On the Field Excursion Bound

TL;DR

The work presents an alternate Friedmann-equation derivation of the Flat Excursion Bound (FLEB) for scalar field excursions in spatially flat FRW cosmologies and shows saturation occurs only when $\ddot a = 0$ and the stress-energy saturates the NEC. It analyzes resilience to slow-roll quantum corrections, showing they are suppressed by $H/M_{Pl}$ and $\sqrt{\varepsilon}$, and extends the bound to higher-derivative kinetic terms, revealing that violations require a superluminal speed of sound ($c_s>1$). The study also examines spacetime curvature effects, proving the bound holds in open/unbounded cases while allowing potential violations in closed universes, and discusses speculative links to swampland bounds on vacua, suggesting a rich interplay between inflationary dynamics, high-energy consistency, and the landscape of vacua. These results reinforce the FLEB as a robust constraint in realistic inflationary scenarios and delineate clear conditions under which higher-derivative corrections or curvature may alter the bound. The findings have implications for the swampland program and the counting of vacua in field space, highlighting a fundamental tension between large field excursions and quantum gravity constraints.

Abstract

In a recent work, Herderschee and Wall (HW) proved a bound on scalar field excursions in spatially flat FRW cosmologies. In this note, we give an alternate proof of their bound using the Friedmann equations, and we prove that it can be saturated only in universes with vanishing acceleration, $\ddot a =0$. We argue that in a realistic (eternal) inflation scenario, the bound is robust against quantum corrections and spacetime curvature, and it can be violated by higher-derivative corrections only at the expense of a superluminal speed of sound. We further speculate on possible connections between the swampland program and the vacuum estimates given in the work of HW.