Understanding ${\Omega }_{\mathrm{gw}}(f)$ in Gravitational Wave Experiments

Abstract

In this paper we provide a comprehensive derivation of the energy density in the stochastic gravitational-wave background ${\Omega }_{\mathrm{gw}}(f)$, and show how this quantity is measured in ground-based detectors such as Laser Interferometer Gravitational-Wave Observatory (LIGO), space-based Laser Interferometer Space Antenna (LISA), and Pulsar Timing Arrays. By definition ${\Omega }_{\mathrm{gw}}(f)\propto S_h(f)$ -- the power spectral density (PSD) of the Fourier modes of the gravitational-wave background. However, this is often confused with the PSD of the strain signal, which we call $S_{\mathrm{gw}}(f)$, and is a detector-dependent quantity. This has led to confusing definitions of ${\Omega }_{\mathrm{gw}}(f)$ in the literature which differ by factors of up to 5 when written in a detector-dependent way. In addition to clarifying this confusion, formulas presented in this paper facilitate easy comparison of results from different detector groups, and how to convert from one measure of the strength of the background (or an upper limit) to another. Our codes are public and on GitHub.

9 pages, 1 figure. Updated with new PTA results in Figure 1, fixed typos

Cited by in corpus (6)

• Upper Limits on the Isotropic Gravitational-Wave Background from Advanced LIGO's and Advanced Virgo's Third Observing Run
• Measuring the primordial gravitational-wave background in the presence of astrophysical foregrounds
• Searching for cosmological gravitational-wave backgrounds with third-generation detectors in the presence of an astrophysical foreground
• Detecting a stochastic gravitational-wave background in the presence of correlated magnetic noise
• Searching for modified gravity in the astrophysical gravitational wave background: Application to ground-based interferometers
• Exploring the multiband gravitational wave background with a semi-analytic galaxy formation model