Tidal Deformation of a Vlasov--Poisson Equilibrium with Cosmological Repulsion: Linear Response and Affine Approximation

We study the deformation of a spatially localized stationary solution of the gravitational Vlasov--Poisson equation by an external trace-free quadratic field, while a positive cosmological repulsion is retained explicitly in physical coordinates. An explicit Kurth-type weak stationary state is used for exact moment identities, and a second smooth compactly supported stationary state is constructed on an inner family of bounded trajectories separated from the exterior radial separatrix by a positive energy gap.Linearization about the smooth state and integration along its stationary characteristics give a finite-time Volterra integral equation. The tensor moment equations determine the initial quadrupole acceleration exactly. Rotational invariance separates the spherical-harmonic degrees; in degree two we derive one- and multi-mode Galerkin systems, obtain orbit formulas that do not differentiate the stationary distribution, and evaluate them by deterministic quadrature. A four-mode calculation quantifies the error of the one-mode response. The equilibrium construction, moment identity, and finite-time linear Volterra theory are rigorous at the stated regularity. Higher-order kinetic transfer is formulated conditionally, under an explicit third-order parameter expansion: for one fixed-axis degree-two polarization its diagonal quadratic source has angular degrees zero, two, and four. A separate affine variational approximation gives an instantaneous Hamiltonian for the mean scale and axialdeformation; it is used as a reduced model, not as an exact nonspherical Vlasov solution.