Lecture 6 General Relativity: Field Equations Dynamics of the Universe: R(t) = ? H(x) = ? Friedmann Equation AS 4022 Cosmology Time and Distance vs Redshift d R −dx x =1+ z = 0 → dt = dt R x H(x) Look - back time : t0 1 1+z −dx dx R D t(z) = dt = = r ∫ ∫ x H(x) ∫ x H(x) t 1+z 1 χ Age : t = t(z → ∞) 0 € Distance : D = R χ r = R Sk (χ) t0 c dt c 1+z R dx c 1+z dx χ(z) = ∫ dχ = ∫ = ∫ 0 = ∫ t R(t) R0 1 R(t) x H(x) R0 1 H(x) Horizon : χ H = χ(z → ∞) Need to know R(t), or R0 and H(x). AS 4022 Cosmology € Einstein’s General Relativity • 1. Spacetime geometry tells matter how to move – gravity = effect of curved spacetime – free particles follow geodesic trajectories • 2. Matter (energy) tells spacetime how to curve – Einstein field equations – nonlinear – second-order derivatives of metric with respect to space/time coordinates AS 4022 Cosmology Einstein Field Equations 1 α 8π G G ≡ R − R α g = T − Λ g µν µν 2 µν c 2 µν µν 2 µ ν gµν = spacetime metric ( ds = gµν dx dx ) Gµν = Einstein tensor (spacetime curvature) Rµν = Ricci curvature tensor α R α = Ricci curvature scalar G = Netwon's gravitational constant Tµν = energy - momentum tensor Λ = cosmological constant AS 4022 Cosmology € Homogeneity and Isotropy homogeneous isotropic not isotropic not homogeneous For cosmology, assume Universe is Homogeneous. Simplifies the equations. : ) AS 4022 Cosmology Cosmological Principle (assumed) + Isotropy (observed) => Homogeneity 1 2 ρ1 = ρ 2 otherise not isotropic for equidistant fidos AS 4022 Cosmology Density Fluctuations vs Scale Homogeneity is a good approximation on large scales. AS 4022 Cosmology Homogeneous perfect fluid density ρ pressure p Einstein field equations: ρ c 2 0 0 0 −c 2 0 0 0 8π G 0 p 0 0 0 1 0 0 G = − Λ µν c 2 0 0 p 0 0 0 1 0 0 0 0 p 0 0 0 1 ---> Friedmann equations : 2 8π G ρ + Λ 2 2 € R˙ = R − k c energy 3 4π G Λ R˙˙ = − ρc 2 + 3p R + R momentum 3c 2 ( ) 3 Note: energy density and pressure decelerate, Λ accelerates. AS 4022 Cosmology € Local Conservation of Energy d[energy] = work d[ρ c 2R3] = −p d[R3] ρ˙ c 2R3 + ρ c 2 (3 R2R˙ ) = −p (3 R2R˙ ) p R˙ ρ˙ = −3 ρ + p = p(ρ) = equation of state c 2 R 8π G Λ Friedmann 1: R˙ 2 = ρ R2 + R2 − k c 2 3 3 8π G Λ 2 R˙ R˙˙ = ρ˙ R2 + 2 R R˙ ρ + 2 R R˙ ( ) 3 ( ) 3 ( ) 8π G ρ˙ R2 Λ R˙˙ = + R ρ + R 3 2 R˙ 3 4π G 3 p Λ R˙˙ = − ρ + R + R = Friedmann 2 3 c 2 3 AS 4022 Cosmology € Newtonian Analogy m G M m 4π E = R˙ 2 − M = R3ρ 2 R 3 m 8π G 2E R R˙ 2 = ρ R2 + 3 m ρ Friedmann equation: € € 8π G ρ + Λ R˙ 2 = R2 − k c2 3 Λ 2E same equation if ρ → ρ + , → −k c2 8π G m € AS 4022 Cosmology € Newtonian Analogy m G M m E = R˙ 2 − 2 R 2 G M V = esc R E > 0 V > Vesc R → ∞ V∞ > 0 € E = 0 V = Vesc R → ∞ V∞ = 0 E < 0 V < Vesc R → 0 € AS 4022 Cosmology Density - Evolution - Geometry R(t) ρ < ρ c Open t k = -1 ρ =ρ c Flat € k = 0 ρ >ρ Closed c k = +1 AS 4022 Cosmology Critical Density • Derive using Newtonian analogy: escape velocity : 3 2 2 2 G M 2 G 4π R ρ 8π G R ρ Vesc = = = R R 3 3 Hubble expansion : R V = R˙ = H R 0 ρ critical density : 2 V 8π G ρ ρ esc = ≡ V 3 H 2 ρ 0 c € 3 H 2 ρ = 0 c 8π G AS 4022 Cosmology € radiation -> matter -> vacuum ρR −4 log ρ radiation : ρR ∝ R −3 ρ matter : ρM ∝ R M € ρΛ vacuum : ρΛ = const 4 3 ρ(x) = ρR x + ρM x + ρΛ € log R € R0 1 x =1+ z = = t R a log R e € 2 / 3 ρM 4 4 t ρR = ρM at x ~ ~ 10 t ~ 10 yr ρR 1/ 2 1/ 3 t ρ € at x ~ Λ z ~ 0.3 t ~ 1010 yr ρM = ρΛ € ρM logt AS 4022 Cosmology € Equation of State ----- w Equation of state : ρ ∝ R−n n = 3(1+ w) pressure p n w ≡ = = −1 energy density ρ c 2 3 Radiation : (n = 4, w =1/3) d[energy] = work 1 2 3 3 2 d c R p d R pR = ρR c [ρ ] = − [ ] 3 2 2 3 2 2 Matter : (n = 3, w = 0) ρ c ( 3 R dR ) + R c dρ = −p ( 3 R dR ) 2 2 R dρ p pM ~ ρM cS << ρM c 1+ = − 2 ≡ −w Vacuum : (n = 0, w = −1) 3 ρ dR ρc 2 1 d[lnρ] pΛ = −ρΛ c w = − −1 3 d lnR Negative Pressure ! ? [ ] n w = −1 3 AS 4022 Cosmology € € Density Parameters critical density : density parameters (today) : 2 3 H0 ρR ρM ρΛ Λ ρc ≡ ΩR ≡ ΩM ≡ ΩΛ ≡ = 2 8π G ρc ρc ρc 3 H0 total density parameter today : Ω0 ≡ ΩR + ΩM + ΩΛ density at a past/future epoch in units of today's critical density : ρ 3(1+w) 4 3 Ω ≡ = ∑Ωw x = ΩR x + ΩM x + ΩΛ x ≡1+ z = R0 /R ρc w in units of critical density at the past/future epoch: 2 4 3 8π Gρ H0 3(1+w) ΩR x + ΩM x + ΩΛ Ω(x) ≡ 2 = 2 ∑Ωw x = 4 3 2 3H H w ΩR x + ΩM x + ΩΛ + (1− Ω0)x Note: radiation dominates at high z, can be neglected at lower z. € AS 4022 Cosmology Hubble Parameter Evolution -- H(z) ˙ 2 2 2 R 8π G Λ k c x =1+ z = R R H ≡ = ρ + − 2 0 R 3 3 R 2 2 2 3 H0 H 4 3 k c 2 ρc = 2 = ΩR x + ΩM x + ΩΛ − 2 2 x 8π G H0 H0 R0 ρ ρ k c 2 Ω ≡ M , Ω ≡ R evaluate at x =1 1 M R → = Ω0 − 2 2 ρc ρc H0 R0 ρ Λ Dimensionless Friedmann Equation: Λ ΩΛ ≡ = 2 2 ρc 3 H H 4 3 2 0 2 = ΩR x + ΩM x + ΩΛ + (1− Ω0 ) x H0 Ω0 ≡ ΩM + ΩR + ΩΛ € Curvature Radius today: k = +1 Ω0 >1 c k Density R k 0 1 € 0 = → = € Ω0 = determines H0 Ω0 −1 Geometry k = −1 Ω0 <1 AS 4022 Cosmology € Possible Universes Vacuum Dominated km/s H ≈ 70 Critical 0 Mpc Cycloid Empty Sub-Critical ΩM ~ 0.3 ΩΛ ~ 0.7 −5 ΩR ~ 8×10 Ω = 1.0 AS 4022 Cosmology € .