On the overlap reduction function of pulsar timing array searches for gravitational waves in modified gravity

Cordes, Nina; Mitridate, Andrea; Schmitz, Kai; Schröder, Tobias; Wassner, Kim

Abstract

Pulsar timing array (PTA) searches for gravitational waves (GWs) aim to detect a characteristic correlation pattern in the timing residuals of galactic millisecond pulsars. This pattern is described by the PTA overlap reduction function (ORF) Γ_ab(ξ_ab), which is known as the Hellings–Downs (HD) curve in general relativity (GR). In theories of modified gravity, the HD curve often receives corrections. Assuming, e.g. a subluminal GW phase velocity, one finds a drastically enhanced ORF in the limit of small angular separations between pulsar a and pulsar b in the sky, ξ_ab → 0. In particular, working in harmonic space and performing an approximate resummation of all multipole contributions, the auto correlation coefficient Γ_aa​ seems to diverge. In this paper, we confirm that this divergence is unphysical and provide an exact and analytical expression for Γ_aa​ in dependence of the pulsar distance L_a and the GW phase velocity v_ph​. In the GR limit and assuming a large pulsar distance, our expression reduces to Γ_aa = 1. In the case of subluminal phase velocity, we show that the regularization of the naive divergent result is a finite-distance effect, meaning that Γ_aa​ scales linearly with fL_a, where f is the GW frequency. For superluminal phase velocity (subluminal group velocity), which is relevant in the case of massive gravity, we correct an earlier analytical result for Γ_ab​. Our results pave the way for fitting modified-gravity theories with nonstandard phase velocity to PTA data, which requires a proper understanding of the auto correlation coefficient Γ_aa.

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