Parameter Estimation in Searches for the Stochastic Gravitational-Wave Background

The stochastic gravitational-wave background (SGWB) is expected to arise from the superposition of many independent and unresolved gravitational-wave signals of either cosmological or astrophysical origin. The spectral content of the SGWB carries signatures of the physics that generated it. We present a Bayesian framework for estimating the parameters associated with different SGWB models using data from gravitational-wave detectors. We apply this technique to recent results from LIGO to produce the first simultaneous 95% confidence level limits on multiple parameters in generic power-law SGWB models and in SGWB models of compact binary coalescences. We also estimate the sensitivity of the upcoming second-generation detectors such as Advanced LIGO or Virgo to these models and demonstrate how SGWB measurements can be combined and compared with observations of individual compact binary coalescences in order to build confidence in the origin of an observed SGWB signal. In doing so, we demonstrate a novel means of differentiating between different sources of the SGWB.

Figure 1

The solid curve denotes the 95% C.L. limit for the power-law SGWB model ${\Omega }_M(f;A,\alpha )=A(f/100\mathrm{Hz}{)}^{\alpha }$, in the $\alpha \mathrm{\text{−}}A$ plane using the latest SGWB measurement with LIGO detectors [34]. The dashed curve denotes the projected sensitivity obtainable by aLIGO at $\mathrm{SNR}=2$, assuming 1 yr of data from two colocated detectors operating at design sensitivity (assuming ${\widehat{Y}}_i=0$).

Figure 2

Top-left: GW spectra for several examples of CBC models are shown. The solid and dot-dash curves were computed using the model described here, for the BNS (black) and BBH (gray) cases, while the dashed curves were computed using the distribution of chirp masses from StarTrack simulation [55] with the same average chirp masses as those used for the solid curves. The solid and dashed curves are nearly identical for low values of $M_c$, indicating that using the average chirp mass is sufficient for the analysis presented here. Top-right: 95% C.L. limits for CBC models for initial LIGO [34] and projected sensitivities for aLIGO assuming $\mathrm{SNR}=2$ and 1 yr of coincident data from colocated interferometers operating at design sensitivity (and ${\widehat{Y}}_i=0$). The region $M_c<2.5M_{\odot }$ corresponds to BNS models, and $M_c>2.5M_{\odot }$ to BBH models. The three different styles (solid, dashed, and dot-dash) correspond to three different choices of star formation rate $R_{*}(z)$ and time-delay probability density function $p(t_d)$, indicating the uncertainty due to these choices at the level of a factor of 2 [17]. Bottom-left: projected 95% C.L. contours for a simulated BNS signal measured with aLIGO. The simulated signal amplitude corresponds to local BNS rate in between the realistic and optimistic rates described in Ref. 43. Combining the stochastic measurement with direct observations of individual BNSs leads to a marginally tighter constraint on the astrophysical parameters describing the BNS population. Bottom-right: Projected 95% C.L. contours for a simulated stochastic BBH signal measured with aLIGO. A rather optimistic local rate $R_{\mathrm{loc}}=0.17{\mathrm{Mpc}}^{-3}{\mathrm{Myr}}^{-1}$ was assumed for this simulation [43], resulting in a relatively narrow contours in the $R_{\mathrm{loc}}-M_c$ plane. However, the signal is still not strong enough to differentiate between a high $R_{\mathrm{loc}}$—low $M_c$ signal and a low $R_{\mathrm{loc}}$—high $M_c$ signal.