Large-$\eta$ Constant-Roll Inflation Is Never An Attractor

TL;DR

This paper demonstrates that large-$$ constant-roll inflation solutions are transient and not true attractors, revealing they are essentially the same as the slow roll attractor under a parameter duality.

Contribution

The paper clarifies that large-$$ constant-roll solutions are transient and equivalent to slow roll attractors, challenging previous claims of them being a new attractor class.

Findings

Large-$$ solutions are transient, not attractors.

Large-$$ solutions are equivalent to slow roll attractors via duality.

Ultra-slow roll solutions also relax to the slow roll 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, recent 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.