The Contraction Budget for the Horizon Problem: E-Folds of Contraction Required in Matter and Stiff Bounces Compared

A matter-like contraction must run for 128.9 e-folds of scale-factor decrease to shrink the comoving Hubble radius by the target factor of 10^28, during which the density grows by a factor of ten to the power 168.0. A radiation-like contraction needs 64.5 e-folds. A stiff contraction needs only 32.2 e-folds, with a density growth of ten to the power 84.0. The stiff budget is smaller in e-folds by the ratio of the shrink rates, a factor of two against matter in the exponent per e-fold, and the advantage compounds in the density growth because the stiff fluid's density also climbs faster per e-fold. Even the cheapest budget, however, demands a contraction lasting tens of e-folds: there is no version of this solution in which a brief contracting episode suffices. The study is a computation from stated assumptions, performed by the accompanying script; it is simulated, not measured.