Hybrid inflation.

Usually inflation ends either by a slow rolling of the inflaton field, which gradually becomes faster and faster, or by a first-order phase transition. We describe a model where inflation ends in a different way, due to a very rapid rolling (‘waterfall’) of a scalar field σ triggered by another scalar field φ. This model looks as a hybrid of chaotic inflation with V (φ) = m φ 2 and the usual theory with spontaneous symmetry breaking with V (σ) = 1 4λ (M − λσ). The last stages of inflation in this model are supported not by the inflaton potential V (φ) but by the ‘non-inflationary’ potential V (σ). Another hybrid model to be discussed here uses some building blocks from extended inflation (Brans-Dicke theory), from new inflation (phase transition due to a non-minimal coupling of the inflaton field to gravity) and from chaotic inflation (the possibility of inflation beginning at large as well as at small σ). In the simplest version of this scenario inflation ends up by slow rolling, thus avoiding the big-bubble problem of extended inflation. PACS number 98.80. Cq On leave from: Lebedev Physical Institute, Moscow. E-mail: linde@physics.stanford. edu 1. There exist three independent ways of classifying inflationary models. The first classification deals with the initial conditions for inflation. The old and new inflationary models were based on the assumption that the Universe from the very beginning was in a state of thermal equilibrium at an extremely high temperature, and that the inflaton field φ was in a state corresponding to the minimum of its temperature-dependent effective potential V (φ) [1, 2]. The main idea of the chaotic inflation scenario was to study all possible initial conditions in the Universe, including those which describe the Universe outside of the state of thermal equilibrium, and the scalar field outside of the minimum of V (φ) [3]. This scenario includes the possibility of new inflation from the state in a thermal equilibrium, but it contains many other possibilities as well. Therefore it can be realized in a much greater variety of models than the new inflationary Universe scenario. In fact, at present the idea of thermal beginning is almost completely abandoned, and all realistic models of inflation from the point of view of the first classification are of the chaotic inflation type [4]. The second classification describes various regimes which are possible during inflation: quasiexponential inflation, power law inflation, etc. This classification is absolutely independent of the issue of initial conditions. Therefore it does not make any sense to compare, say, power law inflation and chaotic inflation, and to oppose them to each other. For example, in [5] it was pointed out that the chaotic inflation scenario, as distinct from the new inflationary Universe scenario, can be realized in the theories with the effective potential e for α ≪ √ 16π. Meanwhile, in [6] it was shown that this inflation is power law. Thus, the inflationary Universe scenario in the theory e describes chaotic power law inflation. Finally, the third classification is related to the way inflation ends. There are two possibilities extensively discussed in the literature: slow rollover versus the first-order phase transition. The models of the first class describe slow rolling of the inflaton field φ, which gradually becomes faster and faster. A particular model of this type is chaotic inflation in the theories φ. The models of the second class should contain at least two scalar fields, φ and σ. They describe a strongly first-order phase transition with bubble production which is triggered by the slow rolling of the field φ. One of the popular models of this type is the extended inflation scenario [7], which is a combination of the Brans-Dicke theory and the old inflationary scenario. There exist other versions of the first-order scenario with two scalar fields, which do not require any modifications of the Einstein theory of gravity, see e.g. [8]. In the beginning it was assumed that the bubbles formed during the first-order phase transition could be a useful ingredient of the theory of the large scale structure formation. However, later it was realized that one should make considerable modifications of the original models in order to avoid disastrous consequences of the bubble production. According to the most recent modification [9], the bubble formation happens only after the end of inflation. In this case, the end of inflation occurs as in the standard slow-rollover scenario. Therefore it would be interesting to find out other possible ways in which inflation may end in the models with several different scalar fields. More generally, one may try to find out other qualitatively new inflationary regimes which may appear due to a combined evolution of several scalar fields. Of course, one should not invent excessively complicated models without demonstrated need.