EMERGENT UNIVERSE : THEORY AND OBSERVATIONS

B. C. Paul*Physics Department

North Bengal UniversitySiliguri -734 013, India

*IUCAA Resource Centre, NBU_____________________________________

Outline of the Talk__________________________________________

Introduction on Emergent Universe (EU) Cosmological Solution by Harrison EU in Non-flat universe & Construction of

Potential for EU EU Model in flat Universe EU with Astronomical Observations Brief Discussions

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy , IU

CA

A (A

ug

. 10

-12

, 20

11

)

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.

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

Why Emergent Universe ?

Emergent Universe

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

INTRODUCTION

)sin()( 222222222 drdrdrtadtds

)1()( 0teata

0)0( 0 aaEver existing universe. No singularity !

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

In 1967, Harrison found a cosmological solution in a 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

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

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

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

The KG equation for field:

The conservation equation for matter is

The Raychaudhuri equation

First Integral Friedmann equation :

It leads to

0)1(3

H

03

VH

)()31(2

1

3

8 2 V

G

a

a

222 )(

2

1

3

8

a

KV

GH

22 )1(4

a

KGH

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

Raychaudhuri equation gives condition for inflation,

For a positive minimum ,

where the time may be infinite.

Va

)31(2

10 2

0 ii taa

2

2

8

3

2

10

i

iiiiGa

KVH

it

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

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 .ia a Const

2

2

4

11

i

iiiaG

,4

11

2

12i

iiiaG

V

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

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 -2

phi

1

2

3

4

5

V

tasVV i ,)(

f

2

1exp)(

f

fif VVVV

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

Determination of the parameters in the potential : Considering the space-time 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 C=D and E = 0

EDeCVV G 2

GG eDeCCGV 2

GG eDCeCGV 22 2

EVVwithC

D

G ooo

ln

1

21 GeCV

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

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

ia

22

22

ii aC

aV

22

12

G

i

ea

V

0014

2

foree

a

GV GG

i

0 G 0

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

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

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

co

smo

log

y … IU

CA

A (A

ug

. 10

-12

, 20

11

)

-10 -8 -6 -4 -2phi

1

2

3

4

5

V

1 23

4

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

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

OUR MODEL : (SM, BCP, NKD, SD, AB, CQG 23, 6927 (2006))

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 universe may contain exotic matter so that energy condition may be violated.

6. The universe is accelerating (Type Ia Supernovae data)

02.002.1 t

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

EOSThe 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

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

Composition of the Emergent Universe :

Using EOS we get,

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

3 0d a

pdt a

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

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

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

AB

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

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

co

smo

log

y … IU

CA

A (A

ug

. 10

-12

, 20

11

)

The suitability of our model studied: Brane World Scenario 1) A. Banerjee, T. Bandyopadhyay, S. Chakraborty, Grav. Cosmo 13, 290 (2007) 2) A. Banerjee, T. Bandyopadhyay, S. Chakraborty, GRG 26, 075017 (2008)

Phantom & Tachyon Field 3) U. Debnath, CQG 25, 205019 (2008)

Non-linear Sigma model 4) A. Beesham, S. V. Chervon, S. Maharaj CQG 26, 075017 (2009)

EU with GB term :

5) BCP & SG (GRG 42, 795,2010)

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

Field Equations

Hubble Parameter in terms of Redshift (z)

dt

dz

zzH

1

1)(

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

Components of matter and dark energy are conserved separately. So we can use energy conservation equation together with EOS to obtain the expression for energy density.

where ‘K’ is a constant of integration and ‘K > 0’ For convenience we re-write the expression for

energy density.

2

2

)1(31

1

1

Aemu

a

K

AA

B

2

2

)1(3

10

A

ssemuemu

a

AA

2

10

A

BKemu

2

0

1

1 emu

s A

BA

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

Finally, using Friedmann equation, we can express Hubble Parameter for the model in terms of redshift parameter.

Here represents baryonic energy density and represents the energy density of the exotic fluid (the suffix ‘0’ signifies present values).

0b

0emu

0 0

0 0

1/223( 1)/23

0

1( ) 1 1

,

1

A

b b

b emu

B K zH z H z

B K

where

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

CONSTRAINT FROM H(Z)-Z DATA : Hubble parameter, as can be seen, is a function of

some variables.

Remember that ‘K’ enters the theory as an integration constant and we are more concerned about finding suitable range of values for A and B for a physically viable cosmological model. For a fixed value of ‘K’ these two parameters can be constrained on ‘A-B’ plane simply minimizing a function.

2/122/)1(3

3

2200

2

111,,,

,

,,,,,,,

00

KB

zKBzzKBAE

where

zKBAEHzKBAHH

A

bb

zH 2

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

2

00

2 )(,,,,,,,,

z

obszH

zHzKBAHHzKBAH

Ho is not important for our analysis so we marginalize over it and obtain probability distribution function in terms of ‘A, B and K’ only.

2

),,,,(exp),,( 0

2

00

zKBAHHPdHKBAL zH

Here is the prior distribution function for present value of the Hubble parameter. We consider Gaussian prior here with .

0HP

8720 H

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

We use Stern data set here to constrain the parameters.

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

Blue(solid)-95%Red (dotted) -99%

Best fit:A=0.122B=-0.0823

JOINT ANALYSIS WITH H(Z)-Z DATA AND BAO PEAK PARAMETER :

The value of the BAO peak parameter (A) is independent of cosmological models and for a flat universe it can be written as :

A

where, and

A= 0.469 ± 0.017

3/2

1

0

3/11

1

)()(

zzE

dz

zE

z

m

2)/(11 BKBbbm 035.01 z

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

We define

and for joint analysis we consider

2

22

017.0

469.0A BAO

222int BAOzHjo

)(0114.00078.0,0500.02757.0 : level confidence 99% Upto

)(0116.00077.0,0306.03053.0 : level confidence % 95 Upto

redBA

BlueBA

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

Blue(solid)-95%Red (dotted) -99%

Best fit:A=0.0094B=-0.1573

JOINT ANALYSIS WITH H(Z)-Z, BAO AND CMB SHIFT PARAMETER :

CMB shift parameter (R) is defined by :

We consider with

lsz

m HzH

dz

0 0/)'(

'R

03.070.1R us gives data WMAP7 .scattering lastof surface the at redshift the is z where ls

2

22

03.0

700.1R CMB

2222CMBBAOzHTot

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

From this analysis we find new constraints :

0.0360 0.0005

0.0083 B 0.0125

0.0328 0.0024

0.0086 0.0120

0.0176, 0.0102

A

A

B

A B

Unto 99% confidence level :

Upto 95% confidence level :

Best Fit value :

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

Blue(solid)-95%Red (dotted) -99%

Best fit:A=0.0102B=-0.0176

SOME COSMOLOGICAL PARAMETERS :

Density parameter

meff

DEeff

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

Effective EOS parameter :IN

DO

-UK

Se

min

ar o

n C

on

fron

ting

Pa

rticle co

smo

log

y … IU

CA

A (A

ug

. 10

-12

, 20

11

)

Black (solid) – Best Fit Blue(dotted) – 95 % Red (dashed)- 99%

Deceleration parameter :IN

DO

-UK

Se

min

ar o

n C

on

fron

ting

Pa

rticle co

smo

log

y … IU

CA

A (A

ug

. 10

-12

, 20

11

)

Black (solid) – Best Fit Blue(dotted) – 95 % Red (dashed)- 99%

Compare with SN Ia data :IN

DO

-UK

Se

min

ar o

n C

on

fron

ting

Pa

rticle co

smo

log

y … IU

CA

A (A

ug

. 10

-12

, 20

11

)

Blue Points – ObservationSolid Line - Theoretical

EU WITH DE, EXOTIC MATTER AND DUST : Let us get back to the original work by

Mukherjee et. al. As was seen in Table. Value of the parameter

‘A’ in the EOS ( ) selects a specific composition of matter energy content of the universe.

For ‘ ‘ the universe contains dark energy, exotic matter and dust.

1/2Bp A

0A

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

EU WITH…. Our aim is to see if we can find a range of values for

the model parameters in an observationally viable scenario !

The Friedmann equation can be written for any ‘A’ as:

where and The as previously shown we can define for Stern

data set

32 2 3( 1)2( 1)

0 1 2( ) (1 ) (1 ) AAconstH z H z z

c

3

( 1) (3( 1)21 2( ) (1 ) (1 )

A Aconstz z z

22 0

0 2

( ( , , , ) ( ))( , , , , ) obs

sternz

H H B K z H zH A B K z

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

CONSTRAINTS FROM OBSERVED HUBBLE DATA (STERN) :

For analysis with Stern data :

0.003 < B < 0.59960.303 < K < 0.63

Within 68.3% confidence

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

Blue (Solid) – 68.3%Pink (Dotted) – 95 %Green (Dashed) – 99%

FROM STERN + BAO DATA :

From joint analysis with Stern and BAO data:

0.009 < B < 0.6060.3126 < K <0.6268

Within 68.3% confidence.

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

Blue (Solid) – 68.3%Pink (Dotted) – 95 %Green (Dashed) – 99%

OTHER COSMOLOGICAL PARAMETERS:

Evolution of density

Evolution of effective EOS parameter :Blue- Best fitRed-68.3%Green-95 %Thick-99%

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

DENSITY PARAMETERS :

Here

The model can permit

Within 68.3% confidence.

1 d e

0.72

For details : Paul, Ghose & Thakur, MNRAS, 413, 686 (2011)

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

Blue– 68.3%Pink– 95.4 %Green– 99.9%

OTHER COMPOSITIONS :IN

DO

-UK

Se

min

ar o

n C

on

fron

ting

Pa

rticle co

smo

log

y … IU

CA

A (A

ug

. 10

-12

, 20

11

)

DENSITY PARAMETERS FOR OTHER CASES:

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

Discussion :

The emergent universe scenario 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.

With GB-term we determine the coupling of the dilaton field which admits EU scenario.

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

CONTD. :

Matter required to realize an EU is determined.

In EU inflationary stage as well as late acceleration can be accommodated

Particle creation may occur at a phase when Hubble parameter varies slowly (early inflationary phase).

Einstein’s static universe permitted in this theory is unstable

We determine the allowed range of values of the EOS parameter using different observational data. Using H(z) vrs. Z data (Stern et al. Data), BAO (Ries et al). We also took the shift parameter predicted from CMBR data. To determine the EOS parameters.

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

We have compared magnitude (μ) vs redshift (z) curves obtained in our model with that obtained from observational data (SNeIa- Wu et al..).

We found that an accelerating universe can be accommodated in EU which is permitted in the presence of exotic matter (decided by EOS parameter A).

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)

THANK YOU

IND

O-U

K S

em

ina

r on

Co

nfro

ntin

g P

article

cosm

olo

gy …

IUC

AA

(Au

g. 1

0-1

2, 2

01

1)