Some remarks on Oscillating Inflation

Nowadays inflation is a widely accepted element of the early cosmology [1]. It gives the possibility of solving many of the shortcomings of the standard hot big bang model and provides the source for the early energy density fluctuations responsible of the large scale structure of the universe observed today. Although there are many models of inflation, the underlying physical ideas are well established. These are characterized by a period of “slow roll” evolution of a scalar field (called inflaton) toward the vacuum potential. During this period the field changes very slowly, so that the kinetic energy φ̇/2 remains smaller than its potential energy V (φ). The energy density associated to the scalar field acts as a “cosmological constant” term, allowing a period of quasi exponential expansion of the scale factor. When the period of inflation ends, the scalar field φ start a phase of rapid coherent oscillations around the vacuum. Very recently ( [2,3]) it has been pointed out that inflation can persist during the coherent oscillations of the inflation field phase. This exciting result is possible when the inflaton potential verifies a simple constrain of curvature far from the core convex part, where the inflaton field can roll slowly. The efficiency of this phenomena could have important implications for GUT scale baryogenesis [4]. In fact, as suggested by Damour and Mukhanov ( [2]),it can be expected that due to the increase of the oscillation frequency, there is the possibility to generate massive particles heavier than ∼ 10GeV . In ref. [2] Damour and Mukhanov estimated the amount of inflation to be ∼ 10 e-fold (powers of the scale factor). They argue that this effect can be more efficient than the parametric resonance effect [5] for the amplification of cosmological perturbations [6]. In ref. [3] Liddle and Mazumdar showed that Mukhanov et al. overestimated the number of e-fold because they have used a slow-roll definition of this object. In their paper, Liddle and Mazumdar found an analytical expression for the number of e-fold of inflation using the appropriate definition finding a number of ∼ 3 e-fold concluding that this effect is not very efficient. The study of adiabatic perturbations in this phase has been made by Taruya [7]. He found a poor amplification in the case of a single scalar field model but anticipated an enormous amplification for multi-field systems. In this letter we review the problem. In particular we find that the analytical expressions used to compare with the numerical estimation are not well defined in the q ∼ 0 region and propose a way to correct these analytical estimations. Furthermore, with this result we study the evolution of the scalar field finding total agreement with the conclusions of ref. [3] for q > 0.2, but a remarkable different result for small q. For this region, the initial conditions are very important. We find that q ∼ 0 gives the leading contribution for oscillating inflation and the dominant part in the amplification of the fluctuations. The letter is organized as follow; first we describe briefly the Damour-Mukhanov model. Then, we make some comments about the initial conditions for this phenomenon and later we propose an improved expression, valid for the leading region of q, which is our main contribution.