Phase-resolved field-space distance criteria in ekpyrotic, bouncing, and cyclic cosmologies
Phys. Rev. D 114, 044016 – Published 6 August, 2026
DOI: https://doi.org/10.1103/qjfz-drq6
Abstract
The inflationary Lyth bound relates the primordial tensor amplitude to the inflaton field excursion. In ekpyrotic, bouncing, and cyclic cosmologies, there is no analogous universal tensor-to-field-distance relation because scalar and tensor perturbations depend on entropy conversion, matching through the bounce, and the specific mechanism that violates or evades the null energy condition. Here, we propose a phase-resolved criterion for the accumulated scalar field trajectory length in a noninflationary smoothing history. The quantity constrained in this study is not, in general, the geodesic distance between the initial and final field-space points. Rather, it is the invariant path length accumulated along the actual cosmological trajectory. Therefore, the resulting inequalities should be understood as sufficient, conservative trajectory length criteria for small-field control, not as model-independent necessary exclusions based on geodesic distance swampland reasoning. We also impose Belinski-Khalatnikov-Lifshitz (BKL) anisotropy suppression as an additional constraint on the ekpyrotic phase. In the canonical phase of the ekpyrotic contraction, we recover the known small-field scaling and embed it in a total trajectory length budget inequality. We impose three requirements: a BKL anisotropy suppression that is parametrized separately, a phenomenological cutoff-corrected distance budget inspired by tower of states logic, and observational conversion windows from residual isocurvature and non-Gaussianity. Furthermore, we propose a new master formula that provides a conditional lower limit on the value of the parameter that depends on the remaining distance available after conversion and the cosmological bounce. We also derive a curvature constraint for scale-invariant entropy perturbations in curved field space which shows that the small total distance and the observed red tilt seem to indicate ultrafast-roll ekpyrosis, sharp turns, short or strongly modified bounces, and/or significant negative sectional curvature of the scalar manifold. Finally, we demonstrate methods for testing the distance budget against observational data. Examples include running, primordial non-Gaussianity, isocurvature, a stochastic gravitational waves background, and late-time dark energy distance constraints. The observational discussion is organized into explicit diagnostic maps, which are used to identify and analyze specific features and patterns in the data. We also apply the formalism to exponential and cyclic ekpyrotic scalar field potentials, providing a mechanism-resolved interpretation of the bounce or crossover contribution for several specific realizations.
