Bounds on field range for slowly varying positive potentials

A bstract In the context of quantum gravitational systems, we place bounds on regions in field space with slowly varying positive potentials. Using the fact that V V Λ s 2, where Λ s (ϕ) is the species scale, and the emergent string conjecture, we show this places a bound on the maximum diameter of such regions in field space: ∆ ϕ ≤ a log (1/ V) + b in Planck units, where a ≤ (d-1) (d-2) d − 1 d − 2, and b is an 𝒪 (1) number and expected to be negative. The coefficient of the logarithmic term has previously been derived using TCC, providing further confirmation. For type II string flux compactifications on Calabi-Yau threefolds, using the recent results on the moduli dependence of the species scale, we can check the above relation and determine the constant b, which we verify is 𝒪 (1) and negative in all the examples we studied.