines). While models with at most moderate precollapse rotation rates (e.g., s11A1O07 or s20A2O09) undergo a pressure-dominated bounce at supernuclear densities, rapidly rotating models (e.g., e20b) experience a centrifugal bounce.

Figure 11

Peak value $|h{|}_{\max }$ of the gravitational wave amplitude at 10 kpc distance for the burst signal (neglecting possibly larger contributions from postbounce convection at later times) for all models versus initial precollapse central angular velocity ${\Omega }_{\mathrm{c},\mathrm{i}}$. The progenitor mass, the EoS, the precollapse differential rotation parameter $A$, and the collapse dynamics are encoded as in Fig. 3.

Figure 12

Relative change $\Delta |h{|}_{\max ,\mathrm{rel}}$ of the peak value $|h{|}_{\max }$ of the gravitational wave amplitude for the burst signal when changing from the Shen EoS to the LS EoS for all models versus the precollapse central angular velocity ${\Omega }_{\mathrm{c},\mathrm{i}}$. The progenitor mass and the precollapse differential rotation parameter $A$ are encoded as in Fig. 3, while the collapse dynamics are not specified, as for some models it depends on the EoS.

Figure 13

Radial profiles of the density ${\rho }_{\mathrm{e}}$ in the equatorial plane at the time of bounce for model s20A3O09 (top panel) and model s40A2O13 (bottom panel) using the Shen EoS (red line) and LS EoS (blue line). In the central parts of the proto-neutron star (for these models at radii smaller than the crossing radius $r\simeq 5.5\mathrm{km}$ and $r\simeq 23.5\mathrm{km}$, respectively, marked by dotted lines) the LS EoS leads to higher densities.

Figure 14

Radial profiles of the weighted density ${\rho }_{\mathrm{e}}{r}^2$ in the equatorial plane at the time of bounce for model s20A3O09 (top panel) and model s40A2O13 (bottom panel) using the Shen EoS (red lines) and LS EoS (blue lines). The vertical lines mark the crossing radius.

Figure 15

Spectrum of the gravitational radiation waveform for model s20A1O05 (top panel), model s20A2O09 (center panel), and model s20A3O13 (bottom panel) using the Shen EoS (red line) and LS EoS (blue line). $\widehat{h}$ is the Fourier transform in frequency space of the waveform amplitude $h$. The dotted lines mark the frequency $f_{\max }$ at the maximum of the waveform spectrum, neglecting low-frequency contributions. The scale of the vertical axis is 1 order of magnitude per major tick mark.

Figure 16

Frequency $f_{\max }$ at the maximum of the waveform spectrum for all models with a given progenitor mass versus the precollapse central angular velocity ${\Omega }_{\mathrm{c},\mathrm{i}}$. Only models that undergo pressure-dominated bounce are shown. The dotted lines mark the average ${\overline{f}}_{\max }$ when using the Shen EoS or LS EoS. The progenitor mass, the EoS, the precollapse differential rotation parameter $A$, and the collapse dynamics are encoded as in Fig. 3.

Figure 17

Location of the gravitational wave burst signals from core bounce for all models (including the e15/e20 models) in the $h_{\mathrm{c}}-f_{\mathrm{c}}$ plane relative to the sensitivity curves of the LIGO, assuming at a distance of 10 kpc. The meaning of the arrows 1, 2, and 3 as well as area 4 are explained in the main text. The dotted line marks the average ${\overline{f}}_{\max }$ of the frequency at the maximum of the waveform spectrum. The progenitor model, the EoS, the initial rotation parameter $A$, and the collapse dynamics are encoded as in Fig. 3.

Figure 18

Location of the gravitational wave burst signals from core bounce in the $h_{\mathrm{c}}-f_{\mathrm{c}}$ plane relative to the sensitivity curves of various interferometer detectors (as color-coded) for an extended set of models with the progenitor s20 using the Shen EoS (dark hues) or LS EoS (light hues). The sources are at a distance of 10 kpc for LIGO, 0.8 Mpc for Advanced LIGO, and 15 Mpc for EURO. The dotted lines mark the average ${\overline{f}}_{\max }$ of the frequency at the maximum of the waveform spectrum for the models when using the Shen EoS or LS EoS. Only the EoS and the collapse dynamics are encoded as in Fig. 3, but not the precollapse differential rotation parameter $A$.

Figure 19

Time evolution of the rotation rate $\beta$ around the time of core bounce for various models of the s20 progenitor series computed with the Shen EoS at fixed precollapse degree $A$ of differential rotation and varying the precollapse central angular velocity ${\Omega }_{\mathrm{c},\mathrm{i}}$. Note that we have augmented this sequence by three extra models s20A3O16 to s20A3O18 (with ${\beta }_{\mathrm{i}}=3.00$, 3.50, and 4.00, respectively) not listed in Table .

Figure 20

Rotation rate ${\beta }_{\mathrm{b}}$ at the time of bounce for all models versus the precollapse central angular velocity ${\Omega }_{\mathrm{c},\mathrm{i}}$. The progenitor mass, the EoS, the precollapse differential rotation parameter $A$, and the collapse dynamics are encoded as in Fig. 3. The lower horizontal line approximately separates pressure-dominated bounce models with and without strong prompt postbounce convection, while upper horizontal line marks the approximate transition between pressure-dominated bounce and centrifugal bounce.

Figure 21

Rotation rate ${\beta }_{\mathrm{pb}}$ in the late postbounce phase for all models versus the precollapse central angular velocity ${\Omega }_{\mathrm{c},\mathrm{i}}$. The progenitor mass, the EoS, the precollapse differential rotation parameter $A$, and the collapse dynamics are encoded as in Fig. 3. As in Fig. 20, the horizontal lines again approximately mark the boundaries between different bounce dynamics.

Figure 22

Peak value $|h{|}_{\max }$ of the gravitational wave amplitude at 10 kpc distance for the burst signal (neglecting possibly larger contributions from postbounce convection at later times) for all models versus the rotation rate ${\beta }_{\mathrm{b}}$ at the time of bounce. At slow to moderately rapid rotation, $|h{|}_{\max }$ is proportional to ${\beta }_{\mathrm{b}}$ to high accuracy (as marked by the dotted line with a slope of 1 in the log-log plot), while for ${\beta }_{\mathrm{b}}\gtrsim 10\%$ centrifugal effects reduce $|h{|}_{\max }$.

Figure 23

Radial profile of the angular velocity $\Omega$ in the equatorial plane at 20 ms after the time of core bounce for a representative subset of the models listed in Table . Note that the inner core is in approximate solid body rotation out to about 10 km, while the outer parts of the proto-neutron star and the postshock region rotate strongly differentially. The dotted lines mark the approximate range for the characteristic angular frequency ${\Omega }_{\mathrm{char}}$.