Modified Gravity: the CMB, Weak Lensing and General Parameterisations

Abstract

We examine general physical parameterisations for viable gravitational models in the $f(R)$ framework. This is related to the mass of an additional scalar field, called the scalaron, that is introduced by the theories. Using a simple parameterisation for the scalaron mass $M(a)$ we show there is an exact correspondence between the model and popular parameterisations of the modified Poisson equation $μ(a,k)$ and the ratio of the Newtonian potentials $η(a,k)$. However, by comparing the aforementioned model against other viable scalaron theories we highlight that the common form of $μ(a,k)$ and $η(a,k)$ in the literature does not accurately represent $f(R)$ behaviour. We subsequently construct an improved description for the scalaron mass (and therefore $μ(a,k)$ and $η(a,k)$) which captures their essential features and has benefits derived from a more physical origin. We study the scalaron's observational signatures and show the modification to the background Friedmann equation and CMB power spectrum to be small. We also investigate its effects in the linear and non linear matter power spectrum--where the signatures are evident--thus giving particular importance to weak lensing as a probe of these models. Using this new form, we demonstrate how the next generation Euclid survey will constrain these theories and its complementarity to current solar system tests. In the most optimistic case Euclid, together with a Planck prior, can constrain a fiducial scalaron mass $M_{0} = 9.4 \times 10^{-30}{\rm eV}$ at the $\sim 20 %$ level. However, the decay rate of the scalaron mass, with fiducial value $ν= 1.5$, can be constrained to $\sim 3%$ uncertainty.