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

Abstract

Abstract 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 $$v<1$$ v < 1 , diverge at small angular pulsar separation. In this paper, we find that the overlap reduction function for the $$v<1$$ 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 $$\ell \sim \sqrt{1-v^2} \, kL$$ ℓ ∼ 1 - v 2 k L , where $$\ell $$ ℓ 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 $$\pi kL\,v^2\,(1-v^2)^2$$ π k L v 2 ( 1 - v 2 ) 2 times a normalization factor, exactly matching the value for the autocorrelation recently derived. Although we focus on the $$v<1$$ v < 1 case, our formulation is valid for any value of v.