Pulse redshift of pulsar timing array signals for all possible gravitational wave polarizations in modified general relativity

TL;DR

This paper derives the pulsar timing redshift for all six gravitational wave polarizations in modified gravity theories, addressing mathematical issues and providing a general, frequency-independent formula with first-order corrections.

Contribution

It offers a comprehensive derivation of pulsar timing redshift for all GW polarizations in modified gravity, including a frequency-independent expression and correction terms.

Findings

Redshift must be split into polarization and interference parts to avoid singularities.

Derived a general formula applicable to any GW frequency without low-frequency assumptions.

Provided first-order correction terms to the standard redshift formula.

Abstract

Pulsar timing arrays (PTA) have the promise to detect gravitational waves (GWs) from sources which are in a unique frequency range of 10^-9 - 10^-6 Hz. This in turn also provides an opportunity to test the theory of general relativity in the low frequency regime. The central concept of the detection of GWs with PTA lies in measuring the time of arrival difference of the pulsar signal due to the passing of GWs i.e. the pulses get red-shifted. In this paper we provide a complete derivation of the redshift computation for all six possible polarizations of GW which arise due to the modifcations to general relativity. We discuss the smoothness of the redshift and related properties at the critical point, where the GW source lies directly behind the pulsar. From our mathematical discussion we conclude that the redshift has to be split differently into polarization part (pattern functions) and interference part, to avoid discontinuities and singularities in the pattern functions. This choice of pattern functions agrees with the formula one uses for interferometers with a single detector arm. Finally, we provide a general expression which can in principle be used for pulsars and GW of any frequency without invoking the low frequency assumption and using said assumption we develop the expression up to first order in the strain and find correction terms to the canonical redshift formula.