The Classical and Quantum Inflaton: the Precise Inflationary Potential and Quantum Inflaton Decay after WMAP

Abstract

We clarify classical inflaton models by considering them as effective field theories à la Ginzburg-Landau. In this approach, the WMAP statement excluding the pure ϕ 4 potential implies the presence of an inflaton mass term at the scale m ∼ 10 13 GeV . Chaotic, new and hybrid inflation models are studied in an unified manner. In all cases the inflaton potential takes the form V ( ϕ ) = m 2 M P l 2 v ( ϕ M P l ) , where all coefficients in the polynomial v ( ϕ ) are of order ( m / M P l ) 0 . If such potential corresponds to supersymmetry breaking, the corresponding susy breaking scale is m M P l ∼ 10 16 GeV which turns out to coincide with the grand unification (GUT) scale. The inflaton mass is therefore given by a see-saw formula m ∼ M GUT 2 / M P l . For red tilted spectrum, the potential which fits the best the present data ( | 1 − n s | ≲ 0.1 , r ≲ 0.1 ) and which best prepares the way for the forthcoming data is a trinomial polynomial with negative quadratic term (new inflation). For blue tilted spectrum, hybrid inflation turns to be the best choice. In both cases, we find an analytic formula relating the inflaton mass with the ratio r of tensor to scalar perturbations and the spectral index n s of scalar perturbations: 10 6 m M P l = 127 r | 1 − n s | where the numerical coefficient is fixed by the WMAP amplitude of adiabatic perturbations. Implications for string theory are discussed. We then review quantum phenomena during inflation which contribute to relevant observables in the CMB anisotropies and polarization and we focus on inflaton decay . The deviation from the scale invariant power spectrum measured by a small parameter Δ turns to be crucial, Δ regulates the infrared too. In slow roll inflation, Δ is a simple function of the slow roll parameters. We find that quantum fluctuations can self-decay as a consequence of the inflationary expansion through processes which are forbidden in Minkowski space-time. We compute the self-decay of the inflaton quantum fluctuations during slow roll inflation: for wavelengths deep inside the Hubble radius the decay is enhanced by the emission of ultrasoft collinear quanta, i.e. bremsstrahlung radiation of superhorizon quanta which becomes the leading decay channel for physical wavelengths H ≪ k p h ( η ) ≪ H / ( η V − ε V ) . The decay of short wavelength fluctuations hastens as the physical wave vector approaches the horizon. Superhorizon fluctuations decay with a power law η Γ in conformal time where Γ is expressed in terms of the amplitude of curvature perturbations Δ R 2 , the scalar spectral index n s , the tensor to scalar ratio r and slow roll parameters. The behavior of the growing mode η η V − ε V + Γ / η features a new scaling dimension Γ. We discuss the implications of these results for scalar and tensor perturbations as well as for non-gaussianities in the power spectrum. The recent WMAP data suggests Γ ≳ 3.6 × 10 −9 .