cosmological principle friedmann-lemaitre equationsweygaert/tim1publication/cosmo... ·...
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Cosmology, lect. 3
Cosmological Principle&
Friedmann-Lemaitre Equations
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Einstein Field Equation
41 82
GR Rg g Tcµν µν µν µνπ
− + Λ = −
41 82
GR Rg T gcµν µν µν µνπ
− = − −Λ
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Cosmological Principle
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A crucial aspect of any particular configuration is the geometry of spacetime: because Einstein’s General Relativity is a metric theory, knowledge of the geometry is essential.
Einstein Field Equations are notoriously complex, essentially 10 equations. Solving them for general situations is almost impossible.
However, there are some special circumstances that do allow a full solution. The simplest one is also the one that describes our Universe. It is encapsulated in the
Cosmological Principle On the basis of this principle, we can constrain the geometry of the Universe and hence find its dynamical evolution.
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“God is an infinite sphere whose centre is everywhere and its circumference nowhere”
Empedocles, 5th cent BC
”all places in the Universe are alike’’Einstein, 1931
Homogeneous Isotropic
Universality Uniformly Expanding
Cosmological Principle:Describes the symmetries in global appearance of the Universe:
The Universe is the same everywhere:- physical quantities (density, T,p,…)
The Universe looks the same in every direction
Physical Laws same everywhere
The Universe “grows” with same rate in - every direction- at every location
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uniform=homogeneous & isotropic(cosmological principle)
Fundamental Tenet
of (Non-Euclidian = Riemannian) Geometry
There exist no more than THREE uniform spaces: 1) Euclidian (flat) Geometry Euclides
2) Hyperbolic Geometry Gauß, Lobachevski, Bolyai
3) Spherical Geometry Riemann
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2 2 2 2 2 2 2 2 2 2( ) sinc kc
rds c dt a t dr R S d dR
θ θ φ = − + +
sin 1
0
sinh 1
c
kc c
c
r kR
r rS kR R
r kR
= +
= =
= −
Distances in a uniformly curved spacetime is specified in terms of the Robertson-Walker metric. The spacetime distance of a point at coordinate (r,,) is:
where the function Sk(r/Rc) specifies the effect of curvatureon the distances between points in spacetime
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Friedmann-Robertson-Walker-Lemaitre(FRLW)
Universe
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Einstein Field Equation
48 GG Tcµν µνπ
= −
( )2
2 , , ,
pT U U pgc
diag c p p p
µ ν µνµν ρ
ρ
= + −
=
, ,RWg R Rµµν λν µν⇒ Γ ⇒
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Einstein Field Equation
48 GG Tcµν µνπ
= −
( )
( )
0 0 2 2 2 20 0 2
1 1 2 2 21 1 2
83 /
82 /
GG G R kc R cc
GG G RR R kc R pc
π ρ
π
→ = + =
→ = + + = −
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Friedmann-Robertson-Walker-LemaitreUniverse
24 3
3 3G pR R R
cπ ρ Λ = − + +
2 2 2 283 3GR R kc Rπ ρ Λ
= − +
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Cosmic Expansion Factor
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Cosmic Expansion is a uniform expansion of space
Cosmic Expansion Factor
0
( )( ) R ta tR
=
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Friedmann-Robertson-Walker-Lemaitre Universe
24 3
3 3G pa a a
cπ ρ Λ = − + +
22 2 2
20
83 3G kca a a
Rπ ρ Λ
= − +
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Friedmann-Robertson-Walker-Lemaitre Universe
24 3
3 3G pa a a
cπ ρ Λ = − + +
22 2 2
20
83 3G kca a a
Rπ ρ Λ
= − +
densitypressure cosmological
constant
curvature term
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Because of General Relativity, the evolution of the Universe is fully determined by four factors:
• density
• pressure
• curvature : present curvature radius
• cosmological constant
( )tρ
( )p t2 2
0/kc R 0, 1, 1k = + −
Λ
• Density & Pressure: - in relativity, energy & momentum need to be seen as one physical quantity (four-vector)
- pressure = momentum flux• Curvature: - gravity is a manifestation of geometry spacetime• Cosmological Constant: - free parameter in General Relativity
- Einstein’s “biggest blunder”- mysteriously, since 1998 we know it dominates the Universe
0R
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24 3
3 3G pa a a
cπ ρ Λ = − + +
22 2 2
20
83 3G kca a a
Rπ ρ Λ
= − +
43Ga aπ ρ= −
2 283Ga a Eπ ρ= +
Relativistic Cosmology Newtonian Cosmology
pΛ
2 20/kc R− ECurvature
Cosmological Constant
Pressure
Energy
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Hubble Parameter
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• Cosmic Expansion is a uniform expansion of space
• Objects do not move themselves:they are like beacons tied to a uniformly expanding sheet:
( ) ( )r t a t x=
( ) ( ) ( )ar t a t x ax H t ra
= = =
( ) aH ta
=
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• Cosmic Expansion is a uniform expansion of space
• Objects do not move themselves:they are like beacons tied to a uniformly expanding sheet:
( ) ( )r t a t x=
( ) ( ) ( )ar t a t x ax H t ra
= = =
( ) aH ta
=
Comoving Position
Comoving PositionHubble Parameter:
Hubble “constant”:H0H(t=t0)
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• For a long time, the correct value of the Hubble constant H0
was a major unsettled issue:
H0 = 50 km s-1 Mpc-1 H0 = 100 km s-1 Mpc-1
• This meant distances and timescales in the Universe had to deal with uncertainties of a factor 2 !!!
• Following major programs, such as Hubble Key Project, the Supernova key projects and the WMAP CMB measurements,
2.6 1 10 2.771.9H km s Mpc+ − −
−=
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• As more accurate measurements of H0become available, gradual rising tension
• CMB determination much lower than
“local values”
• Latest value:strong grav. lensing
H0 = 82.4 +/- 8.3 km/s/Mpc
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1Ht H=
1 10
10
100
9.78
H h km s Mpc
t h Gyr
− −
−
=
⇓
=
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Cosmological Constant&
FRW equations
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Friedmann-Robertson-Walker-Lemaitre Universe
24 3
3 3G pa a a
cπ ρ Λ = − + +
22 2 2
20
83 3G kca a a
Rπ ρ Λ
= − +
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Dark Energy & Energy Density
p p p
ρ ρ ρΛ
Λ
= +
= +
2
8
8
G
cpG
ρπ
π
Λ
Λ
Λ=
Λ= −
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Friedmann-Robertson-Walker-Lemaitre Universe
24 3
3G pa a
cπ ρ = − +
2 220
2 2
/83
kc Ra Ga a
π ρ= −
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Cosmic Constituents
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In addition, contributions by - gravitational waves - magnetic fields, - cosmic rays …
Poor constraints on their contribution: henceforth we will not take them into account !
The total energy content of Universe made up by various constituents, principal ones:
Ωm
Ωrad
Ωv
Ωtot
ΩDM
Ωγ
Ων
baryonic matter
dark matter
photons
neutrino’s
dark/vacuum energy
matter
radiation
Ωb
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LCDM Cosmology• Concordance cosmology
- model that fits the majority of cosmological observations- universe dominated by Dark Matter and Dark Energy
LCDM composition today …
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Cosmic Energy Inventory
Fukugita & Peebles 2004
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Critical Density & Omega
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22 2
20
83G kca a
Rπ ρ= −
Critical Density:
- For a Universe with Ω=0
- Given a particular expansion rate H(t)
- Density corresponding to a flat Universe (k=0)
238crit
HG
ρπ
=
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In a FRW Universe, densities are in the order of the critical density,
22 29 303 1.8791 10
8critH h g cm
Gρ
π− −= = ×
29 2 30
11 2 3
1.8791 10
2.78 10
h g cm
h M Mpc
ρ − −
−
= × Ω
= × Ω
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In a matter-dominated Universe, the evolution and fate of the Universe entirely determinedby the (energy) density in units of critical density:
283crit
GH
ρ π ρρ
Ω ≡ =
Arguably, is the most important parameter of cosmology !!!
Present-day Cosmic Density:
29 2 30
11 2 3
1.8791 10
2.78 10
h g cm
h M Mpc
ρ − −
−
= × Ω
= × Ω
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FRWL Dynamics&
Cosmological Density
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rad m ΛΩ = Ω +Ω +Ω
•The individual contributions to the energy density of the Universe can be figured into the parameter:
- radiation
- matter
- dark energy/ cosmological constant
4 2 4
2 2/ 8
3rad
radcrit crit
T c G TH c
ρ σ π σρ ρ
Ω = = =
m dm bΩ = Ω +Ω
23HΛΛ
Ω =
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Critical DensityThere is a 1-1 relation between the total energy content of the Universe and its curvature. From FRW equations:
2 2
2 ( 1)H Rkc
= Ω− rad m ΛΩ = Ω +Ω +Ω
1 1
1 0
1 1
k Hyperbolic Open Universe
k Flat Critical Universe
k Spherical Close Universe
Ω < = −
Ω = =
Ω > = +
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FRW Universe: CurvatureThere is a 1-1 relation between the total energy content of the Universe and its curvature. From FRW equations:
2 2
2 ( 1)H Rkc
= Ω− rad m ΛΩ = Ω +Ω +Ω
1 1
1 0
1 1
k Hyperbolic Open Universe
k Flat Critical Universe
k Spherical Close Universe
Ω < = −
Ω = =
Ω > = +
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Radiation, Matter & Dark Energy
rad m ΛΩ = Ω +Ω +Ω
The individual contributions to the energy density of the Universe can be figured into the parameter:
- radiation
- matter
- dark energy/ cosmological constant
4 2 4
2 2/ 8
3rad
radcrit crit
T c G TH c
ρ σ π σρ ρ
Ω = = =
m dm bΩ = Ω +Ω
23HΛΛ
Ω =
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Cosmic Constituents:
Evolving Energy Density
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To infer the evolving energy density (t) of each cosmic component, we refer to the cosmic energy equation. This equation can be directly inferred from the FRW equations
23 0p ac a
ρ ρ + + =
The equation forms a direct expression of the adiabatic expansion of the Universe, ie.
dU pdV= −2
3
U c VV a
ρ=
∝
Internal energy
Expanding volume
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To infer (t) from the energy equation, we need to know the pressure p(t) for that particular medium/ingredient of the Universe.
23 0p ac a
ρ ρ + + =
To infer p(t), we need to know the nature of the medium, which provides us with the equation of state,
( , )p p Sρ=
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• Matter:
Radiation:
Dark Energy:
3( ) ( )m t a tρ −∝
4( ) ( )rad t a tρ −∝
3(1 ) 2( ) ( )
1( ) .
wv vt a t p w c
wt cst
ρ ρ
ρ
− +
Λ
∝ ⇐ =
⇓ = −=
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Dark Energy:
Equation of State
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Einstein Field Equation
41 82
GR Rg g Tcµν µν µν µνπ
− + Λ = −
41 82
GR Rg T gcµν µν µν µνπ
− = − −Λ
curvature side
energy-momentum side
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Equation of State4
8vaccT gG
µν µν
πΛ
≡restframe 4
8vaccTG
µν µνηπΛ
≡
00 1, 1iiη η= = −
2pT U U pg
cµ ν µν
µν ρ = + −
restframe:
00 2vac vac
iivac
T c
T p
ρ =
⇒=
42
4
8
8
vacccG
cpG
ρπ
π
Λ=
Λ= −
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Equation of State4
2
4
8
8
vacccG
cpG
ρπ
π
Λ=
Λ= −
2vac vacp cρ= −
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Dynamics
22
34 pGc
φ π ρ ∇ = +
Relativistic Poisson Equation:
2
3 2 0;8
vacvac vac vac
pc G
ρ ρ ρπΛ
+ = − < =
2 0φ∇ < Repulsion !!!
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Dark Energy & Cosmic Acceleration
24 3
3G pa a
cπ ρ = − +
2( )p w cρ ρ=
Nature Dark Energy:
(Parameterized) Equation of State
Cosmic Acceleration:
Gravitational Repulsion:2 1 0
3p w c w aρ= ⇔ <− ⇒ >
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Dark Energy & Cosmic Acceleration
2( )p w cρ ρ=
DE equation of State
: 1 .ww cstρΛ = − =Cosmological Constant:
-1/3 > w > -1: decreases with time
Phantom Energy:
increases with time
3(1 )0( ) ( ) w
w wa a aρ ρ − +=
3(1 ) 1 0ww a wρ − +∝ + >
3(1 ) 1 0ww a wρ − +∝ + <
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Dynamic Dark Energy
2( )p w cρ ρ=
DE equation of State Dynamically evolving dark energy,parameterization:
01
1 ( )( ) ( ) exp 3
a
w w
w aa a da
aφρ ρ
′ + ′= − ′ ∫
0( ) (1 ) ( )aw a w a w w aφ= + − ≈
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23 3( ) wa t t +∝
FRW:
0k =3(1 ) 2( ) ( ) w
v vt a t p w cρ ρ− +∝ ⇐ =
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Acceleration Parameter
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FRW Dynamics:Cosmic Acceleration
Cosmic acceleration quantifiedby means of dimensionless deceleration parameter q(t):
2aaqa
= −
2m
radq Λ
Ω= +Ω −Ω
2mq Λ
Ω≈ −Ω
1; 0;0.5
m
qΛΩ = Ω =
=
0.3; 0.7;0.65
m
qΛΩ = Ω =
= −
Examples:
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Dynamical Evolution
FRWL Universe
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From the FRW equations:
2,0 ,0 0
,02 4 3 20
1( ) rad mH tH a a aΛ
Ω Ω −Ω= + +Ω +
( )0
0 ,0 ,0 2,0 02 1
a
rad m
daH ta
a a Λ
=Ω Ω
+ +Ω + −Ω∫
( )a t Expansion history Universe
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1Ω <
1Ω >
2 13
tH
=
2 13
tH
=
1tH
= ( )202 1
a
rad m
daH ta
a a Λ
=Ω Ω
+ +Ω + −Ω∫
Matter-dominated
Matter-dominated
Hubble time
Age of a FRW universe at Expansion factor a(t)
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While general solutions to the FRW equations is only possible by numerical integration, analytical solutions may be found for particular classes of cosmologies:
• Single-component Universes:- empty Universe- flat Universes, with only radiation, matter or
dark energy
• Matter-dominated Universes
• Matter+Dark Energy flat Universe
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Assume radiation contribution is negligible:
Zero cosmological constant:
Matter-dominated, including curvature
5,0 5 10rad
−Ω ≈ ×0ΛΩ =
1mΩ >
1mΩ <1mΩ =
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2/3
0
( ) ta tt
=
10
m
Λ
Ω =Ω =
00
2 13
tH
=
Albert Einstein and Willem de Sitter discussing the Universe. In 1932 they published a paper together on the Einstein-de Sitter universe, which is a model with flat geometry containing matter as the only significant substance.
0k =
2 2 088 13 3
GGa aa
π ρπ ρ= =FRW:
Age EdS Universe:
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0
( ) ta tt
=
00
m
Λ
Ω =Ω =
00
1tH
=
1k = −
22
20
.kca cstR
= − =FRW:
Age Empty Universe:
Empty space is curved
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1/2
0
( ) ta tt
=
100
rad
m
Λ
Ω =Ω =Ω =
00
1 12
tH
=
In the very early Universe, the energy density is completely dominated by radiation. The dynamics of the very early Universe is therefore fully determined by the evolution of the radiation energy density:
0k =
2 2 02
88 13 3
GGa aa
π ρπ ρ= =
FRW:
Age RadiationUniverse:
41( )rad aa
ρ ∝
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0 0( )( ) H t ta t e −=
01
m
Λ
Ω =Ω =
020
2 20
3 3
3
HH
a a a H a
ΛΛ Λ
Ω = ⇒ =
Λ= ⇒ = FRW:
Age De Sitter Universe: infinitely old
0k =
Willem de Sitter (1872-1934; Sneek-Leiden)director Leiden Observatoryalma mater: Groningen University