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

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 ${\epsilon }_{\mathrm{e}\mathrm{k}}$ that depends on the remaining distance available after conversion and the cosmological bounce.