We give a rigorous and mathematically well defined presentation of the covariant and gauge invariant theory of scalar perturbations of a Friedmann-Lemaître-Robertson-Walker universe for fourth order gravity, where the matter is described by a perfect fluid with a barotropic equation of state. The general perturbations equations are applied to a simple background solution of $R^n$ gravity. We obtain exact solutions of the perturbations equations for scales much bigger than the Hubble radius. These solutions have a number of interesting features. In particular, we find that for all values of $n$ there is always a growing mode for the density contrast, even if the universe undergoes an accelerated expansion. Such behavior does not occur in standard general relativity, where as soon as dark energy dominates, the density contrast experiences an unrelenting decay. This peculiarity is sufficiently novel to warrant further investigation of fourth order gravity models.

We develop a new covariant formalism to treat spherically symmetric spacetimes in metric f(R) theories of gravity. Using this formalism we derive the general equations for a static and spherically symmetric metric in a general f(R) gravity. These equations are used to determine the conditions for which the Schwarzschild metric is the only vacuum solution with vanishing Ricci scalar. We also show that our general framework provides a clear way of showing that the Schwarzschild solution is not a unique static spherically symmetric solution, providing some insight into how the current form of Birkhoff's theorem breaks down for these theories.

Model-independent methods in cosmology have become an essential tool to deal with an increasing number of theoretical alternatives for explaining the late-time acceleration of the Universe. In principle, this provides a way of testing the cosmological concordance (or $\mathrm{\Lambda }\mathrm{CDM}$) model under different assumptions and ruling out whole classes of competing theories. One such model-independent method is the so-called “cosmographic approach”, which relies only on the homogeneity and isotropy of the Universe on large scales. We show that this method suffers from many shortcomings, providing biased results depending on the auxiliary variable used in the series expansion, and is unable to rule out models or adequately reconstruct theories with higher-order derivatives in either the gravitational or matter sector. Consequently, in its present form, this method seems unable to provide reliable or useful results for cosmological applications.

We give a rigorous and mathematically clear presentation of the Covariant and Gauge Invariant theory of gravitational waves in a perturbed Friedmann-Lemaitre-Robertson-Walker universe for Fourth Order Gravity, where the matter is described by a perfect fluid with a barotropic equation of state. As an example of a consistent analysis of tensor perturbations in Fourth Order Gravity, we apply the formalism to a simple background solution of R^n gravity. We obtain the exact solutions of the perturbation equations for scales much bigger than and smaller than the Hubble radius. It is shown that the evolution of tensor modes is highly sensitive to the choice of n and an interesting new feature arises. During the radiation dominated era, their exist a growing tensor perturbation for nearly all choices of n. This occurs even when the background model is undergoing accelerated expansion as opposed to the case of General Relativity. Consequently, cosmological gravitational wave modes can in principle provide a strong constraint on the theory of gravity independent of other cosmological data sets.Comment: 19 pages, 4 figures; v2: corrected to match version accepted for publication in PR