Frequency-domain distribution of astrophysical gravitational-wave backgrounds

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 Gravitational-Wave Transient Catalogue (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.

Figure 1

The differential number of sources per unit frequency, as given in Eq. (4.3) in a toy model in which only the binary formation time $t_{*}$ and chirp mass $M_c$ are varied (see text).

Figure 2

The function $-N_{0}G(s;f)>0$, from Eq. (4.6), is shown for a range of observed frequencies $f$ as indicated in the legend. It scales as $-N_{0}a(f)s^2$ at small $s$ and asymptotes to a $T$-dependent constant at large $s$ (see text for details). An observation time $T=1\mathrm{yr}$ is assumed for illustration.

Figure 3

The probability density $P(x;f)$ as a function of the dimensionless ratio $x=h/h_c$ ($h={\tilde{h}}_f$) for a few representative frequencies $f$. The blue curve is the exact prediction (obtained as the Hankel transform of $G(s;f)$), the violet dashed curve represents the asymptotic expansion (2.34), and the green line indicates the power-law approximation Eq. (2.37) valid at high strain. The amplitude $h$ of the observed DFT is normalized to $h_c=3.69\times {10}^{-23}{\mathrm{Hz}}^{-1/2}$ in all panels.

Figure 4

The GW energy density as a function of frequency for the unsubtracted signal produced by all stellar compact remnants in the Universe. The low- and high-frequency bumps correspond to binary BH and NS mergers, respectively. The upper limit inferred from the Advanced LIGO-Virgo- KAGRA O3 data is indicated on the figure. We also plot the $A+$ sensitivity curve for an observation time $T=1\mathrm{yr}$ and a frequency resolution $\mathrm{\Delta }f=0.25\mathrm{Hz}$.

Figure 5

Distribution of discrete Fourier amplitudes $h=|{\tilde{h}}_f|$ as a function of the dimensionless ratio $x=h/h_c$. The strain normalization is $h_c=5.274\times {10}^{-31}{\mathrm{yr}}^{1/2}$ and the observation time set to $T=1\mathrm{yr}$. The solid (blue) and dotted (green) curves show the distribution $P(x;f)$ of the unsubtracted signal produced by all the cosmological compact binaries without and with lensing. The dashed (red) curve is the distribution $P_r(x;f)$ of the SGWB originating from unresolved sources with $\mathrm{SNR}\le 12$ assuming a single detector of $\mathrm{A}+\text{sensitivity}$ and an observation time $T=1\mathrm{yr}$ (see text for details), except for 2000, which has a threshold of 20, just to show how varying the threshold affects $P_r$.

Figure 6

The SGWB at 50 Hertz when bright sources are subtracted for an observation time of 1 minute or 1 hour.

Figure 7

Top: distribution of the SGWB for various choices of $T_{\mathrm{sub}}$ at fixed observation time $T=1\mathrm{yr}$. Results are shown at a frequency of 50 Hz for illustration. The amplitude of the $h^{-4}$ tail decreases with increasing $T_{\mathrm{sub}}/T$, as in Eq. (4.26), and vanishes for $T_{\mathrm{sub}}=T$, leaving an almost-Rayleigh distribution. Bottom: a validation of Eq. (4.26), obtained by comparing it with the numerically evaluated values of $b$ [using Eq. (b9)].

Figure 8

A histogram of the confirmed LIGO-Virgo- KAGRA confirmed gravitational-wave events, with SNR $\rho \ge 12$, as a function of the SNR. The number of bins is the maximum number such that no bin is empty. Error bars represent Poisson errors. The black curve is a power-law fit to the histogram, with points weighted by the number of events in each bin (95% confidence limits of the exponent for this fit are $-4.083\pm 0.936$), and the purple (dashed) curve is a similar fit, but with the exponent constrained to $-4$.

Figure 9

The function $G(s,f)$ (solid curve) as well as its cubic approximation $G(s;f)=-a(f)s^2+b(f)|s{|}^3$ (dashed curve), with $a(f)$ and $b(f)$ computed from Eqs. (b8) and (b9).

Figure 10

The magnification probability density $\mu \varphi (\mu |z)$ obtained from Eq. (d21) for selected values of the source redshift.