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1.Introduction
The standard cosmology is a successful framework
for interpreting observations. In spite of this fact
there were certain questions which remained
unsolved until 1980s.
For many years it was assumed that any solution of
these problems would have to await a theory of
quantum gravity.
The great success of cosmology in 1980s was the
realization that an explanation of some of these
puzzles might involve physics at lower energies:
“only” 1015 Gev, vs 1019 Gev of quantum gravity.
THE CONCEPT OF INFLATION WAS BORN.
What follows is an outline of the main
features of inflation in his “classical” form;
The reader will find more than one model of
inflation in scientific literature; here we will
refer to the standard inflation which
involves a first order cosmological phase
transition.
2.Classical problems of standard
isotropic cosmology
2.1 The horizon problem
From CBR observations we know that:
T
 10 5
T
On angular scales >> 1°.
Sandard cosmology contains a particle horizon of radius:
t*
R po t *  
a(t*)
dt  2t *
a
(
t
)
0
In the radiation dominated era,when a(t) ~t 1/2. (We will use natural
units, c=1).
In the matter dominated era (a(t) ~t 2/3) :
t*
R po t *  
a (t*)
dt  3t *
a
(
t
)
0
R po(t0)=3t0 ~6000 Mpc h-1
At t= tls (last scattering) Rpo(tls)=3tls
Because of the expansion of the universe the universe at last scattering is now:
t 
a(t )
3t ls 0  3t ls  0 
a(t ls )
 t ls 
2 3
t 
 3t 0  ls 
 t0 
1 3
 3t 0
1
1  zls  2
1

6000
1  zls
h 1 Mpc  100 Mpc
Subtending an angle of about 1°
 The microwave sky shows us homogeneity and isotropy on angular
scales >>1°
 Why do we live in a
nearly
homogeneous
universe even
though some parts
of the universe are
not (or not yet )
causally
connected???
2.2 The flatness problem
From the first Friedmann equation:
8
k
H 2  G  2
3
a
 f (z )
We have (see appendix 1):
 1  1
0


 01  1
 a0 
 
 a 
2
  0  a0  2 
1
1     
( z )
   a  
At the Plank epoch:
4
2
2
 0  eq  a0   aeq   a   a0 

f ( z) 
  
 
 eq   a   a0   aeq   a 
3
Remembering that
a  T 1
 1


 1
 0

Teq T0 2.38  104  b h 2  2.73 104 eV
f ( z) 

 1061
28
T T
1.2  10 eV
We have:
 1 
1
 1  10 61   1

 0 
To get Ω0  1 today requires a FINE TUNNING of Ω in the past.
At the Plank epoch which is the natural initial time, this requires a
deviation of only
1 part in 1061 !!!
However , if Ω =1 from the beginning  Ω =1 forever
But a mechanism is still required to set up such an initial state
3.The idea of inflation
To solve the horizon problem and allow causal contact over the whole of the region
observed at last scattering requires a universe that expands more than linearly (yellow
in the previous figure)
a(t )  t 
 1
In the figure we have a(t )  expt 
ACCELARATED EXPANSION
This is the most general features of what become known as the INFLATIONARY
UNIVERSE.
Equation of state (from the second Friedmann equation):
a
4

G   3 p 
a
3
We want
a  0
p
1

3
The general concept of inflation rests on being able to achieve a negative-pressure
equation of state.
 This can be realized in a natural way using quantum field theory.
4.Basic concepts of quantum
field theory
4.1 The Lagrangian density
1
L          V ( )
2
  Real scalar field
V ()  Potential of the real scalar
field , usually in the form
1
V ( )  m 2  2
2
where m is the mass of the
field in natural units
The restriction to scalar field is not simply for reasons of simplicity but because is expected in
many theory of unification that additional scalar field such as the Higgs field will exist.
The scalar field is in general complex. We will use a real one only for simplicity.
4.2 Energy momentum tensor and equation
of state
 The Lagrangian density written above is obviously invariant under space-time
translations of the origin of the reference system.
 The existence of a global symmetry leads directly to a CONSERVATION LAW,
according to the Noethern’s theorem.(See appendix 2 for details).
The conserved energy-momentum tensor is
Tq   L q 
L
 q
   
From this we read off the energy density and pressure, since:
T44  
T11   p
With the conventions that
1   x
2   y
3   z
 4  ic t
 T11   L 
L
 1
1 
 T44   L 
L
 4
 4  
If we add the requirement of homogeneity of the scalar field:
1   2    3  0
2


 V ( )
2
2

p
 V ( )
2
If
2

 V ()
2
The equation of state is :
p  
This is of the type we need in order to solve the horizon problem! (p< -1/3 ρ).
4.3 Dynamics of the field
From the Euler –Lagrange equation of motion:

L
L

0
    
We now derive the equation of motion for the scalar field.
In order to be correct in general relativity the lagrangian density L needs do take the form of
an invariant scalar times the jacobian
g
g  det g ij
In a Friedmann-Walker-Robertson model:
 g  a 3 (t )
The Euler-Lagrange equation than becomes:
From which it’s not difficult to obtain:
( La 3 ) ( La 3 )


0
   

  3 a 
   2   V  0

a

With the requirements of homogeneity of the field:
  3 a 
  V  0

a

5.Cosmological implications
5.1 Evolution of the energy density
If:
1
 The universe is dominated by the scalar field Φ with Lagrangian L         V ()
2
2
and p= -ρ , that’s to say   V ()
2
 The scalar field is not coupled with anything
From the relation
 
 
d
d
a 3   p a 3
dt
dt
Adding the equation of state for the field (p= -ρ) and solving we have:
and since
2


 V ( )
2
with
2

 V ()
2
  const
   V ()  const
From the first of the Friedmann equation:
8
8
H 2  G  GV ()  const
3
3
 H  const
5.2 Exponential expansion
From the first Friedmann equation:
2
 a 
2
   H  const
a
a  H  a
 a  e Ht
 More then linear expansion:
 this is what we need in order to solve the horizon problem
5.3 Necessity of Cosmological Phase
Transition
 The discussion so far indicates a possible solution of the problems of standard
cosmology, but has a critical, missing ingredient.
 In the period of inflation the dynamics of the universe is dominated by the scalar field
Φ, which has p    as equation of state.
There remains the difficulty of returning to a “normal” equation of state:
 THE UNIVERSE IS REQUIRED TO
UNDERGO A COSMOLOGICAL
PHASE TRANSITION
5.4 Necessity of Reheating
 The exponential expansion produces a universe that is essentially devoid of normal
matter and radiation;
 Because of this the temperature of the universe becomes <<T, if T was the temperature
at the beginning.
 We know that at the end of the inflation the temperature has to be high enough in order
to allow the violation of the barion number and nucleosynthesis.
 A phase transition to a state of 0 vacuum energy, if istantaneous, would transfer the
energy of the field to matter and radiation as latent heat.
 THE UNIVERSE WOULD THEREFORE BE REHEATED
6.The potential of the scalar field
and the SRD approximation
 In order to solve the equation of motion of Φ we have to specify a particular form of the
potential.
 Different forms of V(Φ) have been explored during the years and each of them
produces a different type of expansion of the universe.
Requirements on V(Φ) :
1.In order to have negative perssure:
 
2
 V 
2
8
 a 
   GV 
3
a
  3 a 
  V '    0

a
From this system we derive a(t)
2. THE SRD (SLOW-ROLLING-DOWN) APPROXIMATION:
The solution of the equation of motion become tractable if we make the socalled SRD
approximation:
  3H 


  V '  

  V 
From the equation of motion we have: 3H
 2  V  than becomes a condition on Φ:
 The condition 
'
2
V '2

   2  V ()
9H
V ' 2 m 2p
3  8V
 V
(using the first of Friedmann equations)
V



V '



2
m 2p
V 
  
24
V ' 

V '2

 V ()
3  3H 2
mp
24
In the SRD approximation:
8
H 2  GV
3
   V'

3H
 m p
3.
  V ()
(From Friedmann equation).
  const
 V ()  const
 V ' () 
We will use a potential of the form:
Veff   
4
 b   aT 2 
3
2
In the figure we can see the
temperature dependent potential of
the form written above, illustrated
at various temperatures:
At T>T1 only false vacuum is
available;
At T<T2, once the barrier is small
enough, quantum tunneling can
take place and free the scalar field to move: we have a first order transition to the vacuum state.
It’s important to remark that the energy density difference between the two vacuum states is
V  T 4
7.The Inflation solution of
standard cosmology problems
7.1 The horizon problem
In order to solve the horizon problem we need the horizon of the inflationary epoch to be
now bigger than ours:


dt'
dt'
1
Ht
Horizon during inflation:
OE(t )  a(t ) 
 e  Ht ' 
 const
a
(
t
'
)
H
e
t
t

1 H ( t e t i ) a 0
e
 3t 0
H
ae
Expansion of the horizon
during inflation
If ti<<te
 e H (t i te )  e Ht e
Our horizon (matter dominated expansion)
Growth of inflationary horizon from the end of
inflation up to now
 e Ht e  3t 0
ae
H
a0
If the comoving entropy is conserved, then: a3T3=const
(This is non true when p=p(T,Θ) , that’s to say: when pressure is not only function of the
temperature.This is what happens for example during phase transition at a temperature different from
the critical one)
ae3Te3  a03T03
a
T
 e  e
a0 T0
e
Ht e
 3t 0T0
3 1017 1043 t p  3 1041019109 mp  9 1028
H
Te
From SRD
 H2 
V
m 2p
V 
(1 st Fried.equat.)
If we are dealing with a quantum
field at temperature μ, then en energy
density   4
is expected in the
form of vacuum energy.
Remembering that in natural units:
mpt p  1
H
 

Te m p Te
Where μ 10 15-16 Gev (From GUT
theories

 We define:
f1 
f
H
 2
Te
f1

mp
f2 
 103  104

Te = Temperature at the end of inflation
Its value is strongly dependent on
reheating
Te
A phase transition to a state of zero vacuum energy , if instantaneous, would transfer the energy
To normal matter and radiation (case of perfect reheating)
 the universe would therefore be reheated.
In approximation of “perfect” reheating: Te 
 e Ht e  9  1028103

 f2  1
 Hte  60
Hte  N  e  foldings
It will be proved below that this is also exactly the number needed to solve the flatness
problem
 4
7.2 The flatness problem
As we have already seen, from the first of Friedmann equations we have (see appendix 1 for
details):
 1 (t )  1  (t*)  a(t*) 



 1 (t*)  1  (t )  a(t ) 
2
We take: t*=ti and t=te
Remembering that ρ is nearly constant
during inflation, we have:
a
1
1
 e  1   i  1  i
 ae
Exponential expansion:
a
  i
 ae



2
a(t )  e Ht

  e H (ti te )  e 2 Ht e  e 2 N

N  60
  e1  1   i1  1 e 2 N
We deduce:
e  1
e 2 N  e 120
because of the factor
We would like to have an estimate of the parameter Ω(t) at the present epoch Ω(t0)  Ω0
again the relation
2
 1 (t )  1  (t*)  a(t*)  with tt0



1
te

(
t
)
a
(
t
)
 (t*)  1


1
 0  1  e  ae 
 

1
 e  1  0  a0 
1
1
2
  0  1  i  1 e  2 N
e  eq  ae 
eq 0  aeq 
 aeq 
1
1

0  1  i  1 e 2 N 
a
 e 
4
2
 aeq 


 a0 
 a0 


a 
 eq 
3
2
 ae 


a 
 eq 
2
 aeq 


a
 0 
2
1
1
0  1  i  1 e
2 N
aeqa0
2
e
a
1
 i  1 e
2 N
Te2
T0Teq
If we have perfect reheating:
 
 7 10
10  1000 Gev
2
1
1
 0  1  i  1
e 120 1015 Gev
3 10
4
9 2
53
1
1052 i  1
7.3 Number of e-foldings: criteria for inflation
As we have already seen, successful inflation in any model requires more than 60 e-foldings of the
expansion.The implications of this fact are easily calculated using the SRD equation:
   V ' ( )   V ' ( ) H

3H
3H 2
Using the first of Friedmann equations:
3H 2  8
1
mp
2
V
m 2p
m 2p
V


 

H  
H
8  V '  8

8 e
4
N   Hdt   2  d  2  e2
m p i
mp
ti
te
H (t e  t i )  Hte
 d  
m 2p
8
Hdt
e 
m 2p Hte
4
 mp
 60
 2m p
4
 N > if V’<
A model in which the potential is sufficiently flat (V’<<) that slow-rolling down can
begin will probably achieve the critical 60 e-foldings.
The criterion for successful inflation is thus that the initial value of the field
exceeds the Plank scale (mp)
8.Ending of inflation
 The relative importance of time derivatives of Φ increases as Φ rolls down the potential
and V approaches zero.
The inflationary phase will cease!
 The field will oscillate about the
bottom of the potential, with
oscillation becoming damped

because of the 3H
friction term.
  3H
  V ' ()  0

If the equation of motion remains the one written above (absence of coupling), then:
1.
We will have a stationary field that continues to inflate without end, if V(Φ=0)>0.
2.
We will have a stationary field with 0 energy density.
BUT
If we introduce in the equation the couplings of the scalar field to matter field:
this thing will cause the rapid oscillatory phase to produce particles, leading to
reheating
8.1 Absence of coupling
From the relation:
d
d
( a 3 )   p a 3
dt
dt
It’s not difficult to derive:
  3H (   p)  0
And in presence of the scalar field and radiation:
   3H (    p )   r  4H r  0
pr 
1
r
3
Remembering that:
2

 
 V ( )
2
2

p 
 V ( )
2
 
  V ' ()

  
 
  3H
  V ' ()   4H  0

r
r
Equation of motion  0
  r  0
 r  0
8.2 Adding a term of coupling:
It’s the same thing as varying the equation of motion of the scalar field
  3H  V ' ()
  3H  V ' ()  0
 0
 
 This extra term is often added empirically to represent the effect of particle creation;
 The effect of this term is to remove energy from the motion of Φ and damping it in the
form of a radiation background;
 Φ undergoes oscillations of declining amplitude after the end of inflation and Γ only
changes the rate of damping.
 For more detailed models of reheating see Linde (1989) and Kofman , Linde &
Starobinsky (1997).
We have in this way:
2
 r  4H r  
 

2
2
30
4
Rh
gT
Because of the factor
and also
2  


TRh4 = Temperature of reheating
G=degree of freedom
2
g
30
(harmonic oscillations)
Energy density for relativistic
particles in the case of perfect
reheating
even in the case of perfect reheating TRh is < of the initial one
TGUT
A plot of the exact solution for the scalar
field in a model with a V ()   2
potential.
The top panel shows how the absolute
value of the field falls smoothly with
time during the inflationary phase, and
then starts to oscillate when inflation
ends.
The bottom panel shows the evolution of
the scale factor a(t). We see the initial
exponential behavior flattening as the
vacuum ceases to dominate
The two models shown have different
starting conditions: the former (upper
lines in each panel) gives about 380 efoldings of inflation; the latter only
150.
(From Peacock,1999).
9. Relic fluctuations from Inflation
9.1 Fluctuation spectrum
 During inflation there is a true event horizon, of proper size 1/H
 This fact suggest that there will be thermal fluctuations present, in analogy with black
holes for which the Hawking temperature is:
kTH 
hc
8 2 Rs
Rs 
2GM
c2
 The analogy is close but imperfect, and the characteristic temperature here is:
hcH
kT 
4 2
 The inflationary prediction is of a horizon scale amplitude fluctuation
H
H2


2
The main effect of these fluctuations is to make different parts of the universe have fields
that are perturbed by an amount δΦ with:
 
H
2
 We are dealing with various copies of the same rolling behavior Φ(t) but viewed at different times,
with:
t 



 The universe will then finish inflation at different times, leading to a spread in energy density.
 The horizon scale amplitude is given by the different amounts that the universe have expanded
following the end of inflation:
H

H2
 Ht  H




2
 
H (Indetermination on the scalar field, from
quantum theory of fields. See Peacock,
2
1999 for details)
This plot shows how fluctuations in the scalar
field transform themselves into density
fluctuations at the end of inflation.
Inflation finishes at times separated by t in
time for the two different points, inducing
a density fluctuation
  Ht
9.2 Inflation coupling
From the SRD equation, we know that the number of e-foldings of inflation is:
N   Hdt   H
 If
d
d
  3H 2


V'
V   4
H2
N
 2
H
1 2
3
H 2 3H 3
H3



 N 2
3


V'

Since N ≈ 60 and the observed value of fluctuations
   1015
(Really weak coupling!!!)
 H  105
 If
H
V  m2 2
H 2 3H 3
3H 3





V'
2m 2 
From the first of Friedmann equations:
H2 
V
m 2p
H 
V
mp
3
H
3V 2
2 m
5




10
2m 2 m 3p
m 3p
And since
  mp
From CBR observations
is needed for inflation,
 m  105 mp
This constraints appear to suggest a defect in inflation, in that we should be able to use the
theory to explain why
  105 , rather than using observations to constrain the theory
H
9.3 Gravity Waves
Inflationary models predict a background of gravitational waves of expected
rms amplitude:
hrms 
H
mp
It’s not easy to show from a mathematical point of view how such a prediction
arises.
Here is enough to say that everything comes from the fact that in linear theory any
quantum field is expanded into a sum of oscillators with the usual creation and
annihilation operators.
 The fluctuations of the scalar field are transmuted into density
fluctuations, but gravity waves will survive to the present day.
10. Conclusion
To summarize, inflation:
 Is able to give a satisfactory explanation to the
horizon and flatness problem;
 Is able to predict a scale invariant spectrum, but
problems arises with the amplitude of the
fluctuations predicted (or alternatively with the
coupling constant λ );
 Is strongly linked with quantum field theory.
11.References
•
•
•
•
•
•
Kofman, Linde, Starobinsky,1997:hep-ph/9704452
Linde,1989:Inflation and quantum cosmology, Academic
Press.
Lucchin,1990:Introduzione alla cosmologia, Zanichelli.
Peacock,1999:Cosmological physics, Cambridge
University Press.
Ramond:Quantum field theory.
Weinberg,1972:Gravitation and cosmology, John Wiley
and sons.
Appendix 1
H2 
 
8
k
G cr,0 0
 2
3
 cr,0  0 a
H 2  H 020

 k

0 a 2
H 02 (1   0 )  
H 02  H 02  0
At t=t0

k
 2
0 a
k
a(t 0 ) 2
Substituting this result in the first equation:

a 
H  H 0
 H 02 1   0  0 
0
a
2
2
2
0
And remembering that
  0
 
8
8
H 2  G cr  G cr
 0,cr  H 02 0
3
3
  0  0,cr
 0
It’s not difficult to get the following equation:
 1  1  0  a0  2
  
1
0  1   a 
Appendix 2
Given a lagrangian density L for the field
 a x  and the transformations:
x   x'  x  x
a x   a' x'   a x   a
A
Def:
~a   a' ( x  )   a x  
x    q  q
~a   aq  q
If L is invariant for “A”:
J q  L q 
  J q  0
L
 aq
  a 
And
This is the Noethern’s theorem
A special case: INVARIANCE with respect to SPACE-TIME
TRANSLATIONS
We have:
x    q x q
  q  x q
  q   q
~a  a' ( x ) a x   a' ( x )  a' ( x' ) a' ( x' ) a x 
   ax
  aq   q a
Def
Tq   J q   L q 
L
 q a
  a 
If we take a Lagrangian density
 Tq   L q 
1
L         V ( )
2
L
q
   
CONSERVED
No variations of the
filed