Renormalization group approach in Newtonian cosmology

We apply the renormalization group (RG) method to examine the observable scaling properties in Newtonian cosmology. The original scaling properties of the equations of motion in our model are modified for averaged observables on constant time slices. In the RG flow diagram, we find three robust fixed points: Einstein–de Sitter, Milne, and quiescent fixed points. Their stability (or instability) property does not change under the effect of fluctuations. Inspired by the inflationary scenario in the early Universe, we set the Einstein–de Sitter fixed point with small fluctuations as the boundary condition at the horizon scale. Solving the RG equations under this boundary condition toward the smaller scales, we find a generic behavior of observables such that the density parameter $\Omega$ decreases, while the Hubble parameter $H$ increases for a smaller averaging volume. The quantitative scaling properties are analyzed by calculating the characteristic exponents around each fixed point. Finally we argue the possible fractal structure of the Universe beyond the horizon scale.