Large-$η$ Constant-Roll Inflation Is Never An Attractor

Abstract: Slow roll solutions to inflationary potentials have been widely believed to be the only universal attractor. Over the last few years there has been growing interest in a new class of inflationary models known as Constant-Roll Inflation. Constant roll solutions are a generalization of "ultra-slow roll" dynamics, where the first slow roll parameter is small, but the second slow roll parameter $\eta$ is larger than unity. In Ultra-slow Roll Inflation, the large-$\eta$ solution is a dynamical transient, relaxing exponentially to the attractor de Sitter solution. In the constant roll generalization, papers have concluded that Constant-Roll Inflation represents a new class of non-slow roll attractor solutions. In this paper we show that these attractor solutions are actually the usual slow roll attractor, disguised by a parameter duality, and that the large-$\eta$ solutions, as in the case of ultra-slow roll, represent a dynamical transient.