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

Linear perturbations in viable f(R) theories