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

TL;DR

The paper addresses non-Gaussianities in the gravitational-wave background from a finite SMBHB population in PTAs. It develops analytical expressions for the moments and the argument distribution of timing-residual Fourier coefficients, revealing a persistent, calculable non-Gaussianity (excess kurtosis) in the finite-population regime and a Cauchy-like distribution for coefficient arguments. Through toy and realistic astrophysical models, it shows that Gaussian PTSD analyses can be biased in regimes with few sources or narrow windows, while large-N limits recover Gaussian behavior with residual cross-term effects. The work provides a practical framework for fast, accurate simulations and suggests incorporating non-Gaussian components into inference, with broad applicability to current and future GW detectors.

Abstract

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 interference effects introduce non-Gaussianities in the PTA signal, which are currently unmodeled in PTA analyses.

Figures (8)

• Figure 1: Comparing excess kurtosis as a function of the number of sources numerically (green violin plots) and analytically (dashed blue curve). The vertical green bar indicates the 10% to 90% quantile range of the distribution of the excess kurtosis estimator across the pulsars in the array; the horizontal pink bar indicates the median. Numerical distributions are all of $100$ samples of kurtosis generated from $10^5$ realizations of an SMBHB population with $f_i=f_j=2/T$ and a single pulsar.
• Figure 2: With a single SMBHB source, fourth moments of the Fourier coefficients match the predictions of \ref{['eq:4th-single']} and its cross-pulsar generalization. The data shown here are generated from $10^7$ realizations for frequencies from $1/T$ to $15/T$. The vertical axes have been rescaled so that the predicted value at $f=1/T$ is $1$ in arbitrary units in both cases.
• Figure 3: With many sources, fourth moments of the Fourier coefficients match the predictions of \ref{['eq:auto_many']},\ref{['eq:2nd-moment-product']}, respectively. The data shown here are generated from $10^7$ realizations of an SMBHB population of 1000 sources for frequencies from $1/T$ to $15/T$. The vertical axes have been rescaled so that $\langle|\tilde{a}_i^p|^4\rangle=1$ or $\langle|\tilde{a}_i^{*p}\tilde{a}_i^q|^2\rangle=1$, as appropriate, in arbitrary units at $f=1/T$.
• Figure 4: The distribution of excess kurtosis of all 67 pulsars by frequency. Most frequencies agree fairly well with the prediction of zero in the many-sources limit. At $f=1/T$, the deviation is mostly driven by the correlation between the real and imaginary parts of the coefficients. Color and bar conventions match \ref{['fig:ex-kurt-test']}.
• Figure 5: The ratio of fourth moments to twice the second moments for auto-pulsars (upper) and a fixed-partner subset of cross-pulsars (lower) for an astrophysical model of a background and using the 67 pulsars of the NANOGrav 15yr dataset. At the lowest frequency, a mild disagreement for the auto-pulsar result is likely due to Poisson noise from truncating sources at very low frequencies. The cross-pulsar result matches predictions fairly well across all pairings and frequencies.