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 Dimensionless Friedmann Equation: ρΛ 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 €