A Class of Exact Single-Field Inflationary Solutions beyond Slow Roll

Abstract

We construct exact solutions for single-field inflaton dynamics without invoking the slow-roll approximation. A suitable change of variables reduces the background equation to an Abel equation of the first kind. Although a generic Abel equation is not analytically solvable, we identify a class of inflaton potentials for which the transformed equation admits exact solutions. The resulting framework contains constant-roll inflation as a special case and also accommodates solutions with a constant second Hubble-flow parameter. We analyze the linear local attractor behavior and superhorizon evolution of these rolling backgrounds. Using the public joint CMB likelihood contours in the $(n_s,r)$ plane, we identify compatible parameter regions and show that one rolling branch can also yield $50\leq N_*<60$. Direct numerical evolution of the scalar and tensor modes at representative points validates the local-index predictions to better than $7\times10^{-4}$ in $n_s$ and $2\times10^{-6}$ in $r$. The exact family extends beyond slow roll, although the observationally selected regions displayed here lie close to the slow-roll regime.