Efficient Computation of Overlap Reduction Functions for Pulsar Timing Arrays

TL;DR

This paper derives a general formula for the overlap reduction function in pulsar timing arrays, accommodating arbitrary polarization states and anisotropies in the gravitational-wave background, enhancing GW detection analysis.

Contribution

It introduces a simple, comprehensive formula for the overlap reduction function applicable to various GW polarization modes and anisotropies, extending previous models.

Findings

Derived a general formula for the overlap reduction function.

Provided specific expressions for different GW polarization modes.

Enhanced the analysis framework for gravitational wave detection in pulsar timing arrays.

Abstract

Pulsar timing arrays seek and study gravitational waves (GWs) through the angular two-point correlation function of timing residuals they induce in pulsars. The two-point correlation function induced by the standard transverse-traceless GWs is the famous Hellings-Downs curve, a function only of the angle between the two pulsars. Additional polarization modes (vector/scalar) that may arise in alternative-gravity theories have different angular correlation functions. Furthermore, anisotropy, linear, or circular polarization in the stochastic GW background gives rise to additional structure in the two-point correlation function that cannot be written simply in terms of the angular separation of the two pulsars. In this paper, we provide a simple formula for the most general two-point correlation function--or overlap reduction function (ORF)--for a gravitational-wave background with an arbitrary polarization state, possibly containing anisotropies in its intensity and polarization (linear or circular). We provide specific expressions for the ORFs sourced by the general-relativistic transverse-traceless GW modes as well as vector (or spin-1) modes that may arise in alternative-gravity theories.