Cycles Without End: An Ekpyrotic Braneworld Bounce with a Kinetically Coupled Entropic Mechanism

Cyclic cosmologies replace inflation's single creation event with an eternal sequence of smooth turnarounds, but three problems have historically stood in their way: singular bounces, the blue tilt generic to single-field ekpyrotic contraction, and Tolman's observation that entropy production should make successive cycles longer and larger without bound. We resolve all three within one explicit model. A Randall-Sundrum brane with a timelike extra dimension regularizes the bounce without violating the null energy condition, and a single scalar field's potential carries the universe from a thawing dark-energy plateau through a sign-changing turnaround into a BKL-safe ekpyrotic contraction and back. We show that no correction internal to the brane's Friedmann equation, and no straight two-field trajectory, can rescue the spectral index from the generic ekpyrotic result $n_s\approx3$. We then build the minimal completion that can: a kinetically coupled entropy field, following the general mechanism of Ijjas, Lehners, and Steinhardt, whose coupling steepness has a closed-form, exact-scale-invariance value for this model's ekpyrotic potential, with only a percent-level offset needed to match the measured tilt $n_s=0.9649$. This fixes the local non-Gaussianity ($f_{\rm NL}\approx-0.02$) and tensor-to-scalar ratio ($r\approx7.6\times10^{-26}$). Matching the observed amplitude fixes the conversion coupling and brane tension to unremarkable values ($λ\approx5.7\times10^{-3}$, $ρ_c^{1/4}=10^{13}\,$GeV). Finally, because the brane construction is spatially flat, Tolman's entropy argument does not bound the number of cycles here: we derive $N_{\rm DE}\sim70$-80 e-folds of accelerated expansion needed each cycle to dilute the entropy produced back to a negligible density before the next contraction.