Parameterizing scalar-tensor theories for cosmological probes

TL;DR

This paper develops an analytic framework to understand how certain $f(R)$ gravity models, which mimic $ m{ ext{ extLambda}}$CDM, influence the growth of cosmic structures by deriving a scale- and time-dependent growth parameter and reconstructing the matter power spectrum.

Contribution

It introduces a parameterization of the scalar field mass in $f(R)$ models and derives an analytic expression for the growth parameter $3$ in terms of this mass, enabling the study of structure formation.

Findings

The matter power spectrum shows increased small-scale power at $z \,\sim\, 0$ compared to GR.

The growth parameter $3$ is scale- and time-dependent in these models.

Standard constant-$3$ prescriptions need modification for $f(R)$ models.

Abstract

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.