Finite distance effects on the Hellings-Downs curve in modified gravity

There is growing interest in the overlap reduction function in pulsar timing array observations as a probe of modified gravity. However, current approximations to the Hellings-Downs curve for subluminal gravitational wave propagation, say $v<1$, diverge at small angular pulsar separation. In this paper, we find that the ORF for the $v<1$ case is sensitive to finite distance effects. First, we show that finite distance effects introduce an effective cut-off in the spherical harmonics decomposition at ~ √1-v² \, kL, where is the multipole number, k the wavenumber of the gravitational wave and L the distance to the pulsars. Then, we find that the overlap reduction function in the small angle limit approaches a value given by π kL\,v²\,(1-v²)² times a normalization factor, exactly matching the value for the autocorrelation recently derived. Although we focus on the $v<1$ case, our formulation is valid for any value of v.