Understanding $Omega_mathrm{gw}(f)$ in Gravitational Wave Experiments. (arXiv:1911.09745v1 [gr-qc])

Understanding $Omega_mathrm{gw}(f)$ in Gravitational Wave Experiments. (arXiv:1911.09745v1 [gr-qc])
<a href="http://arxiv.org/find/gr-qc/1/au:+Mingarelli_C/0/1/0/all/0/1">Chiara M. F. Mingarelli</a>, <a href="http://arxiv.org/find/gr-qc/1/au:+Taylor_S/0/1/0/all/0/1">Stephen R. Taylor</a>, <a href="http://arxiv.org/find/gr-qc/1/au:+Sathyaprakash_B/0/1/0/all/0/1">B. S. Sathyaprakash</a>, <a href="http://arxiv.org/find/gr-qc/1/au:+Farr_W/0/1/0/all/0/1">Will M. Farr</a>

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.

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.

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