Mean Square Geodesic Deviation in the Zeldovich Problem on Light Propagation in a Universe with Inhomogeneities

In 1964, Ya.B. Zeldovich formulated the problem of light propagation in the Universe under the influence of inhomogeneities. It is reduced to describing the divergence of two close geodesics in a Riemannian space and is described by the geodesic deviation equation (Jacobi equation) with the curvature along the geodesic line varying randomly. Assuming the curvature to be constant on segments of small but finite length, the problem is reduced to studying the product of random matrices and makes it possible to apply the appropriate well-developed mathematical theory, which, however, did not allow calculating the mean-square growth rate of the geodesic deviation. In our paper, we propose a way to solve this problem by introducing a bilinear quantity, one component of which coincides with the square of the Jacobi field. The system of first-order differential equations for the bilinear quantity is explicitly written out, and the solution, same as the growth rate, is again expressed through the product of matrices. Such a technique can be used in the study of a wide range of problems and is naturally generalized to higher-order moments.