Morphology of the Secondary CMB Anisotropies: the Key to "Smoldering" Reionization

We show how the morphological analysis of the maps of the secondary CMB anisotropies can detect an extended period of ``smoldering'' reionization, during which the universe remains partially ionized. Neither radio observations of the redshifted 21cm line nor IR observations of the redshifted Lyman-alpha forest will be able to detect such a period. The most sensitive to this kind of non-gaussianity parameters are the number of regions in the excursion set, the perimeter of the excursion set, and the genus of the largest (by area) region. For example, if the universe reionized fully at z=6, but maintained about 1/3 mean ionized fraction since z=20, then a 2 arcmin map with 500x500 pixel resolution and a signal-to-noise ratio S/N=1/2 allows to detect the non-gaussianity due to reionization with better than 99% confidence level.