$R^n$ gravity and the chameleon

The Solar System bounds on $R^n$ gravity are often ignored in the literature by invoking the chameleon mechanism. We show that in order for the latter to work, the exponent $n$ must be ridiculously close to unity and, therefore, these theories are severely constrained.

Figure 1

The range $s(n)$ of the effective scalar degree of freedom $\varphi =f^{\prime }(R)$ (in units $c{H}_0^{-1}$) for the theory $f(R)=\alpha R^n$. The parameter $n$ is in the range $1\le n\le 2$ allowed by stability. The function $s(n)$ starts out with vertical tangent at $n=1$.