Beyond Schr\"{o}dinger-Poisson: Nonrelativistic Effective Field Theory for Scalar Dark Matter

TL;DR

This paper develops a systematic framework to derive nonrelativistic Schr"{o}dinger-Poisson equations and their relativistic corrections from the fully relativistic Klein-Gordon-Einstein equations, enhancing the modeling of scalar dark matter beyond traditional approximations.

Contribution

It introduces a method to include relativistic corrections to the Schr"{o}dinger-Poisson system derived from the Klein-Gordon-Einstein equations, applicable in more relativistic regimes.

Findings

Derived explicit relativistic correction equations for scalar dark matter.

Showed the corrected equations extend the validity of the SP system.

Calculated soliton mass-radius relations with relativistic deviations.

Abstract

Massive scalar fields provide excellent dark matter candidates, whose dynamics are often explored analytically and numerically using nonrelativistic Schr\"{o}dinger-Poisson (SP) equations in a cosmological context. In this paper, starting from the nonlinear and fully relativistic Klein-Gordon-Einstein (KGE) equations in an expanding universe, we provide a systematic framework for deriving the SP equations, as well as relativistic corrections to them, by integrating out `fast modes' and including nonlinear metric and matter contributions. We provide explicit equations for the leading-order relativistic corrections, which provide insight into deviations from the SP equations as the system approaches the relativistic regime. Upon including the leading-order corrections, our equations are applicable beyond the domain of validity of the SP system, and are simpler to use than the full KGE case in some contexts. As a concrete application, we calculate the mass-radius relationship of solitons in scalar dark matter and accurately capture the deviations of this relationship from the SP system towards the KGE one.