standard cosmology i longpeople.na.infn.it/~astropar/doc/standard_cosmology_i.pdf · 2009. 9....

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


Some bibliographySome bibliography

• The Early Universe, by E.W. Kolb & M.S. Turner

• Cosmology and Particle Astrophysics, L. Bergstrom & A. Goobar

• Cosmology, S. Weinberg


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


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


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


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


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


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


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


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 ≡


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−=


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


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


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

&&&&

&==−


Normal espansionNormal espansion


Accelerated espansionAccelerated espansion


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.


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.


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


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.


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


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


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


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


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


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.


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 ≈)(


Radiation and matter dominationRadiation and matter domination

standard cosmology i longpeople.na.infn.it/~astropar/doc/standard_cosmology_i.pdf · 2009. 9....

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