Multi-Field Multi-Field Inflation in Inflation in CosmologyCosmology

ByByIftikhar AhmadIftikhar Ahmad

GUCAS, College of Physical GUCAS, College of Physical Sciences, Beijing ChinaSciences, Beijing China

Plan of the Talk• What is cosmology ?What is cosmology ?• Big Bang Cosmology.Big Bang Cosmology.• Problem with Big bang Cosmology.Problem with Big bang Cosmology.• What is Inflation ?What is Inflation ?• Cosmic Microwave BackgroundCosmic Microwave Background• History of Inflation.History of Inflation.• Types of Inflation Models.Types of Inflation Models.• Perturbation spectrum of Multi-Perturbation spectrum of Multi-

field inflation with Small-field field inflation with Small-field potential.potential.

• Conclusion.Conclusion.

What is Cosmology?What is Cosmology?

The Study of the Universe:The Study of the Universe: its structure, origin, evolution, and its structure, origin, evolution, and destiny.destiny.

What is Big Bang What is Big Bang Cosmology?Cosmology?

• The Big Bang The Big Bang Model is a Model is a broadly accepted broadly accepted theorytheory for the for the origin and origin and evolution of our evolution of our universe .universe .

• All matter started All matter started at the one point at the one point with a big bang .with a big bang .

• The The Big BangBig Bang theory predicts theory predicts that the early that the early universe was a universe was a very hot place.very hot place.

• A hot, dense A hot, dense expanding expanding universe, should universe, should be predominantly be predominantly hydrogen, helium.hydrogen, helium.

• Universe is ~75% Universe is ~75% hydrogen, ~25% hydrogen, ~25% helium by mass.helium by mass.

Problems with the Big BangProblems with the Big Bang

• The horizon problemThe horizon problem• The flatness problemThe flatness problem

• The Horizon 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 problemFlatness 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 t

3

2

~1 t

t~1Matter dominated

Radiation dominated

What is Inflation In What is Inflation In Cosmology?Cosmology?

• The rapid The rapid expansion expansion in the in the first 10first 10-35-35 s of s of the Universe.the Universe.

• Inflation is Inflation is simply an epoch simply an epoch during which during which the scale factor the scale factor of the universe of the universe is acceleratingis accelerating

0.. aInflation

Inflation solved the following Inflation solved the following problemsproblems

• The homogeneity on The homogeneity on large scales.large scales.

• The Horizon Problem.The Horizon Problem.• The flatness ProblemThe flatness Problem..• Alen GuthAlen Guth (1981)

proposed an idea which could resolve the horizon problem

Cosmic Microwave Cosmic Microwave BackgroundBackground

• It all began in 1964 when pigeons were accused of roosting and “messing” in a new microwave dish atBell 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:

CMBRCMBR

• CMBR photons emanate from a cosmic CMBR photons emanate from a cosmic photosphere like the surface of the Sun photosphere like the surface of the Sun except that we are inside it looking outexcept that we are inside it looking out

• The cosmic photosphere has a temperature The cosmic photosphere has a temperature which characterizes the radiation that is which characterizes the radiation that is emitted emitted

• Photons in CMBR come from surface of last Photons in CMBR come from surface of last scattering where they stop interacting with scattering where they stop interacting with matter and travel freely through spacematter and travel freely through space

• It has cooled since it was formed by more It has cooled since it was formed by more than 1000 to 2.73 degrees Kthan 1000 to 2.73 degrees K

History of InflationHistory of Inflation

• In 1981 In 1981 Alan GuthAlan Guth proposed model (now it is proposed model (now it is called old inflation ) which based on theory of called old inflation ) which based on theory of supercooling.supercooling.

• In 1982, In 1982, Andrie LindeAndrie Linde , , Andreas Albrecht and Paul Steinhardt proposed a model (Which is proposed a model (Which is now called new inflation model ) in which 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.

Exit Horizon

Horizon

Inflation Radiation dominated

Matter dominated

Provide seeds of CMB Large scale Structure of Large scale Structure of UniverseUniverse

Enter Horizon

t

H

1

Evolution Of Universe

Motion of scalar field

)(V

Slow roll

Oscillating

Reheating

Inflation

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.

i f

Types of Inflation ModelsTypes of Inflation Models

• Large-Field Models.

• Small-Field Models.

• Hybrid Models.

Single-Field model with Small-Field potentialSingle-Field model with Small-Field potential

Consider a potential Consider a potential of typeof typeThen one finds this

The scalar spectral index n

Reference (L. Alabidi and D.H. Lyth, JCAP05 (2006) 016)

Perturbation spectrum of multi-fieldinflation 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 > 2To 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)

With condition

(7)

(8)

Using this condition then first two terms on right side of (7) must vanish so,

Now putting the values of A1, A2, A4 and A5 in above one can gets

Finally we get (9)

When i = j, R(wi) = 0 then it implies that

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 fieldsfor 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)

ConclusionConclusion• 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.

ConclusionConclusion• 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 We have assume that isocurvature perturbation may be neglectedmay be neglected..

Classic EndingClassic Ending