Nonlinear integrated Sachs-Wolfe effect

We discuss the nonlinear extension to the integrated Sachs-Wolfe effect (ISW) resulting from the divergence of the large scale structure momentum density field. The nonlinear ISW effect leads to an increase in the total ISW contribution by roughly two orders of magnitude at $l\sim 1000.$ This increase, however, is still below the cosmic variance limit of the primary anisotropies; at further small angular scales, secondary effects such as gravitational lensing and the kinetic Sunyaev-Zel’dovich effect dominate the nonlinear ISW power spectrum. We show that this second-order nonlinear ISW contribution is effectively the same as the contribution previously described as a lensing effect due to the transverse motion of gravitational lenses and well known as the Kaiser-Stebbins effect in the context of cosmic strings. Because of geometrical considerations, there is no significant three-point correlation function, or bispectrum, between the linear ISW effect and its nonlinear extension. The nonlinear ISW contribution can potentially be used as a probe of the transverse velocity of dark matter halos such as galaxy clusters. Because of the small contribution to temperature fluctuations, of the order of a few tenths of microkelvins, however, extracting useful measurements on velocities will be challenging.