Adiabatic Perturbation of Two-Level Ultralight Dark Matter Soliton Core in Baryonic Dehnen γ Potential (Part I)

We develop a two–level model of ultralight dark matter (ULDM) solitonic core subjected to the adiabatic perturbation due to the baryonic matter. Approximation the dark matter–only core as a Gaussian ground state in a harmonic potential defined by the central core density, and the first radial s–wave excitation (n = 1, l = 0) we project the GPP system onto 2-dimensional Hilbert space. In our formalism, we show that the baryonic Dhenen γ component couples through the overlap integral J_ij .(γ, α) where α = Rc/ab is the ratio of the soliton core radius to the baryon scale radius. The resulting core dynamics is governed by the relative Hamiltonian H_rel(t) = 1/2 ∆(t)σ_z + J(t)σ_x with baryon dependent level splitting ∆(t) and mixing J(t), both linear in enclosed baryonic mass. In the adiabatic limit the soliton follows instantaneous lower eigenstate leading to radial excitation controlled by J_ij (γ, α). We illustrate the predictions using a model dwarf–like galaxy to illustrate the resulting gap and mixing angle between the two states of the model. In (Part–II) companion paper we will consider higher number of excited states and also use this framework with real dwarf spheroidal galaxies, using observed baryonic profiles and stellar kinematics to study the baryon induced shifts in the core radius and infer other ULDM parameters including the dark matter particle mass. We will also study Gaffe–like(γ → 2) distribution model as a limiting case in the Dehnen-γ baryon distribution.