the role of conformal-coupled scalar field in cosmology
DESCRIPTION
The role of conformal-coupled scalar field in cosmology. R. Avakyan G. Harutyunyan E. Chubaryan A. Piloyan Yerevan State University. Let us rewrite conformal respective space in accordance with. And for last action we obtain. where. Corresponding equations:. - PowerPoint PPT PresentationTRANSCRIPT
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.