Lecture 8 Observational Cosmology Parameters of Our Universe The Concordance Model
Time and Distance vs Redshift d dt x =1+ z = R 0 R dt = dx x H(x) Friedmann : H(x) = x 3 + Ω Λ + (1 Ω 0 ) x 2 look - back time : t(z) = t 0 dt = t e radial distance : D 0 (z) = R 0 χ = Ω 0 + Ω Λ ( Ω R = 0) dx = 1 1+z dx x H(x) x x 3 + Ω Λ + (1 Ω 0 ) x 2 1+z 1 t 0 R 0 R(t) c dt = t e 1 x c dx = c 1+z dx x H(x) x 3 + Ω Λ + (1 Ω 0 ) x 2 circumferencial distance : r 0 = R 0 S k (χ) R 0 = c k Ω 0 1 angular diameter distance : D A = r 0 (1+ z) luminosity distance : D L = (1+ z) r 0 1+z 1 1 1/ 2 R χ r D
Angular Diameter Distance Ω Λ +Ω Λ = 1 Ω Λ = 0
D A (z) = r(z) 1+ z D L (z) = ( 1+ z) r(z) r(z) = R 0 S k ( χ(z) ) r(z) / c
Concordance Model Parameters 100 h km/s Mpc 70 km/s Mpc h 0.7 Ω R 4.2 10 5 h 2 8.4 10 5 (CMB photons + neutrinos) Ω B ~ 0.02 h 2 ~ 0.04 (baryons) ~ 0.3 (Dark Matter) Ω Λ ~ 0.7 (Dark Energy) Ω 0 Ω R + + Ω Λ =1.0 Flat Geometry
Our (Crazy?) Universe 70 km/s Mpc ~ 0.3 Ω Λ ~ 0.7 Ω R ~ 8 10 5 Ω 0 =1.0 Vacuum Dominated Empty Critical Sub-Critical accelerating Cycloid decelerating
Evolution of Ω Density at a past/future epoch in units of ρ c = 3 2 /8π G Ω ρ = Ω w x 3(1+w) = Ω R x 4 + x 3 + Ω Λ x 1+ z = R 0 /R ρ c w in units of critical density 3 H 2 /8π G at the past/future epoch : (x) = H 2 0 H Ω x 3 2 M = Ω Λ (x) = H 2 0 H Ω = 2 Λ x 3 x 3 + Ω Λ + (1 Ω 0 )x 2 Ω Λ x 3 + Ω Λ + (1 Ω 0 )x 2 Do we live at a special time? Ω Λ log x
(x) = Ω Λ (x) = Evolution of Ω x 3 Ω R x 4 + x 3 + Ω Λ + (1 Ω 0 )x 2 Ω Λ Ω R x 4 + x 3 + Ω Λ + (1 Ω 0 )x 2
Concordance Model Three main constraints: 1. Supernova Hubble Diagram 2. Galaxy Counts + M/L ratios ~ 0.3 3. Flat Geometry ( inflation, CMB fluctuations) Ω 0 = + Ω Λ =1 2 1 3 concordance model 72 0.3 Ω Λ 0.7
H HST Key Project -1 0 72 ± 3 ± 7 km s Mpc 1 Freedman, et al. 2001 ApJ 553, 47.
Hubble time and radius Hubble constant : H R R Hubble time : R R 0 =100 h km/s Mpc 70 km/s Mpc t H 1 =10 10 h 1 yr 14 10 9 yr ~ age of Universe Hubble radius : R H c = 3000 h 1 Mpc 4 10 9 pc = distance light travels in a Hubble time. ~ distance to the Horizon.
Age vs Hubble time H = R R H = R R t = t0 deceleration decreases age acceleration increases age e.g. matter dominated R t 2 / 3 R = 2 3 H = R R = 2 3 1 t R t t 0 = 2 3 1 = 2 3 t H t H = 1 Age = t 0
Beyond Globular cluster ages: t < t 0 --> acceleration Radio jet lengths: D A (z) --> deceleration Hi-Redshift Supernovae: D L (z) --> acceleration Dark Matter estimates ---> Inflation ---> Flat Geometry CMB power spectra ~ 0.3 Ω 0 1.0 Concordance Model ~ 0.3 Ω Λ ~ 0.7 Ω 0 = + Ω Λ 1.0
Age Constraints Nuclear decay ( U, Th -> Pb ) Decay times for ( 232 Th, 235 U, 238 U) = (20.3, 1.02, 6.45) Gyr 3.7 Gyr = oldest Earth rocks 4.57 Gyr = meteorites ~10 Gyr = time since supernova produced U, Th ( 235 U / 238 U = 1.3 --> 0.33, 232 Th / 238 U = 1.7 --> 2.3 ) Stellar evolution 13-17 Gyr = oldest globular clusters White dwarf cooling ~13 Gyr = coolest white dwarfs in M4
Nuclear Decay Chronology P=parent D=daughter S=stable isotope of D Chemical fractionation changes P/S but not D/S: e.g. concentrate D+S in crystals Samples have same D 0 / S 0 various P 0 / S 0 P decays to D: t /τ P(t) = P 0 e t /τ D(t) = D 0 + P 0 1 e ( ) S(t) = S 0 ( ) = D 0 + P(t) e t /τ 1 D(t) S(t) = D 0 S 0 + P(t) S 0 ( e t /τ 1) D/S D 0 /S 0 Observed slope = ( e t /τ 1) P/S gives age t /τ, typically to ~ 1%
Globular Cluster Ages
12.7± 0.7 Gyr Coolest White Dwarfs Hansen et al. 2002 ApJ 574,155 White dwarf cooling ages --> star formation at z > 5. Cooling times have been measured using ZZ Ceti oscillation period changes.
t 0 = 1 observations : Age Crisis (~1995) dx x x 3 + Ω Λ + (1 Ω 0 ) x 2 = 72 ± 8 km s 1 Mpc 1 t 0 14 ± 2 Gyr old globular clusters t 0 1.0 ± 0.15 Globular clusters older than the Universe? Inconsistent with critical-density matter-only model : t 0 = 2 3 for (,Ω ) Λ = (1,0) ( Ω 0 =1) Strong theoretical prejudice for inflation. Doubts about stellar evolution therory (e.g. convection).