Equations of Motion in an Expanding Universe

Abstract

We make use of an effective field-theoretical approach to derive post-Newtonian equations of motion of hydrodynamical inhomogeneities in cosmology. The matter Lagrangian for the perturbed cosmological model includes dark matter, dark energy, and ordinary baryonic matter. The Lagrangian is expanded in an asymptotic Taylor series around a Friedmann-Lemeitre-Robertson-Walker background. The small parameter of the decomposition is the magnitude of the metric tensor perturbation. Each term of the expansion series is gauge-invariant and all of them together form a basis for the successive post-Newtonian approximations around the background metric. The approximation scheme is covariant and the asymptotic nature of the Lagrangian decomposition does not require the post-Newtonian perturbations to be small though computationally it works the most effectively when the perturbed metric is close enough to the background metric. Temporal evolution of the background metric is governed by dark matter and dark energy and we associate the large-scale inhomogeneities of matter as those generated by the primordial cosmological perturbations in these two components with \(\delta \rho /\rho \le 1\). The small scale inhomogeneities are generated by the baryonic matter which is considered as a bare perturbation of the background gravitational field, dark matter and energy. Mathematically, the large scale structure inhomogeneities are given by the homogeneous solution of the post-Newtonian equations while the small scale inhomogeneities are described by a particular solution of these equations with the stress-energy tensor of the baryonic matter that admits \(\delta \rho /\rho \gg 1\). We explicitly work out the field equations of the first post-Newtonian approximation in cosmology and derive the post-Newtonian equations of motion of the large and small scale inhomogeneities which generalize the covariant law of conservation of stress-energy-momentum tensor of matter in asymptotically-flat spacetime.