Kinetic Initial Conditions for Inflation

Abstract

We consider the classical evolution of the inflaton field $\varphi (t)$ and the Hubble parameter $H(t)$ in homogeneous and isotropic single-field inflation models. Under an extremely broad assumption, we show that the universe generically emerges from an initial singularity in a non-inflating state where the kinetic energy of the inflaton dominates its potential energy, ${\dot{\varphi }}^2\gg V(\varphi )$. In this kinetically-dominated regime, the dynamical equations admit simple analytic solutions for $\varphi (t)$ and $H(t)$, which are independent of the form of $V(\varphi )$. In such models, these analytic solutions thus provide a simple way of setting the initial conditions from which to start the (usually numerical) integration of the coupled equations of motion for $\varphi (t)$ and $H(t)$. We illustrate this procedure by applying it to spatially-flat models with polynomial and exponential potentials, and determine the background evolution in each case; generically $H(t)$ and $|\varphi (t)|$ as well as their time derivatives decrease during kinetic dominance until ${\dot{\varphi }}^2\sim V(\varphi )$, marking the onset of a brief period of fast-roll inflation prior to a slow roll phase. We also calculate the approximate spectrum of scalar perturbations produced in each model and show that it exhibits a generic damping of power on large scales. This may be relevant to the apparent low-$\ell$ falloff in the CMB power spectrum.

Accepted by Physical Review D, 20 pages, 14 figures. v3 contains a significant correction to the proof, submitted as an errata to PRD

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