Bounding field excursions along null geodesics with applications to cosmology

Scalar fields in theories of gravity often inhabit a moduli space of vacua, and coherent spatial or temporal variations in their expectation values can produce measurable gravitational effects. Such variations are expected in contexts ranging from inflationary cosmology to the near-horizon regions of near-extremal black holes, where they can deflect light rays and shift horizons. This work derives a quantitative field excursion bound (FEB) on scalar variations along null geodesics, expressed in terms of the expansion parameter. The bound follows from the Raychaudhuri equation, assuming that all other fields satisfy the null energy condition (NEC). It is saturated in certain spacetimes containing a timelike naked singularity. A possible generalization to semiclassical spacetimes that violate the NEC, but satisfy a strengthened version of the quantum focusing condition (QFC), is proposed. In cosmology, the FEB constrains the extent of large field excursions to be linearly bounded by the number of e-folds, independent of the inflationary model. This has notable implications for anthropic scenarios, where large excursions are often invoked to access favorable vacua.