Resonant phenomena in finite motions of test particles in oscillating dark matter configurations

TL;DR

This paper derives and analyzes nonlinear differential equations describing test particle motions in oscillating dark matter, revealing resonance effects through numerical solutions of Hill equations in scalar dark matter models.

Contribution

It introduces a novel approach to model finite motions of test particles in oscillating dark matter using Hill equations and identifies potential resonance phenomena.

Findings

Radial and orbital oscillations are described by Hill equations.

Numerical integration shows resonance effects in scalar dark matter models.

Comparison with nonlinear systems confirms the Hill equation approximations.

Abstract

Nonlinear differential equations are derived that describe the time evolution of the test particle coordinates during finite motions in the gravitational field of oscillating dark matter. It is shown that in the weak field approximation, the radial oscillations of a test particle and oscillations in orbital motion are described by the Hill equation and the nonhomogeneous Hill equation, respectively. In the case of scalar dark matter with a logarithmic self-interactions, these equations are integrated numerically, and the solutions are compared with the corresponding solutions of the original nonlinear system to identify possible resonance effects.