Weak lensing in the second post-Newtonian approximation: Gravitomagnetic potentials and the integrated Sachs-Wolfe effect

Dark matter currents in the large-scale structure give rise to gravitomagnetic terms in the metric, which affect the light propagation. Corrections to the weak lensing power spectrum due to these gravitomagnetic potentials are evaluated by perturbation theory. A connection between gravitomagnetic lensing and the integrated Sachs-Wolfe (iSW) effect is drawn, which can be described by a line-of-sight integration over the divergence of the gravitomagnetic vector potential. This allows the power spectrum of the iSW-effect to be derived within the framework of the same formalism as derived for gravitomagnetic lensing and reduces the iSW-effect to a second order lensing phenomenon. The three-dimensional power spectra are projected by means of a generalised Limber-equation to yield the angular power spectra. While gravitomagnetic corrections to the weak lensing spectrum are negligible at observationally accessible scales, the angular power spectrum of the iSW-effect should be detectable as a correction to the CMB spectrum up to multipoles of l ≃ 100 with the Planck-satellite.