Clarifying inflation models: Slow roll as an expansion in1/Nefolds

Abstract

Slow-roll inflation is studied as an effective field theory. We find that the form of the inflaton potential consistent with Wilkinson Microwave Anisotropy Probe (WMAP) data and slow roll is V({phi})=NM{sup 4}w(({phi}/{radical}(N)M{sub Pl})), where {phi} is the inflaton field, M is the inflation energy scale, and N{approx}50 is the number of e-folds since the cosmologically relevant modes crossed the Hubble radius until the end of inflation. The inflaton field scales as {phi}={radical}(N)M{sub Pl}{chi}. The dimensionless function w({chi}) and field {chi} are generically O(1). The WMAP value for the amplitude of scalar adiabatic fluctuations {delta}{sub kad}{sup (S)2} fixes the inflation scale M{approx}0.77x10{sup 16}. This form of the potential makes manifest that the slow-roll expansion is an expansion in 1/N. A Ginzburg-Landau realization of the slow-roll inflaton potential reveals that the Hubble parameter, inflaton mass and nonlinear couplings are of the seesaw form in terms of the small ratio M/M{sub Pl}. For example, the quartic coupling {lambda}{approx}(1/N)((M/M{sub Pl})){sup 4}. The smallness of the nonlinear couplings is not a result of fine-tuning but a natural consequence of the validity of the effective field theory and slow-roll approximation. We clarify Lyth's bound relating the tensor/scalar ratio and the value of {phi}/M{sub Pl}. The effectivemore » field theory is valid for V({phi})<<M{sub Pl}{sup 4} for general inflaton potentials allowing amplitudes of the inflaton field {phi} well beyond M{sub Pl}. Hence bounds on r based on the value of {phi}/M{sub Pl} are overly restrictive. Our observations lead us to suggest that slow-roll, single field inflation may well be described by an almost critical theory, near an infrared stable Gaussian fixed point.« less