Seminar

The Early Universe

by Oliver Schmidt

Big Bang Cosmology:Einstein Universe

Friedmann-Lemaître UniverseEinstein-deSitter Universe

OutlineThe observed universe

Metric of the universe

Curvature

Einstein Equation

Cosmological models

•Einstein

•Friedmann-Lemaître

•Einstein-deSitter

Outlook

The observed universe•1011 galaxies

•1011 stars per galaxy

•1012 M per galaxy

•14 Gpc to the edge of the visible universe

•1011 galaxies

•1011 stars per galaxy

•1012 M per galaxy

•14 Gpc to the edge of the visible universe

Composition

2. Matter

3. Radiation

4. Dark matter

5. Vacuum energy of unknown density

( )

−

331

0 10~cm

gtvisibleρ

( )

−

334

0 10~cm

gtradiationρ

The observed universe

The observed universe•1011 galaxies

•1011 stars per galaxy

•1012 M per galaxy

•14 Gpc to the edge of the visible universe

Composition

2. Matter

3. Radiation

4. Dark matter

5. Vacuum energy of unknown density

The universe is expanding!

dHv 0=

( )

−

331

0 10~cm

gtvisibleρ

( )

−

334

0 10~cm

gtradiationρ

The observed universe

The universe is isotropic and homogeneous averaged over large scales!

The Cosmological Principle•The hypersurfaces with constant cosmic standard time are maximally symmetric subspaces of the whole of space-time.

•Not only the metric gμν, but all cosmic tensors such as Tμν, are form-invariant with respect to isometries of these subspaces.

Form invariance of gµν under transformation:

( ) ( )xgxg µ νµ ν ′=

( )xxx ′→

0=+ νµµν εε DDKilling equation: ( ) µµµε xxx −′=

Maximal number of symmetries = Maximal number of Killing Vectors ε =2

)1( +dd

Metric222222222 sin ΦΘ+Θ++−= drdrdrdtcdsMinkowski Metric:

Special Relativity

(~1905)

Metric222222222 sin ΦΘ+Θ++−= drdrdrdtcdsMinkowski Metric:

Schwarzschild Metric: 2222221

222

22 sin

21

21 ΦΘ+Θ+

−+

−−=

−

drdrdrrc

GMdtc

rc

GMds

Metric outside a non-rotating star

Special Relativity

(1916)

(~1905)

Metric222222222 sin ΦΘ+Θ++−= drdrdrdtcdsMinkowski Metric:

Schwarzschild Metric: 2222221

222

22 sin

21

21 ΦΘ+Θ+

−+

−−=

−

drdrdrrc

GMdtc

rc

GMds

Metric outside a non-rotating star

Special Relativity

(1916)

(~1905)

Robertson-Walker Metric: ( ) ( )

ΦΘ+Θ+

−+−= 2222

2

22222 sin

1ddr

kr

drtadtcds

Metric following the cosmological principle

(1935/36)

Robertson-Walker Metric

( )( )

( )( )

Θ

−

−

=

222

22

2

2

sin000

000

001

0

0001

rta

rtakr

ta

gα β

( ) ( )

ΦΘ+Θ+

−+−= 2222

2

22222 sin

1ddr

kr

drtadtcds

Robertson-Walker Metric

( ) ( )

ΦΘ+Θ+

−+−= 2222

2

22222 sin

1ddr

kr

drtadtcds

( )( )

( )( )

Θ

−

−

=

222

22

2

2

sin000

000

001

0

0001

rta

rtakr

ta

gα β

0=k 1=k1−=k

Flat universe Open universe Closed universe

Robertson-Walker Metric

( ) ( ) ( )( )

( )( )

1

0

1

sinh

sin

1

1

01

11

11

2

−===

=

=

−= ∫

−

−

k

k

k

dta

r

r

r

takr

drtatd coord

r

prop

( )2

2222

10

kr

drtadtd

−−== τ

Proper distance:

Cosmological Red Shift:Two light pulses emitted at te and te+δte, observed at t0 and t0+δt0 with a constant coordinate distant dcoord.

( ) ∫∫ −=

10

021

rt

t kr

dr

ta

dt

e( ) ∫∫ −

=+

+

100

021

rtt

tt kr

dr

ta

dt

ee

δ

δ

( ) ( )( )( )000

1

0

0

ta

ta

t

t

ta

t

ta

t ee

e

e ==⇒=⇒δδ

λλδδ

( )( ) 10

1

10 −=−≡eta

taz

λλλ

Curved spacetimes

In General Relativity gravitation is not a force but a property of spacetime geometry.

Curved spacetimes

In General Relativity gravitation is not a force but a property of spacetime geometry.

02

2

=τ

α

d

xd

Geodesic equation

Flat spacetime Curved spacetime

02

2

=Γ+τττ

γβαβγ

α

d

dx

d

dx

d

xd

Christoffel symbols:

∂∂

−∂∂

+∂∂

=Γ αβγ

βαγ

γαβαδα

βγ x

g

x

g

x

gg

2

1

Curved spacetimes( ) 02

2

=Γ+τττ

γβαβγ

α

d

dx

d

dxx

d

xd ( )τδx

( ) ( ) ( ) ( )02

2

=++

+Γ++

τδ

τδ

δτ

δ γγββαβγ

αα

d

xxd

d

xxdxx

d

xxd

Curved spacetimes

0ˆˆ

ˆˆˆ2

ˆ2

=+ βατβτ

α

δτδ

xRd

xdGeodesic deviation:

( ) 02

2

=Γ+τττ

γβαβγ

α

d

dx

d

dxx

d

xd ( )τδx

( ) ( ) ( ) ( )02

2

=++

+Γ++

τδ

τδ

δτ

δ γγββαβγ

αα

d

xxd

d

xxdxx

d

xxd

Curved spacetimes

Riemann curvature:

0ˆˆ

ˆˆˆ2

ˆ2

=+ βατβτ

α

δτδ

xRd

xdGeodesic deviation:

( ) 02

2

=Γ+τττ

γβαβγ

α

d

dx

d

dxx

d

xd ( )τδx

( ) ( ) ( ) ( )02

2

=++

+Γ++

τδ

τδ

δτ

δ γγββαβγ

αα

d

xxd

d

xxdxx

d

xxd

εβ γ

αδ ε

εβ δ

αγ εδ

αβ γ

γ

αβ δα

β γ δ ΓΓ−ΓΓ+∂Γ∂

−∂Γ∂

=xx

R

Ricci curvature: γα γ βα β RR =

Source of curvature

Energy-momentum-stress tensor:

= −

tensor

stress

fluxenergy

T momen

density

tum

energydensity

α β

Tαβ is symmetric!

Source of curvature

Energy-momentum-stress tensor:

= −

tensor

stress

fluxenergy

T momen

density

tum

energydensity

α β

Tαβ is symmetric!

Energy-momentum-stress tensor of a perfect fluid:(heat conduction, viscosity, etc. are negligible)

=

p

p

pT

000

000

000

000ρ

αβ

Einstein Equation

Einstein curvature tensor: RgRG α βα βα β 2

1−=Ricci curvature scalar:

α βα β

αα RgRR ==

α βα βπT

c

GG

4

8=

Einstein Equation

[ ] πρ83 2

2=+= ak

aGtt

Einstein curvature tensor: RgRG α βα βα β 2

1−=

( ) pakaa

aGGGrr π8

12 2

2 =

++−=== ΦΦΘΘ

Ricci curvature scalar:α β

α βα

α RgRR ==

Solving the Einstein equation for a homogeneous isotropic cosmological model of a cosmological perfect fluid yields to

α βα βπT

c

GG

4

8=

Einstein Equation

[ ] πρ83 2

2=+= ak

aGtt

Einstein curvature tensor: RgRG α βα βα β 2

1−=

( ) pakaa

aGGGrr π8

12 2

2 =

++−=== ΦΦΘΘ

( )a

ap+−= ρρ 3

ρπ3

82

2

=+a

akFriedmann equation:

Equation of state: ( )ρpp =

Ricci curvature scalar:α β

α βα

α RgRR ==

Solving the Einstein equation for a homogeneous isotropic cosmological model of a cosmological perfect fluid yields to

α βα βπT

c

GG

4

8=

Standard model

Equation of state

Gas of particles of mass m in thermal equilibrium with T<<m:

( )ρpp =

0=pMatter component with negligible pressure: dust

Equation of state

Gas of particles of mass m in thermal equilibrium with T<<m:

( )ρpp =

0=pMatter component with negligible pressure: dust

Gas of particles of mass m in thermal equilibrium with T>>m:

3

ρ=p

Highly relativistic matter component: radiation

Equation of state

Gas of particles of mass m in thermal equilibrium with T<<m:

( )ρpp =

0=pMatter component with negligible pressure: dust

Gas of particles of mass m in thermal equilibrium with T>>m:

3

ρ=p

Highly relativistic matter component: radiation

Further possibilities: and ρ=pρ−=p

ρν

−=⇒ 1

3p

Einstein model

Static universe

(1917)

Λ−= αβαβαβπ

gTc

GG

4

8

( )Λ+= πρ83

12a

k

Λ: Cosmological constant Λ>0: Repulsive force

ap

Λ−+−=

πρπ

43

3

40

Einstein model

Static universe

(1917)

Λ−= αβαβαβπ

gTc

GG

4

8

( )Λ+= πρ83

12a

k

Λ: Cosmological constant Λ>0: Repulsive force

ap

Λ−+−=

πρπ

43

3

40

ρπρπρ

G

ca

c

Gkp EE

4

41000

2==Λ=⇒=>Λ>

Closed universe with radius aE

Friedmann-Lemaître modelExpanding universe: Λ−= αβαβαβ

πgT

c

GG

4

8Ewith Λ>Λ

Friedmann-Lemaître modelExpanding universe: Λ−= αβαβαβ

πgT

c

GG

4

8Ewith Λ>Λ

0=k 1=k1−=k

( ) ( ) ( ) ( )( ) 3

1

03

0 13cosh4

−Λ

Λ= t

tatta

π ρ ( ) 3

2

tta ∝ ( ) 3

2

tta ∝

( ) t

eta 3

Λ

∝

for small a(t)

for large a(t)

t

a(t)

t

a(t)

t

a(t)

Standard model

( )a

ap+−= ρρ 3ρ

π3

82

2

=+a

ak ρν

−= 1

3p

( ) ( ) ( )( )

ν

ρρ

=

ta

tatt 00

( )( ) ( ) ( )νν ρ

π002

2

3

8tat

ta

kta=

+−

Einstein-deSitter modelAssumptions: Spatially flat (k=0) and dust dominated (ν=3) universe

( ) ( ) ( ) ( ) ( )12

3

13

002

3

3

8

2

3tatttatta +−= ρπ

( ) 3

2

tta ∝

Einstein-deSitter modelAssumptions: Spatially flat (k=0) and dust dominated (ν=3) universe

( ) ( ) ( ) ( ) ( )12

3

13

002

3

3

8

2

3tatttatta +−= ρπ

( ) 3

2

tta ∝

Assuming no change in the equation of state, one finds a time ti with a(ti)=0BIG BANG

( ) ( ) 0 → ∞→tata ( )26

1

tt

πρ =

Einstein-deSitter modelAssumptions: Spatially flat (k=0) and dust dominated (ν=3) universe

( ) ( ) ( ) ( ) ( )12

3

13

002

3

3

8

2

3tatttatta +−= ρπ

( ) 3

2

tta ∝

Assuming no change in the equation of state, one finds a time ti with a(ti)=0BIG BANG

( ) ( ) 0 → ∞→tata ( )26

1

tt

πρ =

Gravitationally bound universe

( ) ( ) ( )03

0max 3

8ˆ tatta ρπ=

( ) ( )03

0

2

3

4ˆ tatt ρπ

=

( ) 3

2

tta ∝

( ) tta ∝

(t small)

(t large)

( ) ( ) 0> → ∞→ cwithcta ta

1=k 1−=k

Standard model

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

0 0,5 1 1,5 2 2,5

t/tmax

a(t)/amax

k=0Einstein-deSitter

k=-1 k=1

Which universe and why?

20

020 3

8

a

kGH −=− ρπ

329

20 10*92,0

8

3

cm

g

G

Hcrit

−=≡π

ρ

critρρ =0critρρ >0 critρρ <0

0=k1=k 1−=kFlat universe Open universeClosed universe

Which universe and why?

20

020 3

8

a

kGH −=− ρπ

329

20 10*92,0

8

3

cm

g

G

Hcrit

−=≡π

ρ

critρρ =0critρρ >0 critρρ <0

0=k1=k 1−=kFlat universe Open universeClosed universe

Observations:

Mpcskm

H72

0 ≈ critm ρρ 3,0≈ critv ρρ 7,0≈critr ρρ 510*8 −≈

Which universe and why?

20

020 3

8

a

kGH −=− ρπ

329

20 10*92,0

8

3

cm

g

G

Hcrit

−=≡π

ρ

critρρ =0critρρ >0 critρρ <0

0=k1=k 1−=kFlat universe Open universeClosed universe

Observations:

Mpcskm

H72

0 ≈ critm ρρ 3,0≈ critv ρρ 7,0≈critr ρρ 510*8 −≈

References•James B. Hartle, Gravity – An introduction to Einstein‘s General Relativity

•Steven Weinberg, Gravitation and Cosmology

•Eckhard Rebhan, Theoretische Physik

•D.W. Scimia, Modern Cosmology and the Dark Matter Problem

•E.R. Harrison, Kosmologie – Die Wissenschaft vom Universum

•P.J.E Peebles, Physical Cosmology

•Charles W. Misner, Gravitation

Any questions?

Thank you!