Gross-Pitaevskii-Poisson equations with a $\xi R \phi^4$ non-minimal coupling term

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 Schr\"odinger-Poisson equations, depending on whether axion self-interactions are taken into account. In this work, a non-minimal gravitational coupling term $\xi R \phi^4$ 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.