For Λ -dominated universes, where Ω 0, Λ ≫ Ω 0,R and Ω 0,M, as well as Ω 0, Λ ≈ 1, and where we now will change our bounds of integration from t i to t and likewise a i to a : ( t − − t i ) H 0 = ∫ ∫ a i a d a ′ ( Ω Ω 0 , Λ Λ a ′ 2 ) ( t − − t i ) H 0 Ω Ω 0 , Λ Λ = ln ⁡ ⁡ | a ′ | | a i a a i exp ⁡ ⁡ ( ( t − − t i ) H 0 Ω Ω 0 , Λ Λ ) = a ( t )