Secondary cosmic microwave background anisotropies in a universe reionized in patches

In a universe reionized in patches, the Doppler effect from Thomson scattering off free electrons generates secondary cosmic microwave background (CMB) anisotropies. For a simple model with small patches and late reionization, we analytically calculate the anisotropy power spectrum. Patchy reionization can, in principle, be the main source of anisotropies on arcminute scales. On larger angular scales, its contribution to the CMB power spectrum is a small fraction of the primary signal and is only barely detectable in the power spectrum with even an ideal, i.e., cosmic variance limited, experiment and an extreme model of reionization. Consequently, patchy reionization is unlikely to affect cosmological parameter estimation from the acoustic peaks in the CMB. Its detection on small angles would help determine the ionization history of the universe, in particular, the typical size of the ionized region and the duration of the reionization process.