  We describe the cosmological evolution predicted by three distinct $f(R)$
theories, with emphasis on the evolution of linear perturbations. The most
promising observational tools for distinguishing $f(R)$ theories from
$\Lambda$CDM are those intrinsically related to the growth of structure, such
as weak lensing. At the linear level, the enhancement in the gravitational
potential provided by the additional $f(R)$ `fifth force' can separate the
theories, whereas at the background level they can be indistinguishable. Under
the stringent constraints imposed on the models by Solar System tests and
galaxy-formation criteria, we show that the relative difference between the
models' linear evolution of the lensing potential will be extremely hard to
detect even with future space-based experiments such as {\it Euclid}, with a
maximum value of approximately 4% for small scales. We also show the evolution
of the gravitational potentials under more relaxed local constraint conditions,
where the relative difference between these models and $\Lambda$CDM could prove
discriminating.
