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Inflation
Università Milano Bicocca,2005
Margutti Raffaella
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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.

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• 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.

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2.Classical problems of standard isotropic
cosmology

• 2.1 The horizon problem
• From CBR observations we know that

On angular scales gtgt 1.
Sandard cosmology contains a particle horizon of
radius
In the radiation dominated era,when a(t) t 1/2.
(We will use natural units, c1).
In the matter dominated era (a(t) t 2/3)
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
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• Subtending an angle of about 1
• The microwave sky shows us homogeneity and
  isotropy on angular scales gtgt1

• Why do we live in a nearly homogeneous universe
  even though some parts of the universe are not
  (or not yet ) causally connected???

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2.2 The flatness problem

• From the first Friedmann equation

We have (see appendix 1)
At the Plank epoch
Remembering that
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We have
To get O0 1 today requires a FINE TUNNING of
O 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 O 1 from the beginning ? O 1
forever But a mechanism is still required to set
up such an initial state
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3.The idea of inflation
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• 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)

• In the figure we have
• ACCELARATED EXPANSION
• This is the most general features of what become
  known as the INFLATIONARY UNIVERSE.
• Equation of state (from the second Friedmann
  equation)

We want
?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.
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4.Basic concepts of quantum field theory

• 4.1 The Lagrangian density

Real scalar field
Potential of the real scalar field , usually in
the form 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.

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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
  Noetherns theorem.(See appendix 2 for details).
• The conserved energy-momentum tensor is
• From this we read off the energy density and
  pressure, since
• With the conventions that

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If we add the requirement of homogeneity of the
scalar field
If
?The equation of state is
This is of the type we need in order to solve the
horizon problem! (plt -1/3 ?).
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4.3 Dynamics of the field

• From the Euler Lagrange equation of motion
• 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
• In a Friedmann-Walker-Robertson model
• The Euler-Lagrange equation than becomes
• From which its not difficult to obtain
• With the requirements of homogeneity of the field

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5.Cosmological implications

• 5.1 Evolution of the energy density
• If
• The universe is dominated by the scalar field F
  with Lagrangian
• and p -? , thats to say
• The scalar field is not coupled with anything
• ?From the relation
• Adding the equation of state for the field (p
  -?) and solving we have
• and since
• with
• ?From the first of the Friedmann equation

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• 5.2 Exponential expansion
• From the first Friedmann equation
• More then linear expansion
• this is what we need in order to solve the
  horizon problem

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• 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 F,
  which has 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

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• 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 ltltT, 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

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6.The potential of the scalar field and the SRD
approximation

• In order to solve the equation of motion of F we
  have to specify a particular form of the
  potential.
• Different forms of V(F) have been explored
  during the years and each of them produces a
  different type of expansion of the universe.
• Requirements on V(F)
• 1.In order to have negative perssure
• 
  From this system we derive a(t)

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• 2. THE SRD (SLOW-ROLLING-DOWN) APPROXIMATION
• The solution of the equation of motion become
  tractable if we make the socalled SRD
  approximation
• From the equation of motion we have
• The condition than
  becomes a condition on F
• (using the first
  of Friedmann equations)

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• In the SRD approximation

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• (From
  Friedmann equation).
• We will use a potential of the form

In the figure we can see the temperature
dependent potential of the form written above,
illustrated at various temperatures At TgtT1
only false vacuum is available At TltT2, 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. Its important to remark that the energy
density difference between the two vacuum states
is
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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
• Horizon during inflation

Our horizon (matter dominated expansion)
Growth of inflationary horizon from the end of
inflation up to now
Expansion of the horizon during inflation
If tiltltte
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If the comoving entropy is conserved, then
a3T3const (This is non true when pp(T,T) ,
thats 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) Rememberin
g that in natural units

• From SRD
• (1 st
  Fried.equat.)
• If we are dealing with a quantum field at
  temperature µ, then en energy density
  is expected in the form of vacuum energy.
• Where µ 10 15-16 Gev (From GUT theories

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• We define
• 
  Te Temperature at
  the end of inflation
• 
  Its value is
  strongly dependent on
• 
  reheating

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 It will be proved below
that this is also exactly the number needed to
solve the flatness problem
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• 7.2 The flatness problem
• As we have already seen, from the first of
  Friedmann equations we have (see appendix 1 for
  details)
• We take tti and tte
• Remembering that ? is nearly constant
• during inflation, we have

Exponential expansion
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We deduce because of the
factor We would like to have an estimate of the
parameter O(t) at the present epoch O(t0) ?
O0 ?again the relation
with t?t0

?te
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If we have perfect reheating
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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
Using the first of Friedmann equations
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• N gt if Vlt
• A model in which the potential is sufficiently
  flat (Vltlt) 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)

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8.Ending of inflation

• The relative importance of time derivatives of F
  increases as F 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
• friction term.

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• If the equation of motion remains the one written
  above (absence of coupling), then
• We will have a stationary field that continues to
  inflate without end, if V(F0)gt0.
• 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

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• 8.1 Absence of coupling
• From the relation
• Its not difficult to derive
• And in presence of the scalar field and
  radiation
• Remembering that

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• 8.2 Adding a term of coupling
• ?Its the same thing as varying the equation of
  motion of the scalar field
• ?We have in this way
• and
  also
  (harmonic oscillations)

• 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 F and damping it in the form of a
  radiation background
• F undergoes oscillations of declining amplitude
  after the end of inflation and G only changes
  the rate of damping.
• For more detailed models of reheating see Linde
  (1989) and Kofman , Linde Starobinsky (1997).

Temperature of reheating Gdegree of
freedom
Energy density for relativistic particles in the
case of perfect reheating
?Because of the factor even in the
case of perfect reheating is lt of the
initial one
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A plot of the exact solution for the scalar field
in a model with a 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
e-foldings of inflation the latter only
150. (From Peacock,1999).
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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
• The analogy is close but imperfect, and the
  characteristic temperature here is
• The inflationary prediction is of a horizon scale
  amplitude fluctuation
• The main effect of these fluctuations is to make
  different parts of the universe have fields that
  are perturbed by an amount dF with

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• We are dealing with various copies of the same
  rolling behavior F(t) but viewed at different
  times, with
• 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
• 
  

(Indetermination on the scalar field, from
quantum theory of fields. See Peacock, 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 in time for
the two different points, inducing a density
fluctuation
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• 9.2 Inflation coupling
• From the SRD equation, we know that the number
  of e-foldings of inflation is
• If
• Since N 60 and the observed value of
  fluctuations
• (Really weak coupling!!!)

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• If
• From the first of Friedmann equations
• And since is
  needed for inflation,

From CBR observations
?This constraints appear to suggest a defect in
inflation, in that we should be able to use the
theory to explain why
, rather than using observations to constrain
the theory
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• 9.3 Gravity Waves
• Inflationary models predict a background of
  gravitational waves of expected rms amplitude
• Its 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.

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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 ? )
• Is strongly linked with quantum field theory.

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11.References

• Kofman, Linde, Starobinsky,1997hep-ph/9704452
• Linde,1989Inflation and quantum cosmology,
  Academic Press.
• Lucchin,1990Introduzione alla cosmologia,
  Zanichelli.
• Peacock,1999Cosmological physics, Cambridge
  University Press.
• RamondQuantum field theory.
• Weinberg,1972Gravitation and cosmology, John
  Wiley and sons.

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Appendix 1
At tt0
?
Substituting this result in the first equation
And remembering that
Its not difficult to get the following equation
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Appendix 2
Given a lagrangian density L for the field
and the transformations
A
Def
If L is invariant for A
And
This is the Noetherns theorem
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A special case INVARIANCE with respect to
SPACE-TIME TRANSLATIONS We have
No variations of the filed
Def
If we take a Lagrangian density
CONSERVED