Gravitational-wave signature of core-collapse supernovae

We calculate the gravitational-wave (GW) signatures of detailed three-dimensional (3D) core-collapse supernova simulations spanning a range of massive stars. Most of the simulations are carried out to times late enough to capture more than 95% of the total GW emission. We find that the $f/g$-mode and $f$-mode of protoneutron star oscillations carry away most of the GW power. The $f$-mode frequency inexorably rises as the protoneutron star (PNS) core shrinks. We demonstrate that the GW emission is excited mostly by accretion plumes onto the PNS that energize modal oscillations and also high-frequency (“haze”) emission correlated with the phase of violent accretion. The duration of the major phase of emission varies with exploding progenitor, and there is a strong correlation between the total GW energy radiated and the compactness of the progenitor. Moreover, the total GW emissions vary by as much as 3 orders of magnitude from star to star. For black hole formation, the GW signal tapers off slowly and does not manifest the haze seen for the exploding models. For such failed models, we also witness the emergence of a spiral shock motion that modulates the GW emission at a frequency near $\sim 100\mathrm{Hz}$ that slowly increases as the stalled shock sinks. We find significant angular anisotropy of both the high- and low-frequency (memory) GW emissions, though the latter have very little power.

Figure 1

Mass density profiles for the various progenitors studied here, from 9.0 (reddish) to 23 (blue) $M_{\odot }$. The accretion of the $\mathrm{Si}/\mathrm{O}$ interface frequently corresponds with the onset of shock revival. Note that the 12.25- and $14\text{−}{M}_{\odot }$ models do not explode and are destined to produce a stellar-mass black hole.

Figure 2

Early postbounce behavior of the mean shock radii (in km) for all the models as a function of time (in seconds after core bounce). All models explode except for the 12.25- and $14\text{−}{M}_{\odot }$ progenitors, which show evidence for a spiral SASI after $\sim 400$ and $\sim 300\mathrm{ms}$ postbounce, respectively, in the form of $\sim 10\mathrm{ms}$ oscillations in the shock radii. Note that the 12.25- and $14\text{−}{M}_{\odot }$ (nonexploding) progenitors also shows a secular $\sim 70\mathrm{ms}$ oscillation in the shock radii.

Figure 3

The gravitational-wave strain, from matter motions, times distance (product in cm) for all the 3D models studied in this paper as a function of time after bounce (in seconds) observed along the positive $x$ axis in the simulation frame. Both cross (blue) and plus (black) polarizations are illustrated. The red star indicates the approximate time of the onset of explosion, defined here as when the stalled shock radius changes concavity at $\sim 150\mathrm{km}$ and accelerates. See Fig. 2.

Figure 4

Same as Fig. 3, but now focusing on the matter strain during the first 250 ms after core bounce to illustrate the prompt convection. Note the stately oscillation present in most models of $\sim 10\mathrm{ms}$ periods visible in the $+$ polarization and absent in the $\times$ polarization. This is a generic feature indicative of the strain polarization geometry. Note that only model 9a has any perturbations on bounce.

Figure 5

In this figure and the next, we provide the gravitational-wave energy spectrogram (to a normalization constant of order unity) at 10 kiloparsecs (kpc) as a function of time after bounce (in seconds) for the 3D models highlighted in this paper. Note that the frequency ranges and durations differ from panel to panel. The colormap provides the spectral density in units of Bethes (${10}^{51}\mathrm{egs}$) per Hertz, where we have integrated over a running 40-ms bin around each time. We have found that the results do not depend significantly upon the width of this window. Though the models in this first set are for 9-, 9.25-, and $9.5\text{−}{M}_{\odot }$, the description in this caption also applies to the 11-, 12.25-, 15.01-, and $23\text{−}{M}_{\odot }$ models in the accompanying figure (Fig. 6). All models show a strong feature out to $\sim 50\mathrm{ms}$ and up to $\sim 2000\mathrm{Hz}$ associated with prompt convection, followed by a short quiescent phase, during which the turbulence between the shock and the protoneutron star grows. Thereafter, all models illustrate high frequency behavior before $\sim 1\mathrm{s}$ (even earlier for the rapidly exploding lower progenitor mass models), which truncates in power as the turbulent accretion which excites it subsides. This “emission haze” is either a superposition of core pulsational $\ell =2$ $p$-modes with a range of node numbers or the GW emission from the supersonic plumes themselves impinging upon the PNS core, or some combination of both. The fundamental $f/g$-mode (early), then the $f$-mode (later), is manifest and significant in all models for the duration of the simulations. We note that the $15.01\text{−}{M}_{\odot }$ model (next figure), which has a relatively lower ejecta mass, sustains turbulent accretion and high frequency haze emission until the end of our simulation, $\sim 2.0\mathrm{s}$ postbounce. We also note that we see higher harmonics/overtones after the first second and after the earlier emission haze abates, most visibly in the 15.01- and $23\text{−}{M}_{\odot }$ models. These later-time PNS pulsational modes are likely the $\ell =2$, $n=1$ $p$-mode. The fundamental $f$-mode frequency grows roughly quadratically with time [89] and is associated with the shrinkage of the PNS as it executes its Kelvin-Helmholtz contraction phase, driven by neutrino losses. Except for the very early phase after bounce, most $g$-modes are suppressed due to both the presence and the growth of PNS convection in the deep core and to their lower frequencies (approximately a few hundred Hertz). Early on, all models show a dark band near $\sim 1000\mathrm{Hz}$ which may be a signature of an avoided crossing and interference with a trapped $g$-mode. See text in Secs. 3a and 3d for a discussion.

Figure 6

Same as Fig. 5, but for models 11-, 12.25-, 15.01-, and $23\text{−}{M}_{\odot }$.

Figure 7

Plots (2D slices of 3D data) of the Mach number distribution in the inner 100 kilometers of the $23\text{−}{M}_{\odot }$ simulation at 574 and 632 ms after bounce. Seen are representative supersonic plumes crashing onto the PNS core. Such plumes not only excite the GW emission, but themselves emit a component of the GW signal. See text in Sec. 3a for a discussion.

Figure 8

Plots of the effective strains as a function of time after bounce (in seconds) for two representative 3D models (the black hole former $14\text{−}{M}_{\odot }$ and late-exploding $23\text{−}{M}_{\odot }$ models). Note that we see a dark red band at low ($\le 50\mathrm{Hz}$) frequencies for the exploding model associated with the asymmetric ejecta at later times (matter “memory”). The nonexploding model shows a spiral SASI signature at $\sim 100\mathrm{Hz}$. This quasiperiodic feature is also seen after $\sim 0.4\mathrm{s}$ in Fig. 2 for both the $14\text{−}{M}_{\odot }$ and the $12.25\text{−}{M}_{\odot }$ black hole formers. Such a feature shows up only for nonexploding models when the shock radius has shrunken below $\sim 100\mathrm{km}$. These features are much less prominent in the gravitational-wave energy spectra because they are low-frequency, and, hence, low-power components. There may be a hint of the SASI itself in the lower left-hand corner of the panel to the right. See text in Sec. 3a for a discussion.

Figure 9

In this figure, we show the spectrogram of the $z$ component of the shock dipole position for four models. Note the focus on lower frequencies. In the nonexploding 12.25- and $14\text{−}{M}_{\odot }$ models, the spiral-SASI modes are clearly seen and manifest gradually increasing frequencies as the PNS core shrinks. The exploding models (for example, the 15.01- and $23\text{−}{M}_{\odot }$ models depicted here) do not show this feature, but do manifest the even lower frequency signals due to the matter memory caused by explosion anisotropy (see also Fig. 8). We note that the $23\text{−}{M}_{\odot }$ model may show a weak SASI signature (analogous with that found in the $25\text{−}{M}_{\odot }$ model in Ref. [72]) that vanishes at $\sim 0.55\mathrm{s}$ and is perhaps indicative of its delayed explosion time. The onset of explosion corresponds with the rise of the low-frequency signal memory in the bright yellow band.

Figure 10

The gravitational-wave energy from matter motions (in $M_{\odot }{\mathrm{c}}^2$) for all the 3D models highlighted in this paper as a function of time after bounce (in seconds). Note that the total GW energy radiated differs by $\sim 3$ orders of magnitude from the least massive, $9M_{\odot }$ to the most massive, $23M_{\odot }$ progenitor. This energy grows by 3 orders of magnitude for the most massive progenitors over the first $\sim 2\mathrm{s}$ of simulation, but is already asymptoting shortly after 1 s postbounce. All models show rapid growth in the first $\sim 50\mathrm{ms}$, associated with the onset of prompt convection driven by the overturn of the shocked mantle dynamically generated at and after bounce as the shock stalls initially into accretion. After this phase, the neutrino-driven turbulence between the shock and the protoneutron star core grows in vigor over a period of $\sim 100\mathrm{ms}$ and excites a spectrum of core pulsational $f$- and some $p$-modes and likely generates a GW component due to the impinging of the plumes onto the PNS that all together constitute the bulk of the gravitational radiation issuing from the supernova. This phase can last from hundreds of milliseconds to $\le 1.5\mathrm{s}$, depending upon progenitor, after which the strains subside to a hum dominated by the fundamental $\ell =2$ $f$-mode and (weakly) overtones. This last phase can last for many seconds. However, the signals from those models destined to leave black holes are still vigorous for a longer period of time, since accretion persists for these models until the black hole forms, after which the signal ceases abruptly. See the text for a discussion.

Figure 11

The gravitational-wave energy from matter motions (in $M_{\odot }{\mathrm{c}}^2$) for all the 3D models in this study versus the model compactness at $1.75M_{\odot }$ [74]. Note the clear linear correlation on the log-log scale, with a power-law slope of $\sim 0.73$. The outliers are the nonexploding models (12.25- and $14\text{−}{M}_{\odot }$, shown with triangle markers) and these models generally take longer to radiate a given amount of GW energy (see Fig. 10). The black hole formers do not as a rule manifest turbulence that is as vigorous as those models that eventually explode. All exploding models show a steep growth in the GW energy associated with the onset of vigorous turbulence. The $23M_{\odot }$, which explodes the latest, has the latest growth spurt in gravitational energy. By contrast, nonexploding models show a slower rate of increase of GW emission, associated with a slower growth rate of turbulence. However, due to the persistently stalled shock and continued accretion, the black hole formers will generate GWs for a longer time and will saturate their total GW emission more slowly. Finally, a spiral SASI mode emerges for the black hole formers when the stalled shock wave retreats significantly (below $\sim 100\mathrm{km}$). This mode modulates the stalled accretion shock radius and the GW emissions of these black hole progenitors quasiperiodically.

Figure 12

Spacetime diagrams illustrating the convective regions in the PNS for eight representative models, identified by the convective luminosity, as a function of time after bounce. Note that convection reaches the PNS core within $\sim 1.6$ to $\sim 4.0\mathrm{s}$, earlier for the low-mass progenitors and later for the $23\text{−}{M}_{\odot }$ model. For all models, PNS convection starts in a shell and then grows to encompass much of the core. The fact that $g$-modes are evanescent in convective regions bears on the suppression of $g$-mode excitation in general, but also on the eventual severing of the interaction/coupling between a trapped $g$-mode interior to it and the PNS surface where the plumes are exciting much of the GW emission. See text for a discussion (Sec. 3d).

Figure 13

The gravitational-wave strain $h_{+}\mathrm{D}$ (cm) as a function of time after bounce (in seconds) for the 9b model, the nonexploding model $12.25M_{\odot }$, and model $23M_{\odot }$ for various, arbitrary, viewing angles illustrated by color. These plots do not include the neutrino memory (Sec. 3f). Note the large anisotropy by viewing angle, which is dominated at late times by the matter memory, which is sourced by the large-scale explosion asymmetry. In the black hole model ($12.25M_{\odot }$), we see no memory, and little anisotropy, at late times. Though we show only the $+$ polarization, the results are consistent for both polarizations. We see that in principle some measure of the degree and angle of the matter explosion asymmetry can be discerned by the sign and magnitude of this memory. However, the neutrino memory, if only weakly correlated with the matter memory, will complicate the extraction of such a feature. Nevertheless, the difference in the frequency spectra of the matter and neutrino memories generically, with the former at a higher average frequency, may allow one someday to disentangle the two.

Figure 14

Same as Fig. 3, but for the gravitational-wave memory due to neutrino emission anisotropy. Because of the cumulative nature of the neutrino GW memory, the strain is both larger and shows less variation on small timescales when compared to the matter-driven strain. We reach strains as high as 1000 cm for the 11- and $23\text{−}{M}_{\odot }$ models. Note that this is consistent with earlier work [93] showing continued growth with time of the memory strain. The strain growth rate also accelerates, showing upwards of 600 cm growth from $\sim 2$ to $\sim 3\mathrm{s}$ for the $11\text{−}{M}_{\odot }$ progenitor. This emphasizes our essential point that models should be carried out to late times to fully capture not only the explosion dynamics but also the emerging GW signatures. The $11\text{−}{M}_{\odot }$ progenitor has the highest neutrino memory strain among those models we present in this paper. Otherwise, we see a general correlation with mass, with the low-mass progenitors having the smallest neutrino memory. The nonexploding 12.25- and $14\text{−}{M}_{\odot }$ models show significant evolution and magnitude in their memory signatures.

Figure 15

Same as Fig. 10, but for the radiated gravitational-wave energy due to the neutrino anisotropy. As seen in Fig. 14, unlike the matter contribution, the neutrino anisotropy continues to contribute to the gravitational signal past 6 s (for our longest simulation), emphasizing the presence of sustained neutrino emission asymmetry. We see again at least 2 orders of magnitude of growth with time after $\sim 50\mathrm{ms}$ in the associated GW energy, this time in the neutrino memory component. The $23\text{−}{M}_{\odot }$ progenitor has neutrino GW memory energies only slightly higher than the matter GW components for the least energetic, lowest-mass progenitors. Importantly, for the same progenitor model, the late time neutrino-sourced GW energies, albeit still growing, are more than 2 orders of magnitude smaller than the corresponding matter components.

Figure 16

Turbulent luminosity (blue, in ${10}^{51}\mathrm{erg}{\mathrm{s}}^{-1}$) at 110 km (except for model $12.25M_{\odot }$, where it is evaluated at 2.5 times the PNS radius), overplotted with both filtered GW strain polarizations (green and black) multiplied by distance (product in cm) as a function of time after bounce (in seconds). We emphasize the clear correlation shown here—the outer turbulent luminosity closely follows the strain. It reproduces both the global time behavior, rising and following at coincident times with the strain, and the episodic “packets” of accretion (for instance, reproducing the bump in the $15.01\text{−}{M}_{\odot }$ strain at $\sim 1.35\mathrm{s}$ and for the $23\text{−}{M}_{\odot }$ strain at $\sim 3.5\mathrm{s}$). From this figure, it is clear that it is the outer downward turbulent convective flux onto the PNS that drives the gravitational-wave signature after the prompt convection phase.

Figure 17

In this figure, we show the correlation between the gravitational-wave energy spectrogram and the accretion rate power in frequency components with $f>25\mathrm{Hz}$ for two representative models. The background spectra in this figure are the same as those plotted in Figs. 5 and 6, but with different color-bar ranges so that the late time structures can be more clearly seen. The orange curve shows the accretion rate “power” in frequency components with $f>25\mathrm{Hz}$. The 25-Hz filter is chosen here because the spectrogram is calculated using a 0.04s-wide sliding window. The accretion rates are measured at a radius of 25 km, and the PNS radii of the two models fall below this value at $\sim 0.9\mathrm{s}$ and $\sim 1.0\mathrm{s}$ after bounce, respectively. Note also that this colormap for the $23\text{−}{M}_{\odot }$ model spectrogram reveals a weak signal below the $f$-mode associated with the avoided crossing that may be the bumped $g$-mode and that seems to continue beyond $\sim 1.5\mathrm{s}$. See text in Sec. 3g for a discussion.

Figure 18

Convective hydrodynamic luminosity within the PNS (blue, in ${10}^{51}\mathrm{erg}{\mathrm{s}}^{-1}$) at its peak value in the inner PNS, overplotted with both gravitational-wave strain polarizations (black and green) multiplied by distance (product in cm) as a function of time after bounce (in seconds) for models 9a, 9b, 9.25, and 9.5. Figure 19 depicts the same quantities, but for models 11, 12.25, 15.01, and $23M_{\odot }$. Note that PNS convection first increases in strength over time and then stays approximately constant (with some minor excursions) or stays high with a secular evolution for the duration of the simulations. Moreover, PNS convection grows in extent over time to encompass more and more of the core, expanding deeper in radius and reaching the center of the PNS after from $\sim 1.7\mathrm{s}$ (the $9\text{−}{M}_{\odot }$ model) to $\sim 4\mathrm{s}$ (the $23\text{−}{M}_{\odot }$ model; see Fig. 19), all the while maintaining its vigor. This behavior shows little or no correlation with the GW strain, which decreases in magnitude with time from an earlier peak and nearly flattens at late time (dominated then by a humming fundamental $f$-mode (see Fig. 5). In particular, at late times, when the strain has subsided substantially, the inner PNS convective luminosity is always still strong, indicating that PNS convection is not the dominant excitation mechanism of CCSN GW emission.

Figure 19

Same as Fig. 18, but for models 11, 12.25, 15.01-, and $23M_{\odot }$.

Figure 20

Gravitational-wave sensitivity curve for all the models as a function of time (in seconds after core bounce), compared with the CCSN GW signals at 10 kpc, including both matter (high frequency, dominant above several hundred Hertz) and neutrino contributions to GW emissions (low frequency, dominant from sub-Hertz to several tens of Hertz). Note that DECIGO, the Einstein Telescope, and the Cosmic Explorer can probe down to the lowest-mass progenitors across three decades in frequency ($\sim 10$ to $\sim 5000\mathrm{Hz}$), whereas aLIGO is sensitive between $\sim 30$ and $\sim 3000\mathrm{Hz}$ only to the most massive progenitors. Our lower limit in frequency is set by the duration of the CCSN simulation. Since our longest simulation is out to 6.2 s, simulations are generally unable to constrain the spectrogram data between 0.1 and 1 Hz. Our upper limit in frequency is set by the Nyquist limit in Table 1.