Solving nonlinear Klein-Gordon equations on unbounded domains via the Finite Element Method

TL;DR

This paper introduces femtoscope, a Python-based FEM tool for solving nonlinear Klein-Gordon equations on unbounded domains, enabling detailed analysis of scalar fields in modified gravity models like chameleon theories.

Contribution

The paper presents femtoscope, a novel FEM and Newton method implementation for Klein-Gordon equations, with new techniques for handling asymptotic boundary conditions on finite domains.

Findings

Femtoscope effectively solves Klein-Gordon equations with asymptotic boundary conditions.

The tool accurately models scalar fields in chameleon and symmetron models.

Application to Earth orbit demonstrates its practical utility.

Abstract

A large class of scalar-tensor theories of gravity exhibit a screening mechanism that dynamically suppresses fifth forces in the Solar system and local laboratory experiments. Technically, at the scalar field equation level, this usually translates into nonlinearities which strongly limit the scope of analytical approaches. This article presents $femtoscope$ $-$ a Python numerical tool based on the Finite Element Method (FEM) and Newton method for solving Klein-Gordon-like equations that arise in particular in the symmetron or chameleon models. Regarding the latter, the scalar field behavior is generally only known infinitely far away from the its sources. We thus investigate existing and new FEM-based techniques for dealing with asymptotic boundary conditions on finite-memory computers, whose convergence are assessed. Finally, $femtoscope$ is showcased with a study of the chameleon fifth force in Earth orbit.