A critical value of the inflationary tensor-to-scalar ratio from inhomogeneous inflation

Abstract

We show that, for a given fixed value of the number of e-folds of the homogeneous solution, inflation succeeds with order unity inhomogeneities in the initial conditions above a characteristic value of the tensor-to-scalar ratio $r$. In practice, we work with an $\alpha$-attractor $T$-model and vary its characteristic scale $\mu$, keeping the initial inhomogeneities in both gradient and kinetic fields of order unity of the inflationary energy scale. Under these conditions, and assuming 100 e-folds for the homogeneous solution, the requirement for 60 e-folds of inflation occurs at a critical characteristic scale $\mu_{crit} \approx 0.02m_{P}$, corresponding to an $r_{crit} \approx 10^{-6}$. Since increasing the amplitude of the inhomogeneities will make inflation less robust and hence require a higher characteristic scale in order for inflation to succeed, for a given number of e-folds achieved by the homogeneous solution $r_{crit}$ is a lower bound.