In this paper we review in detail a number of approaches that have been adopted to try and explain the remarkable observation of our accelerating Universe. In particular we discuss the arguments for and recent progress made towards understanding the nature of dark energy. We review the observational evidence for the current accelerated expansion of the universe and present a number of dark energy models in addition to the conventional cosmological constant, paying particular attention to scalar field models such as quintessence, K-essence, tachyon, phantom and dilatonic models. The importance of cosmological scaling solutions is emphasized when studying the dynamical system of scalar fields including coupled dark energy. We study the evolution of cosmological perturbations allowing us to confront them with the observation of the Cosmic Microwave Background and Large Scale Structure and demonstrate how it is possible in principle to reconstruct the equation of state of dark energy by also using Supernovae Ia observational data. We also discuss in detail the nature of tracking solutions in cosmology, particle physics and braneworld models of dark energy, the nature of possible future singularities, the effect of higher order curvature terms to avoid a Big Rip singularity, and approaches to modifying gravity which leads to a late-time accelerated expansion without recourse to a new form of dark energy.

Over the past decade, f(R) theories have been extensively studied as one of the simplest modifications to General Relativity. In this article we review various applications of f(R) theories to cosmology and gravity — such as inflation, dark energy, local gravity constraints, cosmological perturbations, and spherically symmetric solutions in weak and strong gravitational backgrounds. We present a number of ways to distinguish those theories from General Relativity observationally and experimentally. We also discuss the extension to other modified gravity theories such as Brans-Dicke theory and Gauss-Bonnet gravity, and address models that can satisfy both cosmological and local gravity constraints.

The properties of future singularities are investigated in the universe dominated by dark energy including the phantom-type fluid. We classify the finite-time singularities into four classes and explicitly present the models which give rise to these singularities by assuming the form of the equation of state of dark energy. We show the existence of a stable fixed point with an equation of state w < −1 and numerically confirm that this is actually a late-time attractor in the phantomdominated universe. We also construct a phantom dark energy scenario coupled to dark matter that reproduces singular behaviors of the Big Rip type for the energy density and the curvature of the universe. The effect of quantum corrections coming from conformal anomaly can be important when the curvature grows large, which typically moderates the finite-time singularities.

We review the theory of inflation with single and multiple fields paying particular attention to the dynamics of adiabatic and entropy/isocurvature perturbations which provide the primary means of testing inflationary models. We review the theory and phenomenology of reheating and preheating after inflation providing a unified discussion of both the gravitational and nongravitational features of multi-field inflation. In addition we cover inflation in theories with extra dimensions and models such as the curvaton scenario and modulated reheating which provide alternative ways of generating large-scale density perturbations. Finally we discuss the interesting observational implications that can result from adiabatic-isocurvature correlations and non-Gaussianity.

We derive the conditions under which dark energy models whose Lagrangian densities f are written in terms of the Ricci scalar R are cosmologically viable. We show that the cosmological behavior of f (R) models can be understood by a geometrical approach consisting in studying the m(r) curve on the (r, m) plane, where m ≡ Rf,RR/f,R and r ≡ −Rf,R/f with f,R ≡ df /dR. This allows us to classify the f (R) models into four general classes, depending on the existence of a standard matter epoch and on the final accelerated stage. The existence of a viable matter dominated epoch prior to a late-time acceleration requires that the variable m satisfies the conditions m(r) ≈ +0 and dm/dr > −1 at r ≈ −1. For the existence of a viable late-time acceleration we require instead either (i) m = −r − 1 , ( √ 3 − 1)/2 < m ≤ 1 and dm/dr < −1 or (ii) 0 ≤ m ≤ 1 at r = −2. These conditions identify two regions in the (r, m) space, one for the matter era and the other for the acceleration. Only models with a m(r) curve that connects these regions and satisfy the requirements above lead to an acceptable cosmology. The models of the type f (R) = αR −n and f = R + αR −n do not satisfy these conditions for any n > 0 and n < −1 and are thus cosmologically unacceptable. Similar conclusions can be reached for many other examples discussed in the text. In most cases the standard matter era is replaced by a cosmic expansion with scale factor a ∝ t 1/2 . We also find that f (R) models can have a strongly phantom attractor but in this case there is no acceptable matter era.