An ever-present $Gaia$ snail shell triggered by a dark matter wake

We utilize a novel numerical technique to model star formation in cosmological simulations of galaxy formation - called Superstars - to simulate a Milky Way-like galaxy with $\gtrsim10^8$ star particles to study the formation and evolution of out-of-equilibrium stellar disc structures in a full cosmological setting. In the plane defined by the coordinate and velocity perpendicular to the mid-plane (vertical phase space, $\{Z,V_Z\}$), stars in Solar-like volumes at late times exhibit clear spirals qualitatively similar in shape and amplitude to the $Gaia$ ``Snail shell'' phase spiral. We show that the phase spiral forms at a look back time of $\sim 6$ Gyr during the pericentric passage of a $\sim10^{10}$ $\rm M_{\odot}$ satellite on a polar orbit. This satellite stimulates the formation of a resonant wake in the dark matter halo while losing mass at a rate of $\sim0.5$-$1$ dex per orbit loop. The peak magnitude of the wake-induced gravitational torque at the Solar radius is $\sim 8$ times that from the satellite, and triggers the formation of a disc warp that wraps up into a vertical phase spiral over time. As the wake decays, the phase spiral propagates several Gigayears to present-day and can be described as ``ever-present'' once stable disc evolution is established. These results suggest an alternative scenario to explain the $Gaia$ phase spiral which does not rely on a perturbation from bar buckling or a recent direct hit from a satellite.