Lecture 8
Observational Cosmology
Parameters of Our Universe
The Concordance Model
AS 4022
Cosmology
Time and Distance vs Redshift
d
R 
−dx
 x = 1+ z = 0  → dt =
dt 
R
x H(x)
R
Friedmann : H(x) = H 0 ΩM x 3 + ΩΛ + (1− Ω0 ) x 2
Ω0 ≡ΩM + ΩΛ
look - back time :
t0
t(z) =
1+z
∫ dt =
te
∫
1
χ
( ΩR = 0)
dx
1 1+z
=
∫
x H(x) H 0 1 x
€
dx
ΩM x 3 + ΩΛ + (1− Ω0 ) x 2
radial distance :
t0
D0(z) = R0 χ =
∫
te
1+z
R0
x c dx
c 1+z
c dt = ∫
=
∫
R(t)
x
H(x)
H
0 1
1
dx
ΩM x 3 + ΩΛ + (1− Ω0 ) x 2
1/ 2


c
k
circumferencial distance : r0 = R0 Sk ( χ ) R0 =


H 0  Ω0 −1
angular diameter distance :
DA = r0 (1+ z )
luminosity distance :
DL = (1+ z ) r0
AS 4022
Cosmology
r
D
Angular Diameter Distance
ΩΛ
ΩM
€
ΩM + ΩΛ = 1
€
ΩΛ = 0
€
€
AS 4022
Cosmology
r(z)
DA (z) =
1+ z
DL (z) = (1+ z) r(z)
€
€
r(z) = R0 Sk ( χ (z))
r(z) H0 / c
€
AS 4022
Cosmology
“Concordance Model”
Parameters
km/s
km/s
H 0 ≡ 100 h
≈ 70
Mpc
Mpc
h ≈ 0.7
ΩR ≈ 4.2 ×10−5 h −2 ≈ 8.4 ×10−5 (CMB photons + neutrinos)
ΩB ~ 0.02 h −2 ~ 0.04 (baryons)
ΩM ~ 0.3 (Dark Matter)
ΩΛ ~ 0.7 (Dark Energy)
Ω0 ≡ ΩR + ΩM + ΩΛ = 1.0 → Flat Geometry
AS 4022
Cosmology
Our
(Crazy?)
Universe
Vacuum
Dominated
accelerating
decelerating
km/s
H 0 ≈ 70
Mpc
ΩM ~ 0.3
ΩΛ ~ 0.7
−5
ΩR ~ 8 ×10
Ω0 = 1.0
AS 4022
Cosmology
Critical
Empty
Sub-Critical
Cycloid
Evolution of Ω
Density at a past/future epoch in units of ρ c = 3 H 0 2 /8π G
ρ
Ω≡
= ∑ Ωw x 3(1+w) = ΩR x 4 + ΩM x 3 + ΩΛ
x ≡ 1+ z = R0 /R
ρc w
in units of critical density 3 H 2 /8π G at the past/future epoch :
H02
ΩM x 3
3
ΩM (x) = 2 ΩM x =
H
ΩM x 3 + ΩΛ + (1− Ω0 )x 2
H02
ΩΛ
ΩΛ (x) = 2 ΩΛ =
H
ΩM x 3 + ΩΛ + (1− Ω0 )x 2
Do we live at a special time ?
ΩM
ΩΛ
log x
AS 4022
Cosmology
Evolution of Ω
ΩM x 3
ΩM (x) =
ΩR x 4 + ΩM x 3 + ΩΛ + (1− Ω0 )x 2
ΩΛ
ΩΛ (x) =
ΩR x 4 + ΩM x 3 + ΩΛ + (1− Ω0 )x 2
€
AS 4022
Cosmology
“Concordance” Model
Three main constraints:
1. Supernova Hubble Diagram
2. Galaxy Counts + M/L ratios
ΩM ~ 0.3
3. Flat Geometry
( inflation, CMB fluctuations)
Ω0 = ΩM + ΩΛ = 1
2
1
3
concordance model
H 0 ≈ 72 ΩM ≈ 0.3 ΩΛ ≈ 0.7
AS 4022
Cosmology
HST Key Project
-1
H 0 ≈ 72 ± 3 ± 7 km s Mpc
Freedman, et al.
2001 ApJ 553, 47.
AS 4022
Cosmology
−1
Hubble time and radius
Hubble constant :
 R˙ 
R˙
km/s
km/s
H≡
H 0 ≡   = 100 h
≈ 70
R
Mpc
Mpc
 R 0
Hubble time :
1
tH ≡
= 1010 h −1 yr ≈ 14 ×10 9 yr
H0
~ age of Universe
Hubble radius :
c
RH ≡
= 3000 h −1 Mpc ≈ 4 ×10 9 pc
H0
= distance light travels in a Hubble time.
~ distance to the Horizon.
AS 4022
Cosmology
Age vs Hubble time
R˙
H=
R
R˙
H=
R t=t
0
deceleration decreases age
acceleration increases age
e.g. matter dominated
2R
2/3
˙
R∝t
⇒ R=
3 t
R˙ 2 1
2 1 2
H= =
⇒ t0 =
= tH
R 3t
3 H0 3
AS 4022
Cosmology
tH = 1 H 0
Age = t0
Beyond H0
• Globular cluster ages:
t < t0 --> acceleration
• Radio jet lengths:
DA(z) --> deceleration
• Hi-Redshift Supernovae: DL(z) --> acceleration
• Dark Matter estimates --->
ΩM ~ 0.3
• Inflation ---> Flat Geometry
• CMB power spectra
Ω0 ≈ 1.0
€
• “Concordance Model”
€
AS 4022
Cosmology
ΩM ~ 0.3 ΩΛ ~ 0.7
Ω0 = ΩM + ΩΛ ≈ 1.0
Age Constraints
• Nuclear decay ( U, Th -> Pb )
–
–
–
–
Decay times for (232Th,235U,238U) = (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
– ( 235U / 238U = 1.3 --> 0.33, 232Th / 238U = 1.7 --> 2.3 )
• Stellar evolution
– 13-17 Gyr = oldest globular clusters
• White dwarf cooling
– ~13 Gyr = coolest white dwarfs in M4
AS 4022
Cosmology
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 D0 / S0
various P0 / S0
• P decays to D:
D/S
D0/S0
P(t) = P0 e−t / τ
D(t) = D0 + P0 (1− e−t / τ )
= D0 + P(t) (e t / τ −1)
D(t) D0 P(t) t / τ
=
+
e −1)
(
S(t) S0
S0
AS 4022
Cosmology
S(t) = S0
P/S
Observed slope = (e t / τ −1)
gives age t / τ , typically to ~ 1%
Globular Cluster Ages
AS 4022
Cosmology
Coolest White Dwarfs
12.7± 0.7 Gyr
Hansen et al. 2002 ApJ 574,155
€
White dwarf cooling ages
--> star formation at z > 5.
AS 4022
Cosmology
Cooling times have been
measured using “ZZ Ceti”
oscillation period changes.
Age Crisis (~1995)
∞
H 0 t0 =
∫
1
dx
ΩM x 3 + ΩΛ + (1− Ω0 ) x 2
x
observations :
−1
−1
H 0 = 72 ± 8 km s Mpc
t 0 ≥ 14 ± 2 Gyr old globular clusters
H 0 t 0 ≥ 1.0 ± 0.15
Globular clusters older than the Universe ?
Inconsistent with critical-density matter-only model :
2
H 0 t0 =
3
for (ΩM ,ΩΛ ) = (1,0)
(
)
Strong theoretical prejudice for inflation. Ω 0 = 1
Doubts about stellar evolution therory (e.g. convection).
AS 4022
Cosmology