Frequency-Domain Distribution of Astrophysical Gravitational-Wave Backgrounds

TL;DR

The paper develops a frequency-domain framework for the stochastic gravitational-wave background (SGWB) produced by a collection of discrete, individually coherent sources. It constructs the SGWB statistics from a one-point characteristic function $\psi(q,f)$ of a single source and extends it to a Poisson-summed population via $e^{N_0 G(|q|)}$, enabling the full Fourier-domain PDF $P(\tilde{h}(f))$ and the energy density $\Omega_{\rm gw}(f)$. Key results include analytic forms for the single-source CF, a $\mathrm{d}N_s/\mathrm{d}f \propto f^{-11/3}$ scaling for the source distribution, and a normalization that connects the model to established $\Omega_{\rm gw}$ measurements, with a low-frequency $\Omega_{\rm gw}(f) \propto f^{2/3}$ behavior and lensing effects incorporated via magnification. The work also identifies a $-4$ power-law tail in the SNR distribution for unsubtracted backgrounds and shows how removing bright mergers Gaussianizes the remaining confusion noise, providing a generalizable framework for predicting frequency-domain statistics of discrete GW backgrounds and their observational implications.

Abstract

The superposition of many astrophysical gravitational wave (GW) signals below typical detection thresholds baths detectors in a stochastic gravitational wave background (SGWB). In this work, we present a Fourier space approach to compute the frequency-domain distribution of stochastic gravitational wave backgrounds produced by discrete sources. Expressions for the moment-generating function and the distribution of observed (discrete) Fourier modes are provided. The results are first applied to the signal originating from all the mergers of compact stellar remnants (black holes and neutron stars) in the Universe, which is found to exhibit a $-4$ power-law tail. This tail is verified in the signal-to-noise ratio distribution of GWTC events. The extent to which the subtraction of bright (loud) mergers gaussianizes the resulting confusion noise of unresolved sources is then illustrated. The power-law asymptotic tail for the unsubtracted signal, and an exponentially decaying tail in the case of the SGWB, are also derived analytically. Our results generalize to any background of gravitational waves emanating from discrete, individually coherent, sources.

Figures (4)

• Figure 1: $\frac{dN_s}{df}$ is proportional to $f^{-11/3}$
• Figure 2: A plot of the gravitational-wave energy density, $\Omega_{\rm gw}(f)$ determined by the model described in Ginatetal2020, which follows an $f^{2/3}$ power-law at low frequencies.
• Figure 3: The function $G(s;f)$, normalised as explained in the text, for $f=1000$ Hz. Other frequencies also look like this. $s$ is measured in units of $h_c^{-1}$, where $h_c$ is evaluated at $1$ Hz to be a fixed, frequency-independent, normalisation.
• Figure 4: Source plane optical depth as a function of the source redshift $z$ and magnification $\mu_0$.