Gravitational wave extraction in simulations of rotating stellar core collapse

We perform simulations of general relativistic rotating stellar core collapse and compute the gravitational waves (GWs) emitted in the core-bounce phase of three representative models via multiple techniques. The simplest technique, the quadrupole formula (QF), estimates the GW content in the spacetime from the mass-quadrupole tensor only. It is strictly valid only in the weak-field and slow-motion approximation. For the first time, we apply GW extraction methods in core collapse that are fully curvature based and valid for strongly radiating and highly relativistic sources. These techniques are not restricted to weak-field and slow-motion assumptions. We employ three extraction methods computing (i) the Newman-Penrose (NP) scalar ${\Psi }_4$, (ii) Regge-Wheeler-Zerilli-Moncrief master functions, and (iii) Cauchy-characteristic extraction (CCE) allowing for the extraction of GWs at future null infinity, where the spacetime is asymptotically flat and the GW content is unambiguously defined. The latter technique is the only one not suffering from residual gauge and finite-radius effects. All curvature-based methods suffer from strong nonlinear drifts. We employ the fixed-frequency integration technique as a high-pass waveform filter. Using the CCE results as a benchmark, we find that finite-radius NP extraction yields results that agree nearly perfectly in phase, but differ in amplitude by $\sim 1\%–7\%$ at core bounce, depending on the model. Regge-Wheeler-Zerilli-Moncrief waveforms, while, in general, agreeing in phase, contain spurious high-frequency noise of comparable amplitudes to those of the relatively weak GWs emitted in core collapse. We also find remarkably good agreement of the waveforms obtained from the QF with those obtained from CCE. The results from QF agree very well in phase and systematically underpredict peak amplitudes by $\sim 5\%–11\%$, which is comparable to the NP results and is certainly within the uncertainties associated with core collapse physics.

Figure 1

Top panel (from left to right): $D{h}_{+,e}^{\mathrm{NP}}$ computed using NP extraction at radii $R_{\mathrm{ex}}=(500,700,800,900,1000)M_{\odot }$, $D{h}_{+,e}^{\mathrm{CCE}}$ computed with CCE at ${\mathcal{J}}^{+}$ using world-tube data at $R_{\Gamma }=(500,700,900,1000)M_{\odot }$, and $D{h}_{+,e}^{\mathrm{RWZM}}$ computed using the RWZM formalism at radii $R_{\mathrm{ex}}=(500,700,800,900,1000)M_{\odot }$, all for model A1B3G3. Bottom panels: Absolute difference $\sigma$ of the waves extracted at the various extraction radii and world-tube locations from those extracted at the outermost radius/location. The waveforms converge with increasing extraction radius and world-tube location. For NP extraction, we measure at the outermost detector sphere $R_{\mathrm{ex}}=1000M_{\odot }$ a maximum difference to the next closest detector $R_{\mathrm{ex}}=900M_{\odot }$ of ${\sigma }_4=2\mathrm{cm}$, which corresponds to a percentage error of $\sim 1\%$ relative to the maximum. For CCE, we measure a maximum difference of ${\sigma }_4=1.5\mathrm{cm}$, corresponding to a percentage error of $\sim 0.7\%$ relative to the maximum. For RWZM extraction, we have ${\sigma }_4=14\mathrm{cm}$, corresponding to $\sim 4\%$ relative to the maximum. We note that RWZM is subject to additional high-frequency features and also contains a spurious spike in the bounce peak.

Figure 2

Radial dependence of waveforms computed for model A1B3G5. See caption of Fig. 1 for details. For NP extraction, we measure at the outermost detector sphere $R_{\mathrm{ex}}=1000M_{\odot }$ a maximum difference of ${\sigma }_4=0.4\mathrm{cm}$ to the next closest detector $R_{\mathrm{ex}}=900M_{\odot }$. This corresponds to a percentage error of $\sim 2\%$ relative to the maximum. For CCE, we measure a maximum difference of ${\sigma }_4=1.6\mathrm{cm}$, corresponding to a percentage error of $\sim 6\%$ relative to the maximum. For RWZM extraction, we have ${\sigma }_4=2.4\mathrm{cm}$, corresponding to $\sim 8\%$ relative to the maximum. We note that RWZM is subject to strong additional high-frequency features. We also note that, while CCE shows for this model a larger error due to matter effects than finite-radius ${\Psi }_4$ extraction, it is more accurate at low frequencies (cf. Table ).

Figure 3

Radial dependence of waveforms computed for model A3B3G3. See caption of Fig. 1 for details. For NP extraction, we measure at the outermost detector sphere $R_{\mathrm{ex}}=1000M_{\odot }$ a maximum difference of ${\sigma }_4=2\mathrm{cm}$ to the next closest detector $R_{\mathrm{ex}}=900M_{\odot }$. This corresponds to a percentage error of $\sim 1\%$ relative to the maximum. For CCE, we measure a maximum difference of ${\sigma }_4=1\mathrm{cm}$, corresponding to a percentage error of $\sim 0.2\%$ relative to the maximum. For RWZM extraction, we have ${\sigma }_4=17\mathrm{cm}$, corresponding to $\sim 4\%$ relative to the maximum.

Figure 4

Comparison of waveform amplitudes $D{h}_{+,e}$, their absolute differences $\sigma$ from CCE waveforms, and spectrograms of the power ratio $L_{\mathrm{dB}}$ between waveforms obtained from each extraction method and waveforms obtained via CCE for model A1B3G3. If $L_{\mathrm{dB}}=0$, the waveform of the particular extraction method yields equal power per time and frequency bin with respect to that obtained with CCE. This is indicated by green colors. Blue colors indicate less power, and red colors indicate more power. See text for details. NP extraction and the quadrupole approximation yield remarkable agreement with CCE at frequencies below 2 kHz, at and after core bounce, while the RWZM formalism yields a spurious spike during core bounce and generally contains artificial high-frequency oscillations. This also becomes clear in the spectrograms of the power ratio between RWZM formalism and CCE since $L_{\mathrm{dB}}>0$ over a wide range of time and frequencies (bottom panel). Prior to core bounce, $-8\mathrm{ms}<\mathrm{t}<-1\mathrm{ms}$, NP extraction results in less power compared to CCE, while the QF yields more power.

Figure 5

Comparison of waveform amplitudes $D{h}_{+,e}$, their absolute differences $\sigma$ to CCE waveforms, and spectrograms of the power ratio $L_{\mathrm{dB}}$ between waveforms obtained from each extraction method and waveforms obtained via CCE for model A1B3G5. If $L_{\mathrm{dB}}=0$, the waveform of the particular extraction method yields equal power per time and frequency bin with respect to that obtained with CCE. This is indicated by green colors. Blue colors indicate less power, and red colors indicate more power. See text for details. The waveforms from NP extraction and quadrupole approximation agree well at frequencies below 2 kHz, at and after core bounce, while the RWZM formalism is subject to artificial high-frequency oscillations. The spectrograms of the power ratio between waveforms from the RWZM formalism and CCE in the bottom panel further support this, since $L_{\mathrm{dB}}>0$ in this case. Prior to core bounce, $-8\mathrm{ms}<\mathrm{t}<-1\mathrm{ms}$, NP extraction results in less power compared to CCE, while the QF yields more power.

Figure 6

Comparison of waveform amplitudes $D{h}_{+,e}$, their absolute differences $\sigma$ to CCE waveforms, and spectrograms of the power ratio $L_{\mathrm{dB}}$ between waveforms obtained from each extraction method and waveforms obtained via CCE for model A3B3G3. If $L_{\mathrm{dB}}=0$, the waveform of the particular extraction method yields equal power per time and frequency bin with respect to that obtained with CCE. This is indicated by green colors. Blue colors indicate less power, and red colors indicate more power. See text for further explanation. The waveforms from NP extraction and from the quadrupole approximation agree well below 2 kHz, while the waveform from the RWZM formalism contains artificial high-frequency oscillations, thus leading to a higher power emission than waveforms obtained via CCE (bottom panel). Prior to core bounce, $-10\mathrm{ms}<\mathrm{t}<-2\mathrm{ms}$, NP extraction results in less power compared to CCE, while the QF yields more power.

Figure 7

Comparison of quadrupole to CCE waveforms for model A1B3G3. At and immediately after core bounce, the PV variant leads to a marginally smaller absolute difference from CCE ($\sim 1\%$) than any of the other QF variants.

Figure 8

Comparison of quadrupole to CCE waveforms for model A1B3G5. The PV variant results in the smallest differences of the change $\delta D{h}_{+,e}$ between the first and second peaks (by $\sim 2\%$) compared to CCE.

Figure 9

Comparison of quadrupole to CCE waveforms for model A3B3G3. As for model A1B3G3 (Fig. 7), the PV variant leads to the smallest difference from CCE ($\sim 5\%$).

Figure 10

Convergence of the waveform of model A3B3G3 as computed at ${\mathcal{J}}^{+}$ using CCE. In the bottom panel, we show the absolute differences $\sigma$ between low and medium resolutions, and high and medium resolutions, respectively. Without rescaling any of the difference curves, we observe that they approximately line up at and after core bounce so that the convergence factor is simply given by $C\sim 1$. This indicates a convergence order at and after core bounce of 1. In the prebounce phase, the difference between medium and high resolutions is slightly smaller than the difference between medium and low resolutions, resulting in a slightly larger convergence factor. Given our resolutions, this factor corresponds roughly to second-order convergence.