Lagrangian theory for cosmic structure formation with vorticity: Newtonian and post-Friedmann approximations

We study the nonlinear gravitational dynamics of a universe filled with a pressureless fluid and a cosmological constant $\mathrm{\Lambda }$ in the context of Newtonian gravity, and in the relativistic post-Friedmann approach proposed in paper I [I. Milillo et al., Phys. Rev. D 92, 023519 (2015)]. The post-Friedmann approximation scheme is based on the $1/c$ expansion of the space-time metric and the energy-momentum tensor, and includes nonlinear Newtonian cosmology. Here we establish the nonlinear post-Friedmann framework in the Lagrangian-coordinates approach for structure formation. For this we first identify a Lagrangian gauge which is suitable for incorporating nonzero vorticity. We analyze our results in two limits: at the leading order we recover the fully nonlinear Newtonian cosmological equations in the Lagrangian formulation, and we provide a space-time metric consistent from the perspective of general relativity. We then linearize our expressions and recover the relativistic results at first order in cosmological perturbation theory. Therefore, the introduced approximation scheme provides a unified treatment for the two leading-order regimes, from the small scales described by Newtonian gravity to the large linear scale, where first-order relativistic cosmological perturbation theory gives a very good description of structure formation.