AS 4022 Cosmology
Lecture 7
Dynamics of the Universe
Solutions to the FriedmannEquation for R(t)
AS 4022 Cosmology
Hubble Parameter Evolution -- H(z)
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H 2 ≡˙ R R
2
=8π G
3ρ +
Λ3−
k c 2
R2
H 2
H02 =ΩR x 4 + ΩM x 3 + ΩΛ −
k c 2
H02R0
2 x 2
evaluate at x =1 → 1=Ω0 −k c 2
H02R0
2
€
x =1+ z = R0 R
ρc =3H0
2
8π G
ΩM ≡ρM
ρc
, ΩR ≡ρR
ρc
ΩΛ ≡ρΛρc
=Λ
3H02
Ω0 ≡ ΩM +ΩR +ΩΛ
€
H2
H02 =ΩR x
4 +ΩM x3 +ΩΛ + (1−Ω0 ) x 2
€
R0 =cH0
kΩ0 −1
→
k = +1 Ω0 >1k = 0 Ω0 =1k = −1 Ω0 <1
Dimensionless Friedmann Equation:
Curvature Radius today:
DensitydeterminesGeometry
AS 4022 Cosmology
PossibleUniverses
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H 0 ≈ 70km/sMpc
ΩM ~ 0.3ΩΛ ~ 0.7ΩR ~ 8×10
−5
Ω= 1.0
Empty
CriticalCycloid
VacuumDominated
Sub-Critical
AS 4022 Cosmology
Eternal Static Universe
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Einstein introduced Λ to enable an eternal static universe.
˙ R 2 =8π G ρ +Λ
3
R2 − k c2
˙ R = 0 → Λ =3 k c2
R2 − 8π G ρ
Einstein's biggest blunder. (Or, maybe not.)Static models unstable.Fine tuning.
t
R
VacuumDominated
Empty
Critical
CycloidSub-Critical
AS 4022 Cosmology
Empty Universe (Milne)
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˙ R 2 =8π G ρ +Λ
3
R2 − k c2
Set ρ = 0, Λ = 0. Then ˙ R 2 = −k c2
→ k = −1 ( negative curvature ) ˙ R = c, R = c t
H ≡˙ R
R=
1t
age : t0 =R0
c=
1H 0
Negative curvature drivesrapid expansion/flattening
t
R
VacuumDominated
Empty
Critical
CycloidSub-Critical
AS 4022 Cosmology
Matter-dominatedUniverses
€
R∝ t 2 / 3
Curvature wins
Matter wins
VacuumDominated
Empty
Critical
CycloidSub-Critical
All evolve asR ~ t2/3
at early times.
AS 4022 Cosmology
VacuumDominated
Empty
Critical
CycloidSub-Critical
tR 3/2∝
Critical Universe(Einstein - de Sitter)
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ΩM ≡ρM
ρc
=1
ΩR =ΩΛ = 0 → k = 0 ( flat )
ρ =3 H0
2
8π GR0
R
3
˙ R 2 =8π G
3ρ R2 =
H02 R0
3
R dR R1/ 2 = H0 R0
3 / 2 dt
23
R3 / 2 = H0 R03 / 2 t
RR0
=tt0
2 / 3
, age : t0 =23
1H0
=23
R0
c
Matter decelerates expansion.
AS 4022 Cosmology
VacuumDominated
Empty
Critical
CycloidSub-Critical
Super-CriticalCycloid Universe
€
ΩM >1, ΩR =ΩΛ = 0 → k = +1 ( closed )
˙ R 2 =8π G ρ0 R0
3
3 R− c 2 ˙ R = 0→ R =
8π G ρ0 R03
3 c 2 ≡ Rmax
R =Rmax
2(1− cosα ) H ≡
˙ R R
=2 c
Rmax
sinα1− cosα( )2
t =Rmax
2 c(α − sinα ) "Big Crunch" at t = π
cRmax
Cycloid curveconstructed by rollinga wheel thru angle α.
t0 t
R
€
α
AS 4022 Cosmology
Cycloid Universe
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t =α − sin α( )
R =
1 -
cos
( α)
AS 4022 Cosmology
VacuumDominated
Empty
Critical
CycloidSub-Critical
Sub-CriticalOpen Universe
€
ΩM <1, ΩR =ΩΛ = 0 → k = −1 ( negative curvature )
˙ R 2 =8π G ρ0 R0
3
3 R+ c 2 R →∞ ˙ R → c
R = R∗ ( coshα −1 )
t =R∗
c( sinhα −α )
R∗ ≡4π G ρ0 R0
3
3 c 2 t
R
Matter initially decelerates, butcurvature wins in the end.
AS 4022 Cosmology
Radiation dominated Universe
€
ρR = ρ0R0
R
4
ΩM = ΩΛ = 0
˙ R 2 =8π G
3ρ R2 − k c2
=8π G ρ0 R0
4
3 R2 − k c2 ≈H 0
2 R04
R2
dR R = H 0 R02 dt
12
R2 = H 0 R02 t
RR0
=tt0
1/ 2
, age : t0 =1
2 H 0
=3
32π G ρ0
1/ 2
tR 2/1∝
Neglect curvature at early times.
Radiation decelerates expansion.
AS 4022 Cosmology
Vacuum dominated Universe
1+
1−
€
˙ R 2 =Λ
3R2 − k c2 ΩM = ΩR = 0
H 2 ≡˙ R R
2
=Λ
3−
k c2
R2
RRΛ
= 12
cosh( t / tΛ )exp( t / tΛ )sinh( t / tΛ )
k = +1k = 0k = −1
tΛ ≡ 3/Λ RΛ ≡ c tΛ
Rmin = RΛ
1+ k2
H →1/ tΛ
Λ drives exponential expansion also called inflation
AS 4022 Cosmology
Vacuum Era R(t)
€
H2
H02 =ΩR x
4 +ΩM x3 +ΩΛ + (1−Ω0)x 2
=ΩΛ + (1−Ω0)x 2 +ΩM x3 +ΩR x
4
€
H 2
H02
€
ΩΛ
€
Ω0 >1
€
Ω0 <1
€
x =R0R
€
ΩM x3
€
dt =−dxx H(x)
t = −dx
x H(x)∫
x =R0
R
€
1
€
1
€
H ≈ ΩΛ1/ 2H0
ΩΛ1/ 2H0 t = −
dxx∫
= −ln x = ln(R /R0)
RR0
= exp ttΛ
tΛ =
1ΩΛ
1/ 2H0
AS 4022 Cosmology
Matter Era R(t)
€
H2
H02 =ΩR x
4 +ΩM x3 +ΩΛ + (1−Ω0)x 2
=ΩΛ + (1−Ω0)x 2 +ΩM x3 +ΩR x
4
€
H 2
H02
€
ΩΛ
€
Ω0 >1
€
Ω0 <1
€
x =R0R
€
ΩM x3
€
dt =−dxx H(x)
t = −dx
x H(x)∫
x =R0
R
€
1
€
1
€
H ≈ H0 ΩM1/ 2x 3 / 2
ΩM1/ 2H0 t = − x−5 / 2dx∫
=23x−3 / 2 =
23
RR0
3 / 2
RR0
=tt0
2 / 3
t0 =2
3ΩM1/ 2H0
AS 4022 Cosmology
Radiation Era R(t)
€
H2
H02 =ΩR x
4 +ΩM x3 +ΩΛ + (1−Ω0)x 2
=ΩΛ + (1−Ω0)x 2 +ΩM x3 +ΩR x
4
€
H 2
H02
€
ΩΛ
€
Ω0 >1
€
Ω0 <1
€
x =R0R
€
ΩM x3
€
dt =−dxx H(x)
t = −dx
x H(x)∫
x =R0
R
€
1
€
1
€
H ≈ H0 ΩR1/ 2x 2
ΩR1/ 2H0 t = − x−3dx∫
=12x−2
1x
=RR0
=tt0
1/ 2
t0 =1
2ΩR1/ 2H0
AS 4022 Cosmology
Loitering Universes
€
H2
H02 =ΩM x
3 +ΩΛ + (1−Ω0)x 2
ddx
H 2
H02
= 3ΩM x
2 + 2(1−Ω0)x
dH 2
dx= 0 ⇒ xL =
2(Ω0 −1)3ΩM
€
t 2 / 3€
R
€
et
€
t
VacuumDominated
Empty
Critical
CycloidSub-Critical
€
xL€
H 2
H02
€
ΩΛ
€
Ω0 >1
€
Ω0 <1
€
x
€
ΩM x3
AS 4022 Cosmology
Possible Universes
€
t
€
t 2 / 3€
sinh(t)€
exp(t)€
cosh(t)
€
ΩM
€
ΩΛ
€
ΩM +ΩΛ =1
AS 4022 Cosmology
OurUniverse
€
H0 ≈ 70km/sMpc
ΩM ~ 0.3ΩΛ ~ 0.7ΩR ~ 8 ×10
−5
Ω =1.0
Empty
CriticalCycloid
VacuumDominated
Sub-Critical
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