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We represent the reionization history of the universe as a free function in redshift and study the potential for its extraction from cosmic microwave background (CMB) polarization spectra. From a principal component analysis, we show that the ionization history information is contained in 5 modes, resembling low-order Fourier modes in redshift space. The amplitudes of these modes represent a compact description of reionization observables in the CMB, easily predicted given a model for the ionization fraction. Measurement of these modes can ultimately constrain the total optical depth, or equivalently the initial amplitude of fluctuations to the 1% level regardless of the true model for reionization.
Hu et al. (Thu,) studied this question.