568

CONFORMAL ANALOGS OF THE JORDAN-BRANS-DICKE THEORY. II. (modelsincluding vacuum effects)

R. M. Avagyan, G. H. Harutunyan, and A. V. Hovsepyan

This paper is part of a series based on a modified Jordan tensor-scalar theory of gravitation. Given the

current importance of research on vacuum phenomena in cosmic evolution, we examine several standard

cosmological models with a scalar field and a physical vacuum, including models that have a dominant

scalar field with the vacuum energy taken into account in various conformal representations of the

Jordan theory, as well as models in which ordinary matter that obeys the conventional equations of state

is present. Some noteworthy results are obtained which are, to a certain extent, consistent with currently

available observational data.

Keywords: cosmology: scalar field: vacuum energy

1. Introduction

Existing observational data pose a number of problems for research in cosmology. Besides the singular nature

of the evolution of the universe, there are questions regarding inflationary change in the initial phase of its

development and regarding the phase of accelerated expansion in a later stage [1-4]. In this article we examine

cosmological models in the framework of the so-called “Einstein representation” of the Jordan tensor-scalar theory

[5], where the scalar field is minimally coupled to the tensor field, as well as the “intrinsic representation” of this

theory with a self-consistent scalar field [6]. In the first case, the cosmological constant Λ is responsible for

phenomena associated with the vacuum energy and in the second, a cosmological scalar ( )yϕ is introduced by

analogy with Λ that transforms to the “Einstein” Λ under a certain conformal transformation. Research on vacuum

Astrophysics, Vol. 54, No. 4, December , 2011

0571-7256/11/5404-0568 ©2011 Springer Science+Business Media, Inc.

Original article submitted June 17, 2011; accepted for publication August 24, 2011. Translated from Astrofizika,Vol. 54, No. 4, pp. 631-640 (November 2011).

Academician G. S. Saakyan Department of Theoretical Physics, Erevan State University, Armenia; e-mail: rolavag@ysu.amhagohar@ysu.am ahovs@mail.ru

569

effects at a quantum level shows that they are involved in the phenomenology of the cosmological constant Λ , with

4H~Λ (H is the Hubble constant) in the early De Sitter model, while the vacuum energy induced by the KXD

condensate in later evolutionary times is ~H [7-9].

In accord with these remarks, it initially makes sense to reject a possible contribution from all forms of energy

and to examine the role of Λ in the “Einstein” representation, as well as the role of the cosmological scalar in a

modified version of the Jordan-Brans-Dicke theory. The discussion is broken up into two stages. In the first section

we consider a dominant scalar field in both of the above representations, and in the second, we consider models in

which matter that obeys conventional equations of state is present.

2. Dominant scalar field including the vacuum energy

In the “Einstein” representation of the Jordan theory with the FRW metric for a flat universe, the field

equations are

( ) ( )[ ] ,sin 2222222 ϕθ+θ+−= ddrdrtadtdS (1)

( ) , 03 =Φ adt

d � (2)

, 2

83 2

2

2

Λ+Φπ=��

Ga

a(3)

, 2

18

2 22

2

Λ+Φπ−=+ ����

Ga

a

a

a(4)

where the energy density GπΦ+Λ=ε 82

2� and the pressure GP πΦ+Λ−= 8

2

2�.

Equations (2)-(4) have been derived by applying the variational principle to the action [10].

By analogy with the standard Einstein critical energy density

G

Hco π

=ε8

3 20 (5)

(H0 is the current value of the Hubble constant), we can introduce GHc π=ε 83 2 [11] and then the field equations

(2)-(4) acquire a transparent physical significance. Equation (3) is rewritten in the form

570

, 1 ΛΛ Ω+Ω=

εε

+εε

= ckcc

ck(6)

where 2

2Φ=ε�

ck , Gπ

Λ=εΛ 8,

c

ckck ε

ε=Ω , and

cεε

=Ω ΛΛ .

Thus, the contributions of the scalar field ckΩ and the field created by the Λ -term give a sum of unity

throughout the entire evolution.

Similarly, Eq. (4) can be reduced to

, 3312 ΛΩ+Ω−=+ ckq (7)

where 2aaaq ���= is the so-called dimensionless “slowing down” parameter, or to the more convenient form

. 13

22 ΛΩ+Ω−=+ ck

H

H�(8)

It follows from Eq. (7) (or (8)) that with the mutual compensation of the contributions from the scalar and vacuum

fields, as well as when they are absent, 21−=q , as in the general theory of relativity.

On eliminating ΛΩ , from Eqs. (6) and (8) we get

, 3

8

23 2

2

2 H

G

H

Hck

πΦ−=Ω−=��

(9)

from which, with the natural assumption that ( )Φ= HH , we obtain

G

H

π′

−=Φ8

2� (10)

and Eq. (3) takes the form

. 8

23

22 Λ+

π′

=G

HH (11)

From Eq. (11) we have

( )Λ−π=Φ

232

8H

G

d

dH(12)

and, on integrating,

571

( ) , 82

3

30for 0 ⎟

⎟⎠

⎞⎜⎜⎝

⎛Φ−ΦπΛ=>Λ GchH (13)

and

( ) . 82

3

30orf 0 ⎟

⎟⎠

⎞⎜⎜⎝

⎛Φ−Φπ

Λ=<Λ GshH (14)

From Eq. (10) we then obtain

( )( ), 33

0 0ttcthH −ΛΛ=⇒>Λ (15)

( )( ). 3tg3

0 0ttH −+ΛΛ

=⇒<Λ (16)

The slowing-down parameter q is found using Eq. (7) with Eq. (6):

. 3

131132 ⎟⎠⎞⎜

⎝⎛ Λ−−=+Ω−=

Hq ck (17)

With Eqs. (13) and (14)

( )( ) , 3

320

02 ttcth

q−Λ

+−=⇒>Λ(18)

( )( ) , 3tg

320

0ttq

−Λ−−=⇒<Λ

(19)

which implies that 1→q as ∞→t , i.e., accelerated expansion is possible only if 0>Λ .

Thus, for sufficiently large t ( ∞→t ) it turns out that aaH �= behaves as 3Λ=H , i.e., Λ , which in

this problem plays the role of the vacuum energy density, is proportional to 2H ( 23 H=Λ ) in a late phase of the

development of the universe.

In order to clarify the role of the cosmological scalar in the intrinsic representation of the Jordan theory, as

in the previous problem we ignore the contribution from all forms of matter, leaving only the scalar field and the

572

vacuum effects induced and defined by ( )yϕ .

The equations of the traditional cosmological problem corresponding to the modified action in the JBD theory

[6],

( )∫ −⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡ς−ϕ+−= νμμν xdg

y

yygyR

k

y

cW 4

22

2

1(20)

are

( ), 1

22

23⎟⎟⎠

⎞⎜⎜⎝

⎛ϕϕ−

ς+ϕ=+

�

�

����

y

yy

R

R

y

y

y

y(21)

( ), 2

222

2

2

2

yy

y

y

y

R

R

y

y

R

R

R

R ϕ+ς−−−=+��������

(22)

( ). 3

2

132

2

2

2

yy

y

R

R

y

y

R

R ϕ+−ς=����

(23)

As already noted above, studies at the quantum level indicate that the vacuum energy density is proportional to nH

(where H is the Hubble parameter and n presumably takes on different values in different stages of evolution of the

universe).

Starting with the results obtained for the case of a minimally coupled scalar field for a later stage of

development of the universe ( 23 H=Λ ), it makes sense to assume that in the analogous problem where the

cosmological scalar ( )yϕ plays the role of the vacuum energy density,

( ) , 2Hy α=ϕ (24)

where α is a dimensionless constant.

On introducing the notation yy�=ψ and RRH �= , the system of field equations takes the form

, 2

123

23

22

⎟⎟⎠

⎞⎜⎜⎝

⎛ψ

−ς+

α=ψ+ψ+ψH

HHH

�� (25)

573

, 22

132 222 HHHH α+ψ−⎟⎠⎞⎜

⎝⎛ ς+ψ−ψ−=+ �� (26)

. 32

13 222 HHH α+ψ−ςψ= (27)

Equation (27) implies that

( ),

3293γ≡

ςα−ς+±

=ψH

(28)

so that ς+≤α 293 .

Equations (28) and (26) together imply that

( )( ),

2

112

AH

H −≡γ+

ς+γ−γ=�

(29)

so that

( ) , 1 00

0

ttAH

HH

−+= (30)

( )( ) , 1 100

0

AttAHa

a −+= (31)

( )( ) , 1 0000

AttAHa

a

y

y γγ

−+=⎟⎟⎠

⎞⎜⎜⎝

⎛= (32)

. 1 AQ −= (33)

On evaluating γ and A for large positive ς (as follows from observational data within the confines of the

solar system [6]), we find that

( ) ( ). 3, 32 α−≈ς

α−±≈γ A

574

Negative values of γ are excluded from the discussion, since H > 0 ( 0>aa� ) for an expanding universe.

Positive q is obtained for A < 1, which corresponds to 2>α , while Eq. (28) implies that 3<α ; that is, in

terms of this model an expanding universe is obtained for a vacuum energy density 2Hα=ϕ if 32 <α< .

3. The cosmological problem in the case of a flat universe including vacuum effects

Given that a dusty equation of state ε<<P is generally accepted for late stages of the evolution of the

universe, the system of field equations takes the form

( ) , 23

2

23

83⎟⎟⎠

⎞⎜⎜⎝

⎛∂ϕ∂−ϕ

ς++

ς+π=+

yy

T

y

G

y

y

a

a

y

y ����

(34)

( ), 3

2

832

2

2

2

yy

y

a

a

y

y

y

G

a

a ϕ+−ς+επ=����

(35)

( ), 2

222

2

2

2

yy

y

y

y

a

a

y

y

a

a

a

a ϕ+ς−−−=+��������

(36)

where PT 3−ε= is the trace of the energy-momentum tensor ( ) ikkiik PguuPT −+ε= .

If we assume that Λ=ϕ y and write Eq. (35) in a form similar to Eq. (6), then a natural notation for the

contribution of the vacuum density 23HyΛ≡ΩΛ with the standard estimate of 32≈ΩΛ for it [2] implies that

2Hα=ϕ , which can be used for determining the dynamics of the other physical quantities. As a result, Eq. (34)

can be written in the form Btay =3� (ς+επ

=23

8 0GB ). Using the notation ( )tBfya =3 and ( )tfuaa =� , the system

of field equations (34)-(36) can be written in the form

, 3utf +=� (37)

( ) ( ) , 0232

33 22 =ς+−ς−+α− ftutu (38)

. 2

2112

2

2

22α+−⎟

⎠⎞⎜

⎝⎛ ς+−+−=+

u

t

u

t

u

ft

u

fq

�

(39)

575

Then, differentiating Eq. (38) with respect to t,

( )[ ] ( ) , 0233332 =ς+−+ς−+α− futtuu ��

and using Eq. (37) and introducing the notation tuα−

+=3

32� , we obtain the equation

( ) ( )( )

. 03

1623

3

32

=αα−ς++ς+

α−− t���� (40)

In the absence of ϕ ( 0=α ), Eq. (40) is satisfied by two solutions: 0=� and ( )ς+= 23�� .

In the case of ( )ς+= 23�� ,

( )

( ) ( )00

2

0

0

32

43, 343

; 1, 1

fut

futf

utuu

+++ς=++ς=

++ς=+ς=

�

�

(41)

and the general solution for y(t) and a(t) can be obtained from the equations

( )( )

( ).

3243

,

3243

1

00

2

00

20

ftut

t

f

t

y

y

ftut

ut

f

u

a

aH

+++ς==

+++ς++ς

===

�

�

(42)

For zero constants, integration of Eq. (42) yields a known particular solution [12] which, in fact, can be

obtained from Eq. (42) in the limit of large ς as

( ) ( ) ( ) , , 43120

4320

+ς+ς+ς == taatyy (43)

and here the “slowing down” coefficient 12 += HHq � is given by

( ) , 12

1

2

111

+ς−−=⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

f

f

u

u

Hq

��

(44)

which agrees with Eq. (39); that is, the contribution of the scalar field is also negative, but negligibly small. The

solution 0=� does not satisfy Eq. (39).

A solution for 0≠ϕ is conveniently found by rewriting Eq. (40) in the form

576

, 0=γ−β+�

�

t� (45)

where ( )

( )2-3

16

ας+α=β and

( )α−ς+=γ

3

233, which with the notation tz �= reduces to

. 2 t

dt

zz

zdz =β−−γ (46)

Integrating Eq. (46) yields

,

24

24

42

42

1

2

42

42

1

2

2

2

2

2

⎟⎟

⎠

⎞⎜⎜

⎝

⎛β−γ+γ

β−γ

⎟⎟

⎠

⎞⎜⎜

⎝

⎛β−γ−γ

β−γ

⎥⎥⎦

⎤

⎢⎢⎣

⎡−γ+β−γ

⎥⎥⎦

⎤

⎢⎢⎣

⎡+γ−β−γ

=

t

tct

�

�

(47)

from which we arrive at the obvious conclusion about the existence of a particular solution that has a structure

analogous to the particular solution obtained for 0=α (Eq. (40) implies that �� �=t ):

( )( )

( )( )

( )( )22

22

3

16

3

23

4

9

3

23

2

3

42 α−ς+α−

α−ς+±

α−ς+=β−γ±γ== �

�

�t (48)

in which the second (lower) root does not satisfy Eq. (39).

The “slowing down” parameter q is calculated using the scheme for solving the previous problem:

( ) ( )( )

.

323

138

112323

3

2

1

33

1

2

1

2−

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ς+ς+α−+ς+

α−−−=

α−−

−−=

t

q�

(49)

In the limit 0=α this transforms to Eq. (44). For large ò the parameter

( ) ( )ς+α+

ς+−−=

1612

1

2

1q (50)

becomes positive for ( )ς+>α 23 .

577

The general solution of this problem with 2Hα=ϕ reduces to integrating the equations

, 322

12 2

22−α+−⎟

⎠⎞⎜

⎝⎛ ς+−ψ−= zz

HH

H ��

(51)

( ) ( ) , 23

4

23

2

233

22

2 H

H

zzz

Hm

��

ς+α−

ς+α+

ς+Ω

=++ψ(52)

. 36

1 2 α+−ς+Ω= zzm (53)

Here we have introduced the notation

.yH

G

Hz

a

aH

y

ym

3

8, , ,

20επ=Ωψ===ψ

��

(54)

On solving this quadratic equation for z, from Eq. (53) we can obtain

, 3

31

3

211

ς

⎟⎠⎞⎜

⎝⎛ α−Ω−ς+±

=m

z (55)

and, eliminating 2Hψ� from Eqs. (51) and (52), we can isolate the “slowing down” coefficient 122+==

H

H

a

aaq

�

�

��,

so that

( )[ ] ( ) ( )[ ] ( ), 384

133532323

11 32 zzzzz

zq +α−ς++ς−ς+

ς+−α+= (56)

which, for large ς ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞⎜

⎝⎛ α−Ω−

ς≈

31

6mz approaches the limit

( ) , 12

111

4

5

2

10 α+Ω−−−≈ mq

which for 0=α and 1=Ωm coincides with the Einstein result.

578

If we consider the similar problem with 2Hα=ϕ for a radiation dominated stage of development of the

universe, then, given the equation of state ε=3

1P , Eq. (34) takes the form const3 == DRy� . In this case it is

appropriate to use the notation

. 1

, 2 fd

dyy

R

Dfyd

R

dt =η

=′⇒=η=

Then the problem reduces to integrating the equations

, 21 fRf �+=′ (57)

. 633

1 022 fRfD

fR �� −⎟⎠⎞⎜

⎝⎛ ς−χε

⎟⎠⎞⎜

⎝⎛ α−= (58)

From Eq. (58) we obtain

, 633

14

1

2

1 0 ⎟⎠⎞⎜

⎝⎛ ς+χε

⎟⎠⎞⎜

⎝⎛ α−+±−= f

DfR�

so that

, 633

14121 0 ⎟⎠⎞⎜

⎝⎛ ς+χε

⎟⎠⎞⎜

⎝⎛ α−+±=+=′ f

DfRf �

from which

( ) .

31

3

43

13

21

31

3

1

0

20

0

⎟⎠⎞⎜

⎝⎛ α−χε

⎟⎠⎞⎜

⎝⎛ α−ς+

−η−ηχε

⎟⎠⎞⎜

⎝⎛ α−=

DD

f(59)

The slowing down parameter is then equal to

( ).

13

132

11

00 −η−ηχε

⎟⎠⎞⎜

⎝⎛ α−

+=

D

q(60)

579

q > 0 if ( ) 03

13

20

0 >η−ηχε

⎟⎠⎞⎜

⎝⎛ α−

D; that is, for D > 0, 3<α and for D < 0, 3>α and 30 >α⇒<D . In the limit

of ∞→t , we have 1→q , as in Ref. 13.

4. Conclusion

In this paper we have attempted to construct classical cosmological models that include the vacuum energy.

This is based on the consideration that at a quantum mechanical level, the vacuum energy can be assumed to be

responsible for the cosmological constant Λ in the classical theory of gravitation [14], so that Λ can be regarded

as proportional to Hn, where H is the Hubble constant and n is different in various stages of evolution.

In this paper we have examined different cosmological models in terms of a modified Jordan tensor-scalar

theory. It is assumed that the cosmological scalar ( )yϕ in the lagrangian arises from vacuum effects. The discussion

is broken up into two parts. In the first we examine the cases of a dominant scalar field including the vacuum energy

for a flat universe using the FRW metric. Initially, the problem is represented in terms of the “Einstein” representation

of the Jordan theory, where ( )yϕ transforms into the ordinary cosmological constant Λ . A minimally coupled scalar

field in this representation can be used to write down the field equations in terms of quantities that play the roles

of contributions from the densities of different kinds of energy [15]. In the end, the interpretation the results becomes

much simpler; essentially, they mean, first, that expansion with acceleration is possible only for Λ > 0 and, second,

that in the late stage of development the Hubble constant H is related to Λ as 3Λ=H . Then an analogous

problem is considered, but now in the intrinsic representation of a modified variant of the Jordan theory. Here, based

on the previously obtained relationship between H and Λ , the cosmological scalar is chosen to have the form

( ) 2Hy α=ϕ . Ultimately, expansion with acceleration turns out to be possible if 32 <α< .

In the second section of this paper, we present cosmological models in the intrinsic representation of a

modified Jordan theory, but with matter present, initially a dusty flat universe and then we examine a radiative epoch

of development with ( ) 2Hy α=ϕ . In all cases, the conclusions were similar. Only when a cosmological scalar is

present and interpretable as the vacuum energy density, can accelerated expansion occur in a later stage of evolution

of the universe.

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