Signature of polarized ultralight vector dark matter in pulsar timing arrays

We investigate observational signatures of ultralight vector dark matter with masses $m\ensuremath{\sim}{10}^{\ensuremath{-}24}--{10}^{\ensuremath{-}22}\text{ }\text{ }\mathrm{eV}$ in pulsar timing arrays, taking into account general polarization states of the vector field. We find that vector dark matter induces pulsar timing residuals with nontrivial directional dependence, reflecting the anisotropic property and polarization structure specific to vector dark matter, unlike scalar dark matter. We also derive angular correlation curves of the timing residuals. Intriguingly, circular polarization of the vector dark matter enhances the quadrupole nature of the correlation curve, resulting in a more notable bending of the Hellings-Downs curve. The derived correlation curves offer a useful means to distinguish gravitational wave and dark matter contributions and to probe the nature of dark matter.