Delayed phase mixing in the self-gravitating Galactic disc

The Gaia phase spiral is considered to work as a dynamical clock for dating past perturbations, but some of the previous studies neglected the disc's self-gravity, potentially biasing estimates of the phase spiral's excitation time. We revisit the impact of self-gravitating effects on the evolution of vertical phase spirals and quantify the bias introduced in estimating their excitation time when such effects are ignored. We analysed a high-resolution, self-consistent $N$-body simulation of the MW-Sagittarius dwarf galaxy (Sgr) system, alongside four test particle simulations in potentials constructed from the $N$-body snapshots. In each case, we estimated the winding time of phase spirals by measuring the slope of the density contrast in the vertical angle-frequency space. In the test particle models, the phase spiral begins winding immediately after Sgr's pericentric passage, and the winding time closely tracks the true elapsed time since the Sgr impact. Adding the DM wake yields only a modest (< 100 Myr) reduction of the winding time relative to Sgr alone. By contrast, the self-consistent $N$-body simulation exhibits an initial, coherent vertical oscillation lasting $\gtrsim$ 300 Myr before a clear spiral forms, leading to systematic underestimation of excitation times. An analytical shearing-box model with self-gravity, developed by Widrow (2023), qualitatively reproduces this delay, supporting its origin in the disc's self-gravitating response. Assuming that self-gravity affects phase mixing in the MW to the same degree as the $N$-body model, the lag induced by self-gravity is estimated to be $\sim$ 0.3 Gyr in the solar neighbourhood. Accounting for this delay revises the inferred age of the MW's observed phase spiral to $\sim$0.6-1.2 Gyr, in better agreement with the Sgr's pericentric passage. (shortened for arXiv)