The Matter-Radiation Equality Epoch with a Stiff Spin Component: A Redshift Bound

Matter and radiation contributed equally to the expansion at a redshift near 3400, an epoch that sets the turnover of the power spectrum and the early integrated Sachs-Wolfe contribution to the microwave background. At a bounce temperature of 1 MeV the spin fraction at equality is 6.40e-13, utterly negligible; even at a bounce temperature of 0.01 MeV, already excluded by nucleosynthesis, the fraction is only 6.40e-09 and the equality redshift moves to 3400.0. On every allowed branch the equality epoch is standard to high precision, and the observed equality scale can be used in this framework exactly as in the standard picture, a conclusion this paper establishes rather than presumes. Computed from stated assumptions, not measured.