A post-Newtonian Lagrangian perturbation approach to large-scale structureformation

We formulate Lagrangian perturbation theory to solve the non-linear dynamic s of a self-gravitating fluid within the framework of the post-Newtonian ap proximation in general relativity, using the (3 + 1) formalism. Our formula tion coincides with the Newtonian Lagrangian perturbation theory developed by Buchert for scales much smaller than the horizon scale, and with the gau ge-invariant linearized theory in longitudinal gauge conditions for the lin ear regime. These conditions are achieved by using the gauge-invariant quan tities at the initial time, when the linearized theory is valid. The post-N ewtonian corrections in the solution of the trajectory field of fluid eleme nts are calculated in their explicit forms. Thus our formulation allows us to investigate the evolution of large-scale fluctuations involving relativi stic corrections from the early regime, such as the decoupling time of matt er and radiation, until today. As a result, we are able to show that naive Newtonian cosmology for the structure formation will be a good approximatio n even for perturbations with scales not only inside but also beyond the pr esent horizon scale in longitudinal coordinates. Although the post-Newtonia n corrections are small, it is shown that they have a growing transverse mo de, which is not present in Newtonian theory or in the gauge-invariant line arized theory. Such post-Newtonian-order effects might produce a characteri stic appearance of large-scale structure formation, for example through the observation of anisotropies in the cosmic microwave background radiation ( CMB). Furthermore, because our approach has a straight forward Newtonian li mit, it will also be convenient for numerical implementation based on the p resently available Newtonian simulations. Our results easily allow us to pe rform a simple order estimation of each term in the solution, which indicat es that post-Newtonian corrections cannot be neglected in the early evoluti on of density fluctuations, compared with Newtonian perturbation solutions.