On Pulsar Timing Detection of Ultralight Vector Dark Matter

Ultralight vector dark matter induces metric fluctuations that generate timing residuals in the arrival times of pulsar emissions through two distinct modes: a fast mode, sourced by coherent field oscillations, and a slow mode, arising from interference patterns. These modes enable the detection of vector dark matter with masses $m \sim 10^{-24} - 10^{-22}\ \mathrm{eV}$ and $m \sim 10^{-18} - 10^{-16}\ \mathrm{eV}$, respectively, using pulsar timing arrays. While previous studies have explored the fast mode, they neglect the full statistical treatment of the vector field and a precise treatment of its polarization structure. In this work, we investigate the timing residuals from both modes, fully accounting for the statistical properties of ultralight vector dark matter, assuming equipartition among its three polarization states. The two-point correlation functions of timing residuals that we derive serve as direct tools for identifying vector dark matter signatures as a stochastic background in pulsar timing data.