Finite Populations & Finite Time: The Non-Gaussianity of a Gravitational Wave Background

Strong evidence for an isotropic, Gaussian gravitational wave background (GWB) has been found by multiple pulsar timing arrays (PTAs). The GWB is expected to be sourced by a finite population of supermassive black hole binaries (SMBHBs) emitting in the PTA sensitivity band, and astrophysical inference of PTA data sets suggests a GWB signal that is at the higher end of GWB spectral amplitude estimates. However, current inference analyses make simplifying assumptions, such as modeling the GWB as Gaussian, assuming that all SMBHBs only emit at frequencies that are integer multiples of the total observing time, and ignoring the interference between the signals of different SMBHBs. In this paper, we build analytical and numerical models of an astrophysical GWB without the above approximations, and compare the statistical properties of its induced PTA signal to those of a signal produced by a Gaussian GWB. We show that finite population and windowing effects introduce non-Gaussianities in the PTA signal, which are currently unmodeled in PTA analyses.

💡 Research Summary

This paper investigates the statistical nature of the nanohertz gravitational‑wave background (GWB) as measured by pulsar timing arrays (PTAs), focusing on the consequences of two realistic but often neglected aspects: the finite number of supermassive black‑hole binaries (SMBHBs) that generate the background and the finite observation time of PTA experiments. Conventional PTA analyses treat the GWB as an isotropic, Gaussian, stationary random process, assume that each binary emits only at frequencies that are integer multiples of the total observing span (T), and ignore interference among different binaries. The authors argue that these simplifications are inadequate for an astrophysical GWB that is expected to be produced by a limited population of SMBHBs, especially given recent PTA results that suggest a relatively high GWB amplitude.

Model Construction
The authors model the GWB as a superposition of (N) independent SMBHBs, each characterized by a frequency (f_j), sky direction (\hat\Omega_j), phase (\phi_j), and amplitude (A_j) (which depends on the binary’s chirp mass, redshift, etc.). The number of binaries (N) itself is taken to be a Poisson‑distributed random variable, thereby incorporating population‑size fluctuations. Assuming circular, face‑on binaries, the two GW polarizations are written as cosine and sine functions with a (\pi/2) phase offset. The metric perturbation from the whole ensemble is summed over all sources, and the timing residual for pulsar (p) is obtained by integrating the metric along the Earth–pulsar line of sight. This yields a compact expression for the residual in terms of a complex antenna‑response factor (R_{p,j}).

Finite‑Time Window
Because PTAs observe for a limited time (T), the Fourier transform of the residuals is evaluated on a discrete set of frequencies (f_i=i/T). Each binary’s contribution is multiplied by a real window function (w_{\pm j}=4,\mathrm{sinc}