standard cosmology i longpeople.na.infn.it/~astropar/doc/standard_cosmology_i.pdf · 2009. 9....
TRANSCRIPT
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Astroparticle Course 1
Standard CosmologyStandard Cosmology
�� FriedmannFriedmann--LemaLemaîîtretre--RobertsonRobertson--Walker Walker (FLRW) metric(FLRW) metric
�� Friedman equationFriedman equation
�� Equation of stateEquation of state
�� Universe evolutionUniverse evolution
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Astroparticle Course 2
Some bibliographySome bibliography
• The Early Universe, by E.W. Kolb & M.S. Turner
• Cosmology and Particle Astrophysics, L. Bergstrom & A. Goobar
• Cosmology, S. Weinberg
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Astroparticle Course 3
The cosmological principleThe cosmological principle
We could invoke the “Copernican principle,” that we do not live in a special place in the universe. However, apart from local
inhomogeneities, like stars, planets, galaxies, it really seems
that, on average, matter is quite smoothly distributed
everywhere. This is particolarly true on the very large scales
probed by the microwave background observations where inhomogeneities are of the order of only a few times 10−5. This
observation severely restricts the possible cosmological
theories. It implies that the metric itself should be homogeneous
and isotropic.
On large spatial scales, theOn large spatial scales, the universe universe is homogeneousis homogeneous and isotropicand isotropic
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Astroparticle Course 4
FriedmannFriedmann--LemaLemaîîtretre--RobertsonRobertson--Walker metricWalker metric
Generalizing to 4-dimensional space-time we get the maximally-symmetric Friedmann-Lemaître-Robertson-Walker (FLRW) metric,
ds2
Ω+−
−== 222
1
2)(222 dr
rk
drtadtdxdxgds νµ
µν
Every isotropic, homogeneous three-space can be parameterized (perhaps after performing a coordinate transformation) with coordinates giving a one-parameter family of spaces depending on a scale factor, a(t),
Ω+−
= 222
1
2)(22 dr
rk
drtads
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Astroparticle Course 5
FriedmannFriedmann--LemaLemaîîtretre--RobertsonRobertson--Walker metricWalker metric
Generalizing to 4-dimensional space-time we get the maximally-symmetric Friedmann-Lemaître-Robertson-Walker (FLRW) metric,
ds2
Ω+−
−== 222
1
2)(222 dr
rk
drtadtdxdxgds νµ
µν
Every isotropic, homogeneous three-space can be parameterized (perhaps after performing a coordinate transformation) with coordinates giving a one-parameter family of spaces depending on a scale factor, a(t),
Ω+−
= 222
1
2)(22 dr
rk
drtads
r is dimensionless and k=+1,0,-1 for positive, zero, or negative curvature
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Astroparticle Course 6
Comoving frameComoving frame
In this frame the fluid looks perfectly isotropic. This can happen for constant values of the coordinates r, θ and φ in the FLRW metric. A particle at rest in the comoving frame satisfies the geodesic equation
with the line element given by ds2=dt2. The world line of such a particle corresponds to free fall in the cosmic fluid. At every point a comoving frame can be found where the universe looks maximally isotropic: all observers in the universe will see an isotropic universe (and it will appear that they are all at the “centre” of the universe) from wherever they look, if they are at rest in the local comoving frame.
02
2
=Γ+ds
dx
ds
dx
ds
xd ii νµ
µν
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Astroparticle Course 7
Comoving frameComoving frame
In this frame the fluid looks perfectly isotropic. This can happen for constant values of the coordinates r, θ and φ in the FLRW metric. A particle at rest in the comoving frame satisfies the geodesic equation
with the line element given by ds2=dt2. The world line of such a particle corresponds to free fall in the cosmic fluid. At every point a comoving frame can be found where the universe looks maximally isotropic: all observers in the universe will see an isotropic universe (and it will appear that they are all at the “centre” of the universe) from wherever they look, if they are at rest in the local comoving frame.
02
2
=Γ+ds
dx
ds
dx
ds
xd ii νµ
µν
The metric connection is given by the derivative of
the metric
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Astroparticle Course 8
Conformal timeConformal time
It may seem that we treat time very differently from space in the expression of the FLRW metric. However, we can make a transformation of the time coordinate to conformal time, defined by
Conformal time does not measure the proper time for any particular observer, but it does simplify some calculations. The FLRW metric becomes
Ω−
−−= 22
2
222
1)(2 dr
rk
drdads ττ
a(τ) = a[t(τ)]
)(ta
dtd =τ
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Astroparticle Course 9
EnergyEnergy--momentum tensormomentum tensor
In the early universe the energy density was very smooth, as witnessed by the isotropy of the CMBR. It should therefore be adequate to use the perfect fluid approximation of the cosmic fluid in writing the form of the energy-momentum tensor of cosmological matter,
Since fluid elements will be comoving in the cosmological rest frame, uµ = (1,0,0,0) and
−
−
−=
p
p
pT
ρ
µν
µννµµνρ gpuupT −+= )(
uµ = fluid 4-velocity ρ/p = energy
density/pressure in the fluid rest frame
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Astroparticle Course 10
EinsteinEinstein’’s equations of gravitation (I)s equations of gravitation (I)
Considering the Ricci tensor,
σνα
µσβ
σνβ
µσα
µναβ
µνβα
µναβ ΓΓ−ΓΓ+Γ∂−Γ∂≡R
( )αβσσβασαβµσµαβ gggg ∂−∂+∂=Γ2
1
αµανµν RR ≡
and the Ricci scalar,
which results from a contraction of the Riemann tensor,
µνµν
RgR ≡
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Astroparticle Course 11
EinsteinEinstein’’s equations of gravitation (II)s equations of gravitation (II)
µνµνTconstG ⋅=
By demanding that the Newtonian limit is correctly obtained, the value of the constant can be found:
µνµνπ TGG
N⋅=8
one can form a symmetric tensor with vanishing covariant divergence, the Einstein tensor,
Einstein conjectured that Gµν is proportional to the energy-momentum tensor,
µνµνµνgRRG
2
1−=
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Astroparticle Course 12
The Friedman equation (I)The Friedman equation (I)
a
aR
&&3
00−=ijga
k
a
a
a
a
ijR
++−=2
2
2
22&&&
++−=
22
2
6a
k
a
a
a
aR
&&&
000
== ii RR
µνµν
RgR ≡
αµανµν RR ≡Remembering that
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Astroparticle Course 13
The Friedman equation (II)The Friedman equation (II)
The Friedman equation is the 0-0 component of the Einstein equation,
23
822
a
kG
a
aH N −=
≡ ρ
π&H is the Hubble parameter
and the Friedman equation we get the acceleration equation,
)3(3
4p
G
a
a N +−= ρπ&&
(ρ+3 p)0
From the i-i component of the Einstein equation,
2
2
82
a
kpG
a
a
a
aN
−−=
+ π
&&&
new effect
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Astroparticle Course 14
Mechanical analogyMechanical analogy
By Newton equation
Using uniformity, we obtain the final result:
2a
MGa N−=&&
ka
MGa N −=
22&
3
3
4aM πρ=
23
822
a
kG
a
aH N −=
≡ ρ
π&
dt
adaa
a
MGa N )(22
2
2
&&&&
&==−
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Astroparticle Course 15
Normal espansionNormal espansion
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Astroparticle Course 16
Accelerated espansionAccelerated espansion
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Astroparticle Course 17
0; =νµνT
EnergyEnergy--momentum conservationmomentum conservation
Energy conservation is expressed in GR by the vanishing of the covariant divergence of the energy-momentum tensor,
( ) ( )33 adpad −=ρ
( )pa
a+−= ρρ
&& 3
This equation is actually not independent of the Friedmann and acceleration equations, but is required for consistency.
ρνµρµ
ννµ VVV Γ+∂=;
Very simple physical interpretation: the rate of change of total energy in a volume element of size V=a3 is equal to –pdV.
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Astroparticle Course 18
Equation of stateEquation of state
)(ρpp =
Within the fluid approximation used here, we may assume that thepressure is a single-valued function of the energy density
Many useful cosmological matter sources obey a relation
ρω=p
with ω constant.
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Astroparticle Course 19
Relevant equations of state (I)Relevant equations of state (I)
For a homogeneous and isotropic fluid of particles
),()2(
)(0
3
3
qxfqqq
qdxT A
A
rr
νµµνπ
∑∫=
∑∫==A
A qfqqd
T )()2(
003
3
00π
ρr
∑∫−=−=A
A
i
i qfq
qqdpT )(
)2(3
0
0
2
3
3rr
π
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Astroparticle Course 20
Relevant equations of state (II)Relevant equations of state (II)
qqr
=0
3
ρ=p
For relativistic particles, indipendently of the fA shape,
0≈pnm=ρ
For non relativistic particles ρ is dominated by the rest mass energy, m, which is huge compared to the pressure (proportional to v). Thus, to a good approximation
In any case, a constant ω leads to a great simplification in solving our equations.
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Astroparticle Course 21
Cosmological constantCosmological constant
Adding a cosmological constant term,
-Λgµν, to Einstein’s equation is equivalent to including an energy-momentum tensor of the form
µνµνπ
gG
TN8
Λ=
that is a perfect fluid with
constGN
=Λ
=Λπ
ρ8 ΛΛ −= ρp
so that the equation of state parameter is ω=-1. We say that the cosmological constant is equivalent to vacuum energy.
The Einstein equations applied to a homogeneous and isotropic universe permit only expanding or contracting solutions. At the beginning, this was considered a failure, so that Einstein tried to modify his equation for giving static solutions too.
ρ>0 ⇒ p
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Astroparticle Course 22
Spatial curvatureSpatial curvature
It is sometimes useful to think of any nonzero spatial curvature as yet another component of the cosmological energy budget, obeying
28
3
aG
k
N
kπ
ρ −=
so that the equation of state parameter is ω=-1/3. It is not an energy density, of course; ρk is simply a convenient way to keep track of how much energy density is lacking, in comparison to a flat universe.
28 aG
kp
N
kπ
=
from Friedman equation from the i-i component of the Einstein equation
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Astroparticle Course 23
Critical densityCritical density
It is the energy density corresponding to a flat universe.
cNGH ρ
π
3
82 =
Then, the Friedman equation becomes
Nc
G
H
πρ
8
32
=
23
8
3
8
a
kGG Nc
N −= ρπ
ρπ
122
=−Ha
k
cρ
ρ
H0 = 100 h Km sec-1 Mpc-1 (0.6
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Astroparticle Course 24
Cosmic sum ruleCosmic sum rule
c
ii
ρ
ρ≡Ω
22aH
k
c
kk −=≡Ω
ρ
ρ
If we define the fractions of the critical energy density in each different component by
then the Friedman equation can be recast in the form
1=Ω+Ω+Ω+Ω Λ kRM
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Astroparticle Course 25
Universe evolution (I)Universe evolution (I)
Using the equation of state in the energy-momentum conservation, we get
Neglecting the curvature contribution in the acceleration equation (or for flat space, k=0) and using the equation of state, one gets
)1(3)(
ωρ +−∝ aa
)1(3
2
)(ω+∝ tta
accelerated universe for 1+3ωωωω
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Astroparticle Course 26
Universe evolution (II)Universe evolution (II)
Radiation domination (ω=1/3):4
)(−∝ aaρ
Matter domination (ω=0):
3)(
−∝ aaρVacuum domination (ω=-1):
consta =)(ρ
not accelerated
not accelerated
accelerated (inflation)
Stable matter will be diluted proportionally to the volume factor, a3. For radiation, there is an additional factor of a, since the energy gets red-shifted due to the expansion.
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Astroparticle Course 27
Universe evolution (III)Universe evolution (III)
Radiation domination (ω=1/3):
Matter domination (ω=0):
Vacuum domination (ω=-1):
The more general solution for an arbitrary mixture of matter, radiation and vacuum energy cannot be given in closed form, but one can consider some simple cases.
tta ≈)(
3/2)( tta ≈
tHeta ≈)(
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Astroparticle Course 28
Radiation and matter dominationRadiation and matter domination