Preprint typeset in JHEP style. - HYPER VERSION hep-th/
Introduction to Cosmology
M. Lenc and J. Klusoˇn
Department of Theoretical Physics and Astrophysics
Faculty of Science, Masaryk University
Kotl´aˇrsk´a 2, 611 37, Brno
Czech Republic
Abstract: This paper contains notes devoted to the Introduction to cosmology
Keywords: Cosmology.
Contents
1. Basic Principles 2
1.1 Units 2
2. Brief Introduction to General Relativity 4
2.1 Gravitational Field Equations 7
2.2 Basic principles of Cosmology 13
2.3 Map of Manifolds 13
2.4 Motion of the probe in the FRW Universe 26
2.5 Horizons 33
2.6 More about Proper and Comoving Distances 34
3. Our Universe Today 35
3.1 Matter 35
3.1.1 Hubble Tension 36
3.2 Supernovae and the Accelerating Universe 41
3.3 Dark Energy 42
3.4 Observational Evidence for Dark Energy 45
3.4.1 Luminosity distance 45
3.5 The age of the Universe and the cosmological constant 48
3.6 The Cosmological Constant Problem 50
3.7 The Cosmic Microwave Background 54
4. Early Times in the Standard Cosmology 57
4.1 Review of the building blocks of the standard cosmology and matter 57
4.2 Hot Big Bang 65
4.3 Review of the study of the expansion of the Universe 66
4.4 Epochs of the early Universe 67
4.5 Describing Matter 69
4.6 Particles in Equilibrium 70
4.7 Thermal relics 76
4.8 Status of experimental Search for Dark Matter at the Year 2017 81
4.9 Baryongenesis 82
4.10 Baryon Number Violation 85
4.11 Departure from the Thermal Equilibrium 86
4.12 Neutrino background 86
4.13 Primordial Nucleosynthesis 87
4.13.1 Rate equations 89
1
4.14 Decoupling of matter and radiation 92
4.15 Structure formation and linear perturbation theory 93
5. Inflation cosmology 96
5.1 Problems of the standard Big-Bang model 96
5.2 Problems of the standard scenario 97
5.3 Inflation as a solution 102
5.3.1 The General Idea of Inflation 102
5.4 Many models of inflation 103
5.5 How does the inflation work 106
5.6 Slowly-Rolling Scalar Fields 107
5.7 Solving the problems of standard cosmology 112
5.8 Reheating and Preheating 115
5.9 Quantum fluctuations 116
5.10 Eternal Inflation 120
5.11 Eternal Inflation: Implications 123
5.12 Does Inflation Need a Beginning 125
5.13 Inflation and Observations 125
1. Basic Principles
1.1 Units
We mostly use the natural system of units where the Planck constant, speed of light
and the Boltzman constant are equal to one
¯h = c = kB = 1 . (1.1)
Then the mass M, energy E and temperature T have the same dimensions since
[E] = [Mc2
] = [M] (1.2)
and also we have
[E] = [kBT] = [T] = [M] . (1.3)
Time t and length l have in natural system dimension [M]−1
as follows from the fact
that
[E] = [¯hω] = [ω] = [t−1
] (1.4)
so that [t] = [M]−1
. In the same way we have
[l] = [ct] = [t] = [M]−1
. (1.5)
It is useful to know coeficients that relate various units
2
Quantity SI dimensions Natuaral dimensions Conversions
mass kg M 1GeV = 1.8 × 10−27
kg
length m M−1
1GeV −1
= 0.197 × 10−15
m
time s M−1
1GeV −1
= 6.58 × 10−25
s
energy kg · m2
· s−2
M 1GeV = 5.39 × 10−19
kg · m · s−1
momentum kg · m · s−1
M 1GeV = 5.39 × 10−19
kg · m · s−1
velocity m · s−1
1 = 2.998 × 108
m · s−1
cross section m2
M−2
1GeV −2
= 0.389 × 10−31
m2
force kg · m · s−2
M2
1GeV 2
= 8.19 × 105
Newton
The traditional unit of length in cosmology is Megaparsec
1 Mpc = 3.1 × 1022
m . (1.6)
It is interesting to mention the several units of length that are used in astronomy.
Besides the metric system in use are the astronomical unit (a.u.) which is the average
distance from the Earth to the Sun
1 a.u. = 1.5 × 1011
m (1.7)
Further, there is a light year, the distance that a photon travels in one year
1 year = 3.16 × 107
s , 1 light year = 0.95 × 1016
m (1.8)
parsec (pc)-distance from which an object of size 1a.u. is seen at angle 1arc second
1 pc = 2.1 · 105
a.u. = 3.3 light year = 3.1 × 1016
m (1.9)
It is instructive to give distances of various objects expressed in above units.
10a.u. is the average disance to Saturn, 30a.u. is the same for Pluto, 100a.u. is
the estimate of the maximum distance which can be reached by solar wind (particles
emitted by the Sun). The nearest stars-Proxima and Alpha Centauri are at 1.3pc
from the Sun. The distance to Arcturus, which is the brightest star on the night
sky, is about 36 light years. The distance to Capella, which is one of the most
brightest stars on night sky, is more than 42 light years. The distances to Canopus
and Betelgeuse, which are again one of the most brightest stars on the night sky, are
about 100pc (300 l.y.) and 200pc (642 l.y.) respectively. Crab Nebula-the remnant
of supernova SN1054 is 2kpc (6523 l.y.) away from us.
The next point on the scale of distance is 8kpc. This is the distance from the
Sun to the center of our Galaxy. Our Galaxy is of spiral type, the diameter of its
disc is about 30kpc and the thickness of the disc is about 250pc. The distance to
the nearest dwarf galaxies that are satelites of our Galaxy is about 30kpc. Fifteen
of these satellites are known, the largest of them are Large and Small Magellanic
Clouds are about 50kpc away. It is also interesting to note that only eight Milky
Way satellites were known by 1994.
3
The mass density of the usual matter in usual (not dwarf) galaxies is about 105
higher than the average over Universe.
The nearest usual galaxy-the spiral galaxy M31 in Andromeda constellation- is
800kpc away from the Milky Way. Another nearby galaxy is in Triangulum constellation.
Our Galaxy together with Andromeda and Triangulum galaxies , their satelites
and other 35 smaller galaxies constitute the Local Group which is the gravitationally
bound object consisting of about 50 galaxies.
The next scale is the size of clusters of galaxies which is 1 − 3Mpc. Rich clusters
contain thounsands of galaxies. The mass density in clusters exceeds the average
density over the Universe by a factor of a hundred and even sometimes a thousand.
The distance to the center of the nearest cluster, which is the Vigo constellation, is
about 15Mpc. Clusters of galaxies are the largest gravitationally bound systems in
the Universe.
2. Brief Introduction to General Relativity
As the first step we introduce concept of covariant derivative which is an operator
which reduces to the partial derivative in flat space with Cartesian coordinates, but
transforms as a tensor on an arbitrary manifold.
In flat space in Cartesian coordinates, the partial derivative operator ∂µ is a map
from (k, l) tensor fields to (k, l+1) tensor fields, which acts linearly on its arguments
and obeys the Leibniz rule on tensor products. All of this continues to be true in
the more general situation we would now like to consider, but the map provided by
the partial derivative depends on the coordinate system used. We would therefore
like to define a covariant derivative operator to perform the functions of the
partial derivative, but in a way independent of coordinates. We therefore require
that be a map from (k, l) tensor fields to (k, l + 1) tensor fields which has these
two properties:
1. linearity: (T + S) = T + S ;
2. Leibniz (product) rule: (T ⊗ S) = ( T) ⊗ S + T ⊗ ( S) .
It can be shown that takes the form
µV ν
= ∂µV ν
+ Γν
µλV λ
, (2.1)
where (Γµ)ρ
σ (an n × n matrix known as connection coefficients, where n is the
dimensionality of the manifold, for each µ). Further, we also demand that µV ν
transform as (1, 1) tensor
µ V ν
=
∂xµ
∂xµ
∂xν
∂xν µV ν
. (2.2)
4
This requirement implies following transformation rules for metric coefficients
Γν
µ λ =
∂xµ
∂xµ
∂xλ
∂xλ
∂xν
∂xν
Γν
µλ −
∂xµ
∂xµ
∂xλ
∂xλ
∂2
xν
∂xµ∂xλ
. (2.3)
which of course is not the tensor transformation law; the second term on the right
spoils it. The covariant derivative of a one-form takes the form
µων = ∂µων − Γλ
µνωλ . (2.4)
Then it is clear that the connection coefficients encode all of the information necessary
to take the covariant derivative of a tensor of arbitrary rank. The formula is quite
straightforward; for each upper index you introduce a term with a single +Γ, and
for each lower index a term with a single −Γ:
σTµ1µ2···µk
ν1ν2···νl
= ∂σTµ1µ2···µk
ν1ν2···νl
+Γµ1
σλ Tλµ2···µk
ν1ν2···νl
+ Γµ2
σλ Tµ1λ···µk
ν1ν2···νl
+ · · ·
−Γλ
σν1
Tµ1µ2···µk
λν2···νl
− Γλ
σν2
Tµ1µ2···µk
ν1λ···νl
− · · · . (2.5)
This is the general expression for the covariant derivative.
It turns out that in order to define unique connection on a manifold with a metric
gµν we have to introduce two additional properties:
• torsion-free: Γλ
µν = Γλ
(µν).
• metric compatibility: ρgµν = 0.
This requirement implies that we can express connection coefficients as functions of
metric
Γσ
µν =
1
2
gσρ
(∂µgνρ + ∂νgρµ − ∂ρgµν) . (2.6)
This connection we have derived from the metric is the one on which conventional
general relativity is based . It is known as Christoffel connection or as the LeviCivita
connection.
Now we define the notion of parallel transport that corresponds to the motion
of a vector along a path, keeping constant all the while. The parallel transport is
defined whenever we have a connection. In fact the crucial difference between flat
and curved spaces is that, in a curved space, the result of parallel transporting a
vector from one point to another will depend on the path taken between the points.
More precisely, let us have a curve xµ
(λ) and define the covariant derivative along
the path to be given by an operator
D
dλ
=
dxµ
dλ
µ . (2.7)
5
We then define parallel transport of the tensor T along the path xµ
(λ) to be the
requirement that, along the path,
D
dλ
T
µ1µ2···µk
ν1ν2···νl
≡
dxσ
dλ
σTµ1µ2···µk
ν1ν2···νl
= 0 . (2.8)
This is a well-defined tensor equation, since both the tangent vector dxµ
/dλ and the
covariant derivative T are tensors. This is known as the equation of parallel
transport. For a vector it takes the form
d
dλ
V µ
+ Γµ
σρ
dxσ
dλ
V ρ
= 0 . (2.9)
It is clear that the notion of parallel transport depends on the connection, and
different connections lead to different answers. On the other hand since we consider
the connection that is metric-compatible we obtain that the metric is always parallel
transported
D
dλ
gµν =
dxσ
dλ
σgµν = 0 . (2.10)
Then we show that the inner product of two parallel-transported vectors is preserved.
In fact, if V µ
and Wν
are parallel-transported along a curve xσ
(λ), we have
D
dλ
(gµνV µ
Wν
) =
D
dλ
gµν V µ
Wν
+ gµν
D
dλ
V µ
Wν
+ gµνV µ D
dλ
Wν
= 0 . (2.11)
This means that parallel transport with respect to a metric-compatible connection
preserves the norm of vectors, the sense of orthogonality, and so on.
Now we are going to discuss the geodesics. To begin with we recall that the
tangent vector to a path xµ
(λ) is dxµ
dλ
. The condition that it be parallel transported
is thus
D
dλ
dxµ
dλ
= 0 , (2.12)
or alternatively
d2
xµ
dλ2
+ Γµ
ρσ
dxρ
dλ
dxσ
dλ
= 0 . (2.13)
This is the familiar geodesic equation.
It is important to stress that geodesics in general relativity are the paths followed
by unaccelerated particles. To see this note that we can think about the geodesic
equation as the generalization of Newton’s law f = ma for the case f = 0. In fact it
can be shown that the equation of motion for a particle of mass m and charge q in
general relativity takes the form
d2
xµ
dτ2
+ Γµ
ρσ
dxρ
dτ
dxσ
dτ
=
q
m
Fµ
ν
dxν
dτ
. (2.14)
6
2.1 Gravitational Field Equations
As we know in General Relativity (GR) the metric tensor is dynamical field and the
equations of GR arise as extrema conditions for the action functional. The principle
of equivalence means that all equations have to have the same form in all reference
frames. In other words we require that the action function has to be same in all
reference frames which means that the action is scalar. Since the action is given as
the integral over time of the Lagrangian we find also that the Lagrangian has to be
given as the integral over space section of the spacetime. In summary we postulate
thath the gravity action has the form
Sgr = d4
x
√
−gLgr , (2.15)
where the Lagrangian density Lgr(x) transforms under coordinate transformations
x µ
= x µ
(x) as
L (x ) = L(x) (2.16)
and due to the fact that d4
x −g (x ) = d4
x −g(x) we really see that Sgr does not
change under diffeomorphism transformations. Note that
√
−g means square root
of determinant of the metric gµν where µ, ν = 0, 1, 2, 3 and we work with the metric
with signature (−1, 1, 1, 1). In more details, the fundamental object of the general
relativity is the metric tensor gµν that transforms under coordinate transformations
x µ
= x µ
(x) as
gµν(x ) = gρσ
∂xρ
∂x µ
∂xσ
∂x ν
, (2.17)
where we presume that the transormation x µ
= x µ
(x) is invertible with inverse
xµ
= xµ
(x ).
The simplest possibility is to take the Lagrangian density to be equal to constant
L = −Λ so that
SΛ = −Λ d4
x
√
−g . (2.18)
However this action does not contain the time derivatives of the metric and hence
the dynamics that would follow from this action is trivial. For that reason we should
search more complicated form of the Lagrangian density.
The Lagrange density is a tensor density, which can be written as
√
−g times a
scalar that is function of the metric and its derivatives. The question is the form of
given scalar. Since we know that the metric can be set equal to its canonical form and
its first derivatives set to zero at any one point, any nontrivial scalar must involve at
least second derivatives of the metric. The Riemann tensor is of course made from
second derivatives of the metric, and we argued earlier that the only independent
scalar we could construct from the Riemann tensor was the Ricci scalar R. What we
did not show, but is nevertheless true, is that any nontrivial tensor made from the
metric and its first and second derivatives can be expressed in terms of the metric
7
and the Riemann tensor. Therefore, the only independent scalar constructed from
the metric, which is no higher than second order in its derivatives, is the Ricci scalar.
Hilbert figured that this was therefore the simplest possible choice for a Lagrangian,
and proposed
LH =
√
−gR . (2.19)
The equations of motion should come from varying the action with respect to the
metric. In fact let us consider variations with respect to the inverse metric gµν
,
which are slightly easier but give an equivalent set of equations. Using R = gµν
Rµν,
in general we will have
δS = dn
x
√
−ggµν
δRµν +
√
−gRµνδgµν
+ Rδ
√
−g
= (δS)1 + (δS)2 + (δS)3 . (2.20)
The second term (δS)2 is already in the form of some expression times δgµν
; let’s
examine the others more closely.
Recall that the Ricci tensor is the contraction of the Riemann tensor, which is
given by
Rρ
µλν = ∂λΓλ
νµ + Γρ
λσΓσ
νµ − (λ ↔ ν) . (2.21)
We perform the variation of the Riemann tensor in such a way that we firstly perform
variation of the connection coefficients and then we substitute into this expression.
In fact, after some calculations we find the variation of the Riemann tensor in the
form
δRρ
µλν = λ(δΓρ
νµ) − ν(δΓρ
λµ) . (2.22)
Therefore, the contribution of the first term in (2.20) to δS can be written
(δS)1 = d4
x
√
−g gµν
λ(δΓλ
νµ) − ν(δΓλ
λµ)
= d4
x
√
−g σ gµσ
(δΓλ
λµ) − gµν
(δΓσ
µν) , (2.23)
where we have used metric compatibility. However the integral above is an integral
with respect to the natural volume element of the covariant divergence of a vector;
by Stokes’s theorem, this is equal to a boundary contribution at infinity which we
can set to zero by making the variation vanish at infinity. Therefore this term does
not contribute to the total variation.
In order to calculate the (δS)3 term we have to use the variation
δ(g−1
) =
1
g
gµνδgµν
. (2.24)
and consequently
δ
√
−g = −
1
2
√
−ggµνδgµν
. (2.25)
8
If we now return back to (2.20), and remembering that (δS)1 does not contribute,
we find
δS = d4
x
√
−g Rµν −
1
2
Rgµν δgµν
. (2.26)
However this should vanish for arbitrary variations and consequently we derive Einstein’s
equations in vacuum:
1
√
−g
δS
δgµν
= Rµν −
1
2
Rgµν = 0 . (2.27)
However we would like to get the non-vacuum field equations as well. In other words
we consider an action of the form
S =
1
8πG
SH + SM , (2.28)
where SM is the action for matter, and we have presciently normalized the gravitational
action (although the proper normalization is somewhat convention-dependent).
Following through the same procedure as above leads to
1
√
−g
δS
δgµν
=
1
8πG
Rµν −
1
2
Rgµν +
1
√
−g
δSM
δgµν
= 0 , (2.29)
and we recover Einstein’s equations if we set
Tµν = −
1
√
−g
δSM
δgµν
. (2.30)
In fact (2.30) turns out to be the best way to define a symmetric energy-momentum
tensor.
Einstein’s equations may be thought of as second-order differential equations for
the metric tensor field gµν. There are ten independent equations (since both sides are
symmetric two-index tensors), which seems to be exactly right for the ten unknown
functions of the metric components. However, the Bianchi identity µ
Gµν = 0 which
we prove below represents four constraints on the functions Rµν, so there are only six
truly independent equations. In fact this is appropriate, since if a metric is a solution
to Einstein’s equation in one coordinate system xµ
it should also be a solution in
any other coordinate system xµ
. This means that there are four unphysical degrees
of freedom in gµν (represented by the four functions xµ
(xµ
)), and we should expect
that Einstein’s equations only constrain the six coordinate-independent degrees of
freedom.
It is important to stress that as differential equations, these are extremely complicated;
the Ricci scalar and tensor are contractions of the Riemann tensor, which
involves derivatives and products of the Christoffel symbols, which in turn involve
the inverse metric and derivatives of the metric. Furthermore, the energy-momentum
tensor Tµν will generally involve the metric as well. The equations are also nonlinear,
9
that implies that two known solutions cannot be superposed to find a third. It is
therefore very difficult to solve Einstein’s equations in any sort of generality. Then
in order to solve them we have to perform some simplifying assumptions. The most
popular sort of simplifying assumption is that the metric has a significant degree of
symmetry, and we will talk later on about how symmetries of the metric make life
easier.
We are mainly interested in the existence of solutions to Einstein’s equations
in the presence of “realistic” sources of energy and momentum. The most common
property that is demanded of Tµν is that it represent positive energy densities —
no negative masses are allowed. In a locally inertial frame this requirement can be
written as ρ = T00 ≥ 0. We write it in the coordinate-independent notation as
TµνV µ
V ν
≥ 0 , for all timelike vectors V µ
. (2.31)
This is known as the Weak Energy Condition, or WEC. It seems like a reasonable
requirement however it is very restrictive. Indeed it is straightforward to show that
there are many examples of the classical field theories which violate the WEC, and
almost impossible to invent a quantum field theory which obeys it. Nevertheless, it
is legitimate to assume that the WEC holds in most cases and it is violated in some
extreme conditions. (There are also stronger energy conditions, but they are even
less true than the WEC, and we won’t dwell on them.)
An important property of the energy momentum tensor is that it is conserved.
In the flat background the conservation equation takes the form
∂µTµν
= 0 , (2.32)
where the first equation ∂µTµ0
= 0 expresses the conservation of the energy density
while the remaining three equations ∂µTµi
= 0 defines the conservation of the
momentum density. In general relativity the conservation equation takes the form
µTµν
= 0 . (2.33)
This equation can be proved using the equation of motion for the metric when we
apply the covariant derivative on both sides of this equation
µ
Rµν −
1
2
gµνR = 8πG µ
Tµν . (2.34)
We show that the left side of this equation is identically zero. Note that generally the
matter fields do not have to be on shell since this equation follows from the variation
of the action with respect to the metric. To see this we recall the Bianchi identity
for the Riemann tensor
ρRλ
σµν + νRλ
σρµ + µRλ
σνρ = 0 . (2.35)
10
Now we contract λ and µ indices and by definition Rµ
σµν = Rσν we obtain the identity
ρRσν − νRρσ + λRλ
σνρ = 0 . (2.36)
Then we contract this equation with gρσ
and we obtain
0 = ρRρ
ν − νR + λ
Rλν = 2 µ
(Rµν −
1
2
gµνR) = 0 . (2.37)
which implies that the covariant conservation law of the stress energy-tensor is a
necessary condition for the consistency of the Einstein equation.
On the other hand the stress energy tensor is determined by the matter action.
Clearly when we search the extremum of the action we perform the variation of the
action with respect to the matter fields so that the energy momentum tensor should
be conserved as the consequence of the matter equations of motions as well. Alternatively,
we can presume the evolution of the matter fields on the fixed background
and in this case the energy-momentum tensor should be conserved as well.
To proceed note that the matter action is diffeomorphism invariant so that the
conservation of the energy momentum tensor should follow from the invariance of the
action under general diffeomorphism transformation. In fact, under transformation
x µ
= xµ
+ ξµ
. (2.38)
Then
g µν
(x ) = gρσ ∂x µ
∂xρ
∂x ν
∂xσ
⇒
g µν
(x ) = gµν
(x) + gνλ
(x)∂λξµ
+ ∂λxµ
gλν
(x)
(2.39)
If we expand
g µν
(x ) = g µν
(x + ξ) = g µν
(x) + ∂λg µν
ξλ
= g µν
(x) + ∂λgµν
ξλ
(2.40)
we find the variation gµν
as
δgµν
(x) = g µν
(x) − gµν
(x) = −∂λgµν
(x)ξλ
+ gµλ
∂λξν
+ ∂λξµ
gλν
. (2.41)
Now we proceed to the transformation property of the matter fields. Their form
depends on the character of these fields, whether they are scalars, vectors,..... For
example, in case of the scalar field we find
φ (x ) = φ(x) ⇒ φ (x) − φ(x) = −∂λφξλ
(2.42)
Since the action is invariant under the diffeomorphism invariance we obtain
δξSm =
1
2
d4
x
√
−gTµν( µ
ξν
+ ν
ξµ
) + d4
x
√
−g
δLm
δψ
δψξ = 0 , (2.43)
11
where we also used the fact that the variation of the metric can be written as
g µν
− gµν
= µ
ξν
+ ν
ξµ
(2.44)
Note that the equation (2.43) has to be zero of shell. Let us now presume that
the matter field equations are satisfied which implies that the second term in (2.43)
vanishes. Then using integration by parts we can rewrite (2.43) into the form
δξSm(on shell) = − d4
x
√
−gξµ µ
Tµν = 0 (2.45)
that using the fact that ξµ
is arbitrary implies the conservation of the stress energy
tensor.
We continue with the study of the Einstein equations where we now discuss the
possibility of the introduction of a cosmological constant. In order to introduce it we
add it to the conventional Hilbert action. We therefore consider an action given by
S = d4
x
√
−g(R − 2Λ) , (2.46)
where Λ is some constant. The resulting field equations are
Rµν −
1
2
Rgµν + Λgµν = 0 , (2.47)
and of course there would be an energy-momentum tensor on the right hand side if
we had included an action for matter. Λ is the cosmological constant. In order to
find its meaning it is convenient to move the additional term in (2.47) to the right
hand side, and think of it as a kind of energy-momentum tensor, with Tµν = −Λgµν
(it is automatically conserved by metric compatibility). Then Λ can be interpreted
as the “energy density of the vacuum,” a source of energy and momentum that
is present even in the absence of matter fields. This interpretation is important
because quantum field theory predicts that the vacuum should have some sort of
energy and momentum. In ordinary quantum mechanics, an harmonic oscillator with
frequency ω and minimum classical energy E0 = 0 upon quantization has a ground
state with energy E0 = 1
2
¯hω. A quantized field can be thought of as a collection of
an infinite number of harmonic oscillators, and each mode contributes to the ground
state energy. The result is of course infinite, and must be appropriately regularized,
for example by introducing a cutoff at high frequencies. The final vacuum energy,
which is the regularized sum of the energies of the ground state oscillations of all the
fields of the theory, has no good reason to be zero and in fact would be expected to
have a natural scale
Λ ∼ m4
P , (2.48)
where the Planck mass mP is approximately 1019
GeV, or 10−5
grams. Observations
of the universe on large scales allow us to constrain the actual value of Λ, which turns
12
out to be smaller than (2.48) by at least a factor of 10120
. This is the largest known
discrepancy between theoretical estimate and observational constraint in physics,
and convinces many people that the “cosmological constant problem” is one of the
most important unsolved problems today. On the other hand the observations do
not tell us that Λ is strictly zero, and in fact allow values that can have important
consequences for the evolution of the universe.
2.2 Basic principles of Cosmology
In this section we review basic facts about classical cosmology, following mainly [3].
There are many reviews available on hep-th, see for example [4, 5, 6] 1
. Contemporary
cosmological modes are based on the idea that the Universe is pretty much the same
everywhere-the idea known as Copernican principle. It is clear that this principle
can be applied on the large scales only where local variations of density is averaged
over.
In other words, we can define cosmology as physics on very large scales that
vary from 1kpc which is typical size of galaxies to 1Mpc that is typical intergalactic
distance in the local group of galaxies up to thousands of Mpc that corresponds to
the most distant observed light source which is cosmic microwave background.
It is important to stress that the galaxy distribution in our neighbourhood (about
tens of Mpc) is not regular and it is characterized by clusters of matter. On the
other hand observation on much large distances (hundreds of Mpc) suggests that the
Universe is homogeneous and isotropic with very good approximation. Moreover, this
presumption is even more supported by observation of Cosmic microwave background
that is radiation background of cosmic origin that goes back to 13 billions ago when
matter is transparent to radiation. Today the temperature of CMB is T0 = 2.7255 ±
0, 0006 K with very small variation T
T
∼ 10−5
that means that the early Universe
is homogeneous and isotropic.
The main conclusion of these arguments is that the Universe is spatially homogeneous
and isotropic on the largest scales. Since these claims need more
explanation let us pause in our explanation of cosmology and give some more precise
definition of mathematical claims given above.
2.3 Map of Manifolds
Since we do not have enough time with explanation of the notion of manifold we
presume that reader has enough knowledge regarding this point.
Let M and N be manifolds (generally with different dimensions) and let φ :
M → N be a map. In a natural manner, φ ”pulls back” a function f : N → R on N
to the function f ◦ φ → M → R that is derived by composing f with φ. Similarly,
1
Our metric signature is − + ++. We use units ¯h = c = 1 and define the reduced Planck mass
by Mp = (8πG)−1/2
≈ 1018
GeV .
13
in a natural way, φ maps tangent vectors at p ∈ M to tangent vectors at φ(p) ∈ N.
In other words it defines ma φ∗
: Vp → Vf(p) in following way: For V Vp we define
φ∗
(v) by
(φ∗
(v))(f) = v(f ◦ φ) (2.49)
for all smooth f : N → R. It is easy to see that φ∗
v satisfies the properties of tangent
vector at φ(p). Further, in the coordinate bases of a coordinate system (xν
) at p and
a coordinate system (yµ
) at φ(p) the upper expression takes the form
wµ
(y)
∂
∂yµ
f(y) = vν
(x)
∂
∂xν
f((φ(x))) = vν
(x)
∂f(y)
∂yµ
∂yµ
∂xν
⇒
wµ
(φ(x)) = vν
(x)
∂yµ
∂xν
, (φ∗
v)µ
≡ wµ
.
(2.50)
In the same way we can use φ to ”pull back” one forms at φ(p) to one forms at p.
We define the map (”pull back”) φ∗ : V ∗
φ(p) → V ∗
p by requiring that for v ∈ Vp
(φ∗ω)µvµ
= ων(φ∗
v)ν
, (2.51)
where we used tensor notation. Using the definition of the map φ∗
given in (2.50)
we easily get
(φ∗ω)µ = ων
yν
∂xµ
. (2.52)
We can easily extend the action of φ∗ to map tensors of type (0, l) at φ(p) to tensors
of type (0, l) at p by
(φ∗T)µ1...µl
vµ1
1 . . . vµl
l = Tµ1...µl
(φ∗
v1)µ1
. . . (φ∗
vl)µl
. (2.53)
In the same way we can extend the action of φ∗
to map tensors of type (k, 0) at p to
tensors of type (k, 0) at φ(p) by
(φ∗
T)µ1...µk
(ω1)µ1 . . . (ωk)µk
= Tµ1...µk
(φ∗ω1)µ1 . . . (φ∗ωk)µk
(2.54)
If φ : M → M is diffeomorphism and T is a tensor field on M we can compare T
with φ∗
T. If φ∗
T = T then even though we have moved T via φ it is still the same.
In other words φ is a symmetry transformation for the tensor field T. In the case of
the metric gµν a symmetry transformation-a diffeomorphism φ such that
(φ∗
g)µν = gµν
is called an isometry.
Let us now return to our explanation of basic principles of cosmology. Our first
task is to formulate precisely the mathematical meaning of this assumption. The
evidence comes from the smoothness of the temperature of the cosmic microwave
14
background. In other words, given any two points p and q there is an isometry
which takes p into q. We must mention that there is no necessary relationship
between homogeneity and isotropy; a manifold can be homogeneous but nowhere
isotropic (such as R × S2
in the usual metric) or it can be isotropic around a point
without being homogeneous (such as a cone, which is isotropic around its vertex but
certainly not homogeneous). On the other hand, if a space is isotropic everywhere
then it is homogeneous. On the other hand it should be pointed that, in general,
at each point, at most one observer can see the universe as isotropic. For example,
if ordinary matter fills the universe, any observer in motion relative to the matter
must see an anisotropic velocity distribution of the matter. With this fact in mind
we have to give precise formulation of the notion of isotropy than the clam that
Isotropy is the claim that the Universe looks the same in all directions.: A spacetime
is said to be (spatially) isotropic at each point if there exists a congruence of timelike
curves (observes) with tangent vectors denoted uµ
filling the spacetime and
satisfying the following property. Given any point p and any two unit spatial tangent
vectors sµ
1 , sµ
2 ∈ Vp (In other words vector that are orthogonal to uµ
) there exists an
isometry of gµν that leaves p and uµ
at p fixed but rotates sµ
1 into sµ
2 . Thus, in
an isotropic universe it is impossible to construct a geometrically preferred tangent
vector orthogonal to uµ
. Then we can see that in the case of a homogeneous and
isotropic spacetime the surface Σt of homogeneity must be orthogonal to the tangents
uµ
to the world-lines of the isotropic observers. Now the space-time metric gµν
induces a Riemannian metric hµν(t) on each Σt by restricting the action of gµν at
each p ∈ Σt to vectors tangent to Σt. The induced spatial geometry of the surfaces
Σt is greatly restricted by the following requirements:
• Due to the homogeneity, there must be isometries of hµν that carry any p ∈ Σt
into any q ∈ Σt.
• Due to the isometry it must be impossible to construct any geometrically preferred
vectors on Σt.
Since there is observation evidence for isotropy and the Copernican principle says
that we are not the center of the Universe and therefore observers elsewhere should
also observe an isotropy all cosmological models are based on the existence of homogeneity
and isotropy of manifold. However it is important to stress that this claim
is not certainly true. The Universe is apparently not static, but changing in time.
Therefore the cosmological models are based on the idea that the Universe is homogeneous
and isotropic in space but not in time. This means that the Universe can
be foliated into space-like surfaces such that each slice is homogeneous and isotropic.
Then it is natural to consider our space-time to be R × Σ where R represents the
time direction and Σ is a homogeneous and isotropic three-manifold. Since we may
think of isotropy as invariance under rotation and homogeneity as invariance under
15
translation we get that Σ must be a maximally symmetric space. More precisely, the
homogeneity and isotropy imply that the space has its maximum possible number of
Killing vectors. Therefore we can write the metric in the form
ds2
= −dt2
+ a2
(t)γij(x)dxi
dxj
. (2.55)
Here t is time-like coordinate and (x1
, x2
, x3
) are the coordinates on Σ where γij is
the maximally symmetric metric on Σ. The function a(t) is known as scale factor
that tells us how big the space-like slice Σ is at the moment t. The coordinates used
here in which the metric is free of cross terms dtdxi
and the space-like components
are proportional to a single function of t are known as comoving coordinates and
an observer who stays at constant xi
is also called as “comoving”. Only comoving
observer will think that the Universe looks isotropic.
It is important to stress that these observers, that are at rest to this frame are
geodesic which means that they are free. Note that for these particles (observers) we
have ds2
= −dt2
as follows from the fact that dxi
= 0 which implies that t has the
meaning of the proper time for particles at rest.
We show that the world-line xi
= const obeys the geodesic equation in the metric
(2.55). Note that the geodesic equation takes the form
duµ
dλ
+ Γµ
νλuν
uλ
= 0 , (2.56)
where uµ
is 4− velocity
dxµ
dλ
(2.57)
and where λ is the parameter along the world-line of the particle. To begin with we
calculate the Christoffel symbols
Γµ
νλ =
1
2
gµσ
(∂νgλσ + ∂λgνσ − ∂σgνλ) . (2.58)
For the metric (2.55) we have following non-zero components
g00 = −1 , gij = a2
(t)γij (2.59)
with the inverse components
g00
= −1 , gij
=
1
a2(t)
γij
, (2.60)
where
γij
γjk = δi
k . (2.61)
It can be shown that the only non-zero components of Γµ
νλ are
Γi
0j =
1
2
gik
∂0gjk =
˙a
a
δi
j , Γ0
ij = −a˙aγij , Γi
jk = (3)
Γi
jk , (2.62)
16
where (3)
Γi
jk are the Christoffel symbols for metric γij.
Let us now again consider the equation (2.56). The only non-zero component of
the 4−velocity uµ
= dxµ
dλ
of the particle at rest is
u0
=
dx0
dλ
(2.63)
Now the on-shell condition implies
uµ
uν
gµν = −1 ⇒
dx0
dλ
= 1 . (2.64)
Then clearly (2.56) is obviously satisfied since du0
dλ
= 0 and Γµ
00 for all µ. In other
words the world-lines of particles which are at rest in our reference frame are indeed
geodesic.
As we have shown in introduction the maximally symmetric Euclidean threemetric
γij obey
R
(3)
ijkl = k(γikγjl − γilγjk) , (2.65)
where k is some constant and the superscript on the Riemann tensor reminds to us
that it is associated with the three metric γij not to the metric of entire space-time.
Then the Ricci tensor is
R
(3)
jl = γik
R
(3)
ijkl = 2kγjl . (2.66)
Since the space is maximally symmetric then it will certainly be spherically symmetric
as well. For such a space-time the metric can be put in the form
dσ2
= γijdxi
dxj
= e2β
dr2
+ r2
(dθ2
+ sin2
θdφ2
) . (2.67)
The Ricci tensor for the metric given above has components
R
(3)
11 =
2
r
∂rβ ,
R
(3)
22 = e−2β
(r∂rβ − 1) + 1
R
(3)
33 = [e−2β
(r∂rβ − 1) + 1] sin2
θ .
If we compare these expressions to (2.66) we can solve for β(r):
2
r
∂rβ = 2ke2β
⇒ 2dβe−2β
= 2kr ⇒ β = −
1
2
ln(C − kr2
) ,
e−2β
(r∂1β − 1) + 1 = 2kr2
⇒ e−2β
(r2
ke2β
− 1) + 1 = 2kr2
⇒
⇒ −e−2β
+ 1 = kr2
⇒ C = 1
(2.68)
and the third equation is identically solved. Then we obtain following metric on
space-time:
ds2
= −dt2
+ a2
(t)
dr2
1 − kr2
+ r2
(dθ2
+ sin2
θdφ2
) . (2.69)
17
This form of metric is known as Friedman-Robertson-Walker metric (FRW).
Then the Einstein equations will determine the behavior of the scale factor a(t). We
can also easily see that the metric is invariant under the scaling transformations:
k →
k
|k|
,
r → |k|r ,
a →
a
|k|
.
(2.70)
Therefore it is clear that the only relevant parameter is k/|k| and there are three
cases of interest: k = −1 , k = 0 and k = 1. The case k = −1 corresponds to
constant negative curvature on Σ and is called open, the case k = 0 corresponds no
curvature on Σ and is called flat ; the case k = 1 corresponds to positive curvature
on Σ and is called closed. Now we will examine these possibilities in more details:
• For k = 0 the metric on Σ is
dσ2
= dxidxi
, i = 1, 2, 3 (2.71)
that is simply the Euclidean space. Globally, it could describe R3
or more
complicated manifold, as for example three torus S1
× S1
× S1
.
• For k = 1 we define
r = sin ξ , dr = cos ξdξ (2.72)
and hence the metric on Σ can be written as
dσ2
= dξ2
+ sin2
ξdΩ2
(2.73)
which is the metric of three sphere. In this case the only possible global structure
is actually three sphere.
• The case k = −1 we can write
r = sinh ψ (2.74)
and the metric on Σ is
dσ2
= dψ2
+ sinh2
ψdΩ2
(2.75)
which is the metric of three dimensional space of constant negative curvature.
Globally such a space can extend forever but it can also describe a non-simply
connected compact space.
18
In order to solve the Einstein’s equations of motion we have to calculate the Christoffel’s
symbols for the metric ansatz (2.69). If we denote ˙a ≡ da
dt
then these symbols
are given by
Γ0
11 =
a˙a
1 − kr2
, Γ0
22 = a˙ar2
, Γ0
33 = a˙ar2
sin2
θ ,
Γ1
01 = Γ2
02 = Γ2
20 = Γ3
03 = Γ3
30 =
˙a
a
,
Γ1
22 = −r(1 − kr2
) , Γ1
33 = −r(1 − kr2
) sin2
θ ,
Γ2
12 = Γ2
21 = Γ3
13 = Γ3
31 =
1
r
,
Γ2
33 = − sin θ cos θ , Γ3
23 = Γ3
32 = sin θ .
(2.76)
After simple calculations we can find following nonzero components of the Ricci
tensor
R00 = −3
¨a
a
,
R11 =
a¨a + 2˙a2
+ 2k
1 − kr2
,
R22 = r2
(a¨a + 2˙a2
+ 2k) ,
R33 = r2
(a¨a + 2˙a2
+ 2k) sin2
θ .
(2.77)
Then the Ricci scalar is equal to
R = gµν
Rνµ =
6
a2
(a¨a + ˙a2
+ k) . (2.78)
Since Universe is not empty we are not interested in the vacuum Einstein equations.
Rather we must study the solutions of the Einstein’s equations that contain the
nontrivial right hand side. The standard model with we begin is the Universe filled
by a perfect fluid that is defined as fluids that are isotropic in their rest frame. The
energy momentum tensor for a perfect fluid can be written
Tµν = (p + ρ)UµUν + pgµν , (2.79)
where p and ρ are energy density and pressure as measured in the rest frame and Uµ
is the four-velocity of the fluid. It is clear that if a fluid which is isotropic in some
frame leads to a metric which is isotropic in some frame, the two frames will coincide,
that is the fluid will be in rest frame in comoving coordinates. The four-velocity is
then
Uµ
= (1, 0, 0, 0) , (2.80)
19
and the energy tensor is
Tµν =






ρ 0 0 0
0
0 gijp
0






. (2.81)
If we raise its index we obtain
Tµ
ν = gµκ
Tκν = diag(−ρ, p, p, p) (2.82)
and note that the trace is equal to
T ≡ Tµ
µ = −ρ + 3p . (2.83)
For letter purposes it is also instructive to consider the zero component of the conservation
of the stress energy tensor
0 = µTµ
0 = ∂µTµ
0 + Γµ
µ0T0
0 − Γλ
µ0Tµ
λ =
= −∂0ρ − 3
˙a
a
(ρ + p) .
(2.84)
To proceed it is necessary to choose the equation of state, the relation between ρ and
p. It appears that all perfect fluids relevant to cosmology obey the simple equation
of state
p = wρ , (2.85)
where w is constant independent on time. Then the conservation of energy becomes
˙ρ
ρ
= −3(1 + w)
˙a
a
(2.86)
that can be integrated and we obtain
ρ = a−3(1+w)
. (2.87)
The most interesting examples of cosmological are dust and radiation. Dust is
characterized with w = 0. Examples include ordinary stars and galaxies where the
pressure is negligible in comparison with the energy density. Dust is also known as
matter and Universes whose energy is mostly due to dust are known as matterdominated.
The energy density in matter falls as
ρ ∼ a−3
(2.88)
that can be interpreted as the decrease in the number density of particles as the
Universe expands. (For dust the energy density is dominated by the rest energy that
is proportional to the number density.)
20
The second form of the fluid, Radiation may be used to describe either actual
electromagnetic radiation, or massive particles moving at relative velocities sufficiently
close to the speed of light so that they become indistinguishable from photons.
The stress energy tensor of the radiation can be expressed in terms of the field
strength as
Tµν
=
1
4π
Fµλ
Fν
λ −
1
4
gµν
Fλσ
Fλσ . (2.89)
Then the trace of this stress energy tensor is
T = Tµν
gνµ =
1
4π
Fµλ
Fµλ −
(4)
4
Fλσ
Fλσ = 0 (2.90)
Since this should be also equal to (2.83) we get that
p =
1
3
ρ . (2.91)
An Universe in which most of the energy density is in the form of radiation is known
as radiation-dominated. The energy density in radiation then falls off as
ρ ∼ a−4
. (2.92)
This result implies that the energy density of radiation falls of faster than that in
matter. It is believed that today the energy density of the Universe is dominated by
matter with ρmat/ρrad ∼ 106
. However in the past the Universe was much smaller
and the energy density in radiation would have dominated at very early times.
There is also one important form of energy density that is sometimes considered,
namely that of the vacuum itself. Introducing energy into the vacuum is equivalent
to introducing a cosmological constant so that Einstein’s equations with cosmological
constant are
Eµν = 8πGTµν − Λgµν (2.93)
that is clearly the same form as the equations with no cosmological constant but an
energy-momentum tensor for the vacuum
Tvac
µν = −
Λ
8πG
gµν . (2.94)
This has form of the perfect fluid with
ρ = −p =
Λ
8πG
(2.95)
that implies that w = −1 and from (2.87) we see that the energy density is independent
on a. Since the energy density of matter and the radiation decreases as the
Universe expands, if there is nonzero vacuum energy it tends to wind over the long
term. If this happens we say that the Universe became vacuum-dominated.
21
Now we turn to the Einstein’s equations. Recall that they can be written in the
form
Rµν = 8πG Tµν −
1
2
gµνT . (2.96)
The µν = 00 components is
−3
¨a
a
= 4πG(ρ + 3p) , (2.97)
and the µν = ij equations give
¨a
a
+ 2
˙a
a
2
+ 2
k
a2
= 4πG(ρ − p) . (2.98)
Using (2.97) we simplify (2.98) as
˙a
a
2
=
8πG
3
ρ −
k
a2
. (2.99)
(2.99) together with (2.97) are known as Friedmann equations. More precisely, the
equation (2.99) contains a term proportional to the energy density and a pure GR
term (which we cannot obtain in a Newtonian derivation) proportional to spatial
curvature. On the other hand the equation (2.97) is about of acceleration in the
universe expansion. In order to solve this equation we should know the equation of
state of the matter p = p(ρ).
Now we introduce some terminology considering cosmological parameters. The
rate of expansion is characterized by the Hubble parameter
H =
˙a
a
. (2.100)
The value of the Hubble parameter at present epoch is the Hubble constant, H0.
There is also the deceleration parameter
q = −
a¨a
˙a2
(2.101)
that measures the rate of change of the rate of expanding. Another useful parameter
is the density parameter
Ω =
8πG
3H2
ρ =
ρ
ρcrit
, (2.102)
where the critical density is defined by
ρcrit =
3H2
8πG
. (2.103)
This quantity, that is generally time dependent, is called critical density because the
Friedmann equation (2.99) can be written as
Ω − 1 =
k
H2a2
, (2.104)
22
where generally H is time dependent. The sign of k is therefore determined by
whether Ω is greater than, equal to, or less than one. In other words, we have
ρ < ρcrit ⇒ Ω < 1 ⇒ k = −1 → open ,
ρ = ρcrit ⇒ Ω = 1 ⇒ k = 0 → flat ,
ρ > ρcrit ⇒ Ω > 1 ⇒ k = 1 → closed .
(2.105)
It is useful to know the qualitative behavior of various possibilities of the solutions of
the Friedman equations. Let us for the moment set Λ = 0 and consider the behavior
of Universe filled with fluids of positive energy ρ > 0 and nonnegative pressure p > 0.
Then (2.97) implies that ¨a < 0 . Since we know from observation that the Universe
is expanding (˙a > 0) this means that the Universe is decelerating which could be
intuitively expected since the gravitation attraction of the matter in the Universe
works against the expanding. The fact that the Universe is decelerating means
that it must have been expanding even faster in the past; if we trace the evolution
backward in time, we reach the singularity at a = 0. Notice that if ¨a were exactly
zero, a(t) would be straight line a(t) = Ct (we have chosen the integration constant
that at t = 0, a(0) = 0 and hence H(t) = ˙a
a
= 1
t
so that H−1
0 would determine the
age of the Universe.
The singularity at a = 0 is known as Big Bang. It represents the creation of
Universe from a singular space, not explosion of matter into a pre-existing spacetime.
Since for a → 0 the energy density becomes arbitrary high we do not expect
classical general relativity to give a correct description of nature in this regime.
The future evolution is different for different k. For the open and flat cases
k = −1, 0 the (2.99) implies
˙a2
=
8πG
3
ρa2
+ |k| . (2.106)
Since the right hand side is strictly positive so ˙a never passes through zero. Since
˙a > 0 today it follows that ˙a > 0 for all time. Thus open and flat Universes expand
forever-they are temporally and spatially open. It is however important to keep in
mind that this works on the presumption of nonzero positive energy density. Negative
energy density Universes do not have to expand forever, even if they are open.
The question is how fast these Universes keep expanding? Let us now consider
the quantity ρa3
(recall that this is constant in matter dominated Universe). Using
the conservation of energy (2.84) we get
d
dt
(a3
ρ) = a3
(3
˙a
a
ρ + ˙ρ) = −3pa2
˙a
(2.107)
23
that implies that
d
dt
(a3
ρ) < 0 . (2.108)
This result implies that a2
ρ must go to zero in an ever-expanding Universe where
a → ∞ 2
Then (2.106) implies that
˙a2
→ |k| . (2.109)
(We must stress that it holds for k = −1, 0. Thus for k = −1 an expanding approaches
the limiting value ˙a → 1 while for k = 0 the Universe keeps expanding but
more and more slowly.
For the closed Universe (k = 1) (2.99) implies
˙a2
=
8πG
3
ρa2
− 1 . (2.110)
It is clear that the argument that ρa2
→ 0 as a → ∞ still holds. In this case the
right hand side of the upper equation becomes negative which clearly cannot happen.
Therefore the Universe does not expand indefinitely, a posses an upper bound amax.
As a approaches amax the equation (2.97) implies
¨a = −
4πG
3
(ρ + 3p)amax < 0 (2.111)
and hence ¨a is finite and negative at this point, so a reaches amax and starts decreasing.
Since ¨a < 0 it will inevitably continue to contract to zero- the Big Crunch.
Thus, the closed Universe (on presumption of positive ρ and non negative p) is closed
in time as well as space.
We will now list some of the exact solutions corresponding to only one type of
energy density. For dust-only Universe (p = 0) it is convenient to define a development
angle φ(t), rather than using t as a parameter directly. The solutions are
then, for open Universes;
a =
C
2
(cosh φ − 1) , t =
C
2
(sinh φ − φ) , k = −1 , (2.112)
for flat Universes
a =
9C
4
1/3
t2/3
, k = 0 , (2.113)
and for closed Universes
a =
C
2
(1 − cos φ) , t =
C
2
(φ − sin φ) , k = +1 , (2.114)
2
For example, when a(t) ∼ t we should have ρ ∼ t−4
at least and hence a2
ρ ∼ t−2
→ 0 for
t → ∞.
24
where we have defined
C =
8πG
3
ρa3
= constant . (2.115)
For Universes filled with nothing but radiation, p = 1
3
ρ, we have once again open
Universes,
a =
√
C

 1 +
t
√
C
2
− 1


1/2
, k = −1 (2.116)
flat Universes,
a = (4C )1/4
t1/2
, k = 0 (2.117)
and closed Universes,
a =
√
C

1 − 1 −
t
√
C
2


1/2
, k = +1 (2.118)
where we have defined
C =
8πG
3
ρa4
= constant . (2.119)
Let us now consider the case of nonzero cosmological constant. We start with
Λ < 0. In this case Ω is negative and we get that k = −1. The solution in this case
is
a =
−3
Λ
sin


−Λ
3
t

 . (2.120)
There is also an open (k = −1) solution for Λ > 0 given by
a =
3
Λ
sinh


Λ
3
t

 . (2.121)
A flat vacuum-dominated Universe must have Λ > 0 and the solution is
a ∼ exp

±
Λ
3
t

 (2.122)
while the closed Universe must also have Λ > 0 and satisfies
a =
3
Λ
cosh


Λ
3
t

 . (2.123)
These solutions are a little misleading. In fact the three solutions for Λ > 0 (2.121),(2.122),(2.123)-all
represent the same space-time, just in different coordinates.
This space-time, known as de Sitter space is maximally symmetric as a
space-time. The Λ < 0 solution is also maximally symmetric and is known as antide
Sitter space
25
Before we conclude this section we spend some time with the discussion of the
situation when the matter sector in Universe constitutes more general form of matter.
For example, we can presume that all components of the matter are present. Then
the total density parameter takes the form
Ω =
i
Ωi (2.124)
and the Friedman equation can be written as
Ω − 1 =
k
H2a2
. (2.125)
As in the particular previous example we obtain that the sign of k is determined
whether Ω is greater than, equal to, or less than one. Explicitly, we have
ρ < ρcrit ⇒ Ω < 1 → k = −1 , open ,
ρ = ρcrit ⇒ Ω = 1 → k = 0 , flat ,
ρ > ρcrit ⇒ Ω > 1 → k = 1 , closed .
(2.126)
Since ρi ∼ a−ni
we have
ρi
ρj
=
Ωi
Ωj
= a−(ni−nj)
(2.127)
so that relative amount of energy in different components changes as the Universe
evolves.
2.4 Motion of the probe in the FRW Universe
In order to understand properties of given background it is common strategy to study
the dynamics of the probe in given background. Let us then consider the motion of
particle in the FRW Universe.
Let us consider the action for the massive particle
S = − dλ −ˆgµνuµuν , uµ
=
dxµ
dλ
. (2.128)
where λ is parameter that labels the world-line. We introduce einbain e(λ) so that
the action takes the form
S =
1
2
dλ[
1
ˆgµνuµ
uν
− m2
] , (2.129)
To see the equivalence between these two formulations we perform the variation with
respect to that gives
−
1
2
ˆgµνuµ
uν
− m2
= 0 ⇒ =
1
m
−ˆgµνuµuν (2.130)
26
that inserting back to the action we obtain the original action. Further, the equation
of motion with respect to xµ
gives
−2
d
dλ
(
1
ˆgµνuν
) +
1
∂µˆgρσuρ
uσ
= 0 (2.131)
It is important to stress that the action is invariant under λ = f(λ) so that dλ =
df
dλ
dλ. We can fix the gauge by imposing = 1
m
so that we obtain on-shell condition
ˆgµνuµ
uν
= −1 (2.132)
Note that this relation allows us to write (when ˆg0u = 0)
−1 = (−ˆg00 + ˆgij
dxi
dt
dxj
dt
)(
dt
dλ
)2
⇒
dt
dλ
=
1
ˆg00 − ˆgijvivj
, vi
≡
dxi
dt
(2.133)
Then the equation of motion for xµ
takes the form
duµ
dλ
+ ˆgµν
∂ρˆgνσuσ
uρ
−
1
2
ˆgµν
∂ν ˆgρσuρ
uσ
=⇒
duµ
dλ
+
1
2
ˆgµν
(∂ρˆgνσ + ∂σ ˆgνρ −
1
2
∂ν ˆgρσ)uρ
uσ
= 0 ⇒
d2
xµ
d2λ
+ Γµ
ρσ
dxρ
dλ
dxσ
dλ
= 0 .
(2.134)
It is also interesting to insert the solution of the equation of motion into the action
so that it takes the form
S =
1
2
dλ[
1
ˆgµνuµ
uν
− m2
] =
=
m
2
dt ˆg00 − ˆgijvivj[−(ˆg00 − ˆgijvi
vj
)(
dt
dλ
)2
− 1] =
= −m dt ˆg00 − ˆgijvivj .
(2.135)
It is also interesting to analyse the equation of motion that follows from the original
action
S = −m dλ −gMN ∂λXM ∂λXN (2.136)
The equations of motion have the form
−
∂KgMN
˙XM ˙XN
2 −gMN
˙XM ˙XN
+ ∂λ


gKN
˙XN
−gMN
˙XM ˙XN

 = 0 . (2.137)
27
This is equation of motion for X. Let us denote the variation of the action with
respect to X as δS
δX
. If we multiply given expression with ˙XK
we obtain
δS
δXK
˙XK
= −
∂λgMN
˙XM ˙XN
2 −gMN
˙XM ˙XN
+ ∂λ(
1
−gMN
˙XM ˙XN
) ˙XK
gKM
˙XN
+
+
1
−gMN
˙XM ˙XN
˙XK
∂λgKM
˙XN
+
1
−gMN
˙XM ˙XN
˙XK
gKM
¨XN
= 0
(2.138)
Note that it holds as identity and not as a consequence of the equations of motion.
Let us now consider the flat FRW background
ds2
= −dt2
+ a2
(t)δijdxi
dxj
. (2.139)
so that the action takes the form
S = −m dt 1 − a2δij ˙xi ˙xj .
(2.140)
It is interesting to determine the Hamiltonian formulation of this system
pi =
δL
δ ˙xi
= a2
m
δij ˙xj
1 − a2δij ˙xi ˙xj
. (2.141)
Then we find
H = pi ˙xi
− L =
ma2
1 − a2δij ˙xi ˙xj
= a2 1
a2
piδijpj + m2
(2.142)
using
a2
˙xi
δij ˙xj
=
piδij
pj
m2a2 + piδijpj
(2.143)
Now the equation of motion takes the form
˙xi
= xi
, H =
δij
pj
a−2piδijpj + m2
,
˙pi = {pi, H} = 0 ⇒ pi = ki .
(2.144)
We see that the momentum pi is constant. On the other hand the norm of state
slows since the norm is given as pigij
pj = 1
a2 kiδij
kj.
28
On the other hand let us introduce following variable
Xi
= axi
, ˙xi
=
1
a
( ˙Xi
− HXi
) (2.145)
Using these variables we find the action in the form
S = −m dt 1 − ( ˙Xi − HXi)δij( ˙Xj − HXj) . (2.146)
The meaning of the variables Xi
can be found when we take the non-relativistic limit
where we replace
√
1 − A = 1 − 1
2
A2
so that the action
Snonrel = −m dt + dt
m
2
( ˙Xi
− HXi
)δij( ˙Xj
− HXj
) =
= dt
m
2
˙Xi ˙Xi + . . . ,
(2.147)
where we neglected the remaining terms. Comparing this expression with the standard
form of the non-relativistic Lagrangian we can interpret Xi
= a(t)xi
as the
physical variable even if we mean that both variables are physical.
Now from (2.146) we determine the momenta conjugate to Xi
Pi =
δL
δ ˙Xi
= m
δij( ˙Xj
− HXj
)
(. . .)
(2.148)
and hence the Hamiltonian takes the form
H = ˙Xi
Pi − L =
m
(. . .)
+ PiXi
H = m2 + PiPi + PiXi
H
(2.149)
Using this Hamiltonian we derive the equation of motion
˙Xi
= Xi
, H =
Pi
√
m2 + PiPi
+ Xi
H ,
˙Pi = {Pi, H} = −PiH
(2.150)
The last equation can be integrated as
dPi = −Pi
da
a
⇒ ln Pi = − ln a + ln Ki ⇒ Pi =
Ki
a
. (2.151)
We see that the ”physical” momentum Pi is red shifted as the universe expands. Note
that we can also find the time dependence of Xi
by integrating the first equation
since it takes generally the dorm
˙Xi
= Fi
(t) + G(t)Xi
(2.152)
29
so that we search the solution of the homogeneous equation
˙Xi
= G(t)Xi
⇒ Xi
= Ci
exp( dtG(t)) (2.153)
Note that we have
dtG(t) =
da
dt
1
a
dt =
da
a
= ln a ⇒ e dtG(t)
= eln a
= a . (2.154)
Then we say that Ci
depends on time so we obtain that it has to obey the equation
dCi
dt
= e− dt G(t )
F(t) ⇒
dCi
dt
=
Ki
a
√
m2a2 + KiKi
(2.155)
that can be in principle integrated if we know the time dependence of a. There is a
particulary simple solution corresponding to the particle with zero physical momentum
when Ki = 0. From upper equation we immediately find that Ci
= Ci
= const
and hence
Xi
= Ci
a (2.156)
that is an expected result. The physical interpretation of this result is that particle
slows down with respect to comoving coordinates as the Universe expands (since
a → ∞). In fact this is an actual slowing down, in the sense that a gas of particles
with initially high relative velocities will cool down as the Universe expands.
Very interesting is the case of the particle with null mass which is photon. In
principle we could use the the action for the massive particle written without the
square root and then take the limit m → 0 however we will be more conservative and
consider the standard treatment of the electromagnetic wave in curved background.
We consider the action of free electromagnetic field
S = −
1
4
d4
x
√
−ggµρ
gνσ
FµνFνσ , Fµν = µAν − νAµ = ∂µAν − ∂νAµ (2.157)
Consider now the propagation of a photon in the homogeneous isotropic Universe.
Since the photon wavelength is small compared to the spatial curvature radius even
if the Universe is open or closed. Then we can consider the metric that is spatially
flat
ds2
= −dt2
+ a2
(t)δijdxi
dxj
. (2.158)
Let us introduce conformal time η instead of t that is defined as
dt = adη (2.159)
or equivalently
η =
dt
a(t)
. (2.160)
30
This result can be generally integrated so that we have η = η(t) and we presume
that this relation can be inverted so that t = t(η) and consequently a = a(η). Now
the metric has the form
ds2
= a2
(η)[−dη2
+ δijdxi
dxj
] (2.161)
and we see that the metric element in FRW spacetime is conformally flat in the sense
that
gµν = a2
(η)ηµν , (2.162)
where the Minkowski metric is spanned by coordinates (η, xi
). Then we clearly have
gµν
= a−2
ηµν
,
√
g = a4
(2.163)
and we find that in η, xi
coordinates the action of the electromagnetic field has the
form
S = −
1
4
d4
xηµρ
ηνσ
FµρFνσ . (2.164)
Now it is clear that the solution of the equation of motion for the free electromagnetic
field in the Universe is given as the superposition of the plane waves
A(α)
µ = e(α)
µ eikη−ikx
(2.165)
where k is constant vector, |k| = k and e(α)
µ is the standard polarization vector of
photons with α = 1, 2. Note that k is not the physical frequency as follows from
following arguments. The quantity x = 2π
k
is the coordinate wavelength of a photon
while the physical wavelength at time t is
λ(t) = a(t) x = 2π
a(t)
k
. (2.166)
In the same way we define period η = 2π
k
of electromagnetic wave in conformal
time while the period of the physical time is
T = a(t) η = 2π
a(t)
k
. (2.167)
Then we see that the frequency is equal to
ω(t) =
2π
T
=
k
a(t)
(2.168)
and since we know that the frequency is equal to the magnitude of the physical
momentum of photon we obtain that the physical momentum depends on time as in
case of the massive particle namely
p =
k
a(t)
(2.169)
31
We see that in the expanding universe the scale factor a(t) is growing and hence
the physical wavelength grows. On the other hand the physical momentum is decreasing
function of time. The phenomena when the wavelength is growing during
the expansion of the Universe is named as the redshift. Explicitly, if the photon was
emitted at time ti with physical wave length λi in the physical process as for example
when the electron in the excited state in the atom drops to the ground state which
is certainly physical process. Now we know that the state propagates freely as in
(2.165) and then it is again detected in time t0 where t0 we means the present time
in the reversed physical process when its physical wave length now is
λ(t0) = a(t0)
2π
k
(2.170)
Now expressing 2π
k
using the physical wave length at time of emission we find the
famous relation
λ(t0) =
a(t0)
a(ti)
λi ≡ λi(1 + z(ti)) . (2.171)
The quantity
z(ti) =
a(t0)
a(ti)
− 1 (2.172)
is called redshift. The earlier the object emits the photon then this photon has to
travel longer and consequently a(ti) is smaller and hence object at larger distances
have the larger redshifts.
Note that these formulas are valid in general for all z. Let us now consider
objects that are not in large distance. Then the difference t − t0 is not very large
and we can expand
a(ti) = a(t0) − ˙a(t0)(t0 − ti) (2.173)
Using the present value of the Hubble parameter H0 = ˙a(t0)
a(t0)
≡ ˙a0
a0
we can write
a(ti) = a0[1 − H0(t0 − ti)] (2.174)
so that to the linear order we find following expression for the redshift
z(ti) =
1
1 − H0(t0 − ti)
− 1 H0(t0 − ti) . (2.175)
Finally the travel time is equal to
0 = −dt2
+ a(t)2
dr2
= −dt2
+ (a0 − ˙a0(t0 − t))2
dr2
≈
−dt2
+ a2
0dr2
⇒ (t0 − t) = a0(ri − r0) ≡ R
(2.176)
where R is the physical distance of the object from the our observer. Inserting this
expression into (2.175) we derive famous Hubble law
z = H0R , z 1 . (2.177)
32
The redshift is something that can be measured, we know the rest-frame wavelengths
of various spectral lines in the radiation of distant galaxies, so that we can determine
how much their wavelengths have changed along the path from time ti when they
were emitted to time t0 when they were observed. We therefore know the ratio of
the scale factors at these two times however we do not know the times themselves.
2.5 Horizons
One of the most crucial concepts of the FRW Universe is the existence of horizons.
Suppose a emitter, e sends a light signal to an observer o, who is at r = 0.
Restricting to the radial geodetic (that means that dφ = dθ = 0 we obtain from the
vanishing of the metric elements the equation for null geodetics in the form
ds2
= 0 = a2
(η)(−dη2
+ dr2
) ⇒ η = ±r + r0 , (2.178)
where η is conformal time. Let us presume that the light hits the observer at time η0
that is larger that ηe where ηe is time when this signal was emitted. Since for η = ηo
we have r = 0 we get ηo = r0 and consequently η − ηo = ±r. Since also for ηe this
equation implies
ηo − ηe = re
and we obtain that we should choose the positive sign in front of r since ηo − ηe > 0
and r is positive. Finally we get the relation
ηo − ηe = re . (2.179)
Let us now presume that ηe is bounded from below by ˜ηe; for example ˜ηe might
represent the Big Bang singularity. Then there exist a maximum distance to which
the observer can see, known as a particle horizon distance given by
rph(ηo) = ηo − ˜ηe (2.180)
Similarly, suppose that ηo is bounded from above by ˜ηo. Then there exists a limit
to space-time events which can be influenced by the emitter. This limit is known as
the event horizon distance given by
reh(ηo) = ˜ηo − ηe (2.181)
This horizon distance may be converted to proper horizon distances at cosmic time
t. For example, we have an emitter at time ˜ηe at re = 0. Since dη = dt
a(t)
we obtain
η − ˜ηe =
t
te
dt
a(t )
(2.182)
using also the fact that the proper distance at time t is given by multiplication with
a(t) we get the proper horizon distance as
Dh = a(t)
t
te
dt
a(t )
. (2.183)
33
2.6 More about Proper and Comoving Distances
In this section we discuss in more details two important definitions in cosmology,
which are comoving and proper distances.
Let us start with the definition of comoving coordinates. As we defined previously
comoving coordinated are very natural in cosmology and simply corresponds
to coordinate system where observers, that are static, see Universe as isotropic. It
is important to stress that this isotropy, particularly isotropy of CMB, defines special
local frame of reference known as comoving frame. On the other hand any
observer, that moves with relative velocity with respect to comoving frame see different
regions of the sky either blue or red shifted. Finally velocity of observer with
respect to local comoving frame is known as peculiar velocity. In fact, galaxies
and other clusters of matter generally move with respect to comoving frame however
their peculiar velocities are very small so that they are nearly comoving.
Another important notion is comoving time coordinate is the elapsed time
since the Big Bang and it is proper time of comoving observer. This is the measure of
cosmological time. In summary for two comoving galaxies their comoving distance
remain constant for all times.
Proper distance To begin with let us again write FRW metric when we restrict
ourselves to the radial coordinate only,
ds2
= −dt2
+ a2
(t)dr2
. (2.184)
Then we define Proper distance D, at time t between an observer at the origin and a
distant galaxy as the distance along the surface of constant t, so that dt = 0. Then,
using (2.184) can be easily integrated and we get
D(t) = a(t)R , (2.185)
where we again stress that this is calculated at the constant time t. Performing
derivative of this distance with respect to t we get recession velocity
vrec(t, z) ≡ ˙D(t) = ˙a(t)R(z) =
˙a(t)
a(t)
a(t)R = H(t)D(t) . (2.186)
where R(z) is fixed comoving distance of galaxy observed at redshift z. In more
details, we define Comoving distance with the help of traveling photons that obey
ds = 0. Then the comoving distance between observer at the origin and the galaxy
at the redshift at z(t) we set ds = 0 and integrate to get
dr =
dt
a
⇒ R(tem) =
tob
tem
dt
a(t )
(2.187)
However we can parametrize this distance using observable quantity which is redshifht
that is defined as the ration of the scale factor at the time of observation tobs
34
to the scale factor at the time of emision t
a(tobs)
a(t)
= 1 + z (2.188)
Diferenting this relation with respect to t we get
dz
dt
= −
a(tobs)
a2(t)
˙a(t) ⇒
dz
a(tobs)H(z)
= −
dt
a(t)
, (2.189)
where we parameterize H using z instead of t. Once again, H(z) is the Hubble
constant at the time of the object with redshifht z. Using this relation we finally get
comoving distance of the object at the redshift z in the form
R(z) = −
0
z
dz
a(tobr)H(z )
=
z
0
dz
a(tobs)H(z )
. (2.190)
In other words we see direct relation between comoving distance and redshifht.
Let us return to the expression D(t) = a(t)R and presume that now R depends
on time. Then performing derivation with respect to t we get
˙D = ˙aR + a ˙R = H(t)D(t) + a(t) ˙R (2.191)
that can be interpreted as
vtot = vrec + vpec , (2.192)
where vpec is pecular velocity defined above.
3. Our Universe Today
In this section we will discuss the remarkable properties that have been discovered
in past few years. Most remarkable among them is the fact that the universe is dominated
by a uniformly- distributed and slowly varying source of ”dark energy” which
may be a vacuum energy (cosmological constant), a dynamical field or something
completely different.
3.1 Matter
The inventory of constituencies comprising actual Universe is complicated by the fact
that they are not at all equally visible. In the years before we knew the dark energy
was an important constituent of the Universe and before observations of galaxy and
distributions and CMB anisotrophies observational cosmology measured two numbers:
The Hubble constant H0 and the matter density parameter ΩM . Measuring the
extragalactic distances is very difficult, but most current measurement of the Hubble
constant performed Planck experiment in 2013 gives the value of the cosmological
constant to be equal to
H0 = 67.80 ± 0.77 km/sec/Mpc , (3.1)
35
where
1Mpc = 106
parsec = 3 × 1024
cm . (3.2)
We see that the Hubble parameter in fact has the dimension [t−1
] so that it has the
value
H−1
0 = h−1
· 3 · 107
s = h−1
· 1010
yrs ≈ 1.4 · 1010
yrs , (3.3)
where h is a dimensionless parameter
h = 0.678 . (3.4)
In particle physics units (¯h = c = 1) this is equal to
H0 ∼ 10−33
eV . (3.5)
It is convenient to express the Hubble constant as
H0 = 100 h km/sec/Mpc . (3.6)
It turns out that the scale H−1
0 gives order of magnitude of the age of the Universe
and the distance scale H−1
0 is roughly the size of the observable part equal to
H−1
0 ≈ h−1
· 3000Mpc ≈ 4.3 · 103
Mpc . (3.7)
Note that since ρi = 3H2
0 Ωi/8πG measurement of ρi is often expressed as measurement
of Ωih2
. The Hubble constant provides the rough measure of the scale of
the Universe since in the matter or radiation dominated Universe is t0 ∼ H−1
0 .
3.1.1 Hubble Tension
As we know, Universe is expanding and in fact, its expansion is accelerating. Generally
the rate of expansion is given by function H(z) where the Hubble factor H0 gives
the rate at which the scale factor a(z) in a Friedmann equation is changing today, corresponding
to z = 0. There are several methods how we can find value of H0 but we
are not going into details. The determination of H0 from the cosmic microwave background
corresponds to the early time z = 1000 yields H0 = 67.36±0.54 kms−1
Mpc−1
. On the other hand measurements based on late-time observables z ≤ 10, for example,
on type Ia supernovova, give the value H0 = 73, 3 ± 0.8 kms−1
Mpc−1
. This disagreement
between high-redshift and low-redshift measurements is known as Hubble
tension which was discovered rather recently. Indeed, this tension has existed since
the
first release of results from Planck in 2013 and has grown in significance with
the improvement of the data. The resolution of this tension is not known and it is
remarkable that there are about 102
theories that want to explain Hubble tension
that however require untested modifications of the standard model of particle physics
36
or general relativity. On the other hand there is more conservative possibility that
is based on the idea that the treatment of systematic errors is inadequate.
For years, determinations of ΩM based on dynamics of galaxies and clusters have
leaded to values of ΩM between 0.1 and 0.4. Alteratively, the determination of ΩM is
the same as the determination of the baryons. Recent measurements suggest that
baryons contribute to Ω as
ΩB = 0.05 . (3.8)
In other words baryons constitute rather small fraction of the present energy density
in the Universe. It is also important to stress that the most of the baryons in our
Universe are dark: direct measurements of th mass density of stars give an estimate
Ωstars ∼ 0.005 (3.9)
that is about an order of magnitude smaller than ΩB. The fact that most of the
baryons are dark follows from the dynamics of individual galaxies implies that there
is even matter there. The implied existence this celebrated dark matter is confirmed
by applying the viral theorem to clusters of galaxies, by looking at the temperature
profiles of clusters, by ”weighing” clusters by gravitational lensing and by large-scale
motions of clusters between galaxies. On the other hand there is nothing dramatic
about this observation: baryons may hide in dust and neutral gas clouds, brown
dwarfs etc.
The next form of matter are Photons. They however contribute even smaller
fraction
Ωγ ≈ 6 · 10−4
. (3.10)
From electric neutrality the number density of electrons is about the same 3
as that
of baryons, but then due to their very small mass their contribution to the total mass
fraction is negligible.
The remaining known stable particles are neutrinos. As we will sketch bellow
their number density is calculable in Hot Big Ban theory and these calculations
are confirmed by Big Bang Nucleosynthesis. The number density of each type of
neutrinos is
nνa = 115
1
cm3
, (3.11)
where νa = νe, νµ, ντ . Direct limit on the mass of electron neutrino mνe < 2.6 eV
together with the observations of neutrino oscillations suggests that every type of
neutrino has mass smaller than 2.6 eV . Then the estimation of the energy density
of neutrinos is
ρν,total =
α
mνα nνα < 8 · 10−7 GeV
cm3
(3.12)
3
There are also neutrons whose number is somewhat smaller than the number of protons.
37
that implies
Ων,total < 0.16 . (3.13)
However this estimate does not make use any cosmological date. In fact cosmological
observations give stronger bound
Ων,total < 0.01 . (3.14)
In terms of the neutrino masses this bound reads
mνa < 0.42eV (3.15)
so that every neutrino has to be lighter than 0.14eV . On the other hand atmospheric
neutrino data and further experiments tell that the mass of at least one neutrino must
be larger than 0.02eV . These results suggest that there is window for measuring
neutrino masses by cosmological observations.
We see that most of the energy density in the present Universe is not in the
form of known particles, most energy in the present Universe has to be in something
“unknown”. In fact essentially every known particle in he Standard Model of particle
physics has been ruled out as a candidate for this “unknown” matter. Moreover, there
is a strong evidence that this “something unknown” has two components: clustered
dark energy and unclustered dark energy.
It is believed that Clustered dark matter consists of new stable massive particles.
These make clumps of energy density that encounter for much of the mass
of galaxies and most of the mass of galactic clusters. There are number of ways of
estimating the contribution of non-baryonic dark matter into the total density of the
Universe:
• Composition of the Universe affects the angular anisotropy of cosmic microwave
background (CMB). The present measurements of the CMB anisotropy enable
to estimate the total mass density of dark matter.
• The density of non-baryonic dark matter is crucial for structure formation of
the Universe. If we compare the results of numerical simulations of structure
formation with observational data gives reliable estimate of the mass density
of non-baryonic clustered dark matter.
One of the few things we know about the dark matter is that it must be “cold”not
only is it non-relativistic today, but it must have been that way for a very long
time. The other thing we know about cold dark matter (CDM) is that it should
interact very weakly with ordinary matter, so as to have escaped detection thus far.
In summary the non-baryonic cold dark matter has
ΩCDM ≈ 0.25 . (3.16)
38
There is a direct evidence that dark matter exists in the largest gravitationally bound
objects-clusters of galaxies. There are various methods to determine the gravitating
mass of a cluster and even mass distribution in a cluster, which give consistent results,
for example:
• We measure velocities of galaxies in galactic clusters and make use of the gravitational
virial theorem
Kinetic energy of a gravity= 1
2
Potential energy .
In this way we obtain the gravitational potential and thus the distribution of
the total mass in a cluster.
• The second example of the measurement of masses of clusters use the notion
of intra-cluster gas. Its temperature that is determined from X−ray measurements
is also related to the gravitational potential through the virial theorem.
• The third example of measurement is based on observation of gravitational
lensing of background galaxies by clusters.
Finally, dark matter exists also in galaxies. Its distribution is measured by the
observations of rotation velocities of distant stars and gas clouds around a galaxy.
At present there are many hypotheses considering candidates for this form of
dark matter. One such an idea is that the natural candidates are particles which
participate in weak interactions that of course needs more detailed justification.
Unclustered dark energy
Non-baryonic clustered dark matter is not the whole story. If we use the above
estimates we obtain an estimate for the energy density of all particles
Ωγ + ΩB + Ωµtotal
+ ΩCDM ≈ 0.3 . (3.17)
Since the observation that ΩT ≈ 1 implies that 70 percent of the energy density is
unclustered.
In fact this result nicely fits recent observations. Indeed, it can be shown that
neither relativistic nor non-relativistic matter can lead to the accelerated expansion
of the Universe 4
. In other words the accelerated expansion requires energy stored
in something dramatically different from conventional particles and it has to have
negative pressure. In fact the analysis of the entire set of cosmological date in terms
of dark energy with phenomenological equation of state
p = wρ , w = const (3.18)
4
We will discuss this problem in the next subsection.
39
gives
ΩΛ = 0.72 ± 0.02 (3.19)
(here subscript Λ refers to dark energy) and
−1.2 < w < −0.8 . (3.20)
It is worth noting that the vacuum value, w = −1 is right in the middle of the allowed
region that corresponds to a vacuum energy density
ρΛ ∼ (10−3
eV )4
. (3.21)
Given the significance of these results it is natural to ask what level of confidence we
should have in them. There are potential sources of systematic error and these were
discussed in the original papers [1, 2]. On the other hand the recent measurements
of the cosmic microwave background confirmed the picture outlined above with the
matter density and nonzero cosmological constant.
In summary, the composition of the present Universe is fairly complex. It is challenging
for future physics that most of the energy density comes from species which
particle physicists are unfamiliar with: vacuum or vacuum-like dark energy and nonbaryonic
clumped dark matter. This poses serious problems for both fundamental
physics and cosmology:
• What are the particles of non-baryonic dark matter?
Currently popular option is the lightest supersymmetric particle that is stable
in many supersymmetric extensions of the Standard model. Of course there
are many other options, such as axions, gravitinos and so on. In any case
experimental discovery of the dark matter particle would be great achievement
of both particle physics and cosmology.
• Why there are baryons and no anti-baryons in our Universe?
Alliteratively,what is the origin of matter-antimatter asymmetry of the Universe?
We will discuss this issue later and here we notice only that the solution
of this problem is based on extension of the Standard Model.
• Why the mass density of the non-baryonic dark matter is so similar
to the mass density of baryons?
Both these densities scale as a−3
(t) so their ratio stays constant during most
of the evolution of the Universe. Then it is possible that mechanism which
create baryons and dark matter particles in the early Universe are related to
each other so that the approximate equality of the mass densities is not a
mere coincidence. On the other hand it is difficult to construct corresponding
particle model.
40
• What is the origin of dark energy? If this is vacuum,why vacuum
has non-zero energy density, which, however, is very small by particle
physics standard?
This is one of the most fundamental problems of the microscopic physics. In
natural units the vacuum density is about
ρc ∼ 10−46
GeV 4
. (3.22)
On the other hand we would expect on the basis of the dimensional grounds that
the vacuum energy takes value 1GeV 4
(QCD-scale) or 108
GeV 4
(electroweak
scale). It is great challenge to explain this enormous discrepancy but despite
numerous attempts it remains an open problem.
• Why now?
The energy density of non-relativistic dark matter and dark energy scales differently:
The non-relativistic dark matter scales as a−3
(t) while the latter stays
approximately constant. Hence at early times (small a(t)) the energy density
of non-relativistic matter exceeded by far the dark energy density. Conversely,
future expansion of the Universe will be dominated by dark energy. On the
other hand these energy densities are of the same order of magnitude today.
The question is why is this the case? What is special about the present epoch
of the evolution of the Universe?
3.2 Supernovae and the Accelerating Universe
The first hint that the matter does not dominate the Universe came from the studies
of the Type Ia supernovae that are commonly recognized as ”standard candles”.
The special property of Supernovae Type Ia is that it has nearly uniform intrinsic
luminosity (absolute magnitude M ∼ −19.5). It turns out that they can be detected
at high redshifts (z ∼ 1) that allows in principle a good handle on cosmological
effects.
The importance of the supernovae measurements began to be clear from the
works of two independent groups that observed distant supernovae in order to measure
cosmological parameters: the High-Z Supernova Team and the Supernova Cosmology
Project. These groups obtained the dependence of the redshift on apparent
magnitude. These dates are much better fit by a universe dominated by a cosmological
constant than by a flat matter-dominated model. In fact, the supernova results
alone allow huge range of possible values of ΩM and ΩΛ. On the other hand if we
presume that we know something about one of these parameters the second one will
be tightly constrained and in particular they imply (3.19).
Since these observations are very fundamental one has to ask the question about
the level of confidence of them. In fact there are number of potential sources of
41
systematic error that have been considered by these two research teams. In summary
these results are commonly accepted with their significant predictions considering the
vacuum energy of the Universe.
3.3 Dark Energy
It appears that the most difficult problem to solve is the origin of the dark energy. The
most disappointing possibility would be that the carrier of dark energy is vacuum:
The difficulties with this option will be discussed below.
Another option, more promising from the observational viewpoint is that dark
energy is due to some light field. In fact, there are good reasons to consider the this
dynamical dark matter as an alternative to cosmological constant. Firstly, the dynamical
energy density can evolve slowly to zero so that we can solve the cosmological
constant problem .
The simplest possibility how to describe dark matter is the same kind of source
that is involved in models of inflation in the very early Universe; a scalar field φ
rolling slowly in a potential, something known as quintessence.
As an example, consider a homogeneous scalar field φ(t) in an expanding Universe.
The action of the scalar field is
S = − d4
x
√
−g
1
2
gµν
∂µφ∂νφ + V (φ) , (3.23)
where V (φ) is potential. The equations of motions that follow from the action above
have the form
∂µ[
√
−ggµν
∂νφ] −
√
−g
δV
δφ
= 0 (3.24)
that for homogeneous field in an expanding Universe takes the form
¨φ + 3H ˙φ +
dV
dφ
= 0 . (3.25)
In order to take the back-reaction of this scalar field on the Einstein equations into
account we have to determine the components of the stress energy tensor. In field
theory the stress energy tensor is defined as
Tµν = −
2
√
−g
δSmatter
δgµν
(3.26)
that for the action of the form S = − d4
x
√
−gL takes the form
Tµν = −gµνL + 2
δL
δgµν
, (3.27)
where we have used
δ
√
−g
δgµν
= −
1
2
√
−ggµν . (3.28)
42
More precisely, for the action (3.23) the stress energy tensor takes the form
Tµν = ∂µφ∂νφ − gµν
1
2
gαβ
( αφ)( βφ) + V (φ) . (3.29)
Let us now restrict to the homogeneous case in which all quantities depend only on
cosmological time t and we also set k = 0. A homogeneous real scalar field behaves
as a perfect fluid with
ρ = T00 =
˙φ2
2
+ V (φ) . (3.30)
The other components of the stress energy tensor take the form
Tij = −gij(
1
2
gµν
∂µφ∂νφ + V ) + ∂iφ∂jφ . (3.31)
If we define pressure as
p =
1
3
3
i=1
Tii (3.32)
we get
p =
˙φ2
2
− V (φ) . (3.33)
Thus any state which is dominated by the potential energy of a scalar field will have
negative pressure.
If the slope of the potential V is quite flat we will have solutions for which φ is
nearly constant and only evolving very gradually with time, the energy density in
such a configuration is
ρφ ≈ V (φ) ≈ const. (3.34)
Thus we see that slowly-rolling scalar field is an appropriate candidate for dark energy
with the vacuum equation of state
pφ = −ρφ (3.35)
but the energy density ρφ slowly decreases in time. But this proposal raises several
questions: why the genuine vacuum energy density is zero (constant part of the
potential V0) so that it does not contribute to dark energy density? What is the
physics behind the field φ? Where does the small energy scale, V (φ) ∼ 10−46
GeV
today, come from? All these questions remain unanswered 5
.
In fact, it is important to stress that introducing dynamics opens up the possibility
to bring new problems that depend on form and specific kind of model being
considered. Most quintessence models feature scalar fields φ with masses of order
the current Hubble scale
mφ ∼ H0 ∼ 10−33
eV . (3.36)
5
For certain scalar potentials the fourth question can be explained.
43
In quantum field theory the light scalar fields are unnatural, renormalization
effects tend to drive scalar masses up to the scale of new physics. It is then very
difficult to understand the origin of masses of such a small value when we know
that the scale of new physics is approximately 1011
eV . Moreover, light scalar fields
give rise to long-range forces and time-dependent coupling constant that should be
observable. Therefore we have to invoke additional fine-tunings to explain why the
quintessence field has not already been experimentally detected.
Another possibility, how to explain today acceleration of Universe, is that there
is nothing special about the present era; rather acceleration is just something that
happens from time to time. This can be enforced by oscillating dark energy. In these
models the potential takes the form of a decaying exponential with small perturba-
tions
V (φ) = e−φ
[1 + α cos φ] . (3.37)
Another models of quintessence are k-essence models that are based on presumption
that the scalar field φ has the form
K = f(φ)g( ˙φ2
) , (3.38)
where f, g are functions specified by the model. Unfortunately, in neither the kessence
models nor the oscillating models do we have a compelling particle-physics
motivation for the chosen dynamics and in both cases the behavior still depends
sensitively on the precise form of parameters and interactions chosen.
Given the challenge of the problem it is worthwhile considering the possibility
that cosmic acceleration is not due to some kind of stuff but rather arise from new
gravitational physics.
As a first attempt, consider the simplest correction to the Einstein-Hilbert action,
S =
M2
p
2
d4
x
√
−g R −
µ4
R
+ d4
x
√
−gLM , (3.39)
where µ is a new parameter with units of [mass] and LM is the Lagrangian density
for matter. The equations arising from this action are complicated and it is difficult
to solve them. It is convenient to transform from the action used in (3.39) which
we call the matter frame to the Einstein frame where the gravitational Lagrangian
takes the Einstein-Hilbert form and the additional degrees of freedom ( ¨H and ˙H) are
represented by a fictitious scalar field φ. In terms of the new metric gµν the theory is
that of a scalar field φ(x) minimally coupled to Einstein gravity and non-minimally
coupled to matter with the potential
V (φ) = µ2
M2
p exp

−2
2
3
φ
Mp

 exp


2
3
φ
Mp

 − 1 . (3.40)
44
Yet another option for the explaining the accelerated expansion of our Universe is
that gravity deviates from General Relativity at cosmological distances and time
scales so that the Friedmann equation is not valid at present epoch. Finally, any
modification of the Einstein-Hilbert action must, of course, be consistent with the
classic solar system tests of gravity theory as well as numerous other astrophysical
dynamical tests. In known Lorentz-Invariant examples of such a theory there either
exist ghosts (fields with negative energy unbounded from below) or gravity becomes
strongly coupled at quantum level. A consistent theory of this sort would probably
require “gravitational Higgs mechanism” and violation of Lorentz-invariance but even
this-rather exotic idea- has not yet lead to a consistent model that would be able to
explain the accelerated expansion of the Universe.
In summary, there are many models whose aim is to explain current acceleration
area. All of these models have many problems however it is certainly very important
to study them.
3.4 Observational Evidence for Dark Energy
In this section we briefly review facts considering observational evidence for dark
energy. The first one is based on so named Luminosity distance
3.4.1 Luminosity distance
In 1998 the accelerated expansion of the Universe was reported on the observations of
Type Ia Supernova (SN Ia).This observations are based on the existence of redshift in
the expanding Universe that is related to the fact that the light emitted by a stellar
object becomes red-shifted due the expanding of the Universe. The wavelength λ
increases proportionality to the scale factor a according to the formula
1 + z =
λ0
λ
=
a0
a
, (3.41)
where z is named as redshift and where the subscript zero denotes the quantities
given at present epoch.
Another important concept that is related to the observational tools in an expanding
background is the definition of the distance. In fact there are many ways
how to define distance in expanding Universe. For example, we can consider comoving
distance as a distance measured in comoving variables. It turns out that this
distance does not change during the evolution of the Universe. On the other hand
we can define physical distance that scales proportionally to the scale factor. An
alternative way of defining of distance is through the luminosity distance that plays
a very important role in astronomy, including supernova observations.
Let us consider for a moment Minkowski space-time and define an absolute luminosity
Ls of source that is related to the energy flux F at the distance d from the
45
source by the formula
F =
Ls
4πd2
. (3.42)
We can generalize this relation to the expanding Universe and define the luminosity
distance dL as
d2
L ≡
Ls
4πF
. (3.43)
Let us consider an object with an absolute luminosity Ls located at coordinate distance
χ 6
from an observer located at χ = 0. The energy of object that is emitted
in time interval t1 let is denoted as E1 while the energy that reaches the sphere
at radius χ is written as E0. From the basic principles it is clear that E1 and
E0 are proportional to the frequencies of light at χ = χs and χ = 0 respectively.
In other words, E1 ∼ ν1 , E0 ∼ ν0. We also define the luminosity Ls and L0
through the relations
Ls =
E1
t1
, L0 =
E0
t0
. (3.46)
The speed of light is given by c = ν1λ1 = ν0λ0 where λ1, λ0 are wavelengths at χ = χs
and χ = 0. Then (3.41) implies
λ0
λ1
=
ν1
ν0
=
E1
E0
=
t0
t1
= 1 + z , (3.47)
using also the fact that ν0 t0 = ν1 t1. If we now combine (3.47) and (3.46) we
obtain
Ls
L0
=
E1
E0
t0
t1
= (1 + z)2
. (3.48)
The light travailing along χ direction satisfies the geodetic motion ds2
= −dt2
+
a2
(t)dχ2
= 0 that implies
χs =
χs
0
dξ =
t0
t1
dt
a(t)
=
1
a0H0
z
0
dz
h(z )
, h(z) =
H(z)
H0
, (3.49)
6
Recall that the metric has following form:
ds2
= −dt2
+ a2
(t)[dχ2
+ f2
K(χ)(dθ2
+ sin2
θdφ2
)] , (3.44)
where
fK = sin χ , k = 1 ,
fK = χ , k = 0 ,
fK = sinh χ , k = −1 .
(3.45)
.
46
where we have take t0 as the time at present epoch and consequently χ0 = 0. We
have also used the fact that
1 + z =
a0
a
⇒
dz
dt
= −
a0
˙a
⇒ dt = −
dz ˙a
a0
. (3.50)
Now the form of the metric (3.44) implies that the area of two sphere at t = t0 is
given by S = 4π(a0fK(χs))2
, where χs corresponds to the fact that we observe signal
from the distance χs. Hence the observed energy flux is
F =
L0
4π(a0fK(χs))2
. (3.51)
Using these results we obtain
d2
Ls
=
Ls
4πF
=
Ls4π(a0fK(χs))2
4πL0
= a2
0fK(χs)2
(1 + z)2
. (3.52)
If we combine (3.49) with (3.52) and use the fact that in FRW background fK(χ) = χ
we obtain
dL =
1 + z
H0
z
0
dz
h(z )
. (3.53)
We can invert this result and express H(z) as function of dL(z) and z
H(z) =
d
dz
dL(z)
1 + z
−1
. (3.54)
If we measure the luminosity distance observationally we can determine the expanding
rate of the Universe.
As we know the energy density on the right hand side of the Friedmann equations
includes all components that are presented in Universe, namely non-relativistic
particles, relativistic particles, cosmological constant:
ρ =
i
ρ
(0)
i (a/a0)−3(1+wi)
=
i
(1 + z)3(1+wi)
, (3.55)
where we have used (3.41). Here wi and ρ
(0)
i correspond to the equation of state and
the present energy density of each component.
Then the Friedmann equation takes standard form
H2
= H2
0
i
Ω
(0)
i (1 + z)3(1+wi)
, Ω
(0)
i =
8πGρ
(0)
i
3H2
0
=
ρ
(0)
i
ρ
(0)
c
. (3.56)
Hence the luminosity distance in a flat geometry is given by
dL =
(1 + z)
H0
z
0
dz
i Ω
(0)
i (1 + z )3(1+wi)
. (3.57)
47
The formula above is the basic theoretical ingredient for the direct evidence of the
current acceleration of the Universe that is related to the observation of luminosity
distances of high redshift supernovae.
The Type Ia supernova (SN Ia) can be observed when the white dwarf starts
exceed the mass of the Chandrasekhar limit and explode. The common belief is that
SN Ia are formed in the same way irrespective of where they are in the Universe that
means that they have a common absolute magnitude M independent of the redshift
z. This implies that they can be treated as an ideal standard candle. We do not go
to these details but it is important that using these methods the luminosity distance
of the SN Ia supernovae that was observed is
H0dL 1.16 , for z = 0.83 . (3.58)
On the other hand the theoretical estimate that follows from (3.57) is
H0dL 0, 95 , Ω(0)
m 1 ,
H0dL 1.23 , Ω(0)
m 0.3 , Ω
(0)
Λ 0.7 .
(3.59)
for two-component form of matter. There are of course lot of literature considering
the fitting the estimate date and the form of the matter that is present in Universe.
The conclusion is that the present experimental date suggests the form of the matter
given above.
3.5 The age of the Universe and the cosmological constant
Another important evidence for the existence of the cosmological constant emerges
when we compare the age of the Universe t0 to the age of the oldest stellar populations
ts. It is clear that the consistency demands that t0 > ts. On the other hand it is
difficult to satisfy this condition for a flat cosmological model with normal form of
matter. On the other hand the presence of cosmological constant can resolve this
problem.
To begin with we review the estimates of the oldest stellar objects. It was
estimated that the age of the oldest objects lay in the interval 11 − 13 Gyr. Such
objects are coolest white dwarfs that gradually cool as the age grows. more precisely,
a white dwarf is very hot when it forms. Further, due to the fact that it has no source
of energy it will gradually cool so that it has so low temperature that means it will
no longer emit heat and it will become a cold black dwarf. In fact, the oldest known
white dwarfs still radiate at temperatures of a few thousand kelvins and these objects
put limit on the maximum possible age of the universe.
Consequently the age of the Universe needs to satisfy the lower bound t0 >
11 − 12 Gyr. Let us calculate the age of the Universe from the Friedmann equations
48
where we consider three contributions to the matter: radiation (wr = 1/3), pressureless
dust (wm = 0) and cosmological constant wΛ = −1.
H2
=
8πG
3
ρ −
k
a2
= H2
0 [Ω0
r
a
a0
−4
+ Ω(0)
m
a
a0
−3
+
+Ω
(0)
Λ − k0
a
a0
−2
] , k0 =
k
a2
0H2
0
.
(3.60)
Then using the fact that 1 + z = a0
a
we can determine the age of the Universe as
t0 =
t0
0
dt =
a0
0
da
Ha
= (−dz =
a0da
a2
) =
=
∞
0
dz
H(1 + z)
=
∞
0
dz
H0x[Ω0
rx4 + Ω
(0)
m x3 + Ω
(0)
Λ − k0x2]1/2
,
(3.61)
where x = 1 + z. Since the radiation dominated period is much shorter than the
total age of the Universe it is a natural to neglect its contribution to the formula
above. In other words the integral coming from the region z ≥ 1000 does not affect
too strongly the integral (3.61). Hence we set Ω(0)
r = 0 when we evaluate t0.
Let us start with the case when the cosmological constant is absent (Ω
(0)
Λ = 0).
Since k0 = Ω(0)
m − 1 the integral (3.61) is equal to
t0 =
∞
0
dz
H0x Ω
(0)
m x3 − k0x2
=
∞
0
dz
H0(1 + z)2 1 + Ω
(0)
m z
. (3.62)
For a flat Universe that is characterized with k0 = 0 and Ω0
m = 1 we obtain
t0 =
2
3H0
. (3.63)
As we know the present Hubble parameter is constrained to be
H−1
0 = 9.776h−1
Gyr , 0.64 < h < 0.8 . (3.64)
Then (3.63) gives
t0 = 8 − 10 Gyr . (3.65)
However this does not satisfy the stellar age bound
t0 > 11 − 12 Gyr .
In other words the flat Universe without a cosmological constant suffers from a serious
age problem.
49
For arbitrary Ω(0)
m the equation (3.61) can be integrated and we obtain
H0t0 =
1
1 − Ω
(0)
m
−
Ω(0)
m
2(1 − Ω
(0)
m )3/2
ln



1 − 1 − Ω
(0)
m
1 + 1 − Ω
(0)
m


 (3.66)
that is of course valid for Ω(0)
m < 1 only. Let us consider various limits of the equation
above. For Ω(0)
m → 0 we obtain H0t0 → 1 while for Ω(0)
m → 1 we obtain t0H0 → 2/3.
As we know the observation of the CMB constraints the curvature of the Universe
to be close to be flat |k0| = |Ω(0)
m − 1| 1. However since then Ω(0)
m ≈ 1 in this case
we again obtain
t0 =
2
3H0
8 − 10 Gyr (3.67)
that is again consistent with the time of the stellar age bound.
On the other hand the age problem can be easily solved in a flat Universe (k0 = 0)
with a cosmological constant ΩΛ = 0). In this case the equation (3.61) gives
H0t0 =
∞
0
dz
(1 + z) Ω
(0)
m (1 + z)3 + Ω
(0)
Λ
=
=
2
3 Ω
(0)
Λ
ln


1 + Ω
(0)
Λ
Ω
(0)
m

 ,
(3.68)
where Ω(0)
m + Ω
(0)
Λ = 1. We see that H0t0 → ∞ for Ω(0)
m → 0 and H0t0 → 2/3 for
Ω(0)
m → 1. When Ω(0)
m = 0.3 and Ω
(0)
Λ = 0.7 one has
t0 = 0.964 H−1
0 = 13.1 Gyr , for h = 0.72 . (3.69)
Hence this easily satisfies the constraint t0 > 11 − 12 Gyr that arises from the
observation the oldest stellar populations. Thus the presence of Λ solves the agecrisis
problem.
3.6 The Cosmological Constant Problem
In classical general relativity the cosmological constant Λ is a completely free parameter.
Let us determine corresponding dimension of given constant. Note that it
appears in the action in the form
1
8πG
d4
x
√
−gΛ . (3.70)
Since the dimension of G is [G] = M−2
where M is mass scale and since [d4
x] = M−4
we find from the requirement that the action is dimensionless that the dimension of
Λ is given by the equation
[Λ] = M2
(3.71)
50
while
ρΛ =
1
8πG
Λ (3.72)
has dimension [ρΛ] = M4
as it is expected for the energy density. In fact, Λ is
completely free and its value should be determined by experiment.
The introduction of quantum mechanics changes the situation in some way.
Firstly, the Planck’s constant allows us to define the reduced Planck mass MP ∼
1018
GeV , as well as reduced Planck length
LP = (8πG)1/2
∼ 10−32
cm . (3.73)
Hence the natural guess for the value of the cosmological constant is
Λguess
P ∼ L−2
P , (3.74)
or as an energy density
ρgusss
vac ∼ M4
P = (1018
GeV )4
. (3.75)
We can find support for this guess by thinking about the quantum fluctuation of
vacuum. As we know any quantum field can be considered as collection of infinite
number of harmonic oscillators. From quantum mechanics we know that harmonic
oscillator with frequency ω has the vacuum energy 1
2
¯hω. Since each mode of the
quantum field contributes to the vacuum energy and the net result should be an
integral over all of these modes. Usually we perform an integration over infinite
interval and hence this integral diverges so that the vacuum energy appears to be
infinite. However, the infinity arises from contribution of modes with very small
wavelengths, it is possible to be mistake to include such a modes since we do not
know what happens at these scales. In other words we do not have any justification
whether the quantum field theory approach can be applied in these small scales as
well. To account for our ignorance we should include the cut-off energy above which
we ignore any potential contributions and hope that some more complete theory
could justify this approach. If the cut-off is at the Planck scale we get the value
given above.
However, we claim to have measured the vacuum energy. The observed value is
different from the theoretical estimate:
ρobser
vac ∼ 10−120
ρguess
vac . (3.76)
In other words, we can express the vacuum energy in terms of the mass scale
ρvac = M4
vac (3.77)
so that the observed result is
Mobs
vac ∼ 10−3
eV. (3.78)
51
The discrepancy is thus
Mobs
vac ∼ 10−30
Mguess
vac . (3.79)
In addition to the fact that it is very small to its natural value the vacuum energy
at present posses an additional puzzle. The coincidence between observed vacuum
energy and current matter density. It can be shown that the ratio of vacuum energy
to matter density depends on time as follows from
ΩΛ
ΩM
=
ρΛ
ρM
∼ a3
. (3.80)
As a consequence, at early times the vacuum energy was negligible with respect in
comparison to matter and radiation while at late times matter and radiation are
negligible.
To date the value of the cosmological constant is one of the most mysterious
problems in current physics, perhaps it could be compared with the mysterious radiation
of the black body at the end of 19’ century. On the other hand it is instructive
to consider an example of supersymmetry which relates to the cosmological constant
problem in interesting way. The main idea of supersymmetry is that for each
fermionic degree of freedom there is corresponding bosonic degree of freedom and
vice-versa. For example, for spin 1/2 electron there should be spin 0 electron of the
same mass and charge. The good news is that while bosons contribute positively to
the vacuum energy the fermion contributions is negative. Hence, if the degrees of
freedom exactly match the vacuum energy is zero.
We do not, however, live in supersymmetric state. If supersymmetry exists, then
it must be broken at some scale Msusy. In other words, for physical processes where
the characteristic energy is much smaller than Msusy we do not see any supersymmetry
and this is the case how our word looks like. On the other hand when we
probe physics with energy scale higher with Msusy we can expect that supersymmetry
is restored. More precisely, we can explain this situation as follows. We expect
that SUSY is broken in nature, for example spontaneously broken which means that
there is one ground state. The fluctuation above states gain masses and one expect
that super-partners of known particles, get masses of order Msusy. Then for energies
much smaller than Msusy these particles are not visible, on the other hand for
energies larger than Msusy we can neglect their masses and these particles look like
massless again. Then we say that supersymmetry is restored at higher energies. This
has an consequence for the vacuum energy. Recall that the vacuum energy was defined
as sum over infinite number of oscillators. For modes with energy much larger
that Msusy these modes find their super-partners and hence their contribution to the
vacuum energy vanishes. This is of course does not happen for modes with energy
smaller than Msusy. In other words we can expect that the vacuum energy will be
equal to
ρvac ∼ M4
susy . (3.81)
52
The question is how high Msusy should be. Nice property of SUSY is that it helps
us to understand hierarchy problem- why scale of electroweak symmetry breaking is
much smaller than the scales of quantum gravity or grand unification. For SUSY to
be relevant to the hierarchy problem we need the SUSY breaking scale to be just
above the electroweak breaking scale
Msusy ∼ 103
GeV . (3.82)
Since this is very close to the experimental bound it is now common belief that SUSY
should be discovered soon at Fermilab or CERN, if it is connected to electroweak
physics. However considering relation between SUSY and cosmological constant we
again see that we are in discrepancy with observation:
M(obs)
vac ∼ 10−15
Msusy (Experiment). (3.83)
Of course there exists a possibility that our estimate Mvac ∼ Msusy is incorrect.
For example let us guess following formula
Mvac ∼
Msusy
MP
Msusy . (3.84)
Interestingly, since MP is fifteen orders of magnitude larger than Msusy and Msusy is
fifteen orders of magnitude larger than Mvac this guess gives up the correct answer.
Unfortunately this is simple numerology, we do not know how this formula should
come from.
Another possibility how to explain the value of the cosmological constant is the
presumption that it is simply feature of our local environment. This is the idea
commonly known as anthropic principle.
In order to give this idea concrete meaning let us presume that there are many
different regions of the Universe in which the vacuum energy takes different values.
Then we can expect that we find ourselves in a region which was suitable for our
own existence. Larger value of cosmological constant than we presently observe
would either have led to a rapid re collapse of the universe (if ρvac were negative) or
an inability to form galaxies (if ρvac were positive).
The idea environmental selection is based on certain special conditions and we
do not understand whether these conditions hold in our Universe. In particular we
have to show that there can be a huge number of different domains with slightly
different values of the vacuum energy and that these domains are big enough that
our entire observable Universe is in a single domain. Further we also have to show
that the possible variation of other physical quantities from domain to domain is
consistent with observations.
Recent work in string theory whose pure essence is the currently very popular
idea of String Landscape supports the idea that there are huge number of possible
53
vacuum states rather than a unique one. Unfortunately the detailed discussion of
this idea is beyond the scope of this introduction review.
To conclude, at present, unfortunately,t here is not any theory that could explain
the mysterious facts considering cosmological constant. To find such a theory is one
of the most prominent goals of physical community.
3.7 The Cosmic Microwave Background
Most of the radiation we observe in Universe today is in the form of the almost
isotropic black body spectrum with temperature approximately 2.7K known as Cosmic
Microwave Background (CMB). The small angular fluctuations in temperature
of the CMB reveal a great deal about the constituents of the Universe.
We have seen previously that the radiation gas evolves and sources the evolution
of the expanding Universe. Since the radiation and dusts have different evolution
laws that as we approach earlier and earlier times in the Universe with smaller and
smaller scale factors the ratio of the energy density in radiation to that in matter
grows proportionally to 1/a(t). Furthermore, even particles which are now massive
and contribute to matter used to be hotter, at sufficiently early times were relativistic
and thus contributed to radiation. In summary, we say that the early Universe was
dominated by radiation. More precisely, at early times the CMB photons were easily
energetic enough to ionize hydrogen atoms and therefore the Universe was filled with
a charged plasma. This phase lasted until the photons red shifted enough to allow
protons and electrons to combine during the era of recombination. Shortly after this
time the photons decoupled from the now neutral plasma and free streamed through
the Universe.
More precisely, the concept of an expanding Universe provides us with a clear
explanation of the origin of the CMB. Black body radiation is emitted by bodies
in thermal equilibrium. The present Universe is certainly not in this state, and
so without an evolving space-time we should have no explanation for the origin of
this radiation. However, at early times, the density and energy densities in the
Universe were high enough that matter was in approximate thermal equilibrium at
each point in space, yielding a blackbody spectrum at early times. Then there is
crucial thermodynamic fact about the CMB. A blackbody distribution, such as that
generated at early Universe, is such that at temperature T, the energy flux in the
frequency range [ν, ν + dν] is given by Planck distribution
P(ν, T)dν = 8πh
ν
c
3 1
ehν/kT − 1
dν , (3.85)
where h is Planck’s constant and k is the Boltzmann constant. Under recalling
ν → λν , with λ = constant the shape of the spectrum is unaltered if T → T/λ. We
know that the wave length are stretched with the cosmic expansion and therefore the
frequencies will scale inversely due to the same effect. We then see that the effect of
54
cosmic expanding on an initial blackbody spectrum is to retain its blackbody nature,
but just at lower and lower temperatures
T ∼
1
a
. (3.86)
This is what we mean when we say that the Universe is cooling as it expands.
It is also well known that CMB is not a perfectly isotropic radiation bath. Deviations
from isotropy at the level of one part in 105
have developed over the last
decade into one of our most precise observation tool in cosmology.
It is important to stress that this anisotropy of CMB is divided into two classes.
The first one, known as primary anisotropy, has its origin at the effects on the
surface of the last scattering and before, and secondary anisotropy that is a result
of the interaction between CMB and intervening hot gas or gravitational potentials
that occurs between the last scattering surface and the observer.
The small temperature anisotrophies on the sky are usually analyzed by decomposing
the signal into spherical harmonics via
T
T
=
∞
l=0
l
m=−l
almYlm(φ, θ) , (3.87)
where alm are expansion coefficients and θ and φ are spherical polar angles on the
sky. In other words, we introduced polac coordinates on the last-scattering surface
S2
and spherical harmonics Ylm(φ, θ) are defined as
Ylm(θ, φ) = (−1)
m+|m|
2
2l + 1
4π
(l − |m|)!
(l + |m|)!
Pl|m|(cos θ)eimφ
, (3.88)
where Pl|m|(y) are associated Legendre polynomials of degree l and where l is called
multipole moment of the expansion and m is the azimuthal number. Since T
T
is
real quantity and since Yl0 is real we have that al0 is real. Now we presume that
anisotropies have their origin from the random process with a Gaussian distribution
so that real and imaginary parts of alm are independent. Let us now consider for
each l the (2l + 1) dimensional set
(αl, ¯m| ¯m = 1, 2 . . . , 2l + 1) = (al0,
√
2Re(alm),
√
2Im(alm)|1 ≤ m ≤ l) . (3.89)
Then we define for each l the temperature angular spectrum CTT
l as the square of
the half width of the Gaussian distribution that has the form
fl[αl, ¯m] =
1
2πσ2
l
e
−
α2
l, ¯m
2σ2
l , (3.90)
where σ2
l is defined as
σ2
l = CTT
l =
1
2l + 1
a2
l , a2
l =
l
m=−l
|alm|2
, (3.91)
55
where a2
l is the average over an ensemble of skies which is a collection of skies
observed at different points in the universe for a given l. In other words . . . means
the average with respect to all observers in the universe. As we argued before the
main presumption of cosmology is that the universe is homogenous and isotropic. As
a result there is no preferred observing direction.
However it is important to stress that we can only observe the Universe from
one point so that we cannot perform average over all skies. Then however we assume
ergodic hypothesis that says that the average over the ensemble of all possible skies
is equivalent to the spatial averages over one sky
O = O sky =
1
4π S2
dΩ2O . (3.92)
The fluctuations in the CMB spectrum are useful for the study of cosmology from
many reasons. To understand why, we should show at the first place why they arise.
Matter today in the Universe exists in the form of clusters of starts, galaxies, and
clusters and super-clusters of galaxies. Our understanding how large scale structures
developed is that initially small density perturbations in the otherwise homogeneous
Universe grew through the gravitational instability to the objects we observe today.
Such picture requires that from place to place there were small variations in the
density of matter at the time when CMB firstly decoupled from the photon-baryon
plasma. Then CMB photons propagated freely through the Universe nearly unaffected
by anything except the cosmic expanding itself. However at the time of their
decoupling different photons were released from regions of space with slightly different
gravitational potentials. Since the gravitational potential affects the photon
redshift, photons from some regions redshift slightly more than those from other regions
, giving rise to a small temperature anisotropy in the CMB observed today. In
this sense CMB reflects the initial conditions that ultimately gave rise to structure
in the Universe.
It is important that CMB fluctuations give us the value of Ωtotal. In fact, careful
analysis of all of the features of the CMB power spectrum provide constraints on
essentially all of the cosmological parameters. For example, let us consider recent
result from WMAP. For total density of the Universe they find
0.98 ≤ Ωtotal ≤ 1.08 (3.93)
at 0.95 confidence which is a strong evidence for a flat Universe. Nevertheless, much
tighter constraints on the remaining values can be derived by assuming either an
exactly a flat Universe or a reasonable value of Hubble constant. When for example
we presume a flat Universe, we can derive values for the Hubble constant, matter
density (which then implies the vacuum density from Ωtotal = 1) and baryon density:
h = 0.72 ± 0.05 ,
56
ΩM = 1 − ΩΛ = 0.29 ± 0.07 ,
ΩB = 0.047 ± 0.006 .
(3.94)
If we instead assume that the Hubble constant is given by the value determined by
HST project
H0 = 100 h km sec−1
Mpc−1
, h = 0.71 ± 0.06 (3.95)
we can derive separate tight constraints on ΩM and ΩΛ.
In summary, taking all of the data together we obtain a remarkably consistent
picture of the current constituents of our Universe:
ΩB = 0.04 ,
ΩDM = 0.26 ,
ΩΛ = 0.7 .
(3.96)
There are many mysterious things considering these values. Firstly, the baryon density
is mysterious due to the asymmetry between baryons and antibaryons. Secondly,
the problem with dark matter is that we have never detected it directly and only have
promising ideas as to what it might be. However the biggest mystery is the vacuum
energy, we now try to explain why it is mysterious and what kinds of mechanism
might be responsible for its value.
4. Early Times in the Standard Cosmology
Early times at the in the Standard Cosmology are characterized by very high temperatures
and densities with many particle species kept in (approximate) thermal
equilibrium by rapid interactions. Our goal is then to develop some tools of the
thermodynamics in expanding Universe. In fact, up the mild-1960 it was not clear
whether the early Universe had been hot or cold. This situation changed with the
Pensias and Wilson’s 1964-1965 discovery of 2.7 K microwave background radiation
arriving from the farthest reaches of the Universe since the existence of the microwave
background has been predicted by the hot Universe theory.
4.1 Review of the building blocks of the standard cosmology and matter
For reader’s convenience we review some basics facts considering the standard models
of cosmology.
• The Classical general relativity:
The classical general relativity provides good description of the geometry of
space-time for scales l lP = M−1
P = 10−33
cm or equivalently for energy
scales below the Planck scale MP .
57
• Physical scales are stretched by the scale factor a(t) with respect to the comoving
scales
lphys(t) = a(t)lcom . (4.1)
A physical wavelength redshifts proportional to the scale factor where its time
derivative obeys the Hubble law
dlphys(t)
dt
=
˙a
a
alcom = H(t)lphys(t) =
lphys
dH(t)
. (4.2)
• The equilibrium temperature decreases as the Universe expands as
T(t) =
T0
a(t)
. (4.3)
• The Standard Model of Particle Physics:
The current standard model of particle physics that is experientially tested
with remarkable precision describes the theory of strong (QCD), weak and
electroweak interactions (EW) as a gauge theory based on the gauge group
SU(3)c ⊗ SU(2) ⊗ U(1)Y . (4.4)
The particle content is: three generations of quarks and leptons:
u
d
c
s
t
b
;
νe
e
νµ
µ
ντ
τ
(4.5)
vector Bosons: 8 gluons (massless) that mediate the strong interactions in
QCD, Z0
, W±
that are massive with masses MZ = 91.18 ± 0.02 GeV and
MW = 80.4 ± 0.06 GeV that mediate the electroweak interactions, the photon
(massless)-the mediator of electromagnetic interaction and the scalar Higgs
that was discovered in 2011 at LHC with the mass MH = 125.09 GeV .
• It is known that the couplings associated with strong, weak and electrodynamics
interactions depend on the mass scale that characterize given process. The
current theoretical ideas propose that these couplings are unified in a grand
unified theory (GUT) at the scale
MGUT ∼ 1016
GeV .
Further, the UV scale where the Gravity is eventually unified with the rest of
particle physics is the Planck scale
MP ∼ 1019
GeV .
On the other hand the physics of the Standard Model describes phenomena at
energy scales below MS where
MS ∼ 100 GeV .
58
• The connection between the Standard model of particle physics and early Universe
cosmology is through Einstein’s equations that couple the space-time
geometry to the matter-energy content. We study gravity semi-classically at
energy scales well below the Planck scale. The Standard model of particle
physics is a quantum field theory (QFT) thus the space-time is classical but
with sources that are quantum fields. Semi classical gravity is defined by the
Einstein equations with the expectation value of the energy-momentum tensor
ˆTµν
as sources
Rµν
−
1
2
gµν
R =
ˆTµν
M2
P
, (4.6)
where the expectation value ˆTµν
is taken in given quantum state or density
matrix that is compatible with homogeneity and isotropy so that it has to be
translational and rotational invariant. The ground state of the quantum field
theory is usually the state that solves the classical equations of motion or the
equations of motion with the quantum correction. In this case the vacuum
expectation value of the stress energy tensor corresponds to the classical one.
The general formula above has important in case we study the properties of
the fluctuations above given classical solutions.
As the next step we review basic facts about the Energy scales, time scales and
phase transitions
Energy scales,time scales and phase transitions
In this section we give a brief overview of the main cosmological epochs by focusing
on the energy scales of particle, nuclear and atomic physics.
Energy scales:
• Total Unification
It is expected that Gravitational, strong and electroweak interactions become
unified and described by a single quantum theory at the Planck scale MP ∼
1019
GeV . The most promising approach to this unification is in terms of string
theory however their theoretical consistency is still studied and experimental
confirmation is not available.
• Grand Unification:
Strong and electroweak interactions are expected to become unified at an energy
scale
MGUT ∼ 1016
GeV , TGUT ∼ 1029
K
under large gauge group G, for example SU(5), SO(10) that breaks sponta-
neously
G → SU(3)c ⊗ SU(2) ⊗ U(1)Y
59
at scale below unification. Main arguments for the existence of GUT theories
follow from merging of the running coupling constants of the strong, electromagnetic
and weak interactions for the minimal supersymmetric model and
also the explanation of the small neutrino masses via see-saw mechanism.
• Electroweak:
Weak and electromagnetic interactions are unified in the electroweak theory
based on the gauge group
SU(2) ⊗ U(1)Y .
The weak interactions become short ranged after symmetry breaking phase
transition
SU(2) ⊗ U(1)Y → U(1)em
at the energy scale of the order of the mass of the Z0
, W±
vector bosons corresponding
to temperature
TEW ∼ 100 GeV ∼ 1015
K .
More precisely, at temperature T > TEW the symmetry is restored as a consequence
of the fact that the effective potential of the theory depends on the
temperature as well. For temperature T > TEW the stable minimum of the potential
corresponds to the symmetric phase where all vector bosons are massless
and hence the symmetry is restored. On the other hand for T < TEW the stable
minimum of the potential corresponds to the situation when the vector
bosons W±
, Z0
become massive through Higgs mechanism while photon remains
massless corresponding unbroken U(1) abelian symmetry of quantum
electrodynamics. The temperature TEW determines the temperature scale of
the electroweak phase transition in the early Universe.
• QCD
The strong interaction has a typical energy scale
ΛQCD ∼ 200 MeV .
At this coupling the coupling constant becomes strong αs ∼ O(1) that corresponds
to the temperature scale
TQCD ∼ 1012
K
QCD is asymptomatically free theory that means that the coupling between
quarks and gluons becomes smaller at large energies but diverges at the scale
ΛQCD. For energies below ΛQCD the quantum chromodynamics is strongly
interacting theory and quarks and gluons are bound into mesons and baryons.
60
This phenomenon is interpreted in terms of a phase transition at an energy
scale ΛQCD or TQCD. For T > TQCD the relevant degrees of freedom are weakly
interacting quarks and gluons, while below are hadrons. In the limit when
we can presume that up and down quarks are massless, QCD possesses new
SU(2)L ⊗ SU(2)R chiral symmetry that is spontaneously broken at about the
same temperature scale as the scale of QCD transition. Pions are the Goldstone
bosons that emerge in the breakdown of the chiral symmetry
SU(2)L ⊗ SU(2)R → SU(2)R+L .
The high temperature phase above TQCD where the quarks and gluons are
almost free (because the coupling is small by asymptotic freedom) is a quarkgluon
plasma.
• Nuclear Physics
The low energy scales that are relevant in cosmology are determined by the
binding energy of light elements. For example, the binding energy of deuterium
is ∼ 2 MeV that corresponds to the temperature T ∼ 1010
K. This is the
energy scale that determines the origin of the primordial nucleosynthesis. The
first step in the system of the nuclear reactions that yields the primordial
elements is the formation of deuteron in the reaction
n + p ↔ d + γ .
These nuclear reactions continue and all neutrons end up in nuclei, mainly
helium.
• Atomic physics
A further important low energy scale relevant for cosmology is the binding
energy of hydrogen ∼ 10 eV . This is the energy scale at which free protons
and electrons combine into neutral hydrogen. The large number of photons per
baryons implies that recombination actually takes place at an energy scale of
order 0.3 eV , at about 400000 years after the beginning of the Universe. At this
time when the neutral hydrogen is formed the Universe becomes transparent
since then photons no longer scatter and travel freely. These are the photons
measured by CMB experiments today.
Time Scales:
——————
• Inflation epoch
61
This is (according to current cosmological scenario) the earliest period in the
life of Universe where the scale factor grows exponentially as
a(t) = eHt
.
Current experiments put upper bound on the energy scale of inflation as
H ≤ 1013
GeV .
In order to solve the entropy and horizon problems the inflationary stage hast
to last a time interval δt so that
δtH ∼ 60 ⇒ δt ∼ 10−34
sec .
• Radiation dominated era
The inflationary stage is followed by a radiation dominated era after a short
period of reheating during which the energy stored in the field that drives
inflation decays into quanta of many other fields. These fields reach the state
of thermal equilibrium through the scattering processes.
After the thermal equilibrium is reached we obtain a detailed picture of the
thermal history of the Universe. This description is based on the combination
of the statistical mechanics with the basic principles of QFT: During the first
1000 years of the Universe and after the inflation stage that lasted ∼ 10−34
sec the Universe was radiation dominated. Universe also expands and cools
almost adiabatically. The electroweak transition occurred at the energy scale
T ∼ 100 GeV that corresponds to the time
tEW ∼ 10−12
sec .
The QCD transition occurs at
tQCD ∼ 10−5
sec .
Local Thermal Equilibrium (LTE) and Non equilibrium
Weather some process occurs in or out of a local thermodynamics equilibrium
depends on the comparison of two time scales-the expanding rate and the reaction
rate. To have a contact with standard thermodynamics note that we can
formulate the same problem as the problem of comparing of the cooling rate
(the rate how temperature decreases) and the rate of reaction. In fact the rate
of cooling is related to the rate of the expanding through the formula
˙T
T
= −
1
Ta2
˙a = −H(t) (4.7)
62
as follows from the fact that T ∼ 1
a
. On the other hand collisions as well as
non-collisional processes contribute to establish the equilibrium with a rate Γ.
The local thermodynamic equilibrium is established when
Γ > H(t) (4.8)
In this case the evolution is adiabatic in the sense that the thermodynamics
functions depend slowly on time through the temperature. On the other hand
when the expanding is too fast
H(t) Γ
local thermodynamics equilibrium cannot be established, the temperature drops
too fast for the system to have time to relax.
While a detailed understanding of the relaxation dynamics requires an analysis
of the quantum Boltzmann equations a simple order of magnitude estimate for
a collision rate is given as follows.
The collision rate can be calculated in the standard statistical physics as
Γ ∼< σnv > , (4.9)
where < . . . > means statistical ensemble average and where σ is a scattering
cross section, n is the density of particles that scatter and v is velocity of given
particles. For electromagnetic scattering a typical cross section is of order
σem ∼
α2
Q2
,
where Q2
is transferred momentum and α is the electromagnetic coupling constant.
At high temperature single photon exchange implies the estimate (the
transferred momentum is proportional to the momenta of one photon that is
proportional to the temperature)
σem ∼
α2
T2
.
The density of relativistic degrees of freedom is n ∼ T3
and for v ∼ 1 (This
estimate follows from the fact that particles are ultra-relativistic) we obtain
Γem ∼ α2
T .
In QCD that in the high temperature regime can be treated perturbatively the
estimate of the single gluon exchange can be performed in the similar way and
we get
ΓQCD ∼ α2
sT ,
63
where αs is corresponding coupling constant. We have to compare these estimates
with H. However H2
∼ ρ that in the case of the radiation dominated
era we show that ρ ∼ T4
.Then in this case we find that H ∼ T2
/Mpl so that
ΓQCD
H
=
α2
sMpl
T
> 1 (4.10)
and we obtain that the strong interactions are in LTE for
T ≤ 1016
GeV
In the same way we obtain that electromagnetic interactions are in LTE for
T ≤ 1014
GeV .
It is important to stress for T ≥ α2
Mpl ∼ 1016
GeV all perturbative interactions
should be frozen out and are not effective in maintaining thermal equilibrium.
In other words all known interactions together with any new interactions
that arise from grand unification are not sufficient for maintaining the thermal
equilibrium in the Universe at temperatures greater than 1016
GeV that corresponds
to the time earlier than 10−38
s. In other words Universe is not in
thermal equilibrium at its earliest epoch.
The estimate in case of weak interaction is slightly more involved: a typical
scattering process with an energy transfer E MW has a scattering cross
section
σ ∼ G2
F E2
, E MW
whereas if E MW we have
σ ∼
g4
E2
, E MW .
Then in thermal medium with E ∼ T and with a density of relativistic particles
n ∼ T3
a typical weak reaction rate is
ΓEW ∼ g4
T , T MW
and
ΓEW ∼ G2
F T5
for T MW . In this latter temperature regime the ratio
ΓE
H
∼
T
MeV
3
and hence the weak interactions fall out of LTE for T ≤ 1 MeV .
64
Even if this analysis provides an intuitive estimate for the relaxation time
scales this analysis neglected several important aspects that however have to
be studied on a case-by-case basis. One such an example of subtle effects are
Screening and infrared phenomena: The relaxation rates Γ were calculated on
presumption of an exchange of a vector boson of relativistic degrees of freedom.
In a medium at a high temperature and a density there are important screening
effects that can change these estimates.
4.2 Hot Big Bang
We begin this section with the description of the evolution of the Universe in its hot
stage.
The basic presumption is that it is plausible to extrapolate the evolution of
the Universe back in time using the known microscopic physics (electrodynamics,
nuclear physics, QCD and electroweak theory) and General Relativity. This theory
is called as Hot Big Bang Theory. According to this theory the Universe was
hotter at earlier stages (equivalently, at smaller values of a(t)) and the temperature
scales as a(t)−3
both for non-relativistic and relativistic particles. At high enough
temperatures the Universe was in the phase that is completely different from what
we observe today. Instead of the almost empty space with galaxies here and there
was dense, hot and almost homogeneous plasma that fills the whole Universe. This is
the area whose physical laws are governed by microscopic physics. Note that gravity
plays the role of the spectators of the theory and it is considered as classical. Of
course we consider back-reaction of this matter on the time evolution of the Universe
using the Friedmann equations.
More precisely, the hot Universe theory is based on the phenomena of the phase
transitions and the symmetry breaking. Let us consider for example the simplest
GUT model based on the gauge group SU(5). For temperature T ≥ 1015
GeV there
was no difference between weak, strong and electroweak interactions. The matter in
the Universe was in the form of the dense plasma containing quarks, photons, gluons
etc. Then there was no problem in the transformation of quarks to leptons. In other
words it does not make sense to speak about baryon conservation. At t1 ∼ 10−35
sec
when the temperature has dropped to T ∼ Tc1 ∼ 1014
− 1015
GeV the fist symmetry
breaking phase transition takes place: SU(5) breaks to SU(3)×SU(2)×U(1) where
SU(3) is gauge symmetry of the QCD, theory of the strong interactions. In other
words string interactions were separated from electroweak and leptons. Then at t2 ∼
10−10
sec when the temperature dropped to Tc2 ∼ 102
GeV there was a second phase
transition that broke the symmetry between weak and electromagnetic interactions
SU(3) × SU(2) × U(1) → SU(3) × U(1). As the temperature reduces further to
Tc3 ∼ 102
MeV there was another phase transition with the formation of baryons
and mesons from quarks.
65
4.3 Review of the study of the expansion of the Universe
Let us again analyze the evolution of the Universe. As we have argued before at early
times the Universe was radiation dominated, then matter dominated and presently
dark energy dominated while the curvature term k
a2 was never important.
Deceleration to Acceleration
Since the dark energy dominates at present the Universe accelerates. On the
other hand when matter was dominating the Universe was decelerating. In order to
see when the change in regime occurred we write the Friedmann equations as
˙a2
=
8πG
3
ρa2
=
8πG
3
a2
(ρM + ρΛ) , (4.11)
where we have neglected spatial curvature and also ultra-relativistic matter for the
moment. The reason for this simplification is that the relativistic matter dominates
an expanding of the Universe at much earlier stage. The time derivative of the
equation above implies
2˙a¨a =
8πG
3
˙ρM a2
+ 2(ρM + ρΛ)˙aa =
=
8πGa
3
(−˙aaρM + 2˙aaρΛ)
(4.12)
where we used ˙ρM = −3 ˙a
a
ρM . The expression above is zero when (This event defines
the turning point between decelerating and accelerating phase)
2ρΛ
ρM
= 1 (4.13)
or equivalently
a3
0
a3
≡ (1 + z)3
=
2ΩΛ
ΩM
, (4.14)
where or course ΩM is time-dependent. For expected values ΩΛ = 0.7, ΩM = 0.3 we
have
deceleration → acceleration: z ≈ 0.7
In other words, the Universe was decelerating until fairly recently. Before z ≈ 0.7
the expansion was dominated by the non-relativistic matter.
Radiation domination to matter domination
66
As we know the energy density of ultra-relativistic matter (radiation) scales as a−4
while the energy density of non-relativistic matter scales as a−3
. Then it follows that
the dominant contribution to the energy density of the Universe at very small a (small
t) came from ultra-relativistic matter. Now we estimate zeq at which the equilibrium
between matter and radiation occurred. In other words we would like estimate zeq
when the expansion regime changed from the dominance of ultra-relativistic particles
to the dominance of non-relativistic matter, we write
ρM (t)
ρrad(t)
=
ρM0a3
0a−3
(t)
ρrad0a4
0a−4(t)
=
ρM
ρrad 0
a(t)
a0
, (4.15)
where again the subscript 0 refers to present values. Equilibrium occurs at
ρM (teq)
ρrad(teq)
≈ 1 (4.16)
that gives
a0
a(teq)
≡ 1 + zeq ≈
ρM
ρrad 0
=
ΩM
Ωrad
. (4.17)
Since Ωrad ≈ 10−4
, ΩM ≈ 0.3 we obtain
radiation domination → matter domination : zeq ≈ 3000 .
The corresponding temperature is
Teq = T0(1 + zeq) ≈ 104
K ≈ 1eV . (4.18)
At higher temperatures the expansion of the Universe was dominated by ultrarelativistic
matter. We must to stress that it is important for structure formation that
the most of the part of the lifetime of the Universe is dominated by non-relativistic
matter. This follows from the fact that the expanding rate at both radiation dominated
and vacuum dominated eras is such that gravitational perturbations grow
slowly and only during the matter dominated stage their growth is fast enough so
that the existing structures of the Universe can arise.
4.4 Epochs of the early Universe
There are two important epochs in the evolution of the Universe:Recombination
epoch that is the transition from plasma to neutral gas. This occurs at temperature
T ∼ 3000K, t ∼ 3 · 105
years and nucleosynthesis epoch that occurs at temperatures
T = 1MeV to a few ·10keV . Another event is neutrino decoupling. Briefly,
at high temperatures the neutrino was in thermal equilibrium with the rest of cosmic
plasma. The plasma became transparent for neutrinos at temperature about 1MeV .
This decoupling of neutrinos is very important for nucleosynthesis since it affects the
67
neutron-proton ratio just before nucleosynthesis (Since neutrinos decouples the reaction
that transfers proton into neutrons simply cannot occur) and hence it leads to
the abundances of light elements that need neutrinos for their formations. Further,
the fact that neutrinos decoupled much earlier than photons implies that the present
neutrino-to-photon ration is less than one. This is consequence of the fact that photons
are additionally heated, after neutrino decoupling, due to the annihilations of
e+
with e−
.
If we move further back in time we obtain that the cosmic plasma has more
and more components. At temperatures roughly 0.5MeV there are many electrons
and positrons that are frequently pair created and annihilate: at T > 100MeV
the plasma contains muons and pions. This plasma remains in thermal equilibrium
except possibly for phase transitions
• QCD phase transition
At temperatures above 100MeV (QCD scale) strongly interacting particles
are dissolved into quarks and gluons. This quark-gluon plasma converts into
hadronic matter (mostly pions) during the quark-hadron phase transitions.
Theoretical estimates suggest that the temperature of this phase transition is
about 170MeV .
• Electroweak transition
Briefly, at temperatures well above 100GeV electroweak symmetry is unbroken.
The consequence of this fact is that W and Z bosons are massless. At T ∼
100GeV the phase transition of the electroweak symmetry breaking takes place.
• GUT transition
It is slightly uncertain when we extrapolate back further (equivalently, we go
to higher temperatures), but if we do so we come to the Grand Unification
epoch. The temperature of this epoch is set by GUT scale, TGUT ∼ 1016
GeV .
We expect that at this temperature the Grand Unified phase transition occurs.
On the other hand many models of inflation suggest that the Universe never
had such a high temperature after inflation.
Expansion rate and life-time at radiation domination
Now we will discuss in more details the expansion of the Universe in radiation dominated
stage where we will presume thermal equilibrium of all ultra-relativistic species
7
. In the very early stages of its evolution was filled with an ultra-relativistic gas of
photons, electrons, positrons, etc. At that time the excess of baryons over antibaryons
small fraction (at most 10−19
) of the total number of particles. The matter could be
7
This presumption is not however valid for neutrinos at temperatures below 1MeV .
68
considered as a gas of free particles where their rest masses are small compared to
temperature. In other words the energy density and entropy density corresponds to
the massless species
ρ = 3p =
π2
30
g∗(T)T4
, s =
2π2
45
g∗(T)T3
. (4.19)
where the effective number of particle species g∗(T) is g∗(T) = gB(T)+ 7
8
gF (T) where
gB and gF are the number of boson and fermions species degrees of freedom with
masses m T. For example, for photons gB = 2, gF = 2 for neutrinos and gF = 4
for electrons (Let us sketch the way how to derive the dependence of ρ on T. By
definition
ρ = d3
ke(k)f(
e
T
)
where f( e
T
) is distribution functions and e(k) is an energy. For particles with m T
we can neglect their rest masses so that e = k. After substitution k
T
= m we obtain
ρ = T4
d3
me(m)f(m) ∼ T4
.)
Generally g∗(T) increases with increasing T but rather slowly. This follows from
the fact that at higher temperatures more species are ultra-relativistic (say, electrons
contribute at T > 0.5MeV and do not contribute at lower temperatures.)
Let us now list some time scales that are relevant for the early stage of the
evolution of the Universe:
• Nucleosynthesis
The temperature relevant for nucleosynthesis rages from a few MeV to about
70keV . This era begins at
t ∼ 1 s . (4.20)
and ends at
t ∼ 200 s ∼ 3 min . (4.21)
After this brief introduction we will discuss the properties of the early Universe in
brief details.
4.5 Describing Matter
We try to describe matter a perfect fluid described by an energy-momentum tensor
Tµν = (ρ + p)UµUν + pgµν , (4.22)
where Uµ is the fluid four-velocity, ρ is the energy density at rest frame of the fluid
and p is the pressure in that same frame. By definition the stress energy tensor is
covariantly conserved
µTµν
= 0 . (4.23)
69
In more complicated examples a fluid will be characterized by quantities in addition
to the energy and pressure. Many fluids have a conserved quantity associated with
them and so we will also introduce a number flux density Nµ
which is also conserved
µNµ
= 0 . (4.24)
For non-tachyonic matter Nµ
is a time-like 4-vector and therefore we can write
Nµ
= nUµ
. (4.25)
In the same way we can introduce an entropy flux density Sµ
. This quantity is not
conserved but rather obeys a covariant version of the second law of thermodynamics
µSµ
≥ 0 . (4.26)
It is useful to resolve Sµ
into components parallel and perpendicular to the fluid
4-velocity
Sµ
= sUµ
+ sµ
, (4.27)
where sµUµ
= 0. The scalar s is the rest-frame entropy density that can be written
as
s =
ρ + p
T
. (4.28)
We must also specify an equation of state. Typically we do this in such a way as to
treat n and s as independent variables.
For adiabatic expanding Universe sa3
≈ const eq. (4.19) implies
T(t) ∼
1
a(t)
. (4.29)
We see that the temperature cools during the expansion of the Universe. The background
radiation is a result of the cooling of the hot photon gas during the expansion
of the Universe.
4.6 Particles in Equilibrium
The various particles inhabiting the early Universe can be characterized according to
three criteria: in equilibrium vs. out of equilibrium (decoupled), bosonic vs fermionic
and relativistic (velocities near 1) vs. non-relativistic. In this subsection we will
consider species which are in equilibrium with surrounding thermal bath.
Now we must discuss the conditions under which a particle is in equilibrium
with the surrounding thermal plasma. The particles will be in thermal equilibrium
as long as its interaction rate is larger then the expansion rate of the Universe. In
other words, particles have enough time to share the energy among themselves or
equivalently, equilibrium requires that it should be possible for the products of a given
reaction have the opportunity to recombine in the reverse reaction. If the expanding
70
of the Universe is rapid enough this will not happen. A particle species for which the
interaction rates have fallen below the expanding rate of the Universe is said to have
frozen out or decoupled. The interaction rate of some particle with the background
plasma is Γ where Γ is inverse of the mean time between the reaction of given particle
species with the thermal background. Now the particle will be decoupled from the
thermal bath when the particle has not time enough to react with thermal bath if
Γ H , (4.30)
where the Hubble constant H sets the cosmological timescale.
At the early Universe the particles are in thermal equilibrium (unless they are
very weakly coupled). This can be seen from Friedmann equation when the energy
density is dominated by plasma with ρ ∼ T4
and we have
H2
∼ ρ ⇒ H ∼
√
ρ ∼
T
MP
T (4.31)
so that the Hubble parameter is suppressed with respect to the temperature by a
factor of T/MP . At extremely early times (near the Planck era) the Universe may
be expanding so quickly so that no species are in equilibrium but as the expansion
rate slows the equilibrium becomes possible.
At extremely early times near the Planck era, the Universe may be expanding so
quickly that no species are in equilibrium; as the expansion rate slows, equilibrium
becomes possible. On the other hand the interaction rate Γ for a particle with cross
section σ is typically of the form
Γ = n σv , (4.32)
where n is the number density and v is typical particle velocity. Since n ∼ a−3
the
density of particles will reduce so that the equilibrium can once again no longer be
maintained. In our current Universe no species are in equilibrium with the background
plasma (represented by CMB photons).
Now we review some facts about particles at equilibrium. For a gas of weaklyinteracting
particles we can describe the state in terms of a distribution function
f(p) where the three momentum p satisfies
E(p)2
= m2
+ |p|2
. (4.33)
The distribution function characterizes the density of particles of given momentum.
The number density, energy density and pressure of some species labeled i are given
by
neq
i (T) =
gi
(2π)3
fi(p)d3
p =
gi
2π2
T3
I11
i ( ) ,
71
ρeq
i (T) =
gi
(2π)3
E(p)fi(p)d3
p =
gi
2π2
T4
I21
i ( ) ,
peq
i (T) =
gi
(2π)3
|p|2
3E(p)
fi(p)d3
p =
gi
6π2
T4
I03
i ( ) ,
(4.34)
where
Imn
i ( ) =
∞
xi
ym
(y2
− x2
i )n/2
(ey
1)−1
dy , xi =
mi
T
, (4.35)
and where gi is number of spin states of the particles (massless photons, gγ = 2,
massive vector bosons Z , gZ = 3.) Further, −/+ refers as before to bosons/fermions.
As usual, particles and antiparticles are treated as separate, for spin 1/2 electrons
and positrons we have ge− = ge+ = 2. In thermal equilibrium at a temperature T
the particles will be in either Fermi-Dirac or Bose-Einstein distributions
f(p) =
1
eE(p)/T ± 1
, (4.36)
where the plus sign is for fermions while the minus sign for bosons.
We can do the integrals over the distribution functions in two opposite limits,
particles which are relativistic T m and highly non-relativistic T m. For
relativistic (R) particles that are characterized by condition xi = mi
T
1 the integrals
in (4.35) are
bosons : I11
R (−) = 2ζ(3) , I21
R (−) = I03
R (−) =
π4
15
,
fermions : I11
R (+) =
3ζ(3)
2
, I21
R (+) = I03
R (+) =
7π4
120
,
(4.37)
where ζ is Riemann Zeta function and ζ(3) = 1.202. Then we obtain, for relativistic
bosons, following results:
neq
i =
ζ(3)
π2
giT3
,
ρeq
i =
π2
30
giT4
,
peq
i =
1
3
ρi
(4.38)
and for relativistic fermions
neq
i =
3
4
ζ(3)
π2
giT3
,
ρeq
i =
7
8
π2
30
giT4
,
72
peq
i =
1
3
ρi .
(4.39)
On the other hand non-relativistic (NR) limit, where we have x 1 is the same for
bosons and fermions and we recover the Boltzmann distribution
neq
i = gi
miT
2π
3/2
e−mi/T
ρeq
i = mini ,
peq
i = neq
i T ρeq
i .
(4.40)
independently of whether the particle is bosons or fermions. The results given above
imply several interesting facts. For example, since the densities of relativistic particles
are roughly the same, the relativistic particles remain approximately equal abundances
in equilibrium. We also see that once the particles become non-relativistic,
they become exponentially suppressed with respect to the relativistic species. This
is a result of the fact that it becomes harder for massive particle-antiparticle pairs
to be produced in a plasma with T m.
We would like also mention that although matter is much more dominant than
radiation in the Universe today, since their energy densities scale differently, the early
Universe was radiation dominated. We can write the ratio of the density parameters
in matter and radiation as
ΩM
ΩR
=
ΩM0
ΩR0
a
a0
=
ΩM0
ΩR0
(1 + z)−1
. (4.41)
In the same way as we did above we can determine the redshift of the matter-radiation
equality as
1 + zeq =
ΩM0
ΩR0
≈ 3 × 103
. (4.42)
From the form of the expression above where we compare the densities that scale as
a−3
for matter and a−4
for radiation it is clear that we have made an assumption
that particles that are non-relativistic today were also non-relativistic at zeq. It can
be shown that this presumption is safe.
At any given time not all particles will, be in fact in equilibrium at a common
temperature T. A particle will be in kinetic equilibrium with the background thermal
plasma, i.e when Ti = T only while it is interacting. In other words as long as the
scattering rate
Γ = n < σv > > H . (4.43)
Here < σv > is the velocity averaged cross-section for 2 → 2 processes such as
iγ → iγ , il±
→ il±
(4.44)
73
that maintain good thermal contact between i-particles and the particles (that has
the particle density n) that constitute the background plasma (γ-fotons, l±
-refers
to electrons which are abundant down to T ∼ me and remain strongly coupled to
photons through the Compton scattering through the entire Radiation dominate era
so that Te = T always.) We say that i-particle decouple at the temperature Ti when
the condition
Γ(Ti) ≈ H(Ti) (4.45)
is satisfied. Of course no particle is ever truly decoupled since there are always
some residual interactions. On the other hand it can be shown that their effects are
generally negligible.
If the particle is relativistic at this time (mi < Ti) then it will also be in the
chemical equilibrium with the thermal plasma that is characteristic with the condition
for chemical potentials of the particles i µi , their anti-particles µi and the
chemical potential of photons µγ
µi + µi = µl+ + µl− = µγ = 0 (4.46)
through processes such as
ii ↔ γγ , ii ↔ l+
l−
(4.47)
Then its abundance at decoupling will be just the equilibrium value at the temperature
of decoupling
neq
i (Ti) =
gi
2
nγ(Ti)fB.F , (4.48)
where fB = 1 if i is boson and fF = 3
4
if i is fermion.
Then the decoupled particles i will expand freely without interactions so that
their number in a comoving volume is conserved as nia3
= const and their pressure
and energy density are functions of the scale factor a alone. Even if these particles do
not interact their phase space distribution will retain their equilibrium form (4.36)
with Ti. As long as the particles remain relativistic, Ei and Ti scale as a−1
. Initially
the temperature Ti will track the photon temperature T. However as the Universe
cools below to some mass thresholds (in other words temperature is less than some
mass of particles), these massive particles will become non-relativistic and annihilate.
The annihilation will produce additional photons and other interacting particles that
has an effect of the heating of them. On the other hand Ti is not affected and hence
Ti will drop below T and consequently the faction ni/nγ will decrease below its value
at decoupling.
It can be shown that decoupled photons maintain a thermal distribution even if
they are not in thermal equilibrium. This follows from the fact that the thermal distribution
function redshifts into similar distribution function with lower temperature
proportional 1/a. Then we can speak about an effective temperature of relativistic
74
species that freezes out at a temperature Tf and a scale factor af so that
af Tf = aT(a) ⇒ Trel
(a) = Tf
af
a
. (4.49)
For example, neutrinos decouple at T ≈ 1MeV , shortly thereafter electrons and
positrons annihilate into photons and hence transfer energy and entropy into plasma
leaving neutrinos decoupled. Consequently we expect a neutrino background and
current Universe with a temperature of approximately 2K while the photon temperature
(that arise from the annihilation of electrons and positrons after decoupling of
neutrinos) is about 3K.
Similar effect occurs for particles which are non-relativistic at decoupling however
there is one important difference. For non-relativistic particles the temperature is
proportional to 1
2
mv2
that has the redshift as 1/a2
and we therefore have
Tnon−rel
i (a) = Tf
af
a
2
. (4.50)
The whole picture is as follows: We imagine that the species freeze out while relativistic
or non-relativistic and stay this way afterwards.
Now the notion of the effective temperature allows us to define a corresponding
notion of an effective number of relativistic degrees of freedom that can be defined
as
g∗ =
bosons
gi
Ti
T
4
+
7
8 fermions
gi
Ti
T
4
, (4.51)
where the temperature T is actual temperature of the background plasma assumed
to be in equilibrium while we have taken into account that different species i could
have a thermal distribution with a different temperature that of the photons. Then
the total energy density in all relativistic species comes from adding the contribution
of each species and we obtain a simple formula
ρ =
π2
30
g∗T4
. (4.52)
We can do the same thing for the entropy density. Since the entropy density of
relativistic particles goes as T3
rather T4
, we define the effective number of relativistic
degrees of freedom for entropy as
g∗S =
bosons
gi
Ti
T
3
+
7
8 fermions
gi
Ti
T
3
(4.53)
so that the entropy density of relativistic species is then
s =
2π
45
g∗ST3
. (4.54)
75
For example, in Standard model, we have
g∗ ≈ g∗S



100 for T > 300 MeV
10 for 300 MeV > T > 1 eV
3 for T < 1 MeV
(4.55)
The events that change the effective number of relativistic degrees of freedom are
the QCD phase transition at 300 MeV where quarks and gluons start to form bound
states, and the annihilation of electron-positron pairs at T ≈ 1 MeV .
Thanks to the release of the energy into the background plasma when species
annihilate it is only approximation that the temperature goes as 1/a. It is better to
say that comoving entropy density is conserved so that
s ≈ a−3
(4.56)
which holds in all forms of adiabatic evolutions, entropy is only produced at a process
like a first-order phase transition or out-equilibrium decay. It is expected that the
entropy production from such processes is very small compared to the total entropy
and the adiabatic presumption is excellent approximation for almost the entire early
Universe. If we now combine (4.56) with (4.54) we obtain a better expression for the
evolution of the temperature
T ≈ g
−1/3
∗S a−1
. (4.57)
We see the difference with the naive time dependence T ∼ 1/a. In fact, the temperature
will consistently decrease under adiabatic evolution in an expanding Universe
but it decreases more slowly when the effective number of relativistic degrees of
freedom is diminished.
4.7 Thermal relics
As we know particles typically do not stay in equilibrium forever, they density can
be so low that the interactions become infrequent and the particle freeze out. Since
essentially all of the particles in our current universe belong to this category it is
important to study the relic abundance of decoupled species.
We have seen that relativistic or hot particles have a number density that is
proportional to T3
in equilibrium. Thus a species X that freezes out while still
relativistic will have number density at freeze-out Tf given by
nX(Tf ) ∼ T3
f . (4.58)
Since this is comparable to the number density of photons at that time and since
after this freeze-out both photons and species X have densities that dilute by a
factor a(t)−3
as the Universe expands, we see that the abundance of X particles
today should be comparable to the abundance of CMB photons
nX0 ∼ nγ0 ∼ 102
cm−3
. (4.59)
76
We express this estimate as 102
rather as the precise number since the roughness
of this estimate does not warrant such misleading precision. For example, neutrinos
that are light (mν < MeV ) have a number density of nν = 115cm−3
for each species.
Then a corresponding contribution to the density parameter (if they are heavy enough
to be non-relativistic today)
Ω0,ν =
mν
92 eV
h−2
. (4.60)
Thus, a neutrino with mν ∼ 10−2
eV would contribute Ων ∼ 2 × 10−4
. We see that
this is not large enough to make neutrinos to be dark matter.
Let us now consider species X that is non-relativistic or cold at the time of
decoupling. In this case it is much harder to calculate the relic abundance of a cold
relic than a hot one simply because the equilibrium abundance of non-relativistic
species is changing rapidly with respect to the background plasma. Then we have
to be quite precise following the freeze-out process to obtain a reliable answer. The
direct calculation typically involves very complicated procedure. We rather give here
reasonable approximate expression. If σ0 is annihilation cross-section of the species
X at temperatures T = mX, then the final number density in terms o the photon
density can be determined to be equal to
nX(T < Tf ) ∼
1
σ0mXMP
nγ . (4.61)
Since the particles are non-relativistic at the time of decoupling, they are certainly
non-relativistic today and their energy density is
ρX = mXnX . (4.62)
Then finally we obtain the density parameter
ΩX =
ρX
ρcr
∼
nγ
σ0M3
P H2
0
. (4.63)
Numerically, when ¯h = c = 1 we have 1 cm ∼ 2 × 10−14
GeV so the photon density
today is
nγ ∼ 100 cm−3
∼ 10−39
GeV −3
. (4.64)
The present value of the Hubble constant is
H0 ∼ 10−42
GeV (4.65)
and the Planck mass is
MP ∼ 1018
GeV . (4.66)
Then finally (4.63) gives
ΩX ∼
1
σ0(109 GeV 2)
. (4.67)
77
We see an interesting fact that ΩX does not depend on mX but it depends on the
annihilation cross-section. Let us elaborate more about this result and consider some
weakly interacting massive particle. The annihilation cross-section of these particles,
since they are weakly interacting, should be σ0 ∼ α2
W GF , where αW is weak coupling
constant and GF is the Fermi constant. Using
GF ∼ (3000 GeV )2
, αW ∼ 10−2
(4.68)
and we obtain
σ0 ∼ 10−9
GeV −2
. (4.69)
Then the density parameter of such particles would be
ΩX ∼ 1 . (4.70)
In other words, a stable particle with weak interaction cross section produces relic
density of order of the critical density today and hence provides a perfect candidate
for cold dark matter.
After this introduction let us present the simplest possible scenario, that, of
course, can be refined by more careful calculations.
Let us again assume that there exists a heavy stable particle X and its antiparticle
X. Let us also presume that the dominant process in which these particles
can be destroyed or created is their pair-annihilation or creation with annihilation
products being the particles of the Standard Model. Let us also presume that there
is no asymmetry between X and X in the early Universe, in other words the densities
X and X are equal to each other. However we have to mention that this is actually
a strong assumption that is valid in many, but not all, realistic extensions of the
Standard Model 8
.
Let us outline the overall cosmological behavior of these particles. At hight
temperatures, T MX, the X- particles are in thermal equilibrium with the rest
of cosmic plasma. There are many X − X pairs in the plasma that are continuously
created and annihilate. As the temperature drops below MX, the equilibrium number
density decreases. At some “freeze-out” temperature Tf the number density becomes
so small so that X and X can no longer meet each other during the Hubble time
and their annihilation terminates. After that the number densities of survived X
and X decreases as a−3
(t) and these relic particles contribute to the mass density of
the present Universe. The purpose of the following analysis is to estimate the range
of properties of X particles in which their present mass density is of the order of the
critical density ρc so that X may serve as dark matter candidates.
Let us again assume thermal equilibrium. It is well known that the mean free
path < l > of a particle in a gas depends on the lifetime τann of a non-relativistic
8
In fact, the alternative scenario with the generation of X asymmetry is also interesting since it
might be related to baryon asymmetric the density of dark matter.
78
X-particle as
σann · v · τann · nX =< l > , (4.71)
where v is mean velocity of X particle, σann is the annihilation cross section at
velocity ν and nX = nX is equilibrium number density
nX = gX
mXT
2π
3/2
e−
mX
T . (4.72)
In order to find the life-time of the non-relativistic particle X we have to take some
reasonable value of < l >. It is natural to presume that it is of order 1 in the natural
units < l >∼ 1. Further, it can be also shown that for non-relativistic velocities the
annihilation cross section takes the form
σann =
σ0
ν
, (4.73)
where σ0 is constant. We will discuss its value later. We should now compare the
life-time with the Hubble time, or annihilation rate Γann = τ−1
ann with the expansion
rate H. At T ∼ mX the equilibrium density is of order nX ∼ T3
and Γann
H for not too small σ0. Conversely, the life-time is much smaller than Hubble
time and consequently the annihilation and creation of X − X pairs is rapid and
hence X-particles are in equilibrium with plasma. On the other hand for very small
temperatures T mX the number density nX is exponentially small and Γann H
(τann H−1
). Than it is clear that the thermal equilibrium between X-particles
and background plasma is not maintained. In other words the number density nX
gets diluted only because of cosmological expansion.
The freeze-out temperature Tf is determined by the relation
τ−1
ann ≡ Γann ∼ H , (4.74)
where we can still use the equilibrium formula as X particles are in thermal equilibrium
(with respect to annihilation and creation) just before freeze-out. Then we
find
σ0nX(Tf ) ∼ H ∼
T2
f
M∗
P
, (4.75)
where we have introduced the effective Planck mass
M∗
P =
MP
1.66 g∗(t)
, (4.76)
and hence the expansion rate is equal to
H(t) =
T2
(t)
M∗
P
. (4.77)
79
The solution of the equation (4.75) gives the freeze-out temperature, up to log terms
Tf ≈
mX
ln(m∗
P mXσ0)
. (4.78)
This temperature is quite bit smaller than mX which means that X-particles freeze
out when they are indeed non-relativistic and hence it is natural to call them as cold
dark matter.
At the freeze-out temperature we use (4.75) to get
nX(Tf ) =
T2
f
M∗
P σ0
. (4.79)
It is interesting to note that this density is inversely proportional to the annihilation
cross section. The explanation of this fact is that for higher annihilation cross section
the creation-annihilation processes are longer in equilibrium and less X particles
survive.
In order to estimate eh present density X-particles, it is convenient to consider
ratio nX/s where s is the entropy density
s =
2π2
45
g∗T3
. (4.80)
The point is that during the adiabatic expansion after freeze-out, the entropy density
scales as s ∼ a−3
since in the adiabatic process sa3
= const. In the same way since
we are in the freeze-out regime we have that nXa3
= const we obtain that nX scales
in the same way nX ∼ a−3
. Then, up to a factor of order 1, this ratio at freeze-out is
nX
s
∼
1
g∗(Tf )M∗
P Tf σ0
. (4.81)
At late times, the entropy density, again up to actor of order 1, is equal to the number
density of photons, so the present number density of particles is of order
nX,0
s0
=
nX
s freeze−out
⇒
⇒ nX,0 = s0
nX
s freeze−out
∼ sγ,0
nX
s freeze−out
(4.82)
and the present mass density is
ρX,0 = mXnX,0 =∼ nγ,0
ln(M∗
P mXσ0)
g∗(Tf )M∗
P σ0
, (4.83)
where we have also used (4.78). The formula above is very interesting since we see
that the present mass density depends mostly on one parameter, the annihilation
80
cross section σ0. The dependence on the mass of X-particle is through the logarithm
and g∗(Tf ) is very mild. From this formula we derive the condition that ensure that
X-particles are dark energy candidates, i.e. their present mass density is of order ρc
σ0 ∼
nγ,0
g∗(Tf )M∗
P ρc
ln(M∗
P mXσ0) (4.84)
that leads to the estimate
10−11
σ0 < 10−9
GeV −2
, (4.85)
where the uncertainty in the estimate is a consequence of the way we deal with
various numerical factors. In any case the estimate given above tells us what the
relevant range of mass scales is. To see this note that the annihilation cross section
may be parameterized as
σ0 =
α2
M2
, (4.86)
where α is some coupling constant and M is the mass scale (In the calculation above
M2
= GF .). With α ∼ 10−2
the estimate of the mass scale for σ0 ∼ 10−11
is roughly
M ∼ 1 TeV . (4.87)
In other words, we very mild assumptions we find that the non-baryonic dark energy
matter may naturally originate from the TeV -scale physics. Then it follows
that one natural candidate for the cold dark matter is neutralino. More precisely,
in supersymmetric extensions of the Standard Model the neutralino-that is mixture
of super-partners of photon, Z-boson and neutral Higgs bosons- is the lightest supersymmetric
particle that is often stable with the suitable value of the annihilation
cross section. In fact, the search for both direct and indirect signals from neutralino
dark matter is an active area of experimental research.
The mechanism discussed here is of course not the only one mechanism that is
able to model cold dark matter. Other dark matter candidates include very heavy
relics produced toward the end of inflation, axions, gravitinos, massive gravitons and
so on.
4.8 Status of experimental Search for Dark Matter at the Year 2017
As we argued above, the most compelling dark matter candidate is WIMP. On the
other hand the viable parameter space has recently become smaller with the announcement
in September 2016 by PandaX-II Collaboration and Large Underground
Xenon (LUX) Collaboration. In fact, the search for WIMP particles leads to zero
result.
As we know, physicists don’t know what dark matter is. For that reason we have
to have different experiments to detect it. Natural possibility is to try to produce
81
dark matter in an accelerator, as for example LHC at CERN and then to analyze
its decay product using particle detectors. Second technique is to use instruments
such as the Fermi-Gamma ray Space Telescope to observe dark matter interaction in
and beyond our Galaxy. This is again indirect detection since the telescope observes
particles that are produced by a collision between dark matter particles.
The third techniques that is used in both the LUX and PandaX-II experiments
is known as ”direct detection”. In this case the detector is constructed on Earth with
massive targets. In the case of LUX and PandaX-II the dark matter particles leave
behind trace of light that can be detected with sensors. The basic principle of these
detectors are following. First of all, LUX is located in South Dakota in the US while
PandaX-II is located in Sichuan, China. The basic block of these two experiments
is a time-projection chamber which is a large tank of ultrapure liquid xenon that
is topped with xenon gas. A particle (dark matter or ordinary matter) that enters
the chamber and interacts with a xenon atom in the liquid generates photons and
electrons. The photons produce signal S1 that is detected by photomultiplier at
the top and bottom of the tank. On the other hand electrons are conducted by an
electric field to the gaseous portion of the detector where they induce a second round
of scintillation (the first round of scintillation was performed by external particles
and leaded to the signal S1) and a signal S2. The pattern of S1 and S2 signals
is different when they interact with a dark matter particle than with an ordnary
particle and exactly this difference allows us to distinguish between these particles.
As we know, WIMP particles weigh many times the mass of the proton with
masses between 10 and 1000 GeV and they interact with matter via a weak force
where the strength of this interaction is quantified by its cross section. The LUX and
PandaX-II experiments are sensitive to a wide range of WIMP candidates. However
in their papers from 2016 both LUX and PandaX-II report no WIMP candidates. As
follows from the experiments sensitivities, these null results tell us that if the WIMP
exists and its mass lies between 10 and 1000 GeV its cross section for interacting with
normal matter is less than an extremely small value. These results say us that dark
matter interactions are much rarer than suggested by many popular hypotheses. In
fact, this no result and lack of evidence for supersymmetry at the LHC forces us to
start to doubt about WIMP hypothesis at least in its simplest form.
4.9 Baryongenesis
The symmetry between particles and antiparticles is firmly established in collider
physics. However then we lead to the following question; why the observed Universe
is composed almost entirely of matter with little or no primordial antimatter.
Outside the particle accelerators the antimatter can be seen in cosmic rays in
the form of a anti protons where the ratio of these andirons to protons is
np
np
∼ 10−4
. (4.88)
82
However this ratio is consistent with secondary anti proton productions through
accelerator-like processes
p + p → 3p + p (4.89)
as the cosmic rays stream toward us. In other words there is no evidence for primordial
antimatter in our galaxy. Also let us imagine that we have clusters of matter
and antimatter galaxies. Then we could expect that we could detect background of
γ-radiation from nucleon anti nucleon annihilations with clusters. This background
is not observed and so we conclude that there is negligible antimatter on the scale
of clusters.
All these considerations put an experimental upper bound on the amount of
antimatter in the Universe.
In order to study this problem in more details let us introduce the baryon to
entropy ratio
η ≡
nB
s
=
nb − nb
s
, (4.90)
where nB is the difference between the number of baryons and anti-baryons per unit
volume. The range of η was determined recently as is equal to
η = 6.1 × 10−10
± 0.210−10
. (4.91)
At early times, at temperatures well above 100 MeV ,cosmic plasma contained many
quark-anti quark pairs whose number density was of the order of the entropy density
nq + nq ∼ s , (4.92)
while baryon number density was related to densities of quarks and antiquarks as
follows (baryon number of quarks equals 1/3)
nb =
1
3
(nq − nq) . (4.93)
Hence in terms of quantities characterize the very early epoch, the baryon asymmetry
may be expressed as
η ∼
nq − nq
nq + nq
. (4.94)
We see that there was one extra one extra quark per about 10 billion quark-antiquark
pairs. It is this thiny excess that is responsible for entire baryonic matter in the
present Universe. Thus the natural question arises, as the Universe coolled from early
times to today, what processes, both particle and cosmological, were responsible for
the generation of this very specific baryon assymmetry?
Of course there is no logical contradiction to suppose that this thiny excess of
quarks to antiquarks was built in as an initial condition. Of course, this is not very
satisfactory for physics. Furthermore, inflationary scenario does not provide such an
83
initial condition for Hot Big Bang, rather, inflation theory predicts that the Universe
was baryon-symmetric just after inflation. In other words we would like to explain
the baryon asymmetry dynamically.
As pointed by Sakharov, a small baryon asymmetry may have been produced
in the early Universe from initially symmetric state if three necessary conditions are
satisfied:
• Baryon number (B) violation,
• Violation of C (charge conjugation symmetry) and CP (the composition of
parity and C)
• Departure from thermal equilibrium.
The first condition is clear since when we start from a baryon symmetric Universe,
baryon number violation must take case in order the Universe to evolve into the state
with baryon number violation. In other words, if the baryon number were conserved
that this charge would remain constant during time evolution and hence w we would
not observe the baryon number asymmetry.
The second Sakharov criterion is required since, when C and CP are exact
symmetries it can be shown that the total rate for any processes that produces an
excess of baryons is equal to the rate of the complementary process which produces
an excess of antibaryons and so no net baryon number can be created. CP violation
is present either if there are complex phases in the Lagrangian which cannot be
reabsorbed by field redefinition (explicit symmetry breaking) or if some High scalar
field acquires an VEV which is not real (spontaneous symmetry breaking).
Finally, in order to explain the third equilibrium let us calculate the thermal
equilibrium average of the baryon number operator B at temperature T = 1/β
B T = Tr(e−βH
B) = Tr (CPT)(CPT)−1
e−βH
B =
Tr e−βH
(CPT)−1
B(CPT) = −Tr(e−βH
B) ,
(4.95)
using the fact that (CPT) commutes with H and cyclicity of the trace. Finally, we
have used the fact that B is odd under (PC). Then from the equation above we see
that in the thermal equilibrium the baryon number is equal to zero and there is not
any generation of baryon number.
The first two Sakharov’s conditions may be investigated only within a given
particle model, while the third condition the departure from thermal equilibrium
may be discussed in a more general way.
84
4.10 Baryon Number Violation
At present there are two well understood mechanisms of baryon number non-conservation.
One emerges in Grand-Unified Theories (GUT). Briefly, these GUT describe the
fundamental interactions by means of the unique gauge group G that contains the
Standard Model group
SU(3)C ⊗ SU(2)L ⊗ U(1)Y .
The fundamental idea of GUT is that at energies higher than a certain energy MGUT
the group symmetry is G and that, at lower energies, the symmetry is broken down
to the SM gauge symmetry, possibly through the chain of symmetry breaking. The
motivation for this scenario, whose explanation, however, is beyond the scope of this
review, it the fact that in some models, the (running) gauge couplings of the SM
unify at the scale MGUT 2 × 1016
GeV .
The interesting fact considering GUT is that the baryon number violation emerges
very naturally in it. Briefly, the mechanisms of the baryon number violation is due
to the exchange of super-massive particles. The scale of these new, baryon number
violating interacting is of order 1016
GeV .
Another mechanism of the baryon number violation is related to the triangle
anomaly in the baryonic current. It exists already in the Standard Model and possibly
it operates in all its extensions. The main feature of this mechanism, as applied to
the early Universe, is that it is effective over a wide range of temperatures
100 GeV < T < 1011
GeV .
In summary, realistic mechanism of baryon number non-conservation are rare, but
there are several ways the baryon asymmetry could have been generated. They differ
by the characteristic temperature at which the asymmetry is produced.
The GUT mechanisms operates at extremely high temperatures
T ∼ 1015
− 1016
GeV
The most well developed source of the baryon asymmetry in this context are B- and
CP- violating decays of ultra-heavy particles. At late times the baryon number is
violated by anomalous electroweak processes.
Electroweak baryogenesis is scenario in which the baryon asymmetry is generated
entirely due to the anomalous electroweak processes. Its generation would occur
at temperature of order 100 GeV which is the energy at which these anomalous
processes are switched off. On the other hand the electroweak baryogenesis is still
under development.
In summary, the observed asymmetry may be explained by a number of mechanisms
all of which, however, exist in extensions of the Standard Model only. The
problem is that direct proof that any given mechanism is indeed responsible for the
baryon asymmetry.
85
4.11 Departure from the Thermal Equilibrium
In some scenarios, such as GUT baryogenesis, the third Sakharov condition is satisfied
due to the presence of superheavy decaying particles in a rapidly expanding Universe.
These processes are called as out-of-equilibrium decay mechanisms.
The underlying idea is simple.If the decay rate ΓX of the superheavy particles
X at the time they become non-relativistic (at the temperature T ∼ MX) is much
smaller than the expansion rate of the Universe, then the X particles cannot decay
on the time scale of the expansion and so they remain as abundant as photons for
T ≤ MX. In other words at some temperature T > MX the superheavy particles X
are so weakly interacting so the they decouple from the thermal bath while they are
still relativistic, so that
nX ∼ nγ ∼ T3
(4.96)
at the time of decoupling.
Then we obtain that at temperature T MX they populate the Universe with
an abundance which is much larger than the equilibrium one. This abundance is
precisely the departure from thermal equilibrium needed to produce a final nonvanishing
baryon asymmetry when heavy states X decay in B and CP violating
decays.
It can be shown that the out-of-equilibrium condition requires very heavy states
MX ≤ (1010
− 1016
) GeV , (4.97)
if these heavy particles decay through renormalizable operators.
A different mechanism of the departure from the thermal equilibrium can be
found in the electroweak theory.
A further natural way to depart from equilibrium is provided by the dynamics
of the topological defects.
4.12 Neutrino background
As an example of the previous discussion let us consider the fate of neutrinos in
the expanding Universe. The dynamics of the neutrinos and their reactions with
other components of the matter are governed by the Standard model. Then using
the rules of standard quantum field theory one can calculate the reaction rate Γ of
the neutrinos with the rest of the matter (Roughly speaking the inverse Γ−1
is the
average time between collision of the neutrinos with all form of the matter). When
Γ−1
is larger than H−1
(conversely, when Γ is less than H) there cannot occur the
reactions between the neutrinos and the rest of the matter. We say that in this case
neutrinos effectively decouple from the rest of matter. It can be shown that the
relevant ration is given by
Γ
H
≈
T
1.4MeV
3
=
T
1.6 × 1010K
3
. (4.98)
86
This formula implies that for T ≤ 1.6 × 1010
the neutrinos decouple from the rest of
the matter. On the other hand electrons and positrons can still annihilate at slightly
lower temperature. This process increases the number of the photons. As a result
the photon temperature goes up with respect to neutrino temperature (Remember
that it is natural to speak about two different temperatures for two different species
of particles since they have already decoupled.). We can calculate this increase of
temperature as follows. The increase of T is due to the change of degree of freedom
g and is given by
(aTγ)3
after
(aTγ)3
before
=
gbefore
gafter
=
7
8
(2 + 2) + 2
2
=
11
4
. (4.99)
Let us explain factors given above. In the numerator, one 2 is for electron, one 2
is for positron and the factor 7/8 arises because of fermions. The remaining 2 in
numerator is for photon. In denominator 2 is for photon since they remain after the
annihilation of positrons with electrons. Using the relation above we obtain
(aTγ)after =
11
4
1/3
(aTγ)before =
11
4
1/3
(aTν)before =
=
11
4
1/3
(aTν)after = 1.4(aTν)after .
(4.100)
The first equality is from (4.100), the second follows from the fact that the photons
and neutrinos had the same temperature originally. The third equality follows from
the fact that for decoupled neutrinos aTν are constant. The final result leads to the
prediction that at present the Universe will contain a bath of neutrinos that has
temperature that is lower than of CMBR.
4.13 Primordial Nucleosynthesis
Theory of Big Bang Nucleosynthesis and observations of primordial abundances of
light elements probe the earliest epoch of the evolution of the Universe that is accessible
to observation today. This epoch corresponds to temperatures ranging from
1 MeV to a few 10 keV and age of the Universe from 1 s to 200 s.
Let us briefly review the properties of the matter at this early epoch of the
Universe.
At temperatures above 1 MeV there is a thermal equilibrium with respect to
reactions
p + e ↔ n + νe . (4.101)
As the Universe cools down below T ≈ 1 MeV neutrons are no longer produced
or destroyed, they concentration (relative to protons) ”freezes out”. Alternatively
87
saying, the weak interactions are frozen out and neutrons and protons cannot interconvert.
The equilibrium abundance of neutrinos at this temperature is about 1/6
the abundance of neutrons due to the slightly larger neutron mass.
When we reach a temperature somewhat below 100 keV the Bing-Bang Nucleosynthesis
(BBN) begins 9
. At that point the neutron/proton ration is about 1/7.
Since it is energetically favorable for nucleons to form He the most part of the free
neutrinos are converted into He. For every two neutrons and fourteen protons we
end up with one helium nucleus and twelve protons. In other words 25 % of the
baryons are converted to helium. There are also trace amounts of deuterium and
lithium. Heavier elements are not synthesized in the Big Bang but require supernova
explosions in the later universe. These elements remain in the Universe so their
primordial abundance is measurable today.
It is important to stress that Big Bang Nucleosynthesis serves also as a source
of constraints on particle physics. The fact that the temperature of the Universe
reached at least 1 MeV or so and that the expansion was described by know physics
at this stage constrain significantly some extensions of the Standard models.
The most amazing fact about nucleosynthesis is that, given the Universe is radiation
dominated during the relevant epoch, the relative abundances of the light
elements depend essentially on one parameter, the baryon to entropy ratio
η ≡
nB
s
=
nb − nb
s
, (4.102)
where nB is the difference between the number of baryons and anti-baryons per unit
volume. The range of η was determined recently as is equal to
η = 6.1 × 10−10
± 0.210−10
. (4.103)
Let us be now more specific. We know that at present the Universe is expanding and
filled with radiation that is very cold today (T0 = 2.73K). If we trace the evolution
of the Universe back in time to earlier epochs that were hotter and denser, the
early Universe is a Primordial Nuclear Reactor during its first 20 minutes (≈ 1000).
In fact,when the temperature of the Universe is higher than the binding energy
of nuclei (∼ MeV ) none of the heavy elements (helium and metals) could have
existed in the Universe. The binding energy of the first four light nuclei, H2
, H3
, He3
and He4
are 2.22MeV, 6.92MeV, 7.72MeV and 28.3MeV respectively. Since the
average energy in the thermal ansamble is proportional to the temperature we obtain
that these nuclei could be formed when the temperature of the Universe is in the
range (1 − 30)MeV . Surprisingly, the actual synthesis takes place at much lower
temperature Tnuc = Tn ≈ 0.1MeV . The reason for this delay is the high entropy
9
Note that the nuclear binding energy per nucleon is typically of order 1 MeV so that one could
expect that BBN would occur earlier. However the large number of photons per nucleons at that
time prevent BBG to occur until the temperature drops below 100 keV .
88
of the Universe that implies that the ration of photons to baryons, η−1
is high.
Numerically
η =
nB
nγ
= 5.5 × 10−10 ΩBh2
0.02
, Ωh2
= 3.65 × 10−3 T0
2.73K
3
η10 . (4.104)
Thus, even if the thermal equilibrium is maintained the significant synthesis of nuclei
can occur only at T ≤ 0.3MeV . Then we can expect significant production XA ∼ 1
of nuclear species A at temperature T ≤ TA. However it turns out that the rate of the
nuclear reaction is not high enough to maintain thermal equilibrium between various
species. In order to study non equilibrium abundances in an expanding Universe is
based on rate equations. Let us now review its general concepts.
4.13.1 Rate equations
Consider a reaction in which two particles 1 and 2 interact to form two other particles
3 and 4. For example, let us consider reaction n + νe = p + e that converts
neutrons into protons in the forward direction and proton into neutrinos in the reverse
direction. Another example is the reaction p + e = H + γ where the forward
reaction describes recombination of electron and proton forming a neutral hydrogen
atom with the emission of photon. In general we are interested in how the number
density n1 of particle species 1 changes due to the reaction of the form 1+2 ⇔ 3+4.
Remember that even in case where there is no reaction the number density changes
as n1 ∝ a−3
due to the expansion of the Universe. In other words the quantity that
changes due to the reaction is n1a3
. Further, the forward reaction will be clearly
proportional to the product of the number densities n1n2 while the reverse reaction
will be proportional to n3n4. Hence we can write the equation for the rate of the
change of particle species n1 in the form
1
a3
d(n1a3
)
dt
= µ(An3n4 − n1n2) (4.105)
The left hand side is the relevant rate of change over and above that due to the
expansion of the Universe. On the right hand side the two proportionality constants
have been written as µ and Aµ that generally are functions of time. Usually µ σv
where σ is the cross section for the relevant process and v is relative velocity. The
left hand side has to vanish for system in thermal equilibrium with ni = neq
i where
the superscript eq denotes the equilibrium densities of the different species labeled
with i = 1 . . . 4. If we insert in the above equation the condition ni = neq
i we can
express A as
Aneq
3 neq
4 − neq
1 neq
2 = 0 ⇒ A =
neq
1 neq
2
neq
3 neq
4
(4.106)
and than the rate equation becomes
1
a3
d(n1a3
)
dt
= µneq
1 neq
2 (
n3n4
neq
3 neq
4
−
n1n2
neq
1 neq
2
) . (4.107)
89
On the left hand side we can write d
dt
= aH d
da
that shows that the relevant scale for
this processes is H−1
. Clearly when H
µni
1 the right hand side becomes ineffective
because the factor µ
H
factor. Then we see that the number of particles of species 1
does not change. In other words when the expansion rate of the Universe is large
compared to the reaction rate ( µ
H
1) the given reaction is ineffective in changing
the number of particles. However this result does not mean that the reactions have
reached thermal equilibrium and ni = neq
i . In fact, the opposite situation occurs:
The reactions are not fast enough to drive the number densities towards equilibrium
densities and the number densities ”freeze out” at non-equilibrium values. Of course
the right hand side in (4.107) will also vanish when ni = neq
i that is the extreme limit
of thermal equilibrium.
Using this general formalism we will now apply it to the process of nucleosynthesis
which requires protons and neutrons that combine together to form bound nuclei
of heavier elements like deuterium, helium... The abundance of these elements are
going to be determined by the relative abundance of neutrons and protons in the
Universe. For that reason we should start the discussion with the problem of the
thermal equilibrium between protons and the neutrons in the early Universe. As
long as the inter-conversion between n and p through the weak interaction processes
ν + n ↔ p + e , e + n ↔ p + ν (4.108)
or their decay
n ↔ p + e + ν (4.109)
is rapid with respect to the expansion rate of the Universe thermal equilibrium can
be maintained. Then the equilibrium static physics implies that the equilibrium n/p
ration is equal to
nn
np
=
Xn
Xp
= exp(−Q/T) , (4.110)
where Q = mn − mp = 1.293MeV . For T Q the factor in the exponent is
approaching zero and we obtain Xn ≈ Xp. However when T drops below about
1.3MeV the neutron fraction will drop exponentially on condition that the thermal
equilibrium is still maintained. However to check weather the thermal equilibrium
is maintained we have to compare the expansion rate with the reaction rate. The
expansion rate is
H =
8πGρ
3
, (4.111)
where
ρ =
π2
30
gT4
, (4.112)
where g ≈ 10.75 represents the relativistic degrees of freedom present at these temperatures.
At T = Q this gives H ≈ 1.1s−1
. The reaction rate needs to be computed
90
from weak interaction theory. The neutron to proton conversion rate is approximated
by
λnp ≈ 0.29s−1 T
Q
5
Q
T
2
+ 6
Q
T
+ 12 . (4.113)
At Q = T this gives λ ≈ 5s−1
that is more rapid than the expansion rate. But as
T drops below Q this decreases rapidly and the reaction ceases to be fast enough
to maintain thermal equilibrium. Then we have to work out the neutron abundance
using the equation (4.107).
If we denote n1 = nn, n3 = np and n2, n4 = nl where the subscript l stands for
leptons then the equation (4.107) becomes
1
a3
d(nna3
)
dt
= µneq
l
npneq
n
neq
p
− nn . (4.114)
To proceed we use the fact that µneq
l is equal to the rate of the neutron to proton
conversion λnp. We also use the relation
neq
n
neq
p
= exp(−Q/T) (4.115)
Let us now introduce the fractional abundance
Xn =
nn
(nn + np)
(4.116)
Then the equation (4.114) takes the form
dXn
dt
= λnp((1 − Xn)e−Q/T
− Xn) , (4.117)
where we have used
Xn + Xp = 1 , Xp =
np
nn + np
(4.118)
and also the fact
1
a3
d(nna3
)
dt
=
a3
(nn + np)
a3
dXn
dt
(4.119)
since (nn + np)a3
is constant. This equation can be integrated numerically and
determine how the neutron abundance changes with time. The neutron fraction falls
out of equilibrium when temperature drop below 1MeV and it freezes to about 0.15
at temperature below 0.5MeV . As the temperature decreases further the neutron
decays with a half life of τn ≈ 886.7sec becomes important and starts to reduce
the neutron number density. Then the only way how the neutrons can survive is
through the synthesis of light elements. As the temperature falls further to T =
THe ≈ 0.28MeV significant amount of He could have been produced if the nuclear
reaction rates were high enough. These reactions are all based on D, He and H
91
and do not occur rapidly enough because the mass fraction of D, He and H are still
quite small [10−12
, 10−19
, 5 × 10−19
] at T 0.3MeV . The equilibrium deuterium
abundance passes through unity at temperature of about 0.07MeV which is when
nucleosynthesis can really begin.
The production of still heavier elements-even those like C, O which have higher
binding energies than He is suppressed in the early Universe.
4.14 Decoupling of matter and radiation
In the early hot phase the radiation will be in thermal equilibrium with matter. As
the Universe cools below kBT ( a/10) is the binding energy of atoms the electrons
and ions will combine to form neutral atoms and radiation will decouple from matter.
This occurs at T 3 × 103
K. As the Universe expands further these photons will
continue to exist without any further interaction. We shall now discuss some details
related to the formation of neutral atoms and decoupling of photons.
The relevant reaction is
e + p = H + γ . (4.120)
If the rate of this reaction is faster than the expansion rate then one can calculate
the neutral fraction as follows. Introducing the fractional ionization Xi for each of
the particle species and using the facts that np = ne and np + nH = nB. We also
have Xp = Xe and XH = nH
nB
= 1 − np
nB
= 1 − Xe. The equation that governs the
time evolution of Xe that expresses the equilibrium situation now takes the form
1 − Xe
X2
e
≈ 3.84η
T
me
3/2
exp(B/T) , (4.121)
where η = 2.68 × 10−8
(ΩBh2
) is the baryon-to-photon ratio.We define Te as the
temperature at which 90 percent of the electrons have combined with protons. This
implies np = 0.1nB and hence Xe = Xp = 0.1. This leads to the condition
(ΩBh2
)−1
τ3/2
exp[−13.6τ−1
] = 3.13 × 10−18
, (4.122)
where τ = (T/1eV ). The solution of this equation can be given by iterative procedure.
For ΩBh2
= 1, 0.1, 0.01 we then obtain Tatom = 0.324eV, 0.307eV, 0.292eV .
These results were based on the equilibrium densities. Then it is important
to check that the rate of the reaction p + e ↔ H + γ is fast enough to maintain
equilibrium. It turns out however that this is not fully satisfied and hence we have to
again use the rate equation. The rate equation (4.107) for n1 = ne, n2 = np, n3 = nH
and n4 = nγ and for Xe = ne
ne+nH
takes the form
dXe
dt
= α
β
α
(1 − Xe) − nbX2
e , (4.123)
92
where the recombination rate α is the rate is given by
α = 9.78r2
0c
B
T
1/2
ln
B
T
, (4.124)
where r0 = e2
m2
ec2 is classical electron radius. In (4.123) the ration β/α is given as
β
α
=
meT
2π
3/2
exp[−B/T] (4.125)
Using this result we obtain that the value of Tatom does not change significantly.
4.15 Structure formation and linear perturbation theory
The structure formation is based on the key idea that if there exist small fluctuations
in the energy density in the early Universe, then gravitational instability then leads
in a well understood manner leading to structures like galaxies today. The most
popular model for generating these fluctuations is based on the idea that if the very
early Universe went through the inflation phase then the quantum fluctuations of
the field driving the inflation can lead to energy density fluctuations.
Let us illustrate this idea on the example of the massless scalar field φ minimally
coupled to gravity. The action of the scalar field is
Sφ = −
1
2
d4
x
√
−ggµν
∂µφ∂νφ (4.126)
In spatial flat FRW background this action has the form
Sφ = −
1
2
dxdta3
(t)[−(∂tφ)2
+
1
a2
(∂iφ)2
] (4.127)
so that the equation of motion takes the form
∂t(a3
∂tφ) − a∂i∂i
φ = 0 (4.128)
or equivalently
¨φ + 3H(t) ˙φ −
1
a2
∂i∂i
φ = 0 , (4.129)
where ˙x = ∂tx , ¨x = ∂2
t x. Thanks to the homogeneity and isotropy of space it is
natural to work in the momentum representation where we search for the solutions
in the form
eixk
φk(t) . (4.130)
If we insert (4.130) into (4.129) we obtain ordinary differential equation for φk in the
form
¨φk + 3H(t) ˙φk +
k2
a2
φ = 0 . (4.131)
93
Note that k is a coordinate momentum. The physical momentum at time t is
p =
k
a
(4.132)
and it depends on time.
Looking on (4.131) we see that the second term in it acts as a friction term.
Then we can consider two regimes with the qualitatively different behavior of the
modes φk: Subhorizion modes:
These modes are characterized condition
p =
k
a
H . (4.133)
Modes obeying this property are subhorizon modes since their physical length λ ∼
p−1
is much shorter than the Hubble distance H−1
that is a horizon size in matter
and radion dominated Universe. More precisely, for modes obeying the condition
(4.133) we can neglect the friction term in (4.131) and hence we get
¨φ + ω2
k(t)φ = 0 , ωk(t) =
k
a
(4.134)
This equation has the general solution
φk =
1
a
e
±i
t
t0
dt ωk(t )
(4.135)
since
˙φk = −Hφk + iωkφk ≈ iωkφk ,
¨φk = i ˙ωkφk − ω2
kφk = −iHωkφk − ω2
kφk ≈ −ω2
kφk .
(4.136)
This solution (modulo slowly varying prefactor) describes oscillations with the frequently
experiencing redshift (The frequency is lowered with time).
Superhorizon modes:
These modes are characterized by condition
p =
k
a
H . (4.137)
In this case the last term in (4.131) are negligible and the solutions are
constant mode : φk = const ,
growing mode : φk(t) = K
t
t0
dt
a3(t )
.
(4.138)
94
It is clear that the constant mode is solution of (4.131). The growing mode is solution
as well since
˙φk =
K
a3
, ¨φk = −3Hφk . (4.139)
The gravitational waves obey precisely the same equations as (4.131) so that they
have exactly the same behavior, in particular, for given k one of the superhorizon
modes blows up at small t. It follows that the whole picture of the FRW Universe
with small perturbations is thus self-consistent only if this modes vanishes at finite
times.
Now recall that for radiation dominated and matter dominated Universe H ∼ t−1
while the scale factor behaves as a ∼ t1/2
for radion dominated Universe and a ∼ t2/3
for matter dominated Universe. Then the ratio of physical momentum to H behaves
as
p(t)
H(t)
∝ t1/2
(4.140)
for radiation dominated Universe and
p(t)
H(t)
∝ t1/3
(4.141)
for matter dominated Universe. These results mean that all modes start as superhorizon
and then enter the horizon. In the scalar mode example the requirement
that the growing mode vanishes determines the initial date for each k up to overall
amplitude. Then we have
φk = ck ,
k
a
H , (4.142)
and
φk = ck cos
t
0
dt ωk(t ) ,
k
a
H . (4.143)
For density perturbations the oscillating behavior means that at late enough times
there are sound waves in the primordial plasma with the wave-lengths that are shorter
than the horizon size at each moment of time. Briefly speaking the fate of the
primordial density perturbations is as follows. They stay constant until they enter
the horizon at radiation or matter dominate stage. After that they start to oscillate
and make the sound waves. The amplitudes of these waves grow during the matter
dominated stage due to the gravitational instability. The regions with higher density
tend to gravitationally attract matter and become even more overdense. The dense
regions collapse and form gravitationally bound structures.
Let us now discuss in more details how the simple description given above is
related to the more realistic situation. As long as the fluctuations are small one
can study their evolution by linear perturbation theory. The basic idea of linear
perturbation theory is well defined and simple. We write the metric as
gµν = gFRW
µν + hµν , (4.144)
95
where gFRW
µν is background FRW metric and hµν is small perturbations that propagate
on the background characterized with gFRW
µν . In the same way we perturb the source
energy momentum tensor by
Tµν = TFRW
µν + δTµν , (4.145)
where again TFRW
µν is the stress energy tensor for the background matter that solves
the FRW equations and δTµν are perturbations. If we linearize the Einstein’s equations
one can relate the perturbed quantities by a relation of the form
L(gFRW
µν )hµν = δTµν , (4.146)
where L is second order linear differential operator depending on the background
metric gFRW
µν . As wa argued above due to the fact that the background is maximally
symmetric one can separate out time and space and we can write down the equation
for any given mode labeled with the wave vector k as
L(a(t), k)hµν(t, k) = δTµν(t, k) . (4.147)
Then carefull analysis performed in case of metric perturbations implies that the
linearized equations of motion for gravity perturbations take the forms given in the
toy example of the massless scalar fields studied above. More precisely, it can be
shown,after some simplifications and presumption, that are all well justified, that
perturbed metric can be written in the form
ds2
= a2
(η)[(1 + 2Φ)dη2
− (1 − 2Φ)δabdxa
dxb
] . (4.148)
In other words we obtain one perturbed scalar degree of freedom Φ. Then it can be
shown that the dynamics of the mode Φ is governed by the equations that has the
same form as (4.131).
5. Inflation cosmology
5.1 Problems of the standard Big-Bang model
The standard Big-Bang model suffers from number of problems. Before we enter
in their discussion we review some properties of the Friedmann models at the early
stage of the Universe.
The question is what can we say about the Hubble parameter H = ˙a
a
, the density
ρ and the quantity k?
At the earliest stages of the evolution of the Universe H and ρ could be arbitrarily
large. On the other hand it is believed that for ρ ≥ M4
P effects of quantum gravity
are significant and the quantum fluctuations of metric exceed the classical value of
96
gµν. The standard cosmology where the metric is treated in the classical manner
restricts to the region of phenomena where
ρ ≤ M4
P , T ≤ MP ∼ 1019
GeV, H < MP . (5.1)
We also have to stress that in the expanding Universe thermodynamics equilibrium
cannot be established immediately but only when the temperature T is sufficiently
low. The behavior of the non-equilibrium Universe at densities of order of
the Planck density is very important problem.
Now we come to the list of problems of the standard hot Universe theory
5.2 Problems of the standard scenario
The singularity problem
The Friedmann equations imply that the density of matter in the Universe goes to
infinity as t → 0 and the corresponding solutions cannot be formally continued to
the domain t < 0.
One of the most exciting questions of cosmology is whether anything existed
before t = 0. If there is nothing before t < 0 the question is: where did the Universe
come from?
Studies of the general structure of space-time near a singularity suggest that it is
highly unlikely that this problem could be solved with the framework of the classical
gravitation theory. One hope that these questions could be answered in the context
of string theory. We will review some string theory inspired models in next sections.
However these models are faced with many important and conceptional problems so
that the problem of the birth of the Universe is the most challenging un answered
question in physics.
Flatness Problem
The flatness problem concerns with the observation that the real density of the
Universe, ρ, is known to be very close to the critical density ρc. Recall, that in the
previous section we have studied the Friedmann equation
H2
=
1
3M2
P
ρ −
k
a2
, (5.2)
where now MP ≡ 1√
8πG
∼ 2 · 1018
GeV is the four dimensional Planck mass. Recall
also that H = ˙a
a
where a(t) is the scale factor with the spacetime metric on the form
ds2
= −dt2
+ a2
dΣ , (5.3)
97
where dΣ is comoving volume element of space with k = 0, +1, −1 corresponding to
flat, positively curved and negatively curved spaces respectively. As we known we
can rewrite the Friedmann equation in the form
Ω − 1 =
k
a2H2
, (5.4)
where Ω means the sum of particular Ω’s. Note that for ordinary type of matter,
1
a2H2 will increase with time. To see this we use the continuum equation given by
˙ρ + 3H(ρ + p) = 0 . (5.5)
If we assume an equation of state of the form
p = wρ , (5.6)
for w = const then the continuity equation can be written as
dρ
da
da
dt
+ 3
˙a
a
(1 + w)ρ =
dρ
da
+ 3(1 + w)
ρ
a
= 0 , (5.7)
that implies
ρ ∼ a−3(1+w)
. (5.8)
If we start with Ω ∼ 1 we obtain that k ∼ 0. Then the Friedman equation is
H2
∼ ρ ⇒
˙a
a
∼ a−3(1+w)/2
(5.9)
that implies
daa(1+3w)/2
= t ⇒ a ∼ t
2
3(1+w) . (5.10)
As a consequence we get that
1
a2H2
∼ t2− 4
3(1+w) . (5.11)
This expression grows with time for any w > −1/3-examples include pressureless
dust with w = 0 and radiation with w = 1/3. Looking on the form of the Friedman
equation (5.4) we see that, unless the Universe is exactly flat (k = 0) and, as a
consequence Ω = 1, Ω will rapidly evolve away from Ω = 1. In order to have a value
of Ω close to 1 today, one would therefore expect to need a value of Ω even closer to
1 in the early Universe. This is the famous Flatness problem. That is, how can Ω be
so close to one?
We can argue alternatively as follows. Looking on the form of Friedmann equation
we see that the curvature contribution is
|Ωcurv| ≡
ρcurv
ρc
=
3MP
a2H2
, (5.12)
98
where we have defined the curvature contribution to the Friedmann equation as
|ρcurv| =
3MP
a2
. (5.13)
The present value of the equation (5.12) is
|Ωcurv| < 0.02 . (5.14)
We see that |ρcurv| scales as 1/a2
while the radiation matter and radiation scales
as 1/a3
and 1/a4
respectively. This implies that the curvature contribution to the
Friedman equations was even smaller in the past, for example
nucleosynthesis : |Ωcurv| < 10−16
,
electroweak epoch , |Ωcurv| < 10−26
.
(5.15)
In other words the spatial curvature of the Universe was tiny at the beginning. The
question is, why the initial conditions were so flat? This flatness problem cannot be
solved within Hot Big Bang theory.
The total entropy and total mass problem
The question is why the total entropy S and total mass M of matter in the
observable part of the Universe with Rp is so large. The total entropy S of the
present Universe can be estimate as follows. The size of the observable part of the
Universe is
lH,0 ∼ 2H−1
0 ∼ 1026
m . (5.16)
The entropy inside a sphere of the size lH,0 is roughly of the order of the number of
photons
S ∼ Nγ ∼ nγl3
H,0 . (5.17)
Since the number density of photon is equal to
nγ ∼ T3
γ ∼ (2.7 K)3
, (5.18)
where Tγ is the temperature of the primordial background radiation. Then we finally
obtain
S = 1088
. (5.19)
On the other hand the estimate of the total mass in the observable Universe is
M ∼ l3
H,0ρc ∼ 1055
g . (5.20)
In the Hot Big Bang theory the expansion of the Universe is almost adiabatic so this
huge entropy should be built in as an initial condition. Certainly this initial condition
is very special. Moreover, the condition of naturality, which is the statement that all
dimensionless quantities should be of order 1 implies that such a initial conditions
with huge entropy are rather un-natural.
99
Horizon problem
We known that the region of the Universe look very similar even though, assuming
normal radiation dominated expansion of the early Universe, thay can not have been
in causal contact. In fact, the horizon problem steams from the existence of particle
horizons in FRW cosmologies. Horizons exist because there is only a finite amount
of time since the Big Bang singularity and thus only a finite distance that photons
can travel within the age of the Universe. Consider a photon moving along a radial
trajectory in a flat Universe. In a flat, Universe, we can normalize the sale factor to
be a0 = 1. A radial null path obeys
0 = ds2
= −dt2
+ a2
dr2
(5.21)
so the comoving (coordinate) distance traveled by such a photon between times t1
and t2 is
r =
t2
t1
dt
a(t)
. (5.22)
To get a physical distance as it would be measured by an observer at any time t
simply multiply by a(t). For simplicity, we are in matter dominated Universe for
which
a =
t
t0
2/3
. (5.23)
The Hubble parameter is therefore given by
H =
˙a
a
=
2
3t
= a−2/3
H0 , (5.24)
where H0 is Hubble parameter of today Universe. Note that for a given above we
have that H = 2
3
t and hence H0 = 2
3t0
. Then the photon travels a comoving distance
r = 2H−1
0 (
√
a2 −
√
a1) . (5.25)
The comoving horizon size when a = a∗ is the distance a photon travels since the
Big Bang
rh(a∗) = 2H−1
0
√
a∗ , (5.26)
where we used the fact that at the Big bang we have a1 = 0. The physical horizon
size, as measured on the spatial hypersurface at a∗ is therefore simply
dh(a∗) = a∗rh(a∗) = 2H−1
0 a3/2
∗ = 2H−1
0
H0
H∗
= 2H−1
∗ . (5.27)
The horizon problem is simply the fact that CMB is isotropic to high degree of
precision even though widely separated points on the last scattering surface are
completely outside each other’s horizons. When we look at the CMB we see the
100
Universe at a scale factor aCMB ≈ 1/200. The comoving distance between a point
on the CMB and an observer on Earth is
r = 2H−1
0 (1 −
√
aCMB) ≈ 2H−1
0 . (5.28)
However, the comoving horizon distance for such a point is
rh(aCMB) = 2H−1
0
√
aCMB = 6 × 10−2
H−1
0 . (5.29)
Hence if we observe two widely separated parts of the CMB they will have nonoverlapping
horizons; different patches of the CMB sky were causally disconnected
at recombination. On the other hand they are observed to be at the same temperature
at high precision. This is the core of the famous horizon problem.
Problem of the large-scale homogeneity and isotropy of the Universe
As we argued in introduction all cosmological models are based on the presumption
of absolutely homogeneous and isotropic Universe. Of course Universe is not absolutely
homogeneous and isotropic at now at least on small scale and hence there is
no reason to believe that it was homogeneous at its beginning. The most natural
assumption is that the initial conditions at points that are sufficiently far from one
another were chaotic and uncorrelated. On the other hand it was shown by Collins
and Hawking that class of the initial conditions for which the Universe tends asymptotically
(at large t) to Friedmann Universe is one of measure zero among all possible
conditions. In other words according to this classical analysis Friedmann model is
very unprobable. This is the problem of large scale homogeneity and isotropy.
The galaxy formation problem
We know that Universe contains many inhomogeneities as stars, galaxies and so on.
In order to explain the origin of galaxies one have to presume an existence of initial
inhomogeneities whose spectrum is usually taken to be almost scale invariant. For a
long time the origin of such density inhomogeneities remained obscure.
The baryon asymmetry problem
This is the problem why the Universe is added almost entirely of matter with almost
no antimatter and why on the other hand the number of baryons is much less than
number of photons nB
nγ
∼ 10−9
.
The domain wall problem
101
It is natural to presume that the symmetry breaking occurs independently in all
causally unconnected regions of Universe. Then at all these regions that comprise
Universe at the time of symmetry-breaking phase transition, both field φ = +µ/
√
λ
and the field φ = −µ/
√
λ. Domains filled by the field φ = +µ/
√
λ are separated
from those with the field φ = −µ/
√
λ by domain walls. It can be shown that the
energy density of these walls is so high so that their existence is inconsistent with
cosmological consequences. Since the theories based on the spontaneously breaking
of gauge symmetry are very appealing and since in these theories domain walls arise
in natural way we meet Domain wall problem. In other words how to deal with such
theories in cosmology.
The primordial monopole problems
This problem is closely related to the domain wall problems. Many theories based
on symmetry-braking mechanism can produce another nontrivial structures that are
nontrivial configurations of the scalar and gauge fields and that are stable. However
it can be shown that these objects are very massive. Moreover it can be also shown
that the monopole density at present would be comparable with the baryon density.
Thanks to the enormous massivity these objects we obtain that the Universe filled of
monopoles is 1015
higher than the critical density. This implies that Universe filled
with such matter would have collapsed long ago. The explanation of the mechanism
how to deal with monopoles is one of the most important problems in cosmology.
Unwanted Relics
We have argued that for correct description of the early Universe the models of
particle physics should be present. However these models contain monopoles and
other topological defects. However the energy density of these objects can be very
big and hence the monopole abundance in GUT is serious problem for cosmology if
GUT have anything to do with reality.
5.3 Inflation as a solution
5.3.1 The General Idea of Inflation
The horizon problem is an extremely serious problem for the standard cosmology.
Cosmological inflation is mechanism that can solve this problem.
The main idea is that the Universe undergoes a period of accelerated expansion
defined as a period when ¨a > 0 at early times. The effect of this acceleration is to
quickly expand a small region of space to huge size. At this process the spatial curvature
of the Universe is reduced and consequently we make the Universe extremely
close to flat. In addition, the horizon size is greatly increased so that distant points
on the CMB actually are in causal contact and unwanted relics are diluted, solving
102
the monopole problem. Finally, quantum fluctuations imply that inflation cannot
smooth out the Universe with perfect precision, so there is a spectrum of remnant
density perturbations.
The general idea of inflation is that before Hot Big Bang (but after Planck
era) the Universe was in vacuum-like state and then it went through the era of the
exponential expansion
a(t) = const · e Hinfldt
, (5.30)
where Hinfl is almost constant in time. Due to the exponential expansion a small
patch of the Universe expands to great size. Let us presume that the duration of
inflation tinfl exceeds 140 Hubble times
tinfl >
140
Hinfl
. (5.31)
Let us also presume that the size of the patch is initially at the order Planck size lP =
1
MP
∼ 10−33
cm. Then at the time tinf the size exceeds the present horizon size lH,0 ∼
1028
cm. It is also clear the Universe flattens out, any initial inhomogeneities are
diluted out. In the end of inflation, the Universe becomes spatially flat,homogeneous
and isotropic at exponentially large spatial scales. This solves the horizon and flatness
problems.
A natural way to ensure that the Universe expands exponentially is to assume
that the matter at inflationary stage is in the vacuum-like state characterized with
the energy density ρinfl that is almost constant in time. At some point this energy
density should transform into conventional energy density of hot plasma. This
transformation is called reheating and after reheating the Hot Big Bang era begins.
During reheating, huge entropy is released and this solves the entropy problems.
5.4 Many models of inflation
Before we come to the more detailed study of the question how the inflation works
we give summary of some models of the inflation theory. The common property of
these model is that the matter with suitable equation of state is in the form of the
scalar field(s).
The initial model of inflation (“old inflation model”) was based on idea that
the scalar field φ was initially in a false vacuum with large potential energy. To
end of inflation, a quantum tunneling from the false vacuum to the true vacuum
was performed. However this model has the problem that it leads to an initially
microscopical bubble of the true vacuum which cannot grow to contain our present
observed Universe. Hence the attention shifted to models in which the scalar field φ
slowly rolls during the inflation.
Models of scalar field-driven inflation can be divided into three groups:
• Small-field inflation
103
• Large-field inflation
• Hybrid inflation
Small field inflationary models are based on ideas from spontaneous symmetry breaking
in particle physics. For example, let us consider the scalar field with the potential
in the form
V (φ) =
1
4
(φ2
− σ2
)2
, (5.32)
where we interpret σ as the symmetry breaking scale and λ as a dimensionless coupling
constant. The main idea of the small-field models (”new inflation”) was that
the scalar field starts to roll close to its symmetric point φ = 0. At sufficient high
temperature φ = 0 is a stable ground state of the one-loop finite temperature effective
potential VT (φ). When the temperature drops below to some value that is
smaller than Tc, φ = 0 becomes unstable local minimum of VT (φ) and φ can roll
towards a ground state of the zero temperature potential (5.32) with
φgr = ±σ . (5.33)
The problem of this model is that the slow-roll conditions 10
V
V
2
M2
P 1 ,
V
V
M2
P 1 (5.34)
that for the potential (5.32) take the form
φ2
(φ2 − σ2)2
1
M2
P
,
3φ2
− σ2
(φ2 − σ2)2
1
M2
P
(5.35)
and that have to be valid for inflation to works imply that
σ ∼ MP . (5.36)
However this is in contradiction with the fact that we have to presume that σ is
some symmetry breaking scale of the standard quantum field theory while MP is the
scale of the quantum gravity regime where the approximation of the quantum field
theory in curved space time cannot be valid. The potential (5.32) can be changed
to satisfy the slow-roll conditions however this procedure needs several fine-tuning
of the shape of the potential. A further problem of the slow-roll model is that the
initial field velocity must be constrained to be small which is again fine-tuned initial
condition.
As the alternative to the small-field inflationary models are large-field inflation
models that are also known as chaotic inflation. The simplest example is provided
by a massive scalar field with the potential
V (φ) =
1
2
m2
φ2
. (5.37)
10
Precise definition of these conditions will be given in next section
104
In the chaotic inflation scenario it is presumed that the scalar field rolls towards the
origin from large values of |φ|. The slow roll conditions for the potential (??) takes
the form 11
|φ| MP . (5.38)
Values of |φ| comparable or larger than MP are also required in other realizations
of large-field inflations. The question is whether such a model can consistently be
embedded in a realistic particle physics model, as for example supergravity. In many
these models V (φ) receives supergravity-induced correction terms that destroys the
flatness of the potential for |φ| > MP . The value m ∼ 1013
GeV is required in order
to obtain the observed amplitude of density fluctuations.
With two scalar fields it is possible to construct a class of models which combine
some of the nice features of large-field inflation models which is large set of the initial
conditions that lead to inflation with the small-field inflation where the inflation takes
place at sub-Planckian field values. These models are known as Hybrid inflation. For
example, let us consider two scalar fields φ and ξ with the potential
V (φ, ξ) =
1
4
λξ(ξ2
− σ2
)2
+
1
2
m2
φ2
−
1
2
g2
φ2
ξ2
. (5.39)
In the absence of the thermal equilibrium it is natural to assume that |φ| begins at
large values. For large φ the term
1
2
g2
φ2
ξ2
that serves as an effective mass term for ξ is positive and hence ξ has stable minimum
at ξ = 0. The parameters in (5.39) are chosen such that φ is slowly rolling for values
of |φ| somewhat smaller than MP but the parameters are chosen in such a way
that the potential energy for these fields values is dominated by the first term in
(5.39). The field φ is slowly rolling whereas the potential energy is determined by
the contribution from ξ. Once φ drops to the value
|φ|c =
λξ
g
σ . (5.40)
For this value the effective potential for ξ takes the form
V (φc, ξ) =
λξ
4
(φ2
− 2σ2
)2
(5.41)
that has three extrema
ξ0 = 0 , V (0) = λξσ4
ξ± = ±
√
2σ , V (φ±) = 0 (5.42)
11
Note that the dimensional analysis that implies that V has dimension [V ] = 4 in mass unit
implies that [φ] = 1.
105
that clearly shows that the configuration with ξ = 0 is unstable and decays to the
one of the states ξ± = ±
√
2σ. Since in this case the ground state is not unique we
have a possibility of the formation of topological defects at the end of the inflations.
After the slow-roll conditions break down the period of inflation ends and the
inflation begins to oscillate around its ground state. Since the inflation field φ couples
to other matter fields the energy of the Universe, that at the end of the period of
inflation is stored completely in φ is transferred to the matter fields of the particle
physics Standard model. The description of this process is very complicated,
5.5 How does the inflation work
The key property of the laws of physics that makes inflation possible is the existence
of states of negative pressure. To recognize the effect negative pressure let us again
consider Friedmann equation
¨a = −
4πG
3
(ρ + 3p)a ,
H2
=
˙a2
a2
=
8πG
3
ρ −
k
a2
,
˙ρ = −3H(ρ + p) . (5.43)
Once again, the metric is given by Robertson-Walker form
ds2
= −dt2
+ a2
(t)
dr2
1 − kr2
+ r2
(dθ2
+ sin2
θdφ2
, (5.44)
where k = 0, 1, −1. From the first equation in (5.43) we see that positive pressure
(ρ is always positive) contributes to the deceleration of the Universe while the negative
pressure can cause acceleration. In other words, negative pressure produces a
repulsive form of gravity.
The characteristic property of the inflation is that the physical wavelengths grow
faster than the size of the Hubble radius
dH =
a(t)
˙a(t)
=
1
H
as follows from the fact
˙λphys
λphys
=
1
a(t)λ0
d(a(t)λ0)
dt
=
˙a
a
= H =
˙dH
dH
+ dH
¨a
a
. (5.45)
This equation shows that during inflation when ¨a
a
> 0 the physical wavelengths become
larger than the Hubble radius. However when the physical wavelength becomes
larger than Hubble radius it is causally disconnected from physical processes. The
inflationary era is followed by the radiation dominated and matter dominated stages
where the Hubble radius grows faster than the scale factor and the wavelengths that
106
were outside now re-enter Hubble radius. This is the basic mechanism how the inflation
explains the generation of temperature fluctuations and also the origin of the
emergence of large scale formation: Briefly, quantum fluctuations generated early in
the inflationary stage exit the Hubble radius during inflation and then eventually
re-enter during the matter dominated era.
Remarkably, we can easily find form of the matter that produces negative pres-
sure.
5.6 Slowly-Rolling Scalar Fields
In order the inflation to solve the problems of the standard cosmology it must be
active at extremely early times. Thus we would like to study the earliest times in
the Universe amenable to classical description. It is expected that this is around the
Planck time tP . For that reason we will retain values of Planck mass in the equation
of this section. As we will see there are many models of inflation. In this section we
will restrict ourselves to the study of the model of chaotic inflation.
Consider matter in the form of the scalar field φ that is described with the action
Smatter = − d4
x
√
−g
1
2
gµν
∂µφ∂νφ + V (φ) . (5.46)
In field theory the stress energy tensor is defined as
Tµν = −
2
√
−g
δSmatter
δgµν
(5.47)
that for the action of the form S = − d4
x
√
−gL takes the form
Tµν = −gµνL + 2
δL
δgµν
, (5.48)
where we have used
δ
√
−g
δgµν
= −
1
2
√
−ggµν . (5.49)
More precisely, for the action (5.46) the stress energy tensor takes the form
Tµν = ( µφ)( νφ) − gµν
1
2
gαβ
( αφ)( βφ) + V (φ) , (5.50)
where for the scalar field φ we have αφ = ∂αφ. Let us now restrict to the homogenous
case in which all quantities depend only on cosmological time t and we also set
k = 0. A homogenous real scalar field behaves as a perfect fluid with
ρ = T00 =
˙φ2
2
+ V (φ) . (5.51)
The other components of the stress energy tensor take the form
Tij = −gij(
1
2
gµν
∂µφ∂νφ + V ) + ∂iφ∂jφ . (5.52)
107
If we define pressure as
p =
1
3
3
i=1
Tii (5.53)
we get
p =
˙φ2
2
− V (φ) . (5.54)
Thus any state which is dominated by the potential energy of a scalar field will have
negative pressure.
Note also that the equation of motion for the scalar field are given by
¨φ + 3H ˙φ + V (φ) = 0 , (5.55)
that can be thought of as a usual equation of motion for a scalar field in Minkowski
space but with a friction term due to the expansion of the Universe. The Friedmann
equation with such a field as a sole energy source is
H2
=
8πG
3
1
2
˙φ2
+ V (φ) . (5.56)
The accelerated expanssion occurs if the Universe is dominated by an energy component
that approximates a cosmological constant. In that case the associated expansion
rate will be exponential. From (5.51) we see that for ˙φ2
V (φ) the potential
energy of the scalar field is the dominant contribution to both the energy density
and pressure ant the resulting equation of state is p = −ρ that has the same form as
the state equation for cosmological constant.
More technically, the slow-roll approximation for inflation means that ¨φ term in
(5.55) is neglected with respect to the potential energy. In this case the scalar field
equation of motion and the Friedmann equation become
˙φ = −
V
3H
,
H2
=
8πG
3
V (φ) .
(5.57)
The conditions when we can neglect ˙φ with respect to V (φ) are conveniently characterized
with so named slow roll parameters
=
M2
P
2
V
V
2
, η = M2
P
V
V
, (5.58)
where
8πG = M−2
p . (5.59)
108
It is easy to see that the slow-roll conditions yield inflation. Recall that inflation is
defined by
¨a
a
> 0 (5.60)
that using the fact that
˙H =
¨aa − ˙a2
a2
⇒
¨a
a
= ˙H +
˙a
a
2
or alternatively
¨a
a
= ˙H + H2
. (5.61)
Then the inflation occurs when
˙H
H2
> −1 . (5.62)
But in slow roll
2 ˙HH =
8πG
3
V ˙φ = −
8πG
9
V 2
H
(5.63)
and hence
˙H
H2
= −
4πG
9
V 2
H4
= −
1
16πG
V
V
2
= − (5.64)
which will be small. Smallness of the second parameter η ensures that inflation will
continue for a sufficient period.
It is useful to have a general expression that describes how much inflation occurs
once it has begun. Such a quantity is the number of e-folds defined by
N(t) ≡ ln
a(tend)
a(t)
. (5.65)
Usually we are interested in how many e-folds occur between a given field value φ
and the field value at the end of inflation φend where (φend) = 1. To do this we
express N(t) as
N(t) = ln
a(tend)
a(t)
=
a(tend)
a(t)
da
a
=
=
tend
t
˙a
a
dt =
tend
t
Hdt =
φend
φ
H
d˜φ
˙˜φ
=
= −3
φend
φ
H2 d˜φ
V
= −
1
M2
p
φend
φ
V
V
d˜φ .
(5.66)
The problem of the initial conditions for inflation is very subtle. In case of chaotic
inflation in which we assume that the early Universe emerges from the Planck epoch
109
with the scalar field taking different values in different part of the Universe with
typically Planckian energies.
Let us now consider some examples of the potential that could lead to inflation.
We start with the simple monomial
V = λM4−α
P φα
. (5.67)
For potential above we obtain following slow roll parameters
=
α2
M2
P
2φ2
, η = α(α − 1)
M2
P
φ2
. (5.68)
Inflation starts at a large value of φ and the inflaton then rolls slowly towards the
minimum with increasing and η. Inflation ends when the slow roll conditions are
saturated,
φ ∼ λMP . (5.69)
The number of e-foldings we obtain before this happens is given by
N = ln
a(te)
a(ti)
= Hdt =
da
a
⇒ Hdt = ln(af ) − ln(ai)
te
ti
Hdt =
=
φe
φi
H
dφ
˙φ
= −
φe
φi
3H2
V
dφ = −
1
M2
P
φe
φi
V
V
dφ = −
1
M2
P α
φe
φi
φdφ =
=
φ2
i
2M2
P α
−
1
4
≈
1
2αM2
P
φ2
i
(5.70)
that implies
φi =
√
2αNMP MP . (5.71)
Using this initial value φi we can determine the values of slow roll parameters at ti
i ∼
α
4N
, η ∼
α − 1
N
. (5.72)
Another example of the inflation potential is
V = V0e
− 2
p
φ
MP
(5.73)
with the slow roll parameters
=
1
p
, η =
2
p
. (5.74)
Recall that for this potential we can combine the equation of motion to get
˙φ = −
MP
√
3
V
√
V
=
2
3p
√
V (5.75)
110
that has the solution
V ∼
3M2
4 p2
t2
(5.76)
and hence
H2
∼
p2
t2
⇒ ln a ∼ p ln t ⇒ a ∼ tp
. (5.77)
To gain more insight in the idea of inflation note that in most inflation models
the energy density ρ is approximately constant leading to exponential expanssion of
the scale factor. In fact, using p = −ρ in the Friedmann equation we get
¨a =
8πG
3
ρa (5.78)
that in the approximation of ρ = const can be solved with the ansatz a = eλt
that
inserted in the equation above implies
λ2
−
8πG
3
ρf = 0 ⇒ λ =
8πG
3
ρf , (5.79)
where ρf is constant energy density.
In the original model of inflation the state that drove the inflation involved a
scalar field in a local (but no global) minimum of its potential energy.The scalar
field state employed in the original version of inflation is called a false vacuum since
the state temporally acts as if it were the state of lowest possible energy density.
Classically this state is stable that there is no possibility how the scalar field crosses
a potential energy barrier that separates it from the states of lower energy. However
quantum mechanically this state would decay through tunneling. Initially it was
hoped that this tunneling could successfully ends an inflation but it was soon found
that the randomness of the bubble formation when the false vacuum decayed would
produced large inhomogeneities.
This problem was solved in the new inflation scenario proposed by Linde. In
this theory the inflation is driven by an scalar field with the potential in the form in
the form
V = −
A
2
φ2
+
B
4
φ4
(5.80)
that has minima at φ = 0, V (0) = 0 that is a false vacuum and also minima at
φ± = ± A
B
with V (φ±) = −A2
4B
. This scalar field is called inflaton. If this theory
the inflation is driven by the scalar field on the plateau of the potential energy
diagram (region around the point φ = 0). If this plateau is flat enough, such a state
can be stable enough for successful inflation. Soon after the introduction of the new
inflation scenario it was shown that the inflaton potential need not have either a local
minimum or a gentle plateau: This new scenario is known as a chaotic inflation.
111
5.7 Solving the problems of standard cosmology
To demonstrate the fact that inflation can solve the problems of the standard cosmology
let us again consider the potential with the simplest form
V (φ) =
1
2
m2
φ2
. (5.81)
With this potential the Friedmann equation takes the form
˙φ = −
m2
φ
3H
, H =
m
√
6MP
φ (5.82)
and we find
φ = φ0 −
2
3
m
MP
t (5.83)
and
a = C exp[
m
√
6MP
(φ0t −
√
2MP
2
√
3
t2
)] = a0 exp[
1
4M2
P
(φ2
0 − φ2
)] . (5.84)
The period of time during the solution above is valid ends at t ∼ t at which
a( t) ∼ a(0) exp(
1
2
) . (5.85)
If we take a typical value for m for which < 10−4
we obtain
a( t) ∼ a(0) × 102.7×108
. (5.86)
This has remarkable consequence. A proper distance LP at t = 0 will inflate to a size
10108
cm after a time t ∼ 5 × 10−36
s. As we know the size of observable Universe
today is H−1
0 ∼ 1028
cm. Therefore, only a small fraction of the original Planck
length comprises today’s entire observable Universe.
General arguments
Inflation is not really a theory, but instead it is a paradigm, or class of theories.
Each specific model of inflation makes definitive predictions but the class of the
models as a whole can be tested only by looking for generic features that are common
for all models. Nevertheless, there are number of features of the Universe that seem
to be characterize consequences of inflation. The basic arguments for inflation are
as follows:
• The Universe is big
We know that Universe is very large; the visible part of the Universe contains
about 1090
particles. Most of scientists believe that the creation of Universe
can be explained in scientific terms. Thus we think about the theory that could
112
explain how the Universe got so be so big. Such a theory has to explain the
number of particles, 1090
or more. Simple way to get such a huge number, with
small number as an input, is for the calculation to involve an exponential. The
exponential expansion of inflation can explain this huge number. Moreover,
inflationary cosmology suggests that, even though the observed Universe is
incredible large, it is only a small fraction of the entire Universe.
• The Hubble Expanssion
In standard FRW cosmology the Hubble expanssion is part of the postulates
that define the initial conditions. But the inflation offers the possibility of
explaining how the Hubble expansion began.
• Homogeneity and Isotropy
As we have shown before the degree of uniformity of Universe is starling. The
intensity of the cosmic microwave background radiation is the same in all directions.
The cosmic background radiation was released 400000 years after big
bang after the Universe cooled enough so that the opaque plasma neutralized
into a transparent gas. The cosmic background radiation photons have mostly
been traveling on straight lines since then so they provide an image of what the
Universe looked like at 40000 years after big bang. The observed uniformity of
radiation therefore implies that the observed Universe had become uniform in
temperature by that time. In standard FRW cosmology a simple calculation
shows that the uniformity could be established so quickly if signals could propagate
at about 100 times the speed of light a proposition clearly contradicting
the known laws of physics.
In inflationary cosmology the uniformity is easily explained. It is created initially
on microscopic scales by thermal thermal equilibrium processes and then
inflation takes over and stretches the regions of uniformity to become large
enough to encompass the observed Universe and more.
• Flatness problem
The problem concerns the value of the ration
Ωtot ≡
ρtot
ρ0
, (5.87)
where ρtot is total mass density of the Universe and where ρ0 = 3H2
8πG
is the
critical density that would make the Universe spatially flat (In ρtot the vacuum
energy, it is nonzero, is included.)
There is now general agreement that Ωtot lies in the range
0.1 ≤ Ω0 ≤ 2 , (5.88)
113
but it was very hard to pinpoint the value with more precision. Despite this
large range the value of Ω at early times is highly constrained, since Ω = 1
is an unstable equilibrium point of the standard model evolutions. Thus, if Ω
was exactly equal to one, it would remain exactly one forever. On the other
hand if Ω differs slightly from one in the early Universe, that difference-whether
positive or negative, would be amplified with time. More generally, it can be
shown that Ω − 1 grows as
Ω − 1
t (during the reaiation − dominated era)
t2/3
(during the matter − dominated era)
(5.89)
It was shown that at t = 1s when the processes of big bang nucleosynthesis
were just beginning, Ω must be equal to one to an accuracy of one part of 1015
.
Classical cosmology cannot explain this fact. In the context of modern particle
physics cosmology, where we try to push all thinks all the way back to Planck
scale 10−43
sec the problem becomes even more severe.
While this extraordinary flatness of the early Universe has o explanation in
classical FRW cosmology, it is a natural prediction for inflation cosmology.
During the inflationary period, we have following relation
Ω − 1 ≈ e−2Hinf t
, (5.90)
where Hinf is Hubble parameter during inflation. Thus, as long as there is a
sufficient period of inflation, Ω can start at almost any value and it will be
driven to unity by the exponential expansion. Moreover, recent observation
favored value of Ω0 to be equal to Ω0 = 1.02 ± 0.02 according with recent
WMAP results that is in beautiful agreement with inflation.
• Absence of magnetic monopoles
All grand unified theories predict that there should be, in the spectrum of
possible particles,extremely massive particles carrying a net magnetic charge.
It was shown in the context of the standard cosmology that magnetic monopoles
would be produced so strongly so that they would overweigh everything else
in the Universe by a factor of about 1012
. Such a large mass density would
cause that the Universe would come to its big crunch in about 30.000 years.
Inflation is simplest known mechanism to eliminate monopoles from the visible
Universe even though they are still in the spectrum of possible particles. The
monopoles are eliminated simply due to the fact that inflation diluted them to
a completely negligible level.
• Anisotropy of the cosmic microwave background radiation
114
The process of inflation smooths the Universe completely. On the other hand
the density fluctuations are generated as inflation ends by the quantum fluctuations
of the inflaton field. The general properties of these fluctuations are
that are adiabatic, Gaussian, and nearly scale-invariant.
5.8 Reheating and Preheating
The great strength of inflation is its ability to redshift away all unwanted relics, such
as topological defects. However during this process radiation and dust-like matter
are similarly redshifted away to nothing so that at the end of inflation the Universe
contains nothing but the inflationary scalar field condensate. The question is how
does the matter arise and how is the Universe reheated?
The problem of reheating is very complicated and complex. In fact, the theory
of reheating of the Universe after inflation is the most important application of the
quantum theory of particle creation since almost all matter constituting the Universe
was created during this process.
Now we sketch the standard picture.
Inflation ends when the slow-roll conditions are violated and the field begins
to fall towards the minimum of the potential. Initially all energy density is in the
inflation however now this energy is damped by two possible terms. Firstly, the expanssion
of the Universe naturally damps the energy density. Secondly, the inflation
may decay into other particles, such as radiation or massive particles, both fermionic
or bosonic. To describe this process one introduce a phenomenological decay term
Γφ into the scalar field equation. For example, if we consider the fermions only, then
the rough expression for how the energy density evolves is
˙ρφ + (3H + Γφ)ρφ = 0 . (5.91)
It can be shown that the inflaton undergoes damped oscilations and decays into
radiation that equilibrates rapidly at a temperature known as the reheat temperature
TRH.
More preciselly, early theory of reheating of Universe after inflation were based
on the idea that the homogeneous inflation field can be represented as a collection
of the particles of the field φ. Put differently, we expect that inflation field has the
same form as the ordinary quantum field in the flat spacetime. Then we can model
reheating as a decay of each particle separately and this process can be studied in
the standard perturbative description of particle decay.Typically, it takes thousands
of oscillations of the inflaton field until it decays into usual elementary particles by
this mechanism.
In case of bosons the situation is more complicated since now inflaton oscilations
may give rise to parametric resonance that is characterised by an extremely rapid
decay that results into distributions of products that are far from equilibrium and
115
only much later settles down to an equilibrium distribution at energy TRH. Such
a decay due to the parametric resonance is known as preheating. The parametric
resonance is an example of the coherent field effect that leads to the homogeneous field
decay much faster than would be predicted by perturbative effects. These coherent
effects produce high energy, nonthermal fluctuations that could have significance for
understanding developments at the early Universe, as for example baryogenesis.
5.9 Quantum fluctuations
The key problem is how to test an inflation. The answer is the structure formation.
As we have seen an important reason to involve an inflation is to make the Universe
smooth and flat. However as we observe every day there is a large amount of structure
in Universe. This structure can be traced back to subtle variations in the matter
distribution during the time when the cosmic microwave background was released.
The naive application of inflation in fact excludes such non-uniformity. It is a nice
example of the application of the quantum field theory in curved background that
explains the emergence of non-uniformity.
The main point is that inflation magnifies microscopic quantum fluctuation to
cosmic size and hence provides seeds for structure formations. It is very interesting
that then the details of physics at the highest energy scales is therefore reflected
in the distribution of galaxies and other structures on large scales. More preciselly,
the fluctuations start at their smallest scales and grow larger (in wavelength) as
the Universe expands. Eventually they become larger than the horizon and free.
Intuitively, the different parts of wave can no longer communicate with each other
since light can not keep up with the expanssion of Universe. This is a consequence
of the fact that the scale factor grows faster than the horizon which is a defining
property of an accelerating and inflating Universe. At a later time, when inflation
stops, the scale factor will start to grow slower than the horizon and the fluctuations
will eventually come back within the causal horizon.The fluctuations will then appear
as acoustic waves in the plasma and hence they will affect the CMB.
Let us now study this problem in more details. We assume that metric as well
as the inflaton can be split into a classical background piece and a piece due to
fluctuations according to
gµν = g(0)
µν + hµν(τ, x) ,
φ = φ(0)
+ δφ(τ, x) ,
(5.92)
where for convenience we have introduced conformal time τ such that the metric is
given by
ds2
= a(τ)2
(dτ2
− dx2
) . (5.93)
116
Since the background metric is homogenous it is convenient to Fourier transform the
fluctuation mode δφ as
δφ(τ, x) =
1
(2π)3/2
dkδφkeikx
. (5.94)
Since we can presume that fluctuation are small in magnitude we can neglect the
potential term for the fluctuation mode δφ so that its equation of motion takes to
form
1
√
−g
∂µ
√
−ggµν
∂νδφ = 0 (5.95)
that using the (5.93) takes the form
1
a2
δφ +
2a
a
δφ −
1
a2
∂i∂i
δφ = 0 , (5.96)
where (. . .) = d(...)
dτ
. Finally, using (5.94) we obtain differential equation for mode
δφk
δφk + 2
a
a
δφk + k2
δφk = 0 . (5.97)
If we introduce the rescaled mode µk = aδφk so that
δφk =
µk
a
−
µka
a2
, δφk =
µk
a2
− 2
µka
a2
−
µka
a2
+ 2
µk(a )2
a3
(5.98)
the equation (5.97) can be transformed into
µk + k2
−
a
a
µk = 0 . (5.99)
It can be shown that the metric fluctuations can be reduced to two polarizations
obeying an equation identical to the one for the scalar fluctuations. In what follows
we will consider the scalar fluctuations only.
To proceed let us presume that the conformal factor depend on conformal time
as
a ∼ τ1/2−ν
, (5.100)
where ν is a constant. An important example is a ∼ eHt
with H = const. where the
change of coordinates gives
dτ
dt
=
1
a(t)
= e−Ht
⇒ e−Ht
= −Hτ ⇒ a(τ) = −
1
Hτ
. (5.101)
Comparing with (5.100) we find that −1 = 1/2 − ν ⇒ ν = 3/2. Note also that the
physical range of τ is −∞ < τ < 0. Using now (5.100) the equation for fluctuation
(5.99) takes the form
µk + k2
−
1
τ2
ν2
−
1
4
µk = 0 . (5.102)
117
It is nice that the equation given above has solution known as a Hankel function.
The general solution is given by
fk(τ) =
√
−τπ
2
C1(k)H(1)
ν (−kτ) + C2(k)H(2)
ν (−kτ) , (5.103)
where C1(k) and C2(k) are to be determined by initial conditions.
When we quantize this system we need to introduce oscillators ak(τ) and a†
−k(τ)
such that
µk =
1
√
2k
ak(τ) + a†
−k(τ) ,
πk = µk(τ) +
1
τ
µk(τ) = −i
k
2
ak(τ) − a†
−k(τ) , (5.104)
obey standard commutation relation. It is important to stress that these operators
are time dependent and can be expressed in terms of oscillators at a specific moment
in time using the Bogolubov transformations
ak(τ) = ukak(τ0) + vk(τ)a†
−k(τ0) ,
a†
−k(τ) = u∗
k(τ)a†
−k(τ0) + v∗
k(τ)ak(τ0) ,
(5.105)
where
|uk(τ)|2
− |vk(τ)|2
= 1 (5.106)
Then we can write the quantum field µk as
µk(τ) = fk(τ)ak(τ0) + f∗
k(τ)a−k(τ0) , (5.107)
where
fk(τ) =
1
√
2k
(uk(τ) + v∗
k(τ)) (5.108)
is given in (5.103).
Now we come the key question that is what are the initial conditions? The
ussual choice is to consider the infinite past and choose a state annihilated by the
annihilation operator
ak(τ0) |0, τ0 = 0 , (5.109)
for τ0 → −∞. However there is great debate about this choice in the past and
is commonly known as a Problem of transplanckian physics. However we will not
discuss this issue in this section and we will continue according to common practise.
From (5.104) we get that
πk(τ0) |0, τ0 = −ı
k
2
a†
−k |0, τ0 = −ikµk(τ0) |0, τ0 . (5.110)
118
Since the Henkel functions behave as for τ0 → −∞
H(1)
ν (−kτ) ∼ −
2
kτπ
e−ikτ
,
H(2)
ν (−kτ) ∼ H(1)∗
ν (−kτ) ,
(5.111)
we find that the vacuum choice corresponds to C2(k) = 0 and |C1(k)| = 1.
In summary we have determined the quantum fluctuation and now we would
like to see how they act on CMB. To do this we compute the size of the fluctuation
according to
P(k) =
4πk3
(2π)3
|δφk|2
=
k3
2π2
1
a2
|µk|2
=
k3
2π2
1
a2
|fk|2
=
k3
2π2
1
a2
| − πτ|
4
|H(1)
ν (−kτ)|2
(5.112)
where (. . .) mean the vacuum expectation value with respect to the sate |0, τ0 .
Note that we are working in Heisenberg representation where the quantum mechanical
operators evolve with time while states not.
Now we should calculate (5.112) at late times, namely τ → 0. In this limit the
Hankel function behaves as
H(1)
ν (−kτ) ∼
2
π
(−kτ)−ν
(5.113)
and hence (5.112) for τ → 0 takes the form
P ∼
1
4π2
1
a2
(−τ)1−2ν
k3−2ν
∼
1
4π2
H2
k3−2ν
. (5.114)
For ν = 3/2 and for slow roll when H for τ → 0 is almost constant we can set the
scale of the fluctuations. In fact, we find the well known scale invariant spectrum for
ν = 3/2
P =
1
4π2
H2
. (5.115)
It can be shown that this is more or less the whole story in case of the gravitational,
or tensor, perturbations. The scalar fluctuations obey similar equation
Ps ∼
H
˙φ
2
1
4π2
H2
. (5.116)
Ussualy we express the deviation from the scale invariance by introducing spectral
indices according to
ns − 1 =
d ln Ps
d ln k
= 3 − 2νs ,
nT =
d ln PT
d ln k
= 3 − 2νT ,
(5.117)
119
where νs refers to the scalar perturbations and νT refers to the gravitational, or tensor
perturbations. While not clear from our simplified analysis, the ν s need not be the
sam in the two cases. Observations show that ns is very close to 1 consistent with the
basic idea of inflation. It is extreme important to find any slight deviation from the
scale invariant vale which could give important information about the inflationary
potential.
In fact, the flatness of the spectrum of density fluctuations, together with flatness
of the Universe Ω = 1 constitute the two most robust predictions of inflationary
cosmology. On the other hand there is an important difference between the prediction
of flatness of the Universe and the flatness of the spectrum of perturbations of metric.
It is difficult (though possible) to construct an inflationary model deviating from the
prediction Ω = 1. On the other hand the situation with the flatness of the spectrum
is opposite: It is very difficult (though possible) to construct a model with an exactly
flat spectrum of perturbations of metric. In this sense, existence of a small deviation
of the spectrum of inflationary perturbations from the flat spectrum (i.e. breaking
of the scale invariance of the spectrum) represents an additional robust prediction of
inflation.
5.10 Eternal Inflation
The eternal inflation scenario is based on the discovery of the process of self-reproduction
of inflationary Universe.In fact, this process exists in old inflationary theory and in
the new one but its significance was appreciated after discovery of eternal inflation
in the simplest versions of the chaotic inflation scenario.
In the case of the new inflation, the exponential expansion occurs as the scalar
field rolls from the false vacuum state at the peak of the potential energy towards
to the true vacuum. Remarkably, it was shown very briefly after introduction of this
model that the new inflation scenario is generically eternal. The key point is that,
even though classically the field would roll off the hill, quantum mechanically there
is always an amplitude for it to remain at the top.
The time scale for the decay of the false vacuum is controlled by
m2
= −
∂2
V
∂2φ φ=0
, (5.118)
which is the negative mass-squared of the scalar field when it is at the top of the
hill on the potential. This is a free parameter of each model but m has to be small
compared to Hubble constant or lese the model does not lead to enough inflation.
In other words, for parameters choosen so that the inflation works, the exponential
decay of false vacuum is slower than an exponential expanssion. Even if the
false vacuum is decaying, the expansion outruns the decay and the total volume of
false vacuum actually increases with time rather than decreases. Thus inflation does
not end at all places at once,instead it ends at localized patches, in a succession
120
that continues at infinitum. Each patches essentially a whole Universe so that it
can be said that inflation produces not just one Universe but an infinite number of
Universes.
In the context of the chaotic Universe models the situation is slightly subtle
even if it was shown by A. Linde that these models are eternal as well.We know that
inflation occurs as the scalar field rolls down a hill of the potential energy diagram.
As the field rolls down the hill quantum fluctuations will be superimposed on top
of the classical motion. The best way to think about this is to ask what happens
during one time interval of duration t = H1
(Hubble time) in a region of one
Hubble volume H3
. Suppose that φ0 is the average value of φ in this region at the
start of the interval. By definition of a Hubble time the rate of the expanssion is
given by
a(t + t)/a(t) = eH t
= e . (5.119)
This means that the change of volume is
V (t + t)/V (t) = a3
(t + t)H−3
/(a3
(t)H−3
) = e3
(5.120)
Since e3
≈ 20 we see that volume will expand by a factor 20. Since correlations are
extended typically over one Huble length if follows that in the end of the Hubble
time the initial Hubble size region grows and breaks up into 20 independent Hubble
sized regions.
During the time interval t the classical field φ is rolling down the hill. On the
other hand the classical change in the field φcl during the time interval t is going
to be modified by quantum fluctuations φqu which can drive the field upwards or
downward relative to classical trajectory. For any one of the 20 regions at the end
of the Hubble time we can describe the change of the field as
φ = φcl + φqu . (5.121)
In the crude approximation the fluctuation is treated as a free quantum field. This
fact implies that φqu the quantum fluctuation averaged over one of the 20 Hubble
volumes at the end, will have a Gaussian probability distribution, with a with of
order H/2π. Then there is then a probability that the sum of the two terms on the
right hand side will be positive-that the scalar field will fluctuate up instead down.
As long as the probability is bigger than 1 in 20 then the number of inflating regions
with φ > φcl will be larger at the end of the interval than at the beginning. This
process will then go on forever so inflation will never end.
We see that the condition for an existence of eternal inflation is that the probability
for the scalar field to go up must be bigger than 1/e3
≈ 1/20. It can be shown
that criterion implies the relation
H2
˙φcl
> 3.8 (5.122)
121
The probability that φ is positive tends to increase as one considers larger and
larger values of φ so that sooner or later one reaches the point when the inflation
becomes eternal. In fact for that reason we think that inflation is almost always
eternal.
The eternal inflation follows from the observation that in many models large
quantum fluctuations that are produced during inflation may locally increase the
value of the energy density in some parts of the Universe. These reasons then expand
at a greater rate than their parent domains and quantum fluctuations in them lead
to production of new inflationary domains which expand even faster. This leads to
an eternal process of self-reproduction of the Universe.
In order to understand the process of self-reproduction we should remember that
the processes separated by distances l greater than H−1
proceed independently one
another. This is a consequence of the fact that during an exponential expanssion
the distance between any two objects separated by more than H−1
is growing with
speed exceeding the speed of light. Then an observer in the inflationary Universe can
see only the processes occurring inside the horizon of radius H−1
. In this sense any
inflationary domain of initial radius exceeding H−1
can be considered as a separate
mini-Universe.
In order to study the behavior of such a mini-Universe we should take into
account the quantum fluctuations. Let us consider an inflationary domain of initial
radius H−1
containing sufficient homogeneous field with initial value φ M2
p . From
the basic equation of the inflation model
H =
mφ
√
6
, ˙φ = −m
2
3
(5.123)
we can deduce that during time interval t = H−1
the field inside the domain will
be reduced by φ that follows from the second equation above
φ
t
= −m
2
3
⇒ φ = −m
2
3
H−1
= −
2
φ
, (5.124)
where in the second step we have used the first equation in (5.123). On the other
hand it can be shown that the quantum fluctuation of the field φ is
|δφ(x)| ≈
H
2π
=
mφ
2π
√
6
. (5.125)
Then we see that the magnitude of quantum fluctuation is larger than φ for
mφ∗
2π
√
6
≈
2
φ∗
⇒ φ∗
∼
5
√
m
(5.126)
Then for φ φ∗
the decrease of the field φ due to the classical motion is much
greater than the average amplitude of the quantum fluctuations δφ generated during
122
the same time. On the other hand for φ φ∗
one has δφ(x) φ. Since the
typical wave length of the fluctuation mode is ∼ H−1
it turns out that the whole
domain after the time t = H−1
divides into following number of domain with
almost homogenous field
a( t)H−1
/H−1
= e3HH−1
∼ 20 (5.127)
where the first expression express the physical size of the domain divided wave length.
In summary, we get 20 separated domains of size H−1
, each containing almost homogenous
field φ − φ + δφ. In almost half of these domains the field φ grows by
|δφ(x)| − φ ≈ H/2π rather than decreases. This means that the total volume of
the Universe containing growing field φ increases 10 times. During the next time
interval t = H−1
this process repeats. Thus, after the two time intervals H−1
the
total volume of the Universe containing the growing scalar field increases 100 times.
In other words the Universe enters eternal process of self-reproduction.
One should however be careful with interpretation of this result. There is still
an ongoing debate of whether eternal inflation is eternal only in the future or also in
the past. To see this preciselly where is the problem let us consider any particular
time-like geodetic line at the stage of inflation. For any given observer following this
geodetic the duration ti of the stage of inflation on this geodesic will be finite. On the
other hand eternal inflation implies that if one takes all such geodesics and calculate
the time ti for each of them, then there will be no upper bound for ti. In other words
for each time T there will be such geodesic which experience inflation for the time
ti > T.
Similarly, if we study any particular geodesic in the past time direction, one can
prove that it has finite length. In other words, the inflation n any particular point in
the Universe should have a beginning at some time τi. However there is no reason to
expect that there is an upper bound for all τi on all geodesics. If this upper bound
does not exist, then eternal inflation is eternal not only in the future but also in the
past.
Put differently, there is a beginning for each part of the Universe and there will
be an end for inflation at any particular point. But there will be no end for the
evolution of Universe as a whole in the eternal inflation scenario and at present we
do not have any reason to believe that there was a single beginning of the evolution
of the whole Universe at some moment t = 0 which was traditionally associated with
Big Bang.
If this scenario is correct, then physics alone cannot provide a complete explanation
for all properties of our part of the Universe.
5.11 Eternal Inflation: Implications
Even if the other Universes that are created during the eternal inflation are too
123
remote to imagine observing directly we will see that an eternal inflation has real
consequences in terms of the way we extract predictions from theoretical models.
Firstly, the eternal inflation implies that all hypothesis about initial conditions
for the Universe, such as the Hartle and Hawking no boundary proposal, the tunneling
proposals by Vilekin or Linde become totally divorced from observation. This
follows from the presumption of the eternal inflation with its infinite production of
pocket Universes. Then one can expect that the statistical properties of inflating
region should approach a steady state which is independent on initial condition. Unfortunatelly
there are great problems with the study of this steady state, for example,
the properties of this state seems to depend crucially on the super-Planckian physics
which we do not understand at present. It is however possible that string theory
could be helpful with this study. More preciselly, the same quantum fluctuations
that make eternal inflation possible tend to drive the scalar field further and further
up to potential energy curve so that some attempts that wanted to quantity the
steady state require the imposition of some kind of a boundary condition at large φ.
Even if the Universe forgets the details of its genesis the question, how the
Universe began still remain interesting. To see this note that eternally inflating
Universes continue forever once they start they are apparently not eternal into the
past. 12
The second consequence of the eternal inflation is that the probability of the
onset of inflation becomes totally irrelevant provided that the probability is not
identically zero. In fact, this observation is slightly in the clash with our previous
claim that chaotic inflation gives better result that the new inflation scenario. Even
if the initial conditions necessary for the new inflation scenario cannot be justified
on the basis of the thermal equilibrium as was proposed in original papers, in the
context of the eternal inflation it is sufficient to conclude that the probability for the
required initial conditions is nonzero.
The third consequence of the eternal inflation is the possibility that it offers to
rescue the predictive power of theoretical physics. Here we mean the status of Mtheory.
Even if this theory by itself has uniqueness it appears that the vacuum is far
from unique. Since the predictions will depend on the properties of the vacuum, the
predictive power of M-theory could be limited. Eternal inflation however provides
a possible mechanism to remedy this problem since it might help to constrain the
vacuum state of the real Universe and hopefully significantly enhance the predictive
power of M-theory. We must however stress that this is pure speculation whose
validity is not justified but one can hope that recent works in the context of the
string theory landscape could bring new light on this conjecture.
12
This remark implies that the word “eternal” is not technically correct, we should rather speak
about “semi-eternal” or “future-eternal” Universe.
124
5.12 Does Inflation Need a Beginning
We know that according to the inflation scenario is eternal in the future. Than a
natural question arrives: Is it possible that the inflation is eternal into the past?
There is a nice theorem by Borde, Guth and Vlenkin (2003) that proves that the
answer to this question is no. There is of course no conclusion that an eternally
inflating model must have a unique beginning and no conclusion that there is an
upper bound on the length of all backwards-going geodesics from a given point. In
other words this theorem shows that some new physics would be needed do describe
the past boundary of the inflating region.
5.13 Inflation and Observations
It is very nice that inflation can make prediction which can be tested by cosmological
observations. The inflationary prediction for nearly flat spectrum of density
perturbation is in agreement with both you measurements of the CMB anisotropy
and observations of structures in the Universe.
Let us also give another example where the inflation cosmology gives very nice
explanation of the observation date.
Today,we have three-dimensional map of the distribution of galaxies in space
that contain more than one hundred thousand galaxies.They clearly indicate that the
luminous matter in the Universe is neither uniformly nor randomly distributed. We
see clusters of galaxies,superclusters, filaments and voids that are regions of space
empty of galaxies. The distribution can be quantified in terms of the luminosity
power spectrum.
As we have also seen another observation window in cosmology is the cosmic
microwave background radiation. This radiation is characterised by a surprising
isotropy, in other words it looks the same from all different directions on the sky.
However this radiation has also fractional level of a bit less than 10−4
of anisotropies.
These anisotropies can be characterised in terms of their angular power spectrum.
The sky map (that is clearly two-dimensional of topology of sphere) of anisotropies
is expanded in spherical harmonics Ylm
T
T
(θ, φ) =
∞
l=1
l
m=−l
almYlm(φ, θ) , (5.128)
where θ, φ are the usual angles on the surface of two-sphere. It can be shown that
the angular power spectrum of CMB has characteristic pattern of anisotropies. The
challenge of cosmology is to explain both the overall isotropy of CMB and the specific
patter of anisotropies.
In order to explain these observation structures we have to look to the very early
Universe. The reason is that the Standard Big Bang cosmology that describes the
cosmological evolution at late times where the notion “late times” means the times
125
that includes period of nucleosynthesis and later implies that the length scales that
are currently observed were outside the Hubble radius in the early times and no
causal structure formation scenario is possible.
It is great success of inflationary cosmology that can explains all problems we
listed above and also provides a causal mechanism for the origin of inhomogeneities
in the Universe.
References
[1] B. P. Schmidt et al. [Supernova Search Team Collaboration], “The High-Z Supernova
Search: Measuring Cosmic Deceleration and Global Cur vature of the Universe Using
Type Ia Supernovae,” Astrophys. J. 507 (1998) 46 [arXiv:astro-ph/9805200].
[2] A. G. Riess et al. [Supernova Search Team Collaboration], “Observational Evidence
from Supernovae for an Accelerating Universe and a Cosmological Constant,” Astron.
J. 116, 1009 (1998) [arXiv:astro-ph/9805201].
[3] A. Linde, ”Fizika elementarnich castic i inflaksnaja kosmologija.” Izdavatelstvo
Nauka, 1990.
[4] D. H. Lyth, ”Introduction to Cosmology,” astro-ph/9312022.
[5] A. Linde, ”Lectures on Inflationary Cosmology,” hep-th/9410082.
[6] G. F. R. Ellis and H. van Elst, ”Cosmological Models,” hep-th/9812046.
126