On Super-Planckian Fields at Sub-Planckian Energies

TL;DR

This work analyzes whether a light scalar coupled to gravity can realize super-Planckian excursions without triggering gravitational collapse. Using a Newtonian bound, it shows exterior field variations are limited and interior variations grow only logarithmically with source size, implying exponentially large experiments are needed for large excursions; this bound extends to higher dimensions and to field-space distances. In General Relativity, there is an absolute exterior bound on the scalar excursion, $(\Delta\varphi)_{\rm out}$, with $G\,(\Delta\varphi)_{\rm out}^2\le \xi/(4\pi)\approx 0.1398$, though interior sourcing remains delicate and potentially constrained by energy conditions or quantum corrections. Together, these results indicate that super-Planckian scalar fields are largely unobservable in asymptotically flat space under conventional assumptions, highlighting gravity-induced constraints on such scalars and informing swampland considerations.

Abstract

For a light scalar coupled to gravity, I study the gravitational backreaction associated with large field variations. I show a generic obstruction in sourcing a super-Planckian scalar profile without making the whole experiment collapse into a black hole. In empty space the scalar variation obeys an absolute bound, of order of the Planck scale. A Newtonian analysis suggests that inside its sources the scalar can undergo arbitrarily large variations without causing large gravitational backreactions. However the maximum attainable Δφincreases only logarithmically with the size of the source. The bound straightforwardly generalizes to any number of dimensions, and to moduli space-like cases, where it applies to the invariant length in field space as measured by the kinetic metric.