Fluctuating geometries, q-observables, and infrared growth in inflationary spacetimes

TL;DR

The paper tackles the problem of infrared growth in inflationary spacetimes by constructing gauge-invariant, nonlocal q-observables based on geometric quantities, notably geodesic lengths and cycle-lengths on a torus. Using a perturbative quantum gravity framework with slow-roll dynamics, it analyzes how tensor and scalar fluctuations source large fluctuations in these geometric observables, revealing that certain line- and cycle-length measures become non-perturbative after a modest number of e-folds, signaling a breakdown of perturbation theory for geometry. Through detailed treatment on a toroidal spatial manifold and in satellite- and cycle-based setups, the work demonstrates that IR fluctuations can significantly distort observed spatial distances, consistent with de Sitter instability and self-reproduction, and discusses gauge issues that complicate naive geometric probes. It concludes that while some IR-safe observables may exist for limited regions or late times, many straightforward geometric q-observables fail at long times or large separations, motivating the search for alternative gauge-invariant diagnostics such as distance measures in the space of geometries. The study thus provides both a rigorous framework for gauge-invariant geometric probes in inflation and a cautionary view on using simple geodesic lengths to characterize the IR structure of expanding universes.

Abstract

Infrared growth of geometrical fluctuations in inflationary spacetimes is investigated. The problem of gauge-invariant characterization of growth of perturbations, which is of interest also in other spacetimes such as black holes, is addressed by studying evolution of the lengths of curves in the geometry. These may either connect freely falling "satellites," or wrap non-trivial cycles of geometries like the torus, and are also used in diffeomorphism- invariant constructions of two-point functions of field operators. For spacelike separations significantly exceeding the Hubble scale, no spacetime geodesic connects two events, but one may find geodesics constrained to lie within constant-time spatial slices. In inflationary geometries, metric perturbations produce significant and growing corrections to the lengths of such geodesics, as we show in both quantization on an inflating torus and in standard slow-roll inflation. These become large, signaling breakdown of a perturbative description of the geometry via such observables, and consistent with perturbative instability of de Sitter space. In particular, we show that the geodesic distance on constant time slices during inflation becomes non-perturbative a few e-folds after a given scale has left the horizon, by distances \sim 1/H^3 \sim RS, obstructing use of such geodesics in constructing IR-safe observables based on the spatial geometry. We briefly discuss other possible measures of such geometrical fluctuations.

Figures (2)

• Figure 1: Metric perturbations can lead to significantly different geometries, as illustrated for the sphere.
• Figure 2: Pictured is a torus, made "lumpy" by metric perturbations, along with a representative cycle along which distance ${\cal S}$ can be calculated.