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

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) ₐb (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, ₐb --> 0. In particular, working in harmonic space and performing an approximate resummation of all multipole contributions, the auto correlation coefficient ₐa seems to diverge. In this paper, we confirm that this divergence is unphysical and provide an exact and analytical expression for ₐa in dependence of the pulsar distance Lₐ and the GW phase velocity vₚh. In the GR limit and assuming a large pulsar distance, our expression reduces to 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 ₐa scales linearly with fLₐ, 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 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 ₐa.