Emergent Universe Scenario

B. C. PaulPhysics Department

North Bengal UniversitySiliguri -734 013, India

Collaborators : S. Mukherjee, N. K. Dadhich (IUCAA, Pune)S. D. Maharaj, A. Beesham (South Africa)

_______________________________________Nov. 13, 2007, KITPC, CAS, Beijing

Outline of the Talk___________________________________________

• Introduction on Emergent Universe (EU)• Harrisson Solution• EU in Non-flat universe• Construction of Potential for EU• Our Model in flat Universe• Brief discussion

Emergent Universe

• A universe which is ever-existing, large enough so that space-time may be treated as classical entities.

• No time like singularity.• The Universe in the infinite past is in an almost static

state but it eventually evolves into an inflationary stage.

An emergent universe model, if developed in a consistent way is capable of solving the well known conceptual problems of the Big-Bang model.

In 1967, Harrison found a cosmological solution in closed model with radiation and a positive cosmological constant :

As , the model goes over asymptotically to an Einstein Static Universe,

the expansion is given by a finite number of

e-folds :

• Radius is determined by . However, de Sitter phase is never ending (NO Exit).

2

1

2exp1)(

ii a

tata

t

io

i

oo a

t

a

aN

2ln

• Ellis and Maartens (CQG, 21 (2004) 223) : Considered a dynamical scalar field to obtain

EU in a Closed Universe. In the model, taken up minimally coupled scalar field in a self-interacting potential V( ).

In this case the initial size of the universe is determined by the field’s kinetic energy.

To understand we now consider a model consisting of ordinary matter and minimally coupled homogeneous scalar field in the next section.

ia

• The KG equation for field:

• The conservation equation for matter is

• The Raychaudhuri equation

• First Integral Friedmann equation :

• It leads to

0)1(3

Hw

03

VH

)()31(2

1

3

8 2 Vw

G

a

a

222 )(

2

1

3

8

a

KV

GH

22 )1(4

a

KwGH

• Raychaudhuri equation gives condition for inflation,

• For a positive minimum ,

where the time may be infinite.

Vwa

)31(2

10 2

0 ii taa

2

2

8

3

2

10

i

iiiiGa

KVH

it

• The Einstein Static universe is characterized by

and , we now get

Case I : If the KE of the field vanishes, there must be matter to keep the universe static.

Case II : If only scalar field with non-zero KE, the field rolls at constant speed along the flat potential.

1K .Constaa i

2

2

4

11

i

iiiaG

w

,4

11

2

12i

iiiaG

Vw

• A simple Potential for Emergent Universe with scalar field only is given by

• But drops towards a minimum at a finite value

• Potential is given by

-10 -8 -6 -4 -2phi

1

2

3

4

5

V

tasVV i ,)(

f

2

1exp)(

f

fif VVVV

• Consider - modified action (Ellis et al.CQG 21 (233) 2004)

Define and consider the conformal

transformation , so that

If we now equate we get

2R

24 RRgxdI

lnln6ln61~2

ggRR

R212

gg 2~

R 21ln3

2

34 14

1~2

1~

egRgxdI

• Determination of the parameters in the potential :

Considering the spacetime filled with a minimally coupled scalar field with potential :

• Where A, B, C, D are constants to be determined by specific properties of the EU.

• Potential has a minimum at

• Set the minimum of the potential at the origin of the axis, we choose

• A=C and D = 0. Thus

DCeAVV B 2

BB eCeABAV 2

BB eCeABAV 22 2

DVVwithA

C

B ooo

ln

1

21 BeAV

• By definition EU corresponds to a past-asymptotic Einstein-static model :

• Where is the radius of the initial static model. To determine an additional parameter recall that an ES universe filled with a single scalar field satisfies

• Thus we have

• Basic properties of EU have fixed three of the four parameters in the original potential. The parameter B will be fixed as follows :

• , in the higher derivative context .

2

2

iaV

oa

22

22

ii aA

aV

22

12

B

i

ea

V

0014

2

foree

a

BV BB

i

0 B 0

The corresponding potential in EU is the reflection of the potential that obtained from higher derivative gravity. The different parts of the potential are :(1) Slow-rolling regime or intermediate pre-slow roll phase(2) scale factor grows (slow-roll phase)(3) inflation is followed by a re-heating phase(4) standard hot Big Bang evolution

-10 -8 -6 -4 -2phi

1

2

3

4

5

V

1 23

4

• Leaving the Einstein-Static Regime : ----------------------------------------------------------------------------------------------------

As the universe leaves the ES state, the evolution of the scalar field is determined by the KG equation :

For expansion only if . As the potential drops with increasing we consider the following cases :

(i) decelerated scalar field, friction comes from the expansion.

KE drops, Since

(ii) weakly decelerated scalar field. KE of the scalar field will

increase, slow-rolling regime unlikely.

03

VH

0

V

0

.0,22

likelyVwithperiodrollingslowaVinitiallyV

V

0

OUR MODEL :

In looking for a model of emergent universe, We assume the following features for the universe :

1. The universe is isotropic and homogeneous at large scales. 2. Spatially flat (WMAP results) : 3. It is ever existing, No singularity 4. The universe is always large enough so that classical description of space-time is adequate. 5. The matter or in general, the source of gravity has to be described by quantum field theory. 6. The universe may contain exotic matter so that energy condition may be violated. 7. The universe is accelerating (Type Ia Supernovae data)

02.002.1 t

• EOS

• The Einstein equations for a flat universe in RW-metric

• Making use of the EOS we obtain

• On integration

• 1) If B < 0 Singularity 2) If B > 0 and A > -1 Non-singular (EU)

2

1

BAp

2

2

3a

a

2

2

2a

a

a

ap

03)13(22

2

a

aB

a

aA

a

a

)1(3

2

2

3

3

2

2

13)(

AtB

eB

Ata

• The Hubble parameter and its derivatives are

where

• Here H and are both positive, but changes sign

at .

33

2

2

,,t

tt

t

t

t

t

e

eeH

e

eH

e

eH

H

H

ln

1t

.,13

2

2

3Const

AandB

-10 -8 -6 -4 -2t

10

20

30

40

50

60

a

• Composition of the Emergent Universe :

• Using EOS we get,

• This provides us with indications about the components of energy density in EU

03

a

ap

dt

d

2

21321

1

A

a

KB

Aa

321132

2

2

1322

2 1

1

1

1

2

1

AA aA

K

aA

KB

A

B

321132

2

2

1322

2 1

1

1

1

1

1ppp

aA

AK

aA

AKB

A

Bp

AA

Composition of universal matter for some values of A

Dark Energy

Exotic Matter

Radiation

Dark Energy

Exotic Matter

Cosmic Strings

1 0 1

Dark Energy

Exotic Matter

Stiff matter

0 0

Dark Energy

Exotic Matter

Dust

B

Kunitin

1 2

2

B

Kunitin

nCompositio

3

1

3

1

28

9

a

a2

9

32

1

a

23

8

2

a

3

1

3

2

2

1

48

9

a

24

9

a

64

1

a

3

1

a

3

1

3

1

12

12

A

A3

Discussion :

• The emergent universe scenari are not isolated solutions they may occur for different combination of matter and radiation.

• In the Ellis et al model the universe spends a period in the Einstein Static state before an early inflation takes over. In our model we obtained solution of the Einstein equations, which evolves from a static state in the infinite past to be eventually dominated by cosmological constant.