Correlated gravitational lensing of the cosmic microwave background

Cosmological inhomogeneities gravitationally deflect radiation propagating from distant sources, transforming the spatial and angular correlation functions of intrinsic source properties. For a gaussian distribution of deflections (e.g. from a primordial gaussian density perturbation spectrum or from the central limit theorem) we calculate the probability distributions for geodesic deviations. If the intrinsic variable is also gaussian, e.g. the large scale velocity flow field or cosmic microwave background temperature anisotropies, then distributions and correlation functions of the observed image sky properties can be obtained. Specialising to CMB temperature fluctuations we rederive simply the influence of independent gravitational lensing on the anisotropy angular correlation function and calculate the new effect of lensing correlated with the anisotropies, e.g. arising from the same primordial gravitational perturbation field. Characteristic magnitudes and scales are given in terms of the density power spectrum. The correlated deflection-temperature effect is shown to be negligible.