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

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x =1+ z = R0 R

ρc =3H0

2

8π G

ΩM ≡ρM

ρc

, ΩR ≡ρR

ρc

ΩΛ ≡ρΛρc

=Λ

3H02

Ω0 ≡ ΩM +ΩR +ΩΛ

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H2

H02 =ΩR x

4 +ΩM x3 +ΩΛ + (1−Ω0 ) x 2

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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

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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

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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

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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

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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−

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˙ 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