Search templates for stochastic gravitational-wave backgrounds

Several earth-based gravitational-wave (GW) detectors are actively pursuing the quest for placing observational constraints on models that predict the behavior of a variety of astrophysical and cosmological sources. These sources span a wide gamut, ranging from hydrodynamic instabilities in neutron stars (such as $\mathrm{r}$-modes) to particle production in the early Universe. Signals from a subset of these sources are expected to appear in these detectors as stochastic GW backgrounds (SGWBs). The detection of these backgrounds will help us in characterizing their sources. Accounting for these backgrounds will also be required by some detectors, such as the proposed space-based detector LISA, so that they can detect other GW signals. Here, we formulate the problem of constructing a bank of search templates that discretely span the parameter space of a generic SGWB. We apply it to the specific case of a class of cosmological SGWBs, known as the broken power-law models. We derive how the template density varies in their three-dimensional parameter space and show that for the LIGO 4 km detector pair, with LIGO-I sensitivities, a few hundred templates will suffice to detect such a background while incurring a loss in signal-to-noise ratio of no more than 3%.

Figure 1

The only metric component, $g_{11}$, for a single power-law SGWB plotted as a function of $k$. The template spacing is a minimum near the global maximum at $k={0.5000}_{-0.0625}^{+0.0625}$.

Figure 2

The ambiguity function $M(k,\delta k)$ for a single power-law SGWB plotted as a function of $k$ and the mismatch $\delta k$. For any given $k$, the function attains the maximum possible value of unity when $\delta k=0$. And for any given $\delta k$, the function is a minimum at $k={0.5000}_{-0.0625}^{+0.0625}$, which is consistent with the behavior of the metric depicted in Fig. 1 [27].

Figure 3

The template density $\rho$ (normalized here to have a maximum value of 1) for a BPL parameter space plotted as a function of $k_{\pm }$ and $f_p$, two variables at a time. The fixed parameters in these plots are (clockwise from top) $f_p=50\mathrm{Hz}$, $k_{-}=0.125$, and $k_{+}=-0.125$.

Figure 4

The ambiguity function $M(\mathit{\vartheta },\delta \mathit{\vartheta })$ for a broken power-law SGWB plotted as a function of $f_p$ and the mismatch $\delta f_p$ (both in Hz). Here, we set $k_{\pm }=\mp 2$ and $\delta k_{\pm }=0$. Note how quickly $M$ asymptotes to a value of unity for large $f_p$, implying a small number of templates for $f_p>\sim 200\mathrm{Hz}$. The ambiguity function is within 99% for: (a  $f_p$ in [50, 120] Hz, if $|\delta f_p|\le 4\mathrm{Hz}$, (b) $f_p$ in [120, 150] Hz, if $|\delta f_p|\le 10\mathrm{Hz}$, and (c) $f_p>150\mathrm{Hz}$, if $|\delta f_p|\le 20\mathrm{Hz}$.

Figure 5

$M(\mathit{\vartheta },\delta \mathit{\vartheta })$ for BPL spectra plotted as a function of $k_{+}$ and $k_{-}$ for four values of $f_p=50$, 100, 190, 450 Hz (clockwise from top left). The mismatch values are $\delta k_{\pm }=0.25$ and $\delta f_p=2.5\mathrm{Hz}$. As $f_p$ increases, note how little $M$ varies with $k_{+}$ (for any given $k_{-}$). This is indicative of the template spacings being larger along $k_{+}$ than along $k_{-}$. This is expected since most of the contribution to the CC statistic arises at lower frequencies, where the $k_{-}$ index has support (especially, when $f_p>\sim 200\mathrm{Hz}$).

Figure 6

$M(\mathit{\vartheta },\delta \mathit{\vartheta })$ for a broken power-law SGWB plotted as a function of $k_{-}$ and $f_p$ for four values of $k_{+}=-4,-2,-1$, 0 (clockwise from top left). The mismatch values are $\delta k_{\pm }=0.25$ and $\delta f_p=2.5\mathrm{Hz}$.

Figure 7

$M(\mathit{\vartheta },\delta \mathit{\vartheta })$ for a broken power-law SGWB plotted as a function of $k_{+}$ and $f_p$ for four values of $k_{-}=0$, 1, 2, 4 (clockwise from top left). The mismatch values are $\delta k_{\pm }=0.25$ and $\delta f_p=2.5\mathrm{Hz}$.