Stochastic gravitational wave background phenomenology in a pulsar timing array

Pulsar timing offers an independent avenue to test general relativity and alternative gravity theories. This requires an understanding of how metric polarizations beyond the familiar transverse tensor ones imprint as a stochastic gravitational wave background and correlate the arrival time of radio pulses from a pair of millisecond pulsars. In this work, we focus on an isotropic stochastic gravitational wave background and present a straightforward, self-contained formalism for obtaining the power spectrum and the overlap reduction function, the relevant physical observable in a pulsar timing array, for generic gravitational degrees of freedom featuring both transverse and longitudinal modes off the light cone. We additionally highlight our consideration of finite pulsar distances, which we find significant in two ways: first, making all the modes well defined, and second, keeping the small scale power that is contained by pulsars of subdegree separations in the sky. We discuss this for tensor, vector, and scalar polarizations, for each one focusing on the angular power spectrum and the overlap reduction function for an isotropic stochastic gravitational wave background. Our results pave the road for an efficient numerical method for examining the gravitational wave induced spatial correlations across millisecond pulsars in a pulsar timing array.