Reionization Optical Depth

The CMB optical depth to Thomson scattering τ encodes the integrated electron column density between the observer and the last scattering surface, and is therefore sensitive to the history of reionization. Planck's measurement of τ = 0.054 ± 0.007 implies reionization was largely complete by z ~ 7.7. This relatively low optical depth is in mild tension with earlier WMAP measurements that suggested higher τ, and it sets tight constraints on when and how rapidly the universe was reionized. Models that invoke a rapid reionization driven purely by post-collision star-forming galaxies at z ~ 6–9 can reproduce the Planck τ, but they predict a Lyman-alpha emitter visibility function and a 21 cm signal that are in tension with some observations. The precise value of τ and its implications for the reionization epoch therefore sit at the intersection of several independent constraints that do not fully converge.

Successive Collision Theory predicts a reionization optical depth consistent with the Planck measurement while simultaneously explaining the extended reionization history suggested by 21 cm and Lyman-alpha observations. In SCT, reionization had an early onset driven by pre-existing ionizing populations from the collision debris, but proceeded slowly at first because the pre-existing sources were distributed and their ionizing flux built up gradually as the debris field evolved. This extended reionization history — beginning at z ~ 15 and completing at z ~ 6 — produces a total integrated optical depth consistent with τ ~ 0.054 while distributing the electron scattering events over a longer time window than models with rapid late reionization. The slow early phase contributes relatively little to τ because the electron fraction is low, while the rapid completion phase at z ~ 7–8 contributes the bulk of the optical depth, matching the Planck constraint.

The hereditary time transmission mechanism introduces a small correction to the optical depth calculation in SCT. The proper time experienced by CMB photons as they propagate through the nested frame hierarchy differs fractionally from the coordinate time of the FLRW background, modifying the effective path length through the ionized intergalactic medium and thus the accumulated optical depth. This correction is at the level of a fraction of a percent — well below the current measurement precision — but it constitutes a unique SCT prediction: the effective τ inferred from CMB polarization (which samples photon scattering in proper time) should differ by a tiny but systematic amount from the τ inferred from the kinematic Sunyaev-Zeldovich effect (which samples momentum transfer in coordinate time). Detecting this differential would provide a direct measurement of the hereditary time correction to the propagation of photons through the cosmic hierarchy.