Over the last few years much attention has been given to the study of modified gravity theories in order to find a more natural explanation for the late time acceleration of the Universe. Nevertheless, a comparison of the matter power spectrum predictions made by these theories with available data has not yet been subjected to a detailed analysis. In the context of f (R) theories of gravity we study the predicted power spectra using both a dynamical systems approach for the background and solving for the matter perturbations without using the quasi-static approximation, comparing the theoretical results with several SDSS Data. The importance of studying the first order perturbed equations by assuming the correct background evolution and the relevance of the initial conditions are also stressed. We determine the statistical significance in relation to the observational data and demonstrate their conflict with existing observations.

We study f (R) gravity models in the language of scalar-tensor theories. The correspondence between f (R) gravity and scalar-tensor theories is revisited since f (R) gravity is a subclass of Brans-Dicke models, with a vanishing coupling constant (ω = 0). In this treatment, four f (R) toy models are used to analyze the early-universe cosmology, when the scalar field φ dominates over standard matter. We have obtained solutions to the Klein-Gordon equation for those models. It is found that for the first model (f (R) = βR n ), as time increases the scalar field decreases and decays asymptotically. For the second model (f (R) = αR + βR n ) it was found that the function φ(t) crosses the t-axis at different values for different values of β. For the third model f (R) = R − ν 4 R , when the value of ν is small the potential V (φ) behaves like the standard inflationary potential. For the fourth model f (R) = R − (1 − m)ν 2 R ν 2 m − 2Λ , we show that there is a transition between 1.5 < m < 1.55. The behavior of the potentials with m < 1.5 is totally different from those with m > 1.55. The slow-roll approximation is applied to each of the four f (R) models and we obtain the respective expressions for the spectral index ns and the tensor-to-scalar ratio r.

One of the exact solutions of f (R) theories of gravity in the presence of different forms of matter exactly mimics the ΛCDM solution of general relativity at the background level. In this work we study the evolution of scalar cosmological perturbations in the covariant and gauge -invariant formalism and show that although the background in such a model is indistinguishable from the standard ΛCDM cosmology, this degeneracy is broken at the level of first -order perturbations. This is done by predicting different rates of structure formation in ΛCDM and the f (R) model both in the complete and quasi -static regimes.