Modified Gravity: the CMB, Weak Lensing and General Parameterisations
  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 $\mu(a,k)$ and the ratio of the Newtonian potentials
$\eta(a,k)$. However, by comparing the aforementioned model against other
viable scalaron theories we highlight that the common form of $\mu(a,k)$ and
$\eta(a,k)$ in the literature does not accurately represent $f(R)$ behaviour.
We subsequently construct an improved description for the scalaron mass (and
therefore $\mu(a,k)$ and $\eta(a,k)$) 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 $\nu = 1.5$, can be constrained to $\sim 3%$ uncertainty.
