Transcript Multi-Field Inflation in Cosmology

Multi-Field Inflation in
Cosmology
By
Iftikhar Ahmad
GUCAS, College of Physical Sciences,
Beijing China
Plan of the Talk
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What is cosmology ?
Big Bang Cosmology.
Problem with Big bang Cosmology.
What is Inflation ?
Cosmic Microwave Background
History of Inflation.
Types of Inflation Models.
Perturbation spectrum of Multi-field inflation
with Small-field potential.
• Conclusion.
What is Cosmology?
The Study of the Universe:
its structure, origin, evolution, and destiny.
What is Big Bang Cosmology?
• The Big Bang Model
is a broadly accepted
theory for the origin
and evolution of our
universe .
• All matter started at
the one point with a
big bang .
• The Big Bang
theory predicts that
the early universe was
a very hot place.
• A hot, dense
expanding universe,
should be
predominantly
hydrogen, helium.
• Universe is ~75%
hydrogen, ~25%
helium by mass.
Problems with the Big Bang
• The horizon problem
• The flatness problem
• The Horizon Problem:
• The horizon problem tells
us the large scale
homogeneity and isotropy
of the universe must be a
part of initial conditions
but Hot Big Bang theory
is unable to explain it.
Flatness problem
The flatness problem
is simple that during
radiation or matter
domination aH is
decreasing function
of time.
In ‘standard’
cosmology: =1
means universe is flat
If 1,  moves
quickly away from 1
after big bang Today
universe is close to
flat
 1
Inflation
 1 ~ t
t
2
3
 1 ~ t
Matter dominated
Radiation dominated
What is Inflation In Cosmology?
• The rapid expansion
in the first 10-35 s of
the Universe.
• Inflation is simply
an epoch during
which the scale
factor of the
universe is
accelerating
Inflation a  0
..
Inflation solved the following problems
• The homogeneity on
large scales.
• The Horizon Problem.
• The flatness Problem.
• Alen Guth (1981)
proposed an idea which
could resolve the
horizon problem
Cosmic Microwave Background
• It all began in 1964 when
pigeons were accused of
roosting and “messing” in a
new microwave dish at
Bell Labs in New Jersey, when
Arno Penzias and Robert
Wilson, then at Bell Labs,
noticed a small discrepancy in
their microwave instruments
that indicated an excess of
radiation coming in from
space. Not content to ignore
it, they soon made one of the
profound discoveries of the
20th century:
CMBR
• CMBR photons emanate from a cosmic photosphere
like the surface of the Sun except that we are inside it
looking out
• The cosmic photosphere has a temperature which
characterizes the radiation that is emitted
• Photons in CMBR come from surface of last scattering
where they stop interacting with matter and travel freely
through space
• It has cooled since it was formed by more than 1000 to
2.73 degrees K
History of Inflation
• In 1981 Alan Guth proposed model (now it is called
old inflation ) which based on theory of supercooling.
• In 1982, Andrie Linde , Andreas Albrecht and Paul
Steinhardt proposed a model (Which is now called new
inflation model ) in which (φ) is roll slowly, field φ is call
the inflaton.
• Linde(1983) proposed one of the most popular
inflationary model is chaotic inflation
• After this there are so many inflation model has been
proposed.
Enter Horizon
Exit Horizon
1
H
Provide seeds of CMB Large
scale Structure of Universe
Horizon
Inflation
Radiation
dominated
Matter
dominated
Evolution Of Universe
t
Motion of scalar field
V ( )

Slow roll
Oscillating
Inflation
i
f
Reheating

The new inflationary potential is shown. The potential curvature is very flat in order to
permit the field to slow roll down the hill to yield enough e-folds of inflation during that
time. Inflation begins at some φi and ends at φf when the field begins to evolve rapidly to
its stable symmetry–breaking state φ = v, around which the field oscillates until reheating.
Types of Inflation Models
• Large-Field Models.
• Small-Field Models.
• Hybrid Models.
Single-Field model with Small-Field potential
Consider a potential of type
Then one finds
this
The scalar spectral index n
Reference (L. Alabidi and D.H. Lyth, JCAP05 (2006) 016)
Perturbation spectrum of multi-field
inflation with small-field potential
• The multi-field inflation model relaxes
the difficulties suffered by single field
inflation models, and thus may be
regarded as an attractive implementation
of inflation.
• Small-field models are typically
characterized by V ′′(φ) < 0.
• We take a potential for multi-fields
• Where i and µi are the parameters
describing the height and tilt of
potential.
The Scalar Spectrum of Spectral Index for p > 2
To find number of e-folds using Eq. (1)
(1)
After putting the vales and simplifications one gets
(2)
under approximation conditions
(3)
Using this condition in Eq.(3) we get
(4)
Sasaki -Stewart formulism
(5)
Using this formula (5) we can find the spectral index
For sake of simplicity of our result
we make some substitution like
(6)
(7)
With condition
Using this condition then first two terms on right side of (7) must
vanish so,
(8)
Now putting the values of A1, A2, A4 and A5 in above one can gets
Finally we get
When i = j, R(wi) = 0 then it implies that
(9)
In this case the scalar spectrum multi-field will be the same as that of
its corresponding single (Reference Alabidi Land Lyth D H,
JCAP05(2006)016)
then R(wi) is always positive so spectrum is more redder than its
corresponding single field. When
then R(wi) is always negative therefore the spectrum is less red than its
corresponding single field. However, for more general cases, it seems
that dependent of parameters of fields and initial conditions, there is
not a definite conclusion.
In the case
of two scalar
fields
for i = 1, 2; j
= 1, 2
The Scalar Spectrum for p = 2

Equation (8) is only valid for p>2,sowe need to separately calculate the case
Using Eq(9) to find number of e-folds
(10)
The spectral index
(11)
(12)
From equation of motion we get
(13)
With the help of Eq.(13) one gets
(15)
Further if we have same µ i, we
will obtain
(14)
Conclusion
• For p>2 we found that the spectrum may be redder or
bluer than of its corresponding single field.
• The result is dependent of the value of fields and their
effective masses with this value at the horzion-crossing.
• Our result is different from that of multi-field inflation
with power law potential, in which the definite conclusion
that the spectrum is redder or bluer than of its
corresponding single field may be obtained.
• When the effective masses of all fields are equal, the
spectrum will be the same with that of its corresponding
single field.
Conclusion
• By studying the spectrum behavior for p = 2 , it is noted
that the spectral index lies between that of the single field
with largest µ k and that of the single field with smallest µ
k.
•
In this case we observed that spectrum may be redder or
bluer then of its corresponding single field φk.
• then the definite conclusion that the spectrum is more
redder or bluer than of its corresponding single field φk
may not obtained.
• But when we fixed μk = μj = μ0, then the spectrum will be
the same with that of its single field φk.
• We have assume that isocurvature perturbation may be
neglected.
Classic Ending