Optical Depth and Scalar Amplitude

The optical depth at reionization \(\tau\) and the scalar amplitude \(A_s\) exhibiy a well-known degeneracy. As discussed on the Photon Optical Depth page, reionization uniformly reduces the amplitude of the power spectra by a factor of \(e^{-2\tau}\). Simultaneously, the size of the oscillations in the CMB power spectra is directly proportional to amplitude of the primordial scalar perturbations, \(A_s\). Consequently, observations tighly constrain the combined quantity \(A_s e^{-2\tau}\), rather than \(A_s\) and \(\tau\) individually. Using Plank 2018 data, this value is calculated to be \(\approx 1.88\times10^{-9}\).

This degeneracy can be broken by examining how an increase in \(\tau\) boosts the power at large angular scales (low multipoles), specifically in the E polarization spectrum. This rise in power occurs because additional scatterings during reionization generate polarized radiation, leaving a distinct signature at scales corresponding to the size of the horizon at reionization, the so-called EE polarization bump. For more details, see the Photon Optical Depth page.