Skewness in the Hellings-Downs curve

TL;DR

The paper tackles non-Gaussian fluctuations in the Hellings-Downs correlation arising from a finite population of discrete gravitational-wave sources in Pulsar Timing Arrays. It extends the variance framework of Allen (2023) to third order, deriving the skewness for a single source and for the many-source interference regime, and introduces a three-point-average function to define cosmic skewness that survives in the large-N limit. It shows that the total and cosmic skewness track HD-sign and persist due to a geometric three-point coupling, offering a new observable beyond the standard Gaussian HD analysis. The findings motivate incorporating higher-order moments into PTA analyses, potentially via a hierarchical Bayesian model that treats the overlap-reduction function as a non-Gaussian random variable with mean anchored to the HD curve and fluctuations governed by cosmic variance and skewness.

Abstract

Recent Pulsar Timing Array datasets provide compelling evidence for a nano-Hertz gravitational-wave background, but robust detection requires characterizing statistical fluctuations of the Hellings-Downs (HD) correlation expected from a finite population of discrete sources. Building on the variance calculation of Allen (2023), we derive the third central moment (skewness) of the HD correlation for a single unpolarized point source and an ensemble of many interfering point sources in the confusion-noise regime. To isolate the intrinsic non-Gaussianity of the background, we extend the pulsar-averaging formalism to third order by introducing a three-point averaged correlation function, which allows us to define the cosmic skewness. We find that the skewness remains non-zero in the large-source-number limit and is controlled by a new geometric three-point function. These results suggest that incorporating higher-order moments could provide additional information on source discreteness beyond standard Gaussian analyses.

Figures (8)

• Figure 1: Comparison of single-source statistical quantities. The dashed line represents the HD curve $\mu_\text{u}(\gamma)$. The solid lines show the unpolarized single-source standard deviation $\sigma_\text{u}$ and the cubic root of the unpolarized single source skewness numerator $\kappa_\text{u}$.
• Figure 2: The unpolarized single-source skewness $\mathcal{S}_\text{u}(\gamma)$ as a function of the angular separation $\gamma$.
• Figure 3: Statistical quantities for a large number of sources where $\mathcal{H}_2^2 \gg \mathcal{H}_4$. We assume the interfering source model and set the pulsar term $\chi=1$. The dashed line represents the many-point mean $\mu$ (equivalent to the single-source mean without normalization), the solid lines show the many-point standard deviation $\sigma$ and the cubic root of the skewness numerator $\kappa$.
• Figure 4: The many-point skewness $\mathcal{S}$ as a function of the angular separation $\gamma$.
• Figure 5: Comparison of cosmic fluctuations. The solid line represents the cubic root of the cosmic skewness numerator $\kappa_\text{cosmic}$, and the dashed line shows the square root of the cosmic variance $\sigma_\text{cosmic}$.