  The recent compelling observation of the nanohertz stochastic gravitational
wave background has brought to light a new galactic arena to test gravity. In
this paper, we derive a formula for the most general expression of the
stochastic gravitational wave background correlation that could be tested with
pulsar timing and future square kilometer arrays. Our expressions extends the
harmonic space analysis, also often referred to as the power spectrum approach,
to predict the correlation signatures of an anisotropic polarized stochastic
gravitational wave background with subluminal tensor, vector, and scalar
gravitational degrees of freedom. We present the first few nontrivial
anisotropy and polarization signatures in the correlation and discuss their
dependence on the gravitational wave speed and pulsar distances. Our results
set up tests that could potentially be used to rigorously examine the isotropy
of the stochastic gravitational wave background and strengthen the existing
constraints on possible non-Einsteinian polarizations in the nanohertz
gravitational wave regime.
