Chameleons with field-dependent couplings

Certain scalar-tensor theories exhibit the so-called chameleon mechanism, whereby observational signatures of scalar fields are hidden by a combination of self-interactions and interactions with ambient matter. Not all scalar-tensor theories exhibit such a chameleon mechanism, which has been originally found in models with inverse power runaway potentials and field-independent couplings to matter. In this paper we investigate field theories with field-dependent couplings and a power-law potential for the scalar field. We show that the theory indeed is a chameleon field theory. We find the thin-shell solution for a spherical body and investigate the consequences for Eöt-Wash experiments, fifth-force searches and Casimir-force experiments. Requiring that the scalar field evades gravitational tests, we find that the coupling is sensitive to a mass scale which is of order of the Hubble scale today.

Figure 1

Numerical field profile for $(n,k)=(10,1)$ and $m_{c}R={10}^{-3}$ together with the analytical approximation (dashed line). The analytical approximation is seen to be a very good match to the actual solution.

Figure 2

The thin-shell profile for the Earth when $(n,k)=(10,1)$ and $m_{c}R={10}^6$.

Figure 3

The effective coupling for a spherical body with constant density. When the body is very small the field inside the body is the same as the background, ${\varphi }_b$, leading to a big coupling. Then as the radius gets bigger the field inside the body starts moving away from the background and the coupling decreases. Finally when we reach $m_{c}R>1$, the field inside the body settles at ${\varphi }_c$ and develops a thin shell such that the coupling starts to decrease like $1/R^3$.

Figure 4

The thin-shell profile for the Earth when $(n,k)=(10,1)$ and $m_{c}R={10}^6$ together with the analytical approximation (dashed line). The horizontal line shows ${\varphi }_b$, the minimum in the background. The error between the numerical solution and the analytical approximation is less than 10% in the whole range.

Figure 5

General behavior of the chameleon pressure $\frac{F_{\varphi }}{A}$ as a function of the plate separation $d$.

Figure 6

PPN constraints on chameleon theories coming from experimental bounds on the Eddington parameter in light-deflection experiments. The shaded areas show the regions of parameter space that are allowed by the current data. The solid horizontal black lines indicate the cases where $M$ and $\sigma$ take “natural values.” The solid vertical lines show when $M_{\beta }=H_0$. The dashed black lines indicate when $|{\beta }_{,{\varphi }_c}|M_{\mathrm{pl}}=1$ for ${\rho }_c=\mathcal{O}(1\mathrm{g}/{\mathrm{cm}}^3)$. The amount of allowed parameter space increases with $n$.

Figure 7

Constraints on chameleon theories coming from Eöt-Wash bounds on deviations from Newton’s law. The shaded areas show the regions of parameter space that are allowed by the current data. The solid horizontal black lines indicate the cases where $M$ and $\sigma$ take natural values. The solid vertical lines show when $M_{\beta }=H_0$. The dashed black lines indicate when $|{\beta }_{,{\varphi }_c}|M_{\mathrm{pl}}=1$ for ${\rho }_c=\mathcal{O}(1\mathrm{g}/{\mathrm{cm}}^3)$. The amount of allowed parameter space increases with $n$.

Figure 8

Constraints on chameleon theories coming from experimental fifth-force searches (the Irvine experiment). The shaded areas show the regions of parameter space that are allowed by the current data. The solid horizontal black lines indicate the cases where $M$ and $\sigma$ take natural values. The solid vertical lines show when $M_{\beta }=H_0$. The dashed black lines indicate when $|{\beta }_{,{\varphi }_c}|M_{\mathrm{pl}}=1$ for ${\rho }_c=\mathcal{O}(1\mathrm{g}/{\mathrm{cm}}^3)$.

Figure 9

Constraints on chameleon theories coming from experimental searches for the Casimir force. The shaded areas show the regions of parameter space that are allowed by the current data. The solid horizontal black lines indicate the cases where $M$ and $\sigma$ take natural values. The solid vertical lines show when $M_{\beta }=H_0$. The dashed black lines indicate when $|{\beta }_{,{\varphi }_c}|M_{\mathrm{pl}}=1$ for ${\rho }_c=\mathcal{O}(1\mathrm{g}/{\mathrm{cm}}^3)$. The amount of allowed parameter space increases with $n$.

Figure 10

Combined constraints on chameleon theories. The shaded areas show the regions of parameter space that are allowed by the current data. The solid horizontal black lines indicate the cases where $M$ and $\sigma$ take natural values. The solid vertical lines show when $M_{\beta }=H_0$. The dashed black lines indicate when $|{\beta }_{,{\varphi }_c}|M_{\mathrm{pl}}=1$ for ${\rho }_c=\mathcal{O}(1\mathrm{g}/{\mathrm{cm}}^3)$. The amount of allowed parameter space increases with $n$.