Thermodynamic approach to warm inflation

We study the thermodynamic behavior of a decaying scalar field coupled to a relativistic simple fluid. It is shown that if the decay products are represented by a thermalized bath, its temperature evolution law requires naturally a new phenomenological coupling term. This “energy loss” term is the product between the enthalpy density of the thermalized bath and the decay width of the scalar field. We also argue that if the field φ decays “adiabatically” some thermodynamic properties of the fluid are preserved. In particular, for a field decaying into photons, the radiation entropy production rate is independent of the specific scalar field potential V (φ), and the energy density ρ and average number density of photons n scale as ρ ∼ T 4 and n ∼ T 3. To illustrate these results, a new warm inflationary scenario with no slow roll is proposed. PACS number(s): 98.80.Cq, 05.40.+j Typeset using REVTEX Electronic address: [email protected] Electronic address: [email protected] 1 In the new inflationary scenario accelerated expansion and reheating are separated into two distinguished periods. The first one is an exponential growth of the scale factor with the Universe evolving to a supercooled state. Due to this adiabatic expansion the temperature of the Universe decreases nearly 10 orders of magnitude [1]. At the end of this supercooling process the Universe is reheated. The field rapidly oscillates about the global minimum of its potential, and the energy density of the inflaton field is completely or almost completely converted into radiation in less than one expansion Hubble time. As a matter of fact, either on its early [2] or modern version based on parametric resonance (sometimes called preheating) [3,4], the reheating is a very fast and extremely nonadiabatic mechanism. In principle, if a sustained radiation component during inflation is allowed, the supercooling and subsequent reheating could be supressed or at least weakened by many orders of magnitude [5]. The first field motivated scenario based on this idea is the isothermal or warm inflationary picture as proposed by Berera [6]. Like in new inflation, the warm picture starts from a high temperature phase transition with the universe evolving through a de Sitter inflationary period dominated by the scalar field potential. However, in the course of the expansion, energy is continuously drained from the field φ to the thermal bath. The whole process is described by an “energy loss” term, Γφ̇. At the level of the scalar field equation of motion, it contributes like an additional viscosity Γφ̇, which may dominate the term 3Hφ̇ corresponding to the redshift of φ̇ (the momentum of field) by the expansion. In the warm picture, the persistent thermal contact during inflation implies that the scalar field evolves in a sort of over damped regime. As a result, the Universe approaches a state where the dilution of the thermalized bath (due to expansion) may continuously be compensated by the production of particles from the decaying scalar field, thereby guaranteeing the constancy of the temperature. If this scenario works during an interval of time long enough, thermal fluctuations may also produce the primordial spectrum of density perturbations [7,8](see also [9] for an updated and more detailed analysis). More recently, this nice picture has severely been criticized in terms of its physical plausibility by Yokoyama and Linde [11]. The basic argument is that Γφ̇, is not sufficiently 2 strong in the regime where it should describe warm inflation. This coupling seems to be only a small contribution in a sub-leading thermal correction, and as such, it is not expected to play a prominent role in the inflationary process [11](see, however, Refs. [9] and [10]). On the other hand, as discussed in an extended framework [12], the basic idea of warm inflation is so attractive that it cannot be discarded without a more detailed investigation or even further attempts to solve the above pointed out difficulties. In particular, it is not so neat that an “energy loss” term like Γφ̇ is a realistic approximation for describing the energy dissipated by the φ field to a thermalized bath. In what follows we apply basic thermodynamics arguments for studying the decay of the scalar field coupled to a thermalized bath which is represented by a relativistic simple fluid. We recall that thermodynamics has often been used when the underlying microphysics of a given phenomenon has not been completely clarified. The basic reason is that it yields relations among the macroscopic quantities whose validity is independent of the microphysics on which they ultimately depend. Therefore, it is interesting to consider some thermodynamic criteria on the question related to the plausibility of warm inflation. As we shall see, this macroscopic treatment clearly suggests a new form to the coupling term, which depends explicitly on the created component through its enthalpy density. It has a quite simple form, namely, Γφ(ρ+p), where Γφ is the decay width of the scalar field, and may alter significantly the studies of the reheating period, as well as the original and extended warm inflationary pictures [13]. We will limit our analysis to homogeneous and isotropic FRW flat universes. Following standard lines, the energy content is a mixture of a coupled scalar field plus a relativistic simple fluid representing the thermalized bath. The total energy stress of this self-gravitating mixture obeys the Einstein field equations (EFE). In addition, since the scalar field works like a source of particles, the particle flux of the relativistic bath (N = nu) must satisfy a balance equation of the type N ;μ = Ψ. The basic equations are: 8π mpl ( φ̇ 2 + V (φ) + ρ )