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.

We present a phase-plane analysis of cosmologies containing a baryotropic fluid with an equation of state p ␥ ϭ(␥Ϫ1) ␥ , plus a scalar field with an exponential potential Vϰexp(Ϫ) where 2 ϭ8G. In addition to the well-known inflationary solutions for 2 Ͻ2, there exist scaling solutions when 2 Ͼ3␥ in which the scalar field energy density tracks that of the baryotropic fluid ͑which for example might be radiation or dust͒. We show that the scaling solutions are the unique late-time attractors whenever they exist. The fluid-dominated solutions, where V()/ ␥ →0 at late times, are always unstable ͑except for the cosmological constant case ␥ϭ0). The relative energy density of the fluid and scalar field depends on the steepness of the exponential potential, which is constrained by nucleosynthesis to 2 Ͼ20. We show that standard inflation models are unable to solve this ''relic density'' problem. ͓S0556-2821͑98͒05408-3͔ PACS number͑s͒: 98.80.Cq

We present a detailed investigation of chaotic ination models which feature two scalar elds, such that one eld (the inaton) rolls while the other is trapped in a false vacuum state. The false vacuum becomes unstable when the magnitude of the inaton eld falls below some critical value, and a rst or second order transition to the true vacuum ensues. Particular attention is paid to the case, termed`Hybrid Ination' by Linde, where the false vacuum energy density dominates, so that the phase transition signals the end of ination. We focus mostly on the case of a second order transition, but treat also the rst order case and discuss bubble production in that context for the rst time.False vacuum dominated ination is dramatically dierent from the usual true vacuum case, both in its cosmology and in its relation to particle physics. The spectral index of the adiabatic density perturbation originating during ination can be indistinguishable from 1, or it can be up to ten percent or so higher. The energy scale at the end of ination can be anywhere between 10 16 GeV, which is familiar from the true vacuum case, and 10 11 GeV. On the other hand reheating is prompt, so the reheat temperature cannot be far below 1 0 11 GeV. Cosmic strings or other topological defects are almost inevitably produced at the end of ination, and if the inationary energy scale is near its upper limit they contribute signicantly to large scale structure formation and the cosmic microwave background anisotropy.Turning to the particle physics, false vacuum ination occurs with the inaton eld far below the Planck scale and is therefore somewhat easier to implement in the context of supergravity than true vacuum chaotic ination. The smallness of the inaton mass compared with the inationary Hubble parameter still presents a diculty for generic supergravity theories. Remarkably however, the diculty can be avoided in a natural way for a class of supergravity models that follow from orbifold compactication of superstrings. This opens up the prospect of a truly realistic, superstring derived theory of ination. One possibility, which w e show t o be viable at least in the context of global supersymmetry, is that the Peccei-Quinn symmetry is responsible for the false vacuum.

We review the relation between the inflationary potential and the spectra of density (scalar) perturbations and gravitational waves (tensor perturbations) produced, with particular emphasis on the possibility of reconstructing the inflaton potential from observations. The spectra provide a potentially powerful test of the inflationary hypothesis; they are not independent but instead are linked by consistency relations reflecting their origin from a single inflationary potential. To lowest-order in a perturbation expansion there is a single, now familiar, relation between the tensor spectral index and the relative amplitude of the spectra. We demonstrate that there is an infinite hierarchy of such consistency equations, though observational difficulties suggest only the first is ever likely to be useful. We also note that since observations are expected to yield much better information on the scalars than on the tensors, it is likely to be the next-order version of this consistency equation which will be appropriate, not the lowest-order one. If inflation passes the consistency test, one can then confidently use the remaining observational information to constrain the inflationary potential, and we survey the general perturbative scheme for carrying out this procedure. Explicit expressions valid to next-lowest order in the expansion are presented. We then briefly assess the prospects for future observations reaching the quality required, and consider simulated data sets motivated by this outlook.

Macroscopic fundamental and Dirichlet strings have several potential instabilities: breakage, tachyon decays, and confinement by axion domain walls. We investigate the conditions under which metastable strings can exist, and we find that such strings are present in many models. There are various possibilities, the most notable being a network of (p, q) strings. Cosmic strings give a potentially large window into string physics.