For radiation-dominated universes, where Ω 0,R ≫ Ω 0,M and Ω 0,Λ, as well as Ω 0,R ≈ 1 : t H 0 = ∫ ∫ 0 a d a ′ Ω Ω 0 , R a ′ − − 2 t H 0 Ω Ω 0 , R = a ′ 2 2 | 0 a ( 2 t H 0 Ω Ω 0 , R ) 1 / 2 = a ( t )