Series expansion of the overlap reduction function for scalar and vector polarizations for gravitational wave search with pulsar timing arrays

In our previous work \cite{PTA2} we calculated the overlap reduction function for the tensor polarization without employing the short wavelength approximation, this was done by obtaining a power series of nested sums which is valid for all gravitational wave frequencies and pulsar distances. In this work we generalize the power-series expansion method to vector and scalar polarizations. We have compared our expression for the breathing and vector modes with previous literature. We present for the first time analytic expressions for the overlap reduction function of the longitudinal mode for all angles between the pulsar pairs.

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