Finite distance effects on the Hellings-Downs curve in modified gravity

TL;DR

This paper investigates how finite distance effects influence the overlap reduction function in pulsar timing arrays for subluminal gravitational waves, revealing a cutoff in spherical harmonics and a specific small-angle limit behavior.

Contribution

It introduces a finite distance correction to the Hellings-Downs curve for subluminal gravity, providing a new formulation valid for any wave velocity.

Findings

Finite distance effects cause a cutoff at rom spherical harmonics decomposition.

The overlap reduction function approaches a specific value in the small angle limit.

The formulation applies to any gravitational wave velocity, not just subluminal.

Abstract

There is growing interest in the overlap reduction function in pulsar timing array observations as a probe of modified gravity. However, current approximations to the Hellings-Downs curve for subluminal gravitational wave propagation, say $v<1$, diverge at small angular pulsar separation. In this paper, we find that the ORF for the $v<1$ case is sensitive to finite distance effects. First, we show that finite distance effects introduce an effective cut-off in the spherical harmonics decomposition at $\ell\sim \sqrt{1-v^2} \, kL$, where $\ell$ is the multipole number, $k$ the wavenumber of the gravitational wave and $L$ the distance to the pulsars. Then, we find that the overlap reduction function in the small angle limit approaches a value given by $\pi kL\,v^2\,(1-v^2)^2$ times a normalization factor, exactly matching the value for the autocorrelation recently derived. Although we focus on the $v<1$ case, our formulation is valid for any value of $v$.