Dependence of density perturbations on the coupling constant in a simple model of inflation.

In the standard inflationary scenario with inflaton potential V (Φ) = M4 − 1 4λΦ 4, the resulting density perturbations δρ/ρ are proportional to λ1/2. Upper bounds on δρ/ρ require λ <∼ 10 −13. Ratra has shown that an alternative treatment of reheating results in δρ/ρ ∝ λ−1, so that an upper bound on δρ/ρ does not put an obvious upper bound on λ. We verify that δρ/ρ ∝ λ−1 is indeed a possibility, but show that λ <∼ 10 −13 is still required. (a) Electronic address (internet): [email protected]. (b) Electronic address (internet): [email protected]. (c) Electronic address (internet): [email protected]. The inflationary paradigm [1–4] explains many mysteries of large scale cosmology. It also provides a source of density fluctuations which act as the seeds for structure formation, and predicts that these fluctuations have a Harrison-Zel’dovich spectrum [5–8]. The main problem with the standard inflationary scenario is that it requires very small self-couplings of the inflaton field Φ in order to produce mass fluctuations with the correct amplitude of δρ/ρ ≃ 10−5 at horizon crossing. This is because δρ/ρ ∝ λ1/2, where λ is the quartic self-coupling of Φ. It turns out that δρ/ρ <∼ 10 −5 requires λ <∼ 10 −13. Many models have been constructed which attempt to make such small couplings arise naturally. However, Ratra argues that a very small coupling may not be necessary [9]. He finds that the dependence of δρ/ρ on λ is sensitively dependent on “reheating,” that is, on how the transition from the inflationary era to the radiation-dominated era is modelled. In the standard inflationary scenario, the reheating transition takes place in a few Hubble times. In Ratra’s alternative scenario, reheating is instantaneous (which means, in practice, much less than a Hubble time). In this case Ratra finds that δρ/ρ is proportional to λ−1, a dramatically different result. Since, as Ratra points out, the reheating process is quite complicated, involving nonequilibrium thermodynamics of a quantum field in curved space, we should be cautious about adopting a specific model of it unless we are convinced that its predictions are robust. It is therefore extremely important to check this point, and to see whether or not a small δρ/ρ can result from a coupling which is larger than λ ≃ 10−13. We have reanalyzed Ratra’s results for the simple potential V (Φ) = M4 − 4λΦ 4 , (1) where M is a constant, and Φ = 0 at the start of inflation. Of course, this potential is unbounded below and must be modified for Φ > ΦMAX, where ΦMAX = (4/λ) 1/4M and is defined via V (ΦMAX) = 0. This potential was originally intended to mock up a Coleman-Weinberg potential in a gauge theory (in which case λ ∼ g4, where g is the gauge coupling). This possibility was subsequently discarded (since λ ∼ g4 is much too large), but the prediction for δρ/ρ from the potential of Eq.(1) was thoroughly analyzed in both the standard scenario and in Ratra’s alternative scenario, and therefore provides a good test case. Ratra has also analyzed several other possible potentials, but we will not do so here. All of our results will apply strictly to the potential of Eq.(1); we will have nothing to say about Ratra’s other models, although it would be interesting to compare his results for an exponential potential with those of, for example, Ref. [10]. Ratra’s analysis includes a complete rederivation of the fluctuation amplitude and 2 spectrum, making use of gauge noninvariant variables followed by careful identification of the gauge variant modes. However, the final result can (necessarily) be derived using the more standard gauge invariant formalism of Bardeen [11]. In fact, we can simply use the final formula of Bardeen, Steinhardt, and Turner (BST) [8], without reference to its long derivation. Many other analyses have confirmed this formula, except for small differences in the overall normalization. These will not be relevant, however. The BST formula for δρ/ρ for a perturbation with wavenumber k which first crossed out of the horizon at time tc and then reentered during the matter dominated era is δρ ρ ≃ 1 5π H2 Φ̇(tc) . (2) Here H is the Hubble parameter during inflation, related to M via H = (8π/3)M/MPl, where MPl is the Planck mass. The field Φ(t) is treated as a classical, spatially uniform, background field; quantum fluctuations in Φ are what ultimately result in the density fluctuations of Eq.(2). Clearly, to compute δρ/ρ we need to compute Φ̇(tc). To do so, we use the equation of motion Φ̈ + 3HΦ̇− λΦ = 0 (3) which follows from the potential of Eq.(1). This equation is easy to solve in the slowrollover approximation, where we neglect Φ̈. When this approximation is valid we find