Averaging out Inhomogeneous Newtonian Cosmologies: III. The Averaged Navier-Stokes-Poisson Equations

TL;DR

This paper develops a rigorous framework for averaging the Navier-Stokes-Poisson equations in Newtonian cosmology, addressing the mathematical properties of averages and deriving the full averaged system for cosmological fluid dynamics.

Contribution

It introduces a detailed formulation of averaging procedures for cosmological fluid fields and derives the complete averaged Navier-Stokes-Poisson equations in terms of fluid kinematic quantities.

Findings

Averaging procedures require consistent measurement device parameters.

Derived formulae for averaging derivatives of fluid fields.

Established the full system of averaged equations for cosmological fluids.

Abstract

The basic concepts and hypotheses of Newtonian Cosmology necessary for a consistent treatment of the averaged cosmological dynamics are formulated and discussed in details. The space-time, space, time and ensemble averages for the cosmological fluid fields are defined and analyzed with a special attention paid to their analytic properties. It is shown that all averaging procedures require an arrangement for a standard measurement device with the same measurement time interval and the same space region determined by the measurement device resolution to be prescribed to each position and each moment of time throughout a cosmological fluid configuration. The formulae for averaging out the partial derivatives of the averaged cosmological fluid fields and the main formula for averaging out the material derivatives have been proved. The full system of the averaged Navier-Stokes-Poisson equations in terms of the fluid kinematic quantities is derived.