Reheating after inflation.

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. PACS numbers: 98.80.Cq, 04.62.+v On leave of absence from Institute of Astrophysics and Atmospheric Physics, Tartu EE-2444, Estonia On leave of absence from Lebedev Physical Institute, Moscow 117924, Russia 1. The theory of reheating of the Universe after inflation is the most important application of the quantum theory of particle creation, since almost all matter constituting the Universe at the subsequent radiation-dominated stage was created during this process [1]. At the stage of inflation all energy was concentrated in a classical slowly moving inflaton field φ. Soon after the end of inflation this field began to oscillate near the minimum of its effective potential. Gradually it produced many elementary particles, they interacted with each other and came to a state of thermal equilibrium with some temperature Tr, which was called the reheating temperature. An elementary theory of reheating was first developed in [2] for the new inflationary scenario. Independently a theory of reheating in the R inflation was constructed in [3]. Various aspects of this theory were further elaborated by many authors, see e.g. [4]. Still, a general scenario of reheating was absent. In particular, reheating in the chaotic inflation theory remained almost unexplored. The present paper is a short account of our investigation of this question [5]. We have found that the process of reheating typically consists of three different stages. At the first stage, which cannot be described by the elementary theory of reheating, the classical coherently oscillating inflaton field φ decays into massive bosons (in particular, into φ-particles) due to parametric resonance. In many models the resonance is very broad, and the process occurs extremely rapidly (explosively). Because of the Pauli exclusion principle, there is no explosive creation of fermions. To distinguish this stage from the stage of particle decay and thermalization, we will call it pre-heating. Bosons produced at that stage are far away from thermal equilibrium and typically have enormously large occupation numbers. The second stage is the decay of previously produced particles. This stage typically can be described by methods developed in [2]. However, these methods should be applied not to the decay of the original homogeneous inflaton field, but to the decay of particles and fields produced at the stage of explosive reheating. This considerably changes many features of the process, including the final value of the reheating temperature. The third stage is the stage of thermalization, which can be described by standard methods, see e.g. [1, 2]; we will not consider it here. Sometimes this stage may occur simultaneously with the second one. In our investigation we have used the formalism of the time-dependent Bogoliubov transformations to find the density of created particles, n~k(t). A detailed description of this theory will be given in [5]; here we will outline our main conclusions using a simple semiclassical approach. 2. We will consider a simple chaotic inflation scenario describing the classical inflaton scalar field φ with the effective potential V (φ) = ± 2 mφφ 2 + λ 4 φ. Minus sign corresponds to spontaneous symmetry breaking φ → φ + σ with generation of a classical scalar field σ = mφ √ λ . The field φ after inflation may decay into bosons χ and fermions ψ due to the interaction terms − 2 gφχ and −hψ̄ψφ. Here λ, g and h are small coupling constants. In case of spontaneous symmetry breaking, the term − 2 gφχ gives rise to the term −g2σφχ2. We will assume for simplicity that the bare masses of the fields χ and ψ are very small, so that one can write mχ(φ) = gφ, mψ(φ) = |hφ|. Let us briefly recall the elementary theory of reheating [1]. At φ > Mp, we have a stage of inflation. This stage is supported by the friction-like term 3Hφ̇ in the equation of motion for the scalar field. Here H ≡ ȧ/a is the Hubble parameter, a(t) is the scale factor of the Universe. However, with a decrease of the field φ this term becomes less and less important, and inflation ends at φ <∼ Mp/2. After that the field φ begins oscillating near the minimum of V (φ) [6]. The amplitude