Series expansion of the overlap reduction function for scalar and vector polarizations for gravitational wave search with pulsar timing arrays

TL;DR

This paper extends the power series expansion method to compute the overlap reduction function for scalar and vector gravitational wave polarizations in pulsar timing arrays, providing new analytic expressions especially for the longitudinal mode.

Contribution

It generalizes the previous tensor-only approach to include scalar and vector modes, offering comprehensive analytic formulas for all polarization types and angles.

Findings

Derived power series for scalar and vector overlap reduction functions.

Compared new formulas with existing literature for validation.

Provided first analytic expressions for longitudinal mode overlap reduction.

Abstract

In our previous work \cite{PTA2} we calculated the overlap reduction function for the tensor polarization without employing the short wavelength approximation, this was done by obtaining a power series of nested sums which is valid for all gravitational wave frequencies and pulsar distances. In this work we generalize the power-series expansion method to vector and scalar polarizations. We have compared our expression for the breathing and vector modes with previous literature. We present for the first time analytic expressions for the overlap reduction function of the longitudinal mode for all angles between the pulsar pairs.