Statistical properties of astrophysical gravitational-wave backgrounds

We investigate how a stochastic gravitational-wave background, produced from a discrete set of astrophysical sources, differs from an idealized model consisting of an isotropic, unpolarized, and Gaussian background. We focus, in particular, on the different signatures produced from these two cases, as observed in a cross-correlation search. We show that averaged over many realizations of an astrophysical background, the cross-correlation measurement of an astrophysical background is identical to that of an idealized background. However, any one realization of an astrophysical background can produce a different signature. Using a model consisting of an ensemble of binary neutron star coalescences, we quantify the typical difference between the signal from individual realizations of the astrophysical background and the idealized case. For advanced detectors, we find that, using a cross-correlation analysis, astrophysical backgrounds from many discrete sources are probably indistinguishable from an idealized background.

Figure 1

The discrete overlap reduction function ${\gamma }_N(f)$ for five different values of $N$. The overlap reduction function $\gamma (f)$ from a Gaussian isotropic background is shown by the black dashed line.

Figure 2

(a) Ten realizations of the discrete overlap reduction function ${\gamma }_N(f)$ with $N={10}^4$ events (blue). The mean of the blue curves is shown in red. The standard overlap reduction function $\gamma (f)$ is shown in dashed black. (b) Using a simulation of 1000 realizations of $N={10}^4$ background sources, we calculate the standard deviation of ${\gamma }_N(f)$ at each frequency bin. The blue curves represent $\pm$ one standard deviation about the mean, which is shown in red. The dashed black corresponds to $\gamma (f)$. (c) Variation in the overlap reduction function. We plot ${\sigma }_{\gamma }(f)$—the standard deviation of the discrete overlap reduction function as a function of frequency. Each color represents a different value of $N$. The magnitude of ${\sigma }_{\gamma }(f)$ is approximately constant, which implies that the fractional error grows as ${\gamma }_N(f)$ becomes smaller at higher frequencies.

Figure 3

(a) Histogram of the fractional bias in a stochastic search $R$ [see Eq. (41)] due to the discreteness of a non-Gaussian background. Each color corresponds to a different value of $N$. (b) The standard deviation ${\sigma }_R$ of the left-hand-side histograms as a function of $N$ (blue). The dashed lines include all events whereas the solid lines exclude events loud enough to resolve individually with matched filtering (${\rho }_t>8$); see Eq. (36). The dashed red line corresponds to the number of events required to produce a stochastic signal that can be detected with $\mathrm{SNR}=2$ [see Eq. (42)] using Advanced LIGO with an observational period of $t_{\mathrm{obs}}=1\mathrm{yr}$. Note that ${\sigma }_R$ does not depend on $t_{\mathrm{obs}}$ whereas $\mathrm{SNR}\propto t_{\mathrm{obs}}^{1/2}$.