Revisiting chameleon gravity - thin-shells and no-shells with appropriate boundary conditions

TL;DR

This paper analytically investigates chameleon scalar fields around spherical bodies, identifying conditions for thin-shell and no-shell solutions, and assesses their compatibility with experimental gravity constraints.

Contribution

It derives analytic solutions for chameleon fields, clarifies the existence of no-shell solutions, and analyzes their implications for local gravity tests and model constraints.

Findings

Existence of no-shell solutions where the field is nearly frozen inside the body.

Effective coupling in no-shell solutions matches thin-shell case under certain conditions.

Field must be close to the potential extremum at the body's center for consistency with experiments.

Abstract

We derive analytic solutions of a chameleon scalar field $\phi$ that couples to a non-relativistic matter in the weak gravitational background of a spherically symmetric body, paying particular attention to a field mass $m_A$ inside of the body. The standard thin-shell field profile is recovered by taking the limit $m_A*r_c \to \infty$, where $r_c$ is a radius of the body. We show the existence of "no-shell" solutions where the field is nearly frozen in the whole interior of the body, which does not necessarily correspond to the "zero-shell" limit of thin-shell solutions. In the no-shell case, under the condition $m_A*r_c \gg 1$, the effective coupling of $\phi$ with matter takes the same asymptotic form as that in the thin-shell case. We study experimental bounds coming from the violation of equivalence principle as well as solar-system tests for a number of models including $f(R)$ gravity and find that the field is in either the thin-shell or the no-shell regime under such constraints, depending on the shape of scalar-field potentials. We also show that, for the consistency with local gravity constraints, the field at the center of the body needs to be extremely close to the value $\phi_A$ at the extremum of an effective potential induced by the matter coupling.