A Note on a Density Cutoff ∆ and Singularity-Free Cosmology

Abstract

In classical general relativity a spatially flat expanding universe filled with matterof positive energy density reaches infinite density at a finite time in the past. A widelystudied way to avoid this is to assume that the energy density cannot exceed a maximalvalue. We write this maximal density as ρ∆ = ∆ and study the resulting effectiveFriedmann equation H2 = 8πG3 ρ (1 − ρ/ρ∆). For a fluid with constant equation of statew > −1 we prove that the scale factor has a strictly positive minimum, we give theexact solution, and we show that the curvature scalar stays bounded. The classicalequations are recovered as ∆ → ∞. The equation is the effective equation known fromloop quantum cosmology; this note is expository, and its purpose is to set out theargument and the proofs in a self-contained way. It does not derive the cutoff from afundamental theory.