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

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 $V(\varphi )=N{M}^{4}w(\frac{\varphi }{\sqrt{N}{M}_{\mathrm{Pl}}})$, where $\varphi$ is the inflaton field, $M$ is the inflation energy scale, and $N\sim 50$ 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 $\varphi =\sqrt{N}{M}_{\mathrm{Pl}}\chi$. The dimensionless function $w(\chi )$ and field $\chi$ are generically $\mathcal{O}(1)$. The WMAP value for the amplitude of scalar adiabatic fluctuations $|{\Delta }_{kad}^{(S)}{|}^2$ fixes the inflation scale $M\sim 0.77\times {10}^{16}$. 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 $M/M_{\mathrm{Pl}}$. For example, the quartic coupling $\lambda \sim \frac{1}{N}(\frac{M}{M_{\mathrm{Pl}}}{)}^4$. 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 $\varphi /M_{\mathrm{Pl}}$. The effective field theory is valid for $V(\varphi )\ll M_{\mathrm{Pl}}^4$ for general inflaton potentials allowing amplitudes of the inflaton field $\varphi$ well beyond $M_{\mathrm{Pl}}$. Hence bounds on $r$ based on the value of $\varphi /M_{\mathrm{Pl}}$ 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.