Revisiting chameleon gravity: Thin-shell and no-shell fields with appropriate boundary conditions

Abstract

We derive analytic solutions of a chameleon scalar field $\ensuremath{\phi}$ that couples to a nonrelativistic 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 ${m}_{A}{r}_{c}\ensuremath{\rightarrow}\ensuremath{\infty}$, 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 ${m}_{A}{r}_{c}\ensuremath{\gg}1$, the effective coupling of $\ensuremath{\phi}$ 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 ${\ensuremath{\phi}}_{A}$ at the extremum of an effective potential induced by the matter coupling.