Gross–Pitaevskii–Poisson Equations from a ξRϕ⁴ Non-Minimal Scalar-Curvature Coupling

In cosmological scenarios where the Peccei–Quinn symmetry is broken after inflation, small-scale axion field inhomogeneities can undergo gravitational collapse, leading to the formation of bound structures. The dynamics of these systems are commonly described using cosmological perturbation theory applied to the Einstein–Klein–Gordon equations. In the non-relativistic regime, this description reduces to the Gross–Pitaevskii–Poisson or Schrödinger–Poisson equations, depending on whether axion self-interactions are included. In this work, we extend the axion’s relativistic action by introducing a non-minimal scalar-curvature coupling of the form $\xi R{\varphi }^4$, which effectively induces a gravitationally mediated pairwise interaction. By performing a perturbative expansion and subsequently taking the non-relativistic limit, we derive a modified set of evolution equations governing the early stages of axion structure formation.