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

12 pages, Lecture given at `The Density Perturbation in the Universe', Demokritos Center, Athens, Grece, June 2004We clarify classical inflaton models by considering them as effective field theories `a la Ginzburg-Landau. In this approach, the WMAP statement excluding the pure phi^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(phi)=m^2 M_{Pl}^2 v(phi/M_{Pl}), where all coefficients in v(varphi) are of order one.If such potential corresponds to supersymmetry breaking, the susy scale is sqrt{m M_{Pl}}~10^{16}GeV which coincides with the GUT scale. For red tilted spectrum new inflation turns to be favoured while 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 r (tensor/scalar ratio) and the scalar spectral index ns: 10^6 m/M_{Pl} = 127 sqrt{r|1-n_s|} where the numerical coefficient follows from the WMAP amplitude. We 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 is measured by a small parameter Delta.Delta regulates the infrared divergences too.In slow roll inflation, Delta is a simple function of the slow roll parameters. The quantum fluctuations can selfdecay in the inflationary expansion through processes forbidden in Minkowski. We compute the self-decay of the inflaton quantum fluctuations during slow roll:superhorizon fluctuations decay as eta^{Gamma} in conformal time where the new scaling dimension Gamma is expressed in terms of the amplitude, n_s and r. we discuss the implications for scalar and tensor perturbations as well as for non gaussianities