Gross-Pitaevskii-Poisson equations with a $ξR φ^4$ non-minimal coupling term

TL;DR

This work introduces a non-minimal coupling term $\xi R \phi^4$ in the axion action to encode gravitationally mediated pair interactions and derives the resulting modified Einstein–Klein–Gordon system. Through a careful Newtonian-limit reduction, the real scalar field $\phi$ is mapped to a complex field $\psi$, yielding a Gross–Pitaevskii–type equation with an effective nonlinear coupling that now includes curvature through $R$, and a background/perturbation framework that couples to the metric potentials $\Phi$ and $\Psi$. The analysis recasts the single-field dynamics into an $N$-body Schrödinger picture, presenting an explicit Hamiltonian with external and contact interaction terms, thereby enabling the calculation of a partition function for axions with gravity-mediated interactions. In the limit of vanishing non-minimal coupling and perturbations ($\xi=0$, $\Phi=\Psi=0$), one recovers the standard GP–Poisson or Schrödinger–Poisson equations used to model axion miniclusters and axion stars, while the non-minimal term potentially broadens the mass and stability range of bound configurations. The framework provides a pathway to link microscopic axion physics with macroscopic structure formation and offers avenues for astrophysical constraints on $\xi$ via DM phenomenology.

Abstract

In scenarios where the Peccei-Quinn symmetry breaks after inflation, small-scale axion inhomogeneities may gravitationally collapse into bound structures. The evolution of these systems is typically modeled through cosmological perturbation theory applied to the Einstein-Klein-Gordon equations. In the non-relativistic regime, this framework reduces to the Gross-Pitaevskii-Poisson or Schrödinger-Poisson equations, depending on whether axion self-interactions are taken into account. In this work, a non-minimal gravitational coupling term $ξR φ^4$ is included into the axion's relativistic action as a way to introduce a gravitationally mediated pairwise interaction. By performing a perturbative expansion and subsequently taking the non-relativistic limit, an alternative set of equations that govern the early stages of structure formation is obtained.