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

TL;DR

This paper derives new correlation functions for gravitational-wave backgrounds with arbitrary polarization and dispersion, accounting for source interference, and analyzes their implications for testing gravity theories with pulsar timing arrays.

Contribution

It provides a closed-form harmonic analysis of correlation functions considering interference and dispersion, revealing fundamental limits in distinguishing gravity models.

Findings

Interference modifies correlation shapes but preserves key multipole moments.

Higher-order multipoles are truncated by pulsar distances and dispersion.

A fundamental degeneracy limits distinguishing modified gravity from GR.

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.