The Sensitivity of Ligo to a Stochastic Background, and its Dependance on the Detector Orientations

TL;DR

This paper analyzes how the sensitivity of a global interferometer network to a stochastic gravitational-wave background depends on detector orientations, deriving an analytic overlap reduction function $\gamma(f)$ and two optimal configuration classes. It shows that for nearby detector pairs, a 45°-to-joining-arc configuration (configuration II) often optimizes broadband SB sensitivity, while for longer baselines configuration I (arms along the joining arc) can be preferable; the optimal choice also depends on phase delays and frequency. A detailed optimal data-analysis framework for SB searches with networks, including correlated noise considerations, is presented, and the LIGO/VIRGO implications are quantified: the planned LIGO orientations are within a few percent of optimal, while VIRGO/GEO orientations would dramatically reduce SB sensitivity. The work yields a projected 90% broadband upper limit on the SB energy density of about $\Omega_g^{(90\%)} \sim 5\times10^{-10}h_{75}^{-2}(\hat{\tau}/10^7\,\mathrm{s})^{-1/2}$ for advanced detectors, underscoring the importance of network geometry in SB constraints and detections.

Abstract

We analyze the sensitivity of a network of interferometer gravitational-wave detectors to the gravitational-wave stochastic background, and derive the dependence of this sensitivity on the orientations of the detector arms. We build on and extend the recent work of Christensen, but our conclusion for the optimal choice of orientations of a pair of detectors differs from his. For a pair of detectors (such as LIGO) that subtends an angle at the center of the earth of $\,\alt 70^\circ$, we find that the optimal configuration is for each detector to have its arms make an angle of $45^\circ$ (modulo $90^\circ$) with the arc of the great circle that joins them. For detectors that are farther separated, each detector should instead have one arm aligned with this arc. We also describe in detail the optimal data-analysis algorithm for searching for the stochastic background with a detector network, which is implicit in earlier work of Michelson. The LIGO pair of detectors will be separated by $\sim 3000 \, {\rm km}$. The minimum detectable stochastic energy-density for these detectors with their currently planned orientations is $\sim 3\%$ greater than what it would be if the orientations were optimal.