Systematic study of the stochastic gravitational-wave background due to stellar core collapse

Stellar core collapse events are expected to produce gravitational waves via several mechanisms, most of which are not yet fully understood due to the current limitations in the numerical simulations of these events. In this paper, we begin with an empirical functional form that fits the gravitational-wave spectra from existing simulations of stellar core collapse and integrate over all collapse events in the Universe to estimate the resulting stochastic gravitational-wave background. We then use a Gaussian functional form to separately fit and model a low-frequency peak in the core-collapse strain spectra, which likely occurs due to prompt convection. We systematically study the parameter space of both models, as well as the combined case, and investigate their detectability by upcoming gravitational-wave detectors, such as Advanced LIGO and the Einstein Telescope. Assuming realistic formation rates for progenitors of core-collapse supernovae, our results indicate that both models are 2–4 orders of magnitude below the expected sensitivity of Advanced LIGO, and 1–2 orders of magnitude below that of the Einstein Telescope.

Figure 1

${\mathrm{\Omega }}_{\mathrm{GW}}(f)$ for various parameter choices for the high-frequency model of an SGWB produced by stellar core collapse. $a$ and $b$ are in units of Hz, while $\xi$ has units of ${\mathrm{m}}^2/\mathrm{s}$. Overall, this model has three free parameters: $a$, $b$, and $\xi$. In general, the GW energy density increases as $a$ decreases; at frequencies above a few Hz, it essentially scales proportionally to $a^{-6}$. As $b$ increases, the rate at which the exponential term in Eq. (9) suppresses the GW energy density is decreased; this leads to an increase in the overall energy density and pushes the peak of the distribution to higher frequencies. Finally, the GW energy density scales proportionally to $\xi$. The effect of each parameter on the spectrum is further illustrated in Fig. 2. Also shown are the $\mathrm{SNR}=2$ sensitivities of Advanced LIGO [99] and Einstein Telescope (ET) [67], assuming one year of exposure and two colocated detectors.

Figure 2

Left: ${\mathrm{\Omega }}_{\mathrm{GW}}/\xi$ at 100 Hz versus $a$ for several choices of $b$. Right: ${\mathrm{\Omega }}_{\mathrm{GW}}/\xi$ at 100 Hz versus $b$ for several choices of $a$. Overall, ${\mathrm{\Omega }}_{\mathrm{GW}}$ tends to increase as $a$ decreases and as $b$ increases. See the text for more details.

Figure 3

Gravitational-wave signal from the Ott et al. $\mathrm{s}27{\mathrm{F}}_{\mathrm{HEAT}}1.05$ simulation [74]. In particular, this signal is from a polar observation of a cross-polarized gravitational wave. The original $f|\tilde{h}(f)|$ data is plotted in gray and the fit to this data (with $a=5\mathrm{Hz}$ and $b=63\mathrm{Hz}$) of our $f|\tilde{h}(f)|$ model (Eq. (7)) is shown in black.

Figure 4

Plus-polarized gravitational wave signal from an equatorial observation computed in the Abdikamalov et al. A4O06.5 simulation [102]. The original $f|\tilde{h}(f)|$ data is plotted in gray, and the fit to this data (with $a=82\mathrm{Hz}$ and $b=248\mathrm{Hz}$) of our $f|\tilde{h}(f)|$ model (Eq. (7)) is shown in black.

Figure 5

Plus-polarized gravitational wave signal from the Yakunin et al. B20-WH07 simulation [75] produced by changing mass quadrupole moment (as opposed to neutrino emission) is plotted. The original $f|\tilde{h}(f)|$ data is plotted in gray, and the fit of our $f|\tilde{h}(f)|$ model (with $a=6\mathrm{Hz}$ and $b=11\mathrm{Hz}$, Eq. (7)) is plotted in black.

Figure 6

Gravitational wave signal from an equatorial observation computed by Müller et al. in the u8.1 simulation [74]. The original $f|\tilde{h}(f)|$ data are plotted in gray, and the fit to these data (with $a=75\mathrm{Hz}$ and $b=201\mathrm{Hz}$) by our $f|\tilde{h}(f)|$ model (Eq. (7)) is shown in black.

Figure 7

Gravitational wave signal from the Ott et al. $\mathrm{s}27{\mathrm{F}}_{\mathrm{HEAT}}1.00$ simulation [74]. In particular, this signal is from a polar observation of a plus-polarized gravitational wave. The original $f|\tilde{h}(f)|$ data is plotted in gray, and the fit to this data (with $a=5\mathrm{Hz}$ and $b=56\mathrm{Hz}$) to our $f|\tilde{h}(f)|$ model in Eq. (7) is shown in black. Note the low-frequency structure at $\approx 30–100\mathrm{Hz}$, which deviates from the functional form of Eq. (7), necessitating a modified functional form to describe this spectrum accurately.

Figure 8

${\mathrm{\Omega }}_{\mathrm{GW}}(f)$ for various parameter choices for the low-frequency model of an SGWB produced by stellar core collapse. $\mu$ and $\sigma$ are in units of Hz, while $A^{\prime }$ is in units of $\mathrm{m}/{\mathrm{kg}}^{1/2}$. In Eq. (13), $\sigma$ defines the width of the Gaussian peak and has no effect on the peak amplitude of the strain spectrum; however, when it is integrated over redshift to calculate ${\mathrm{\Omega }}_{\mathrm{GW}}$, a wider peak results in an increased GW energy density. A larger value of $\sigma$ also allows for some amount of GW energy density at frequencies above $\mu$, while low values of $\sigma$ result in a sharp cutoff at $f=\mu$. Finally, ${\mathrm{\Omega }}_{\mathrm{GW}}(f)$ goes as $A^{\prime 2}$, so this is simply a scaling parameter [111].

Figure 9

Top left: 95% confidence expected sensitivity contours for Advanced LIGO and the Einstein Telescope in the $\xi -a$ plane, after marginalizing over the $b$ parameter. Top right: 95% confidence expected sensitivity contours for Advanced LIGO and the Einstein Telescope in the $\xi -b$ plane, after marginalizing over the $a$ parameter. Bottom left: Posterior distribution of $\xi$ after marginalizing over all other parameters; expected sensitivities at 95% confidence are shown as dashed vertical lines. Bottom right: Sensitivity in $\xi$ is translated into sensitivity in the $E_{\nu }-⟨q⟩$ plane, assuming ${\lambda }_{\mathrm{CC}}\approx 0.01M_{\odot }^{-1}$. More details are provided in the text [111].

Figure 10

Top left: 95% confidence expected sensitivity contours for Advanced LIGO and the Einstein Telescope in the $A^{\prime }-\mu$ plane, after marginalizing over the $\sigma$ parameter. Top right: 95% confidence expected sensitivity contours for Advanced LIGO and the Einstein Telescope in the $A^{\prime }-\sigma$ plane, after marginalizing over $\mu$. Bottom left: Posterior distribution of $A^{\prime }$ after marginalizing over all other parameters; expected sensitivities at 95% confidence are shown as dashed vertical lines. Bottom right: Sensitivity in $A^{\prime }$ is translated into sensitivity in the $A-{\lambda }_{\mathrm{CC}}$ plane. More details are provided in the text [111].

Figure 11

Expected 95% sensitivity contours for Advanced LIGO (black) and the Einstein Telescope (gray) in the $b-a$ plane (left) and the $A^{\prime }-\xi$ plane (right). The contours in the $\sigma -\mu$ plane span most of the area and do not provide significant insight; thus, they are not shown here. More details are provided in the text.