de Sitter space and the equivalence between $f(R)$ and scalar-tensor gravity

It is shown that, when $f^{\prime \prime }\ne 0$, metric $f(R)$ gravity is completely equivalent to an $\omega =0$ scalar-tensor theory with respect to perturbations of de Sitter space, contrary to previous expectations. Moreover, the stability conditions of de Sitter space with respect to homogeneous and inhomogeneous perturbations coincide in most scalar-tensor theories, as is the case in metric $f(R)$ gravity.