Harmonic Analysis on Correlation for Gravitational-Wave Backgrounds of Arbitrary Polarization from Interfering Sources in Generic Dispersion Relation

Abstract

The Hellings-Downs (HD) correlation serves as the fundamental benchmark for detecting the gravitational-wave background (GWB) in pulsar timing arrays (PTAs) within General Relativity (GR). However, this canonical signature relies on the idealization of a continuum of sources without interference. In realistic astrophysical scenarios dominated by supermassive black hole binaries (SMBHBs), interference between discrete sources induces intrinsic deviations in the spatial correlation, which may mimic or obscure signatures of modified gravity. In this work, we derive the closed-form spatial correlation functions for a GWB with arbitrary polarization and generic GW dispersion relations, in the presence of source interference. Through a rigorous harmonic analysis, we demonstrate that source interference modifies the correlation shape but strictly preserves the lowest non-vanishing multipole moment characteristic of each polarization, specifically the quadrupole for tensor, dipole for vector, and monopole for scalar modes. The truncation at higher-order multipoles is governed by the interplay between pulsar distances and dispersion effects. Furthermore, we quantify the statistical degeneracy between interference-induced variation and modified gravity signatures. We conclude that access to only a single realization of the Universe imposes a fundamental theoretical limit on distinguishing modified gravity from GR using spatial correlations alone.

Figures (3)

• Figure 1: Distributions of the normalized Legendre coefficients $w_l^{\rm P}$ for all polarization modes, derived from 10,000 stochastic realizations across distinct dispersion parameters $\eta$ and pulsar configurations. The left and right columns illustrate results for the finite-distance ($fD_a = fD_b = 10$) and infinite-distance ($fD_a = fD_b = \infty$) cases, respectively. Blue, grey, and orange violins denote $\eta = 0.8$, $1.0$, and $1.2$, respectively. For comparison, the dash-dotted lines in the first and second rows represent the analytical predictions for the TT and ST modes.
• Figure 2: Distributions of pulsar auto-correlations for all polarization modes, derived from 10,000 stochastic realizations across distinct dispersion parameters $\eta \in \{0.8, 1.0, 1.2\}$. All distributions are computed for a fixed pulsar configuration of $fD_a = 10$. Blue, grey, and orange violins denote the cases for $\eta = 0.8$, $1.0$, and $1.2$, respectively. For reference, the black dash-dotted line denotes the analytical predictions for the TT and ST modes.
• Figure 3: Comparison of the function $(I_l^P)^2(fD, \eta)$ between the finite-distance regime $fD=10$ and the infinite-distance limit. The vertical dashed lines denote the dispersion parameters $\eta = 0.8, 1.0$, and $1.2$ from left to right.