Astroparticle Physics

17.12 Chapter 12 371The related mass densities for the two scenarios are estimated to beϱv 1 = Λ 1c 28πG ≈ 10−52 m −2 × 9 × 10 16 m 2 /s 2 kg s 28π × 6.67 × 10 −11 m 3≈ 5.37 × 10 −27 kg/m 3 = 5.37 × 10 −30 g/cm 3 ,ϱv 2 = Λ 2c 28πG ≈ 5.37 × 1095 kg/m 3 = 5.37 × 10 92 g/cm 3 .2. Starting fromH 2 = 8πG3 (ϱ + ϱ v) ⇒ H 2 − 1 3 Λc2 = 8πG3 ϱand since H = Ṙ/R one obtains (for ϱ = const)ṘR = √13 Λc2 + 8πG3 ϱ ⇒ R = R i exp(√13 Λc2 + 8πG3 ϱt ),which represents an expanding universe.3. The Friedmann equation extended by the Λ term for a flat universe isṘ 2R 2 − 1 3 Λc2 = 8πG3 ϱ.For 1 3 Λc2 ≫ 8πG3ϱ this equation simplifies to√ √Ṙ 1 1R = 3 Λc2 =3 Λ 0c 2 (1 + αt) = a √ √11 + αt with a =3 Λ 0c 2 ,∫ln R = a √ 1 + αt dt + const = a 23 α (1 + αt)3/2 + const ,( )2aR = exp3α (1 + αt)3/2 + const ;boundary condition R(t = 0) = R i( )2aR(t = 0) = exp3α + const = R i ⇒2a3α + const = ln R i ⇒ const = ln R i − 2a3α ,( 2aR = R i exp3α (1 + αt)3/2 − 2a ) ( )2a= R i exp3α3α [(1 + αt)3/2 − 1]with a =√13 Λ 0c 2 .For large t this result shows a dependence likeR ∼ exp(βt 3/2) with β = 2a√ α.3