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arXiv · 2605.12579

Phase-resolved field-space distance criteria in ekpyrotic, bouncing and cyclic cosmologies

Marcin Postolak
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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 non-inflationary 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 BKL (Belinski-Khalatnikov-Lifshitz) 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 parameterized 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 $\epsilon_{\rm ek}$ that depends on the remaining distance available after conversion and the cosmological bounce.

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Marcin Postolak. 2026-05-12. Phase-resolved field-space distance criteria in ekpyrotic, bouncing and cyclic cosmologies. https://doi.org/10.1103/qjfz-drq6

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