Cosmological Perturbation Theory, 26.4.2011 version

18 ADIABATIC AND ISOCURVATURE PERTURBATIONS IN A SIMPLIFIED UNIVERSE44so thatandWe definew =ρ rρ = 11 + y13(1 + y)ρ mρ =y1 + y1 + w = 4 + 3y3(1 + y)y ≡aa eq= ρ mρ r(18.2)ρ r + p rρ + p = 44 + 3yc 2 s = 43(4 + 3y)ρ m + p mρ + p= 3y4 + 3y(18.3)1 − 3c 2 s = 3y4 + 3y . (18.4)whereThe Friedmann equation isH 2 ≡ 1 ( ) da 2a 2 dη= 8πG3 ρa2 = 8πG3 (1 + y)ρ r0a 4 0 a−2⇒dy√ 1 + y= 2Cdη ,The solution isAt the time of matter-radiation equality,√2πGC ≡3 ρ a 2 0r0 .a eqy = C 2 η 2 + 2Cη . (18.5)y = y eq = C 2 η 2 eq + 2Cη eq = 1 ⇒ Cη eq = √ 2 − 1, (18.6)so we can write the solution aswhereη 3 ≡( ) η 2 ( ) ηy = + 2 , (18.7)η 3 η 3η eq(√ )√ = 2 + 1 η eq (18.8)2 − 1is the time when y = 3 (ρ m = 3ρ r ).The Hubble parameter isAt matter-radiation equality it has the valueH ≡ a′a = y′y = η + η 3η 3 η + 1 . (18.9)2η2 H eq = 2√ 2√2 + 11η eq= 4 − 2√ 2η eq= 2√ 2η 3. (18.10)At early times, η ≪ η 3 , the universe is radiation dominated, so thaty ≈ 2ηη 3≪ 1 ⇒ a ∝ η ⇒ H = 1 η ∝ a−1 . (18.11)At late times, η ≫ η 3 , the universe is matter dominated, so thaty ≈( ηη 3) 2≫ 1 ⇒ a ∝ η 2 ⇒ H = 2 η ∝ a−1/2 . (18.12)