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

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) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mrow> <mml:mi>a</mml:mi> <mml:mi>b</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>ξ</mml:mi> <mml:mrow> <mml:mi>a</mml:mi> <mml:mi>b</mml:mi> </mml:mrow> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , 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, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>ξ</mml:mi> <mml:mrow> <mml:mi>a</mml:mi> <mml:mi>b</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>v</mml:mi> <mml:mrow> <mml:mrow> <mml:mi>ph</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> . In the GR limit and assuming a large pulsar distance, our expression reduces to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mrow> <mml:mi>a</mml:mi> <mml:mi>a</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . 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 .