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Clarifying inflation models: Slow roll as an expansion in 1 / N efolds

2006; American Physical Society; Volume: 73; Issue: 2 Linguagem: Inglês

10.1103/physrevd.73.023008

1550-7998

D. Boyanovsky, H. J. de Vega, N. Sánchez,

Galaxies: Formation, Evolution, Phenomena

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