The Redshift of Recombination Under a Bounce-Modified Expansion History: A Sub-Percent Calculation

Recombination, the formation of neutral hydrogen at a redshift near 1090, is located by the microwave background with remarkable precision: the standard value carries an uncertainty of about a quarter of a unit in redshift. With the stated sensitivity, a one percent residual change in the expansion rate would move the last-scattering redshift to 1093.2, a shift of 3.3 units, more than ten times the observational uncertainty of 0.25. Conversely, the spin term predicted at recombination for bounce temperatures above the nucleosynthesis bound corresponds to a shift orders of magnitude below the uncertainty. The instrument is sharp; the predicted signal on the allowed branch is far beneath it; and stating both numbers together is the correct, complete result. Computed from stated assumptions, not measured.