Inhomogeneous universes admitting a post-Newtonian approximation

We rigorously construct a linear post-Newtonian approximation applicable to cosmological solutions of the Einstein-Euler system. Our approach centres on solving the initial value problem. Firstly, we provide a construction for families of suitably near-Newtonian initial data. These data uniformly satisfy the constraints required by the Einstein equations, and describe a finite number of fluid inhomogeneities superimposed upon a spatially homogeneous background, allowing for any value of the cosmological constant. Secondly, we develop a consistent approximation scheme for the families of solutions launched by these data. Solving the equations determined by this scheme, we obtain a linear post-Newtonian approximation for each family of solutions, valid locally in time and globally in space.