A Comprehensive Perturbative Formalism for Phase Mixing in Perturbed Disks. II. Phase Spirals in an Inhomogeneous Disk Galaxy with a Nonresponsive Dark Matter Halo

We develop a linear perturbative formalism to compute the response of an inhomogeneous stellar disk embedded in a nonresponsive dark matter (DM) halo to various perturbations like bars, spiral arms, and encounters with satellite galaxies. Without self-gravity to reinforce it, the response of a Fourier mode phase mixes away due to an intrinsic spread in the vertical (O-z), radial (O-r), and azimuthal (O-f) frequencies, triggering local phase-space spirals. The detailed galactic potential dictates the shape of phase spirals: phase mixing occurs more slowly and thus phase spirals are more loosely wound in the outer disk and in the presence of an ambient DM halo. Collisional diffusion due to scattering of stars by structures like giant molecular clouds causes superexponential damping of the phase spiral amplitude. The z-v(z) phase spiral is one-armed (two-armed) for vertically antisymmetric (symmetric) bending (breathing) modes. Only transient perturbations with timescales (TP) comparable to the vertical oscillation period (T-z similar to 1/O-z) can trigger vertical phase spirals. Each (n, l, m) mode of the response to impulsive (T-P < T = 1/(nO(z) + lO(r) + mO(f))) perturbations is power-law (similar to T-P/T) suppressed, but that to adiabatic (T-P > T) perturbations is exponentially weak (similar to exp[-(similar to T-P/T)(a)]) except for resonant (T -> infinity) modes. Slower (T-P > T-z) perturbations, e.g., distant encounters with satellite galaxies, induce stronger bending modes. Sagittarius (Sgr) dominates the solar neighborhood response of the Milky Way (MW) disk to satellite encounters. Thus, if the Gaia phase spiral was triggered by a MW satellite, Sgr is the leading contender. However, the survival of the phase spiral against collisional damping necessitates an impact similar to 0.6-0.7 Gyr ago.