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Reheating after inflation occurs due to particle production by the oscillating inflaton field. In this paper we briefly describe the perturbative approach to reheating, and then concentrate on effects beyond the perturbation theory. They are related to the stage of parametric resonance, which we called preheating. It may occur in an expanding universe if the initial amplitude of oscillations of the inflaton field is large enough. We investigate a simple model of a massive inflaton field φ coupled to another scalar field χ with the interaction term g 2 φ 2 χ 2 . Parametric resonance in this model is very broad. It occurs in a very unusual stochastic manner, which is quite different from parametric resonance in the case when the expansion of the universe is neglected. Quantum fields interacting with the oscillating inflaton field experience a series of kicks which, because of the rapid expansion of the universe, occur with phases uncorrelated to each other. Despite the stochastic nature of the process, it leads to exponential growth of fluctuations of the field χ. We call this process stochastic resonance. We develop the theory of preheating taking into account the expansion of the universe and backreaction of produced particles, including the effects of rescattering. This investigation extends our previous study of reheating after inflation [1]. We show that the contribution of the produced particles to the effective potential V (φ) is proportional not to φ 2 , as is usually the case, but to |φ|. The process of preheating can be divided into several distinct stages. In the first stage the backreaction of created particles is not important. In the second stage backreaction increases the frequency of oscillations of the inflaton field, which makes the process even more efficient than before. Then the effects related to scattering of χ-particles on the oscillating inflaton field terminate the resonance. We calculate the number density of particles nχ produced during preheating and their quantum fluctuations χ 2 with all backreaction effects taken into account. This allows us to find the range of masses and coupling constants for which one can have efficient preheating. In particular, under certain conditions this process may produce particles with a mass much greater than the mass of the inflaton field. PACS: 98.80.CqIfA-97-28 SU-ITP-97-18 hep-ph/9704452

For higher-derivative f (R) gravity where R is the Ricci scalar, a class of models is proposed which produce viable cosmology different from the LambdaCDM one at recent times and satisfy cosmological, Solar system and laboratory tests. These models have both flat and de Sitter spacetimes as particular solutions in the absence of matter. Thus, a cosmological constant is zero in flat space-time, but appears effectively in a curved one for sufficiently large R. A 'smoking gun' for these models would be small discrepancy in values of the slope of the primordial perturbation power spectrum determined from galaxy surveys and CMB fluctuations. On the other hand, a new problem for dark energy models based on f (R) gravity is pointed which is connected with possible overproduction of new massive scalar particles (scalarons) arising in this theory in the very early Universe.PACS numbers: 04.50.+h, 95.36.+x, 98.80.-k

The theory of reheating of the Universe after inflation is developed. We have found that typically at the first stage of reheating the classical inflaton field φ rapidly decays into φ-particles or into other bosons due to a broad parametric resonance. Then these bosons decay into other particles, which eventually become thermalized. Complete reheating is possible only in those theories where a single particle φ can decay into other particles. This imposes strong constraints on the structure of inflationary models, and implies that the inflaton field can be a dark matter candidate.

Recent observations of Type 1a supernovae indicating an accelerating universe have once more drawn attention to the possible existence, at the present epoch, of a small positive Λ-term (cosmological constant). In this paper we review both observational and theoretical aspects of a small cosmological Λ-term. We discuss the current observational situation focusing on cosmological tests of Λ including the age of the universe, high redshift supernovae, gravitational lensing, galaxy clustering and the cosmic microwave background. We also review the theoretical debate surrounding Λ: the generation of Λ in models with spontaneous symmetry breaking and through quantum vacuum polarization effects -mechanisms which are known to give rise to a large value of Λ hence leading to the "cosmological constant problem." More recent attempts to generate a small cosmological constant at the present epoch using either field theoretic techniques, or by modelling a dynamical Λ-term by scalar fields are also extensively discussed. Anthropic arguments favouring a small Λ-term are briefly reviewed. A comprehensive bibliography of recent work on Λ is provided.

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Behaviour of a weekly self-interacting scalar eld with a small mass in the de Sitter background is investigated using the stochastic approach (including the case of a double-well interaction potential). Existence of the de Sitter invariant equilibrium quantum state of the scalar eld in the presence of interaction is shown for any sign of the mass term. The stochastic approach is further developed to produce a method of calculation of an arbitrary anomalously large correlation function of the scalar eld in the de Sitter background, and expressions for the two-point correlation function in the equilibrium state, correlation time and spatial physical correlation radius are presented. The latter does not depend on time that implies that the characteristic size of domains with positive and negative values of the scalar eld remains the same on average in the equilibrium state in spite of the expansion of the t = const hypersurface of the de Sitter space-time.

We introduce a new cosmological diagnostic pair {r, s} called Statefinder. The Statefinder is dimensionless and, like the Hubble and deceleration parameters H(z) and q(z), is constructed from the scale factor of the Universe and its derivatives only. The parameter r(z) forms the next step in the hierarchy of geometrical cosmological parameters used to study the Universe after H and q, while the parameter s(z) is a linear combination of q and r chosen in such a way that it does not depend upon the dark energy density ΩX (z). The Statefinder pair {r, s} is algebraically related to the the dark energy pressure-to-energy ratio w = p/ε and its time derivative, and sheds light on the nature of dark energy/quintessence. Its properties allow to usefully differentiate between different forms of dark energy with constant and variable w, including a cosmological constant (w = −1). The Statefinder pair can be determined to very good accuracy from a SNAP type experiment.