The Exponent Race in a Contracting Universe: For Which Equations of State Does Matter Catch the Torsion Term?

The critical equation of state is w = 1.0: at exactly stiff matter the torsion-to-matter ratio remains at its normalised value 1.0e-06 at every scale factor, as the script verifies. Below the critical value, crossing is guaranteed and its location is computed. For dust, equality is reached at a scale factor of 1.000e-02 relative to the reference epoch, by which point the matter density has grown by a factor of ten to the power 6.0. For radiation the crossing sits at 1.000e-03. As w approaches the critical value the crossing retreats sharply: at w = 0.99 it lies at 1.000e-200, with a matter density gain of ten to the power 1194.0, reflecting the near-degeneracy of the two exponents. The divergence of the crossing scale as w tends to 1 is the quantitative form of the statement that stiff matter and torsion grow together. The study is a computation from stated assumptions, performed by the accompanying script; it is simulated, not measured.