A practical theorem on gravitational-wave background statistics

Inspiralling supermassive black-hole binaries (SMBHBs) are expected to be the main source of the nanohertz gravitational-wave background (GWB) targeted by pulsar timing arrays (PTAs). We provide a simple and general analytic expression for the probability distribution function (PDF) of the GWB characteristic strain squared $h_c^2$ in the limit of a large but finite effective number of sources, $N$, relevant for the lowest-frequency bands where PTAs are most sensitive. Explicitly, we show that for $N \gg 1$, the PDF of the rescaled variable $y \equiv h_c^2/\overline{h_c^2}$ takes the universal self-similar form $P(y) \simeq N^{1/3} \mathcal{P}(N^{1/3} (y -1))$, where $\mathcal{P}$ is the reflected map-Airy distribution. The effective number of in-band sources $N$ is fully specified by the mean $\overline{h_c^2}$ and the cubic shot-noise strain scale $\overline{h_0^3}$, a new summary statistic of the GWB that depends only on the local properties of the SMBHB population. This result is universal: it applies to any population of SMBHBs, regardless of whether they are circular or eccentric, and of the mechanism dominating orbital hardening. We explicitly quantify the accuracy of the large-source-count PDF for a simple but physically realistic SMBHB model, and outline its practical application to PTA data analysis.