On the Field Excursion Bound

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.