Kinetic Initial Conditions for Inflation

Abstract

We consider the classical evolution of the inflaton field $\phi(t)$ 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, $\dot{\phi}^2 \gg V(\phi)$. In this kinetically-dominated regime, the dynamical equations admit simple analytic solutions for $\phi(t)$ and $H(t)$, which are independent of the form of $V(\phi)$. 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 $\phi(t)$ 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 $|\phi(t)|$ as well as their time derivatives decrease during kinetic dominance until $\dot{\phi}^2\sim V(\phi)$, 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.