Small scale integrated Sachs-Wolfe effect

The integrated Sachs-Wolfe (ISW) effect can be an important factor in the generation of cosmic microwave background anisotropies on all scales, especially in a reionized curvature- or {Λ}-dominated universe. We present a simple analytic treatment of the ISW effect, which is analogous to thick last scattering surface techniques for the Doppler effect, that compares quite well with the full numerical calculations. The power spectrum of temperature fluctuations due to the small scale ISW effect has a wave number dependence k^-5 times that of the matter power spectrum.