An Inverse Lyth Bound for Non-Attractor Inflation

Abstract

I present a novel physics result generated entirely autonomously by ChatGPT 5.6 Sol. The Lyth bound relates an observable primordial tensor amplitude to a lower limit on the field excursion during single-field slow-roll inflation. We show that non-attractor evolution obeys a complementary upper bound. For a canonically normalized scalar field, the nonconstant superhorizon curvature mode is anti-damped whenever the second Hubble-flow parameter satisfies $\epsilon_2\equiv d\ln\epsilon/dN<-3$. The same condition requires the inflaton kinetic energy to decrease sufficiently rapidly that the field-space distance traversed during a non-attractor interval obeys $\Delta\phi_{\rm NA} / M_{\text{P}} < \left(2\sqrt{2\epsilon_{\rm in}}/3\right) \left(1-e^{-3\Delta N/2}\right) < \left(2\sqrt{2\epsilon_{\rm in}}/3\right)$.The result is independent of the potential and does not require the slow-roll approximation. If $\epsilon_2=-p<-3$ is constant, the stronger bound $\Delta\phi/M_{\text{P}}<2\sqrt{2\epsilon_{\rm in}}/p$ is saturated asymptotically. In ultra-slow-roll inflation, $p=6$, giving $\Delta\phi<\sqrt{2\epsilon_{\rm in}}M_{\text{P}}/3$. We derive an exact relation between field excursion and amplification of the velocity of the nonconstant curvature mode, and give the corresponding scalar-power relation in the quasi-de Sitter limit. In contrast with the usual Lyth relation, arbitrarily large non-attractor amplification approaches a finite field distance. This provides a model-independent field-range constraint on canonical single-field mechanisms for amplifying primordial fluctuations.