Early growth of cosmological inhomogeneity at the horizon scale

The weakly nonlinear evolution of collisionless matter in the expanding Universe is examined within general relativity. An approximate solution, whose perturbation scale is comparable to or larger than the Hubble radius, is obtained by series expansion with respect to the background scale factor. The solution represents that the early density contrast is controlled by the initial data at each Lagrangian position and the subsequent growth is modified by the presence of the magnetic part of the Weyl curvature, which is produced by the nonlinear coupling, even if it vanishes in the initial condition. The effect of the gravitomagnetic part on the dynamics is demonstrated in a concrete example. The dynamical role becomes less important as the perturbation scale increases beyond the horizon scale