Introductory Review of Cosmic Inflation

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Introductory Review of Cosmic Inflation

• Shinji Tsujikawa (???? ) hep-ph/0304257
• Research Center for the Early Universe,
  University of Tokyo

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Content

• Ingredients for standard big-bang cosmology
• What is inflation
• Problems of the standard big-bang cosmology
• Introducing the scalar field
• Scalar field dynamics for inflation
• Density perturbation and the origin of large
  scale structure
• Reheating after inflation

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Ingredients for Standard big-bang cosmology

• Our tool General Relativity
• Based on the cosmological principle
• Metric Friedmann-Robertson-Walker metric
• Energy-momentum tensor for perfect fluid

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Ingredients for Standard big-bang cosmology

• Then the Einstein equations yield
  is the Planck energy
  hereafter c1
• The equation of state for
  radiation p0 for matter
• When K0 (flat infinite universe), the solution
  for radiation dominantmatter dominant
• In these simple cases the universe is expanding
  deceleratedly.

Scalar field
Horizon
Monopole
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Ingredients for Standard big-bang cosmology

• Introducing Hubble parameter
  and critical density
  
• we can rewrite the Friedmann equation as
• And we can further define , thus
  we have

Flatness Problem
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What is inflation?

• Inflation means a stage in the early universe,
  with accelerated expansion
• It is also equivalent to r3plt0 negative
  pressure!
• It is also equivalent to where the Hubble
  parameter
• Without inflation, the standard big-bang
  cosmology would suffer from several severe
  problems. A.H.Guth first noted that introducing
  inflation would provide an efficient solution to
  these problems (A.H.Guth, Phys. Rev. D 23, 347,
  1981).

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Problems of the standard big-bang cosmology -
Flatness problem

• Friedmann equation can be rewritten aswhere
  . In standard big-bang, for either
  radiation or matter dominant,
  always decreases.
• is unstable it tends to shift away
  from unity with the expansion of the universe
• WMAP , very close to one.
• We require
  . This is an extremely
  fine-tuning of initial conditions.

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Problems of the standard big-bang cosmology -
Flateness problem

• With inflation since
• term increases during
  inflation, W rapidly approaches unity. As long
  as the inflationary expansion is sufficient, W
  stays of order unity even in the present epoch.

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Problems of the standard big-bang cosmology -
Horizon problem

• Particle horizon is the largest distance that
  can have casual contact at time t
• Radiation dominant matter dominanttherefore
  the horizon is much smaller in the past.
• In fact the causally contacted surface of the
  last scattering surface only corresponds
  physical scale to the angle of order
• Observationally, however, we see photons which
  thermalize to the same temperature
  horizon in all regions in the CMB sky.

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Problems of the standard big-bang cosmology -
Horizon problem

• If there is an inflation period in the early
  universe, the scale factor a(t) would grow
  drastically, while the particle horizon would
  nearly stay unchanged. Then before inflation, the
  scale could be much smaller than horizon.
• Therefore the isotropy in the CMB spectrum and
  the large scale structure can be solved.

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Problems of the standard big-bang cosmology -
Horizon problem

• Another form of the Horizon problem the origin
  of large scale structure.
• Comoving wavelength l. Physical wave lenghth
  al
• Perturbation scale larger than horizon can not be
  amplified and therefore can not form structure.
• Larger scale perturbations enter the horizon
  later, and has less time to evolve, to form
  structure.

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Problems of the standard big-bang cosmology -
Horizon problem

• Therefore it is practically impossible to
  generate a scale-invariant perturbation spectrum
  between the big bang and the time of the last
  scattering in the standard big-bang cosmology
• COBE and WMAP have seen nearly scale-invariant
  perturbation spectrum .
• WMAP

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Problems of the standard big-bang cosmology -
Horizon problem

• If there is inflation
• Early stage of inflation, the scale of
  perturbations is smaller than horizon, which
  form the seeds of large scale structure.
• Perturbations grow out of horizon
  Scale of perturbation and are frozen.
• After inflation standard big-bang stage. The
  perturbations enter
  horizonhorizon again. Then the frozen
  perturbations continue to evolve into structure.

spectrum
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More about the origin of the large scale structure

• Hubble radius 1/Ht. Comoving Hubble
  length 1/aH.
• Hubble radius (Hubble length) can be a good
  estimator of the particle horizon, both being t
• Later we shall not distinguish horizon and
  Hubble length
• Standard big-bang cosmology aH always decreases,
  then comoving Hubble length increases all the
  time.
• Inflation aH increases, comoving Hubble length
  decreases.

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More about the origin of the large scale structure

• Early stage of inflation llt1/aH, causality works
  to generate small quantum fluctuations, which
  form the seeds of large scale structure.
• Then lgt1/aH, perturbations are frozen
• After inflation standard big-bang stage. 1/aH
  increases. llt1/aH again, then causality works
  again. Then the frozen perturbations continue to
  evolve into structure.
• The small perturbation imprinted during
  inflation appears as large-scale perturbations
  after this the second horizon crossing.

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Problems of the standard big-bangcosmology--
Monopole problem

• According to the view of particle physics, the
  breaking of supersymmetry (SUSY) leads to the
  production of many unwanted relics such as
  monopoles, cosmic strings, and topological
  defects.
• String theory gravitinos, Kaluza-Klein particle,
  etc.
• Their energy density decrease as a matter
  component, much slower than radiation energy
  density. In radiation-dominant era, these massive
  relics could be the dominant materials in the
  universe, which contradicts with observations.

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Problems of the standard big-bangcosmology--
Monopole problem

• If there is inflation
• Provided that these unwanted relics are produced
  before inflation, their energy densities would
  decrease drastically with the fast increase of
  scale factor a(t). Thus unwanted relics can be
  red-shifted away.
• We still have to worry about those relics
  produced after inflation. Generally if the
  reheating temperature is sufficiently low ,the
  thermal production of unwanted relics, such as
  gravitinos, can be avoided.

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Scalar fields in particle physics and cosmology

• To obtain inflation, we need materials with the
  unusual property of a negative pressure.
• It is normally imagined that inflation begins at
  the Planck scale. Therefore we have to seek for a
  quantum theory to describe the materials for
  inflation scalar field (spin-0)
• Because of Planck scale, it is most suitable to
  adopt a quantum theory of gravity to describe
  inflation.
• Unfortunately this theory has not come yet.
• Our approach is a semi-classical one we do not
  quantize the gravity field. Quantum Field Theory
  Classical background

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Scalar fields in particle physics and cosmology

• As yet, there has been no direct observation of a
  fundamental scalar particle (such as Higgs), but
  they play a crucial role in particle physics
  theory in bring about mass through spontaneous
  symmetry broken (SSB).
• Earlier inflationary models simply use the Higgs
  field for the Grand Unified Theory (GUT) such as
  SU(5) and first order transitions.
• However they can not meet the requirements of
  cosmology.

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Scalar fields in particle physics and cosmology

• In inflationary cosmology scalar fields are
  introduced in a more phenomenological way.
• Anyway, we look for guidance about the likely
  form of the scalar field potential in particle
  theory, hoping that in the end, it will belong
  to a complete Theory of Everything (TOE).
• Recent trend is to construct inflationary models
  based on superstring or supergravity models.
• The scalar field responsible for inflation is
  often called inflaton.

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Scalar field dynamics

• The standard way to specify a particle theory is
  via its lagrangian. The lagrangian of a single
  scalar field with potential V is
• Then the energy-momentum tensor can be written
  as
• Assuming the f is spatially homogeneous, or
  noting the fact that spatially gradient terms
  , the energy-momentum tensor take the form
  of a perfect fluid with

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Scalar field dynamics

• Substituting the expression of r and p into the
  basic two equations, we have
• During inflation we require r3plt0, which yields
  . Therefore a flat potential is
  required.
• Once inflation gets under way, then the curvature
  term in the Friedmann equation becomes less and
  less important. Normally it is assumed negligible
  from the start.
• Different inflationary models give different
  potentials.

Chaotic
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Slow-roll approximation
spectrum

• The standard technique for analyzing inflation is
  the slow-roll approximation
• Defining the so-called slow roll parameterone
  can verify that slow-roll approximation are valid
  when eltlt1,
  hltlt1
• e and h are functions of V, therefore it is easy
  to see where inflation might occur. Inflation
  ends when e and h grow gt1

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Relation between inflation and slow-roll

• Slow-roll approximation is a sufficient condition
  for inflation. This can be qualitatively seen in
  the first approximated equation of motion.
• Another way to see this explicitly slow-roll
  requiresconsequently
• One the other hand we havethe definition of
  inflation is recovered.

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Amount of inflation

• We need sufficient amount of inflation to solve
  the flatness problem, horizon problem, ets.
• Usually we define
• To solve the flatness problem, we require Ngt70
• Similar value of N is required to solve the
  horizon problem.

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A simple example Chaotic inflation

• This model is described by the quadratic of
  quartic potential
• Substituting the form of V into previous
  equations, we have
• We have a exponentially expanding solution

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A simple example Chaotic inflation

• The slow-roll parameter reads therefore the
  inflationary period ends around
  ,after which the universe enters a
  reheating stage.
• The total amount of inflation is approximately
• In order to lead to sufficient inflation Ngt70,
  we require the initial value to be

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A simple example Chaotic inflation

• Detailed analysis should have more dependence on
  Quantum Field theory (QFT).
• The detailed form of the inflation potential V
  should be corrected by loop correction in
  perturbation theory and renormalization theory.
• The inflaton mass m can have dependence on f,
  whose form can be calculated from the
  Renormalization Group Equation (RGE).
• Another important type of inflationary model is
  using multi-field, such as hybrid models of
  A.D.Linde (Phys. Lett. B, 259, 38, 1991)

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Basic picture of density perturbation and the
origin of large scale structure

• At early stage of inflation, vacuum fluctuation
  of the inflaton field is generated. Quantum
  Field Theory
• After the first horizon crossing, the fluctuation
  grows as classical one, which forms the origin of
  large scale structure
• After the second horizon crossing, the
  fluctuation evolve fully classically, which forms
  todays universe.
• Here I shall briefly show the first two stages
  how quantum fluctuation forms the origin of large
  scale structure.
• Inflationary models have most predictive power in
  this aspect. Therefore the observation can kill
  many inflationary models through observing the
  large scale structure.

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Vacuum fluctuation in quantized scalar field

• The lagrangian of a scalar field in arbitrary
  spacetime (metric) is
• The field equation can be obtained by
• For a flat spacetime, we adopt the Lorentz
  metric, then we have
• For non-interacting, or free field, we havethen
  the field equation is
  , namely the Klein-Gordon equation

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Vacuum fluctuation in quantized scalar field

• Inflaton we need the Robertson-Walker metric
  instead of the Lorentz metric.
• Then the field equation for the inflaton
  issplit the field into an unperturbed part and
  a perturbation
  
  f(x,t)f(t)df(x
  ,t)
• and given a Fourier component, we have

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Vacuum fluctuation in quantized scalar field

• After canonical quantization for , we
  have
• where and are the annihilation and
  creation operator for an inflation with momentum
  k, respectively.
• satisfies
• Because annihilation operator annihilates the
  vacuum, we have the form of the vacuum
  fluctuation,which is purely a quantum effect.

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Vacuum fluctuation in quantized scalar field

• From slow-roll approximation, we can deduce that
  H varies very slowly during inflation.
• Then we can seek a solution ignoring the
  variation of H
• Here L is the comoving box size for
  normalization. And k is the wavenumber.
  Remembering that 1/k and 1/aH means the comoving
  wavelength and the comoving Hubble length
  respectively, the epoch kaH just means the
  crossing of horizon.

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The spectrum of perturbation

• A few Hubble times after horizon-crossing we have
  kltltaH therefore
• The spectrum of density perturbation can be
  defined as
• The spectrum of primordial curvature perturbation
  is given by
• Because we are dealing with slow-roll inflation,
  H varies very slowly, the above expressions can
  be evaluated at the epoch of horizon exit kaH

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The spectrum of perturbation

• Using slow-roll approximation, we can express
  as where e is the slow-roll parameter.
• Now we can define the effective spectral index
  n(k) asthis is equivalent to the power-law
  behavior that is assumed when defining the
  spectral index in the normal way,

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The spectrum of perturbation

• A simple calculation evaluated at kaH can show
  that
• Because slow-roll requires eltlt1 and hltlt1, we draw
  an important conclusioninflation predicts the
  spectrum is close to scale-invariant.
• WMAP strong support for the inflation scenario.

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The spectrum of perturbation

• Similarly we have
• Different models have different e and h,
  therefore sufficiently precise measurement of the
  spectrum, n(k) and would discriminate
  different models.
• WMAP

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Reheating Recovering the Hot Big Bang

• Reheating the period of inflationary expansion
  gives way to the standard Hot Big Bang evolution
• Typically reheating would have little impact on
  the predictions on density perturbation from the
  inflationary scenario.
• However, reheating is crucial to our
  understanding of whether baryogenesis can be
  brought about successfully whether gravitinos
  might be over producedwhether topological
  defects can be produced after inflation.

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Reheating Recovering the Hot Big Bang

• There are typically three periods of the
  reheating process1. non-inflationary scalar
  field dynamics,2. decay of inflation
  particles,3. thermalization of decay product.
• The theory of the second stage has recently
  gained important developments, which led to a
  significant change of view since books such as
  Kolb and Turners The Early Universe (1990).

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Reheating Recovering the Hot Big Bang

• Once inflation is over, slow-roll approximation
  is no longer valid. Recalling the general
  equation of motion for inflaton f

• The scalar field begins to oscillate about the
  minimum of the potential.
• Then the equation of motion can be rewritten as
  the equation for the time-average energy density
  

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Reheating Recovering the Hot Big Bang

• If the particle decay is slow (e.g. if the only
  decay channels are into fermions), one can insert
  a phenomenological term directly into the above
  equation such an equation can be used to
  describe the coherent oscillation of inflaton,
  slowly producing fermions
• Recently it was found that the inflaton may decay
  into bosonic particles, allowing a decay by
  parametric resonance. This permits an extremely
  rapid decay of the inflaton particles.
• This dramatically rapid decay has been termed
  preheating to distinguish it from the old
  scenario of reheating, which is now believed to
  happen later than preheating.

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Reheating Recovering the Hot Big Bang

• The occupation number generated by parametric
  resonance (preheating) are huge, so that bosons
  created are far from thermal equilibrium.
• Fermions and the Pauli exclusion principle.
• Decay and thermalization the bosonic particles
  produced in preheating should decay, interact,
  and finally reach thermal equilibrium. The
  details will be strongly dependent on the field
  theory adopted.
• After reheating the universe is on its way of
  standard big-bang cosmology again.

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Summary and discussion

• Success of inflation it solves a number of
  cosmological problems such as flatness, horizon,
  and monopole problems, and at the same time it
  generates the seed for nearly scale-invariant
  large scale structure.
• Cosmological scenarios alternative to
  inflationpre-big-bang (M.Gasperini and
  G.Veneziano, Astropart.Phys. 1, 317, 1993)and
  cyclic model (P.J.Steinhardt and N.Turok,
  Phys.Rev.D, 65, 126003, 2002)
• Problem the origin of inflaton? What is the
  state of the universe before inflation?
• Future high-precision observation is expected to
  reveal the detailed nature of inflation from
  theoretical side extensive works are
  consctructing viable models based on string and
  supergravity theories.

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Recommendation for references

• J.V.Narlikar and T.Padmanabhan, Gravity, Gauge
  theories, and Quantum Cosmology. D.Reidel,
  1986.This book provides a careful introduction
  to the details of quantum cosmology. To do so the
  authors have described the ingredients of Gauge
  Field Theories and General Relativity before
  begin the discussion for quantum cosmology,
  including the inflationary scenario.

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Recommendation for references

• E.W.Kolb and M.S.Turner, The Early Universe,
  Addison-Wesley, 1990This is classic book
  described ideas across the whole range of what
  had become known as particle cosmology or
  particle astrophysics, including such topics as
  topological defects, inflationary cosmology, dark
  matter, axions, and even quantum cosmology.

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Recommendation for references

• A.R.Liddle and D.H.Lyth, Cosmological Inflation
  and Large-Scale Structure, Cambridge University
  Press, 2000.As a recent textbook, it provides a
  modern and unified overview of the inflationary
  scenario and the origin of density perturbation.
  Its discussion is very clear, and carefully
  compares predictions with observations.

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Thank you very much for your attention!