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!