Seminar The Early Universe - Institute of Nuclear...
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Seminar
The Early Universe
by Oliver Schmidt
Big Bang Cosmology:Einstein Universe
Friedmann-Lemaître UniverseEinstein-deSitter Universe
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OutlineThe observed universe
Metric of the universe
Curvature
Einstein Equation
Cosmological models
•Einstein
•Friedmann-Lemaître
•Einstein-deSitter
Outlook
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The observed universe•1011 galaxies
•1011 stars per galaxy
•1012 M per galaxy
•14 Gpc to the edge of the visible universe
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•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
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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
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The observed universe
The universe is isotropic and homogeneous averaged over large scales!
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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
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Metric222222222 sin ΦΘ+Θ++−= drdrdrdtcdsMinkowski Metric:
Special Relativity
(~1905)
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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)
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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)
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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
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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
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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
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Curved spacetimes
In General Relativity gravitation is not a force but a property of spacetime geometry.
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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
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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
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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
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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 =
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Source of curvature
Energy-momentum-stress tensor:
= −
tensor
stress
fluxenergy
T momen
density
tum
energydensity
α β
Tαβ is symmetric!
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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
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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=
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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=
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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
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Equation of state
Gas of particles of mass m in thermal equilibrium with T<<m:
( )ρpp =
0=pMatter component with negligible pressure: dust
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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
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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
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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
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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
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Friedmann-Lemaître modelExpanding universe: Λ−= αβαβαβ
πgT
c
GG
4
8Ewith Λ>Λ
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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)
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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=
+−
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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 ∝
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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 =
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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
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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
![Page 36: Seminar The Early Universe - Institute of Nuclear Physicstheorie.ikp.physik.tu-darmstadt.de/nhc/pages/lectures/rhiseminar04... · Seminar The Early Universe by Oliver Schmidt Big](https://reader031.fdocuments.in/reader031/viewer/2022021822/5b15112c7f8b9a467c8d884d/html5/thumbnails/36.jpg)
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
![Page 37: Seminar The Early Universe - Institute of Nuclear Physicstheorie.ikp.physik.tu-darmstadt.de/nhc/pages/lectures/rhiseminar04... · Seminar The Early Universe by Oliver Schmidt Big](https://reader031.fdocuments.in/reader031/viewer/2022021822/5b15112c7f8b9a467c8d884d/html5/thumbnails/37.jpg)
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 −≈
![Page 38: Seminar The Early Universe - Institute of Nuclear Physicstheorie.ikp.physik.tu-darmstadt.de/nhc/pages/lectures/rhiseminar04... · Seminar The Early Universe by Oliver Schmidt Big](https://reader031.fdocuments.in/reader031/viewer/2022021822/5b15112c7f8b9a467c8d884d/html5/thumbnails/38.jpg)
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 −≈
![Page 39: Seminar The Early Universe - Institute of Nuclear Physicstheorie.ikp.physik.tu-darmstadt.de/nhc/pages/lectures/rhiseminar04... · Seminar The Early Universe by Oliver Schmidt Big](https://reader031.fdocuments.in/reader031/viewer/2022021822/5b15112c7f8b9a467c8d884d/html5/thumbnails/39.jpg)
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
![Page 40: Seminar The Early Universe - Institute of Nuclear Physicstheorie.ikp.physik.tu-darmstadt.de/nhc/pages/lectures/rhiseminar04... · Seminar The Early Universe by Oliver Schmidt Big](https://reader031.fdocuments.in/reader031/viewer/2022021822/5b15112c7f8b9a467c8d884d/html5/thumbnails/40.jpg)
Any questions?
Thank you!