Tensor-to-Scalar Ratio Through a Bounce: How Many E-Folds of Post-Bounce Expansion the BICEP Bound Requires

This paper computes the tensor-to-scalar ratio that survives a torsion-regulated bounce as a function of the post-bounce expansion history, and finds the minimum expansion the current gravitational-wave bound demands. Tensor modes entering the bounce with a ratio r = 0.28 at the pivot scale are diluted as r = 0.28 exp(-2.4 N), where N counts the e-folds over which scalar growth continues after the tensor amplitude has frozen. At a fiducial N = 1.35 the surviving ratio is r = 0.011, a factor 3.3 below the BICEP/Keck and Planck bound of 0.036. Satisfying that bound at all requires N of at least 0.855; with no dilution the entering value of 0.28 is excluded outright, and at N = 2 the ratio falls to 0.002304. The bounce therefore does not automatically solve the tensor problem that large-field inflation faces: it solves it only if the post-bounce phase provides roughly one e-fold or more of differential scalar growth, a concrete condition on the expansion history that can be checked in any specific model. The study is computed from stated assumptions, not measured.