Parameterizing scalar-tensor theories for cosmological probes
  We study the evolution of density perturbations for a class of $f(R)$ models
which closely mimic $\Lambda$CDM background cosmology. Using the quasi-static
approximation, and the fact that these models are equivalent to scalar-tensor
gravity, we write the modified Friedmann and cosmological perturbation
equations in terms of the mass $M$ of the scalar field. Using the perturbation
equations, we then derive an analytic expression for the growth parameter
$\gamma$ in terms of $M$, and use our result to reconstruct the linear matter
power spectrum. We find that the power spectrum at $z \sim 0$ is characterized
by a tilt relative to its General Relativistic form, with increased power on
small scales. We discuss how one has to modify the standard, constant $\gamma$
prescription in order to study structure formation for this class of models.
Since $\gamma$ is now scale and time dependent, both the amplitude and transfer
function associated with the linear matter power spectrum will be modified. We
suggest a simple parameterization for the mass of the scalar field, which
allows us to calculate the matter power spectrum for a broad class of $f(R)$
models.
