The role of conformal-coupled scalar field in cosmology

R. Avakyan G. Harutyunyan E. Chubaryan A. Piloyan

Yerevan State University

4

2

1dxLgRFgW m

Let us rewrite conformal respective space in accordance with4V

gF

Fg

0

)(

And for last action we obtain

40 2

1dxLgRFgW m

where

,3 02

2

0 FF

F

FF

Corresponding equations:

.2

1

ggT s

,0 g),(

2

1

0

sm TTF

G

Cosmological scalar in the theory of JBD.Following by analogy of a introducing of cosmological constant in GR we rewrite action of Theory of JBD:

xdLy

yyyRy

cg

cW m

42

,,4

)(216

1

y

yykT

y

23

2

23

yy

y

yy

y

yy

y

yTT

y

kG

2)21(23

2

1

23 2

,,

2

,,

,

0~~

, g ),~~

(~~

0 T

y

kgG

,,,,~~

2

1~ gg

where

k

y

y

y

2

)23( 0,,

xdLgRk

yg

cW m

4,,

0 ~~2

12

~

2~1~

gy

yg

0

~ .)23(

)2(20

G

yIn Einstein frame

23

)31()(

1 33

kRy

dt

d

R

Field equations in Jordan proper frame

02

2

22

2

2

33

y

y

y

y

y

y

R

R

y

k

R

k

R

R

0

2

2

22

2

2

22

y

y

y

y

y

y

R

R

y

y

y

k

R

k

R

R

R

R

Field equations in Einstein frame

dty

yd

a

c

03, ,

2

,0

22

2,

2

13

y

k

a

k

a

a

2

,0

22

2,

2

,

2

12

y

k

a

k

a

a

a

a

)1(3 n

,)/()( 00naat

n = 0, 3, 4, 6

In Einstein frame:a)the solutions for vacuum model (n=0),1

,)(0

00 ttHeaa 300

0

H

b)for others models (n=3, 4, 6)

)()(2/ 00

22tt

n

nttnn nn eA

Ee

Aa

n

n

aE 000

nnn

n EaaA 02/

032

nn

when 0

)(32

1 000

2/

0

ttn

a

an

In the Einstein frame, when k = 0, we have

3)(

a

ct

38

)1(2 0002

030

nyqHac

eventually we have for Einstein frame

4//()/(2

)/)(4/3()1(32

062

00

000

yacaa

aayq

n

n

when 5.00,0,0 qaa Expansion in power series of t give (n=0, 3, 6 )

,23

220

nDz

zq

30

33

21

a

Ez n

The existence of transition from decelerated phase to acceleration mean that in any point q is going to zero. Modern state is according to P=0 case. It is can be seen from eq., that for certain conditions we can have above mentioned transition.

-P ,2)(/31

3

060

0

EaCq

0P ,//3

)/3(

2

3

2

1303

60

60

0

aEaC

aCq

)24(,P,2//3

3606

60

0

aEaC

q

The solutions in the proper frame of the JBD theory with a dominating non minimally coupled scalar field. It is shown that only by advent of the cosmological scalar, with corresponding to the usual cosmological constant in the Einstein frame, the possibility arises for the evolution phase with transition uniform and further accelerated expansion of the Universe during cosmological time scales.

)23(3,2/)1(2/)1(

ycyc

y

y

0)23( ycc

Then a) Consider the case 0q=0 only , but in that case the theory of JBD is unclear. Note, that when

2/32/3

yyP )3/1( 02 sc

b) Let us consider the case 0

cycc .

2

0

2/2/2/3

2/2/

2 )3()3(

)1)(3()1)(3(31

ycycy

ycycy

R

RR

Eventually in a result of analysis deceleration coefficient for expanding Universe in the framework of the Theory of JBD we have:

a)if dominated non-minimally coupled scalar field it is possible only decelerated expansion

b) in the presence of cosmological constant in the case of non-minimally coupled scalar field arises a possibility of evolution of the Universe with transition from zer0 accelerated phase to accelerated phase in time order of 10 mld. yr. Lets construct the cosmological model non-minimally coupled scalar field in the presence of cosmological constant in the by taking into account the radiation.

The solutions in the Jordan proper frame

,3 constDRy ,Ff

0

04

2

002

4

9,

4

k

y

D

yk ,43

21

3

2

2 FF

d

dF

1

2

)1(

3

3

2

2

2

0

0

f

f

f

fy

D

D

kf

q

The solutions in the Einstein frame

nn a

yE 00

0

8

0

2

3

4

y

cc

)1(3 n

644

6

33

3

2

1

a

c

a

Ea

c

q

The condition of accelerated expansion

6

2

4

40

00

6

2

04

40

00

888

a

c

a

a

ya

c

ya

a

y

So we have:The presence of cosmological constant takes a way for possibility of accelerated expansion of the Universe also in radiation epoch

The action which correspond to following eq.:

4422

122

1

122

1gdLRW m

matTRgR

12

61

2

1

036

32 R

pgUUpT

ggg

mat

)(

12336

1

3

2 422

where

pH

q 36

12

for accelerated expansion we have

pG

H3

8

32

2

G

Hkrit

8

3 2

Then

For the values of , typical in cosmology, the factor is expressed as follows:

2

21

3

kq

H

vacuum model 1

21

6

kq

H

matter dominated 0

1q radiation dominated 1/ 3

1q scalar field dominated 0,0 P

21

3

kq

H

stiff fluid 1

41302

3aaH

a)In the case of scalar field domination or in the case of

3

1P

ca

cc

c

cHbcta 1

22,,2

for too large a we have

6

b)in the case of radiation (3

1 )

c) matter dominated era30,0

ap

a

a

q

3

610

0

as can be seen for a2

1 q

For accelerated solution we get for following times

20

23

00

24

ttt

q = - 1

d) P

a

c

a

c 221

tt

ecec 32

32

1

00

t

ea 30

and q=1

Accelerated expansion can be possible only in the case of and P=0 when the energy

P

density satisfied to G

H

8

3

2

1 2

andG

H

8

3

2

1 2

Conclusion

When we considered JBD theory in the Jordan proper frame and Einstein frame acceleration was possible only when we introduced cosmological constant by analogy of GR. For last action, which we can get by certain conformal transformation, acceleration is possible in vacuum and dust cases.