Stochastic gravitational-wave background from cosmological supernovae

Based on new developments in the understanding of supernovae (SNe) as gravitational-wave (GW) sources we estimate the GW background from all cosmic SNe. For a broad range of frequencies around 1 Hz, this background is crudely comparable to the GW background expected from standard inflationary models. While our estimate remains uncertain within several orders of magnitude, the SN GW background may become detectable by second-generation space-based interferometers such as the proposed Big Bang Observatory (BBO). By the same token, the SN GWs may become a foreground for searches of the inflationary GWs, in particular, for sub-Hz frequencies where the SN background is Gaussian and where the BBO will be most sensitive. SN simulations lasting far beyond the usual cutoff of about 1 s are needed for more robust predictions in the sub-Hz frequency band. An even larger GW background can arise from a hypothetical early population of massive stars, although their GW source strength as well as their abundance are currently poorly understood.

Figure 1

Spectrum of GW background from Eq. (4) for a simulation from Ref. 7. Solid lines: the spectrum is continuously extended with the low-$f$ tail according to Eq. (3). The upper curve is obtained from Fig. 9 of Ref. 7, whereas the lower is the same shifted downwards by a factor 100, which is considered a more realistic estimate. Horizontal lines: GW stochastic spectrum produced during slow-roll inflation and evaluated from Eq. (6) of Ref. 41. We assume $T/S=0.3$ for the ratio of the tensorial and scalar contributions to the cosmic microwave background radiation (CMBR) anisotropy and consider two values $\pm {10}^{-3}$ of the running of the tensorial power-law index. Dotted lines: sensitivities of the indicated detectors. The BBO sensitivity is approximate and may change slightly in the final design.

Figure 2

Neutrino luminosity $L_{\nu }(t)$, anisotropy $q(t)$, and GW strain $h(t)$ times distance from anisotropic neutrino emission only, see Eq. (1), for model s15r of Ref. 6, as functions of time after bounce. We also show the GW strain $h(t)$ times distance from anisotropic neutrino emission, using the average anisotropy $⟨q⟩=0.45\%$.

Figure 3

GW source spectra: Solid and dotted ragged lines are total and neutrino contribution, respectively, for model s15r of Ref. 6 at distance $D=10\mathrm{kpc}$. The straight solid line is the low-$f$ tail Eq. (3) with $h_{\infty }$ from Eq. (1), using $q(t)$ and $L_{\nu }(t)$ of Fig. 2, leading to $f|\tilde{h}(f)|\simeq 2.21\times {10}^{-22}$. The dashed line is the schematic spectrum for PopIII stars of Eq. (8).

Figure 4

GW background for model s15r with rotation (colored bands and solid lines) and model s11nr180 without rotation (dashed line) from Ref. 6. The source spectra have been continuously extended using the zero-$f$ tail, Eq. (3), for $f\lesssim 1\mathrm{Hz}$, except for the solid lines. The dark shaded band and lower solid lines for model s15r reflect the plausible range 6–36 of the enhancement factor to correct for the limited time of the simulation. For model s11nr180 the lower enhancement factor of 6 was assumed. We always use $\alpha =0$ in Eq. (7), except for the light shaded band and upper solid line which show the difference between $\alpha =0$ and $\alpha =2$ (for enhancement factor 36) for model s15r.

Figure 5

GW source spectra from mass motions plus anisotropic neutrino emission for the proto-neutron star model pns180 of Ref. 6 at a distance of 10 kpc. The darker (upper on the left) curve is for the full simulation with the fluctuating $q(t)$ integrated over 1.2 s, whereas for the lighter (lower on the left) curve $q(t)$ was replaced with $⟨q⟩\sim 3\times {10}^{-5}$.

Figure 6

Neutrino luminosity $L_{\nu }(t)$, anisotropy $q(t)$, and GW strain $h(t)$ times distance from anisotropic neutrino emission only, see Eq. (1), for model pns180 of Ref. 6, as functions of time after bounce. We also show the GW strain $h(t)$ times distance from anisotropic neutrino emission, using the average anisotropy $⟨q⟩=3\times {10}^{-5}$.

Figure 7

GW background resulting from the source spectrum for the proto-neutron star model pns180 of Ref. 6 shown in Fig. 5 without continuation to $f\lesssim 1\mathrm{Hz}$. We use $\alpha =0$ in Eq. (7) and an enhancement factor of 3 (about $1/3$ of the neutrino energy is emitted until simulation end). The solid darker (upper on the left) line thus corresponds to the darker line in Fig. 5 [fluctuating $q(t)$], whereas the solid lighter (lower on the left) line is for $q(t)$ replaced by $⟨q⟩$ (lighter line in Fig. 5). Other notation as in Fig. 4.

Figure 8

GW background for our standard PopIII core-collapse case and event rates $4.4\times {10}^{-4}{\mathrm{s}}^{-1}<R_{\mathrm{III}}<0.2{\mathrm{s}}^{-1}$. For the shaded band the source spectrum shown in Fig. 3 was cut off at $f\le 1\mathrm{Hz}$ and in this sense does not rely on the zero-frequency limit. The solid lines are obtained from continuing the source signal by the zero-frequency limit.