Bounding field excursions along null geodesics with applications to cosmology

TL;DR

The paper establishes a field excursion bound (FEB) for scalar fields along null geodesics by combining the Raychaudhuri equation with the null energy condition, linking coherent moduli variations to gravitational focusing. It demonstrates the FEB in a naked timelike singularity spacetime and then extends the bound to inflationary contexts, proving that large field excursions along light rays are linearly limited by the number of e-folds $\Delta N$, independently of the inflationary model. A potential generalization to semiclassical spacetimes violating NEC is proposed via a strengthened quantum focusing condition (SQFC), yielding a quantum FEB that depends on the quantum expansion $\Theta$ and the generalized entropy. The work discusses phenomenological implications for inflationary landscapes and anthropic scenarios, showing that achieving large excursions would require very long inflation or NEC-violating quantum effects, and suggests directions for future exploration of the SQFC and higher-derivative corrections.

Abstract

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.

Figures (3)

• Figure 1: Schematic depiction of the initial and final time slices, denoted by $\Sigma_{i}$ and $\Sigma_{f}$, respectively. The cone corresponds to the past lightcone of $p$, which is denoted as $P$.
• Figure 2: An illustration of the spacelike surfaces $B$ and $B'$ bounded by spacelike surfaces $\sigma$ and $\sigma'$ related by evolution along the $k_{\mu}$ null vector.
• Figure 3: An illustration of how the derivative measures how the entropy changes relative to an infinitesimal change in $\sigma$ localized in an area $\mathcal{A}$ around $y_{1,\perp}$.