A phase-space representation of Friedmann-Lemaitre universes containing both dust and radiation and the inevitability of a big bang

We classify all non-static Friedmann–Lemaître universes containing dust and radiation (or, as limiting cases, vacuum), and exhibit their histories as orbits in a three-dimensional (phase-) space ℘, which faithfully reflects the topology of the set of states. As coordinates in ℘ we take the dimensionless parameters Ω, ω and λ representing, respectively, the energy density of the dust, the radiation and the vacuum (corresponding to the cosmological constant). Three open subspaces of ℘ contain, respectively, the orbits representing oscillating, inflectional and bouncing universes; their boundaries and edges contain the orbits of all the other universes. The classification proceeds from initial values |(Ω₀, ω₀)| in terms of (a, λ0) diagrams (a = R/R0). An alternative set of ‘invariant’ parameters is also introduced, characterizing whole models rather than states of models. Finally we show that present-day redshift and density data exclude all models not having a big bang, without using assumptions about the origin of the microwave background radiation and without a priori assumptions about the value of the cosmological constant.