Bulk Viscous Cosmological Models with Variable G and Λ

G en eral Relativ ity an d G ravi tation , Vol. 30, No. 4 , 1998

Bulk Viscous Cosmological Models with Variable

G and L

Tarkeshwar Singh,1 ,2 A. Beesham 2 ,3 and W. S. Mbokazi2

Rece ived Ju ly 21, 1997 . Rev . version Novem ber 6, 1997

Einstein ’ s equat ions with variab le grav itat ional and cosmological ª con-

stant sº are considered in the presence of bulk viscosity for the spat ially

¯ at homogeneous and isotropic universe in a way which conserves the

energy m om entum tensor. A solut ion is found in which the cosmological

t erm varies inversely with the square of t ime. Our approach is compared

with that of Arbab.

KEY WORDS : Cosm ology ; bulk viscosity ; variab le G and L

1. INTRODUCTION

One of the outstanding problems in cosmology is the cosmological constant

problem [1,2]. Since its introduct ion, its signi® cance has been studied from

time to time by various workers [3± 5]. In modern cosmological theories the

cosmological constant remains a focal point of interest. A wide range of

observat ions now compellingly suggest s that the universe possesses a non-

zero cosmological constant [6]. In the context of quantum ® eld theory a

cosmological term corresponds to the energy density of vacuum. It was

suggested that the universe might have been born from an exited vacuum

¯ uctuat ion which triggered oŒin¯ at ionary expansion, followed by super-

1 Perm anent address: Department of Mat hemat ics, Shri Ram deobaba Kam la Nehru

Engineering College, Kat ol Road, Nagp ur-440 013, India.

E-mail: rknec@bom 2.vsnl.net .in2

Departm ent of Applied Mat hemat ics, University of Zululand, P rivat e Bag X1001,

Kwa-Dlangezwa 3886, South Africa3 E-mail: ab eesham @pan.uzulu.ac.za

5 7 3

0001-7701/ 98/ 0400-0573$15.00/ 0 1998 P lenum Publishing Corporation

5 7 4 S in g h , B e e sh a m a n d M b oka z i

cooling and subsequent reheating with the release of locked up vacuum

energy. The cosmological term, which is a measure of the energy of empty

space, provides a repulsive force opposing the gravit ational pull between

the galaxies. If the cosmological term exists, the energy it represents counts

as mass because, as Einstein showed, mass and energy are equivalent .

If the cosmological term is large enough, it s energy plus the matter in

the universe could add up to the number that in¯ ation predicts. Unlike

the case of standard in¯ ation, a universe with a cosmological term would

expand faster with time because of the push from the cosmological term

[7]. But recent research suggest s that the cosmological term corresponds

to a very small value of the order of 10 - 58 cm - 2 [8].

It has been suggested by Linde [4] that L is a funct ion of temperature

and related to the spontaneous symmetry breaking process. Therefore, it

could be a function of time in a spat ially homogeneous expanding universe

[9]. The latest measurements of the Hubble parameter [10,11] point to an

int rinsic fragility of the standard (photon conserving) frw (Friedmann±

Robertson± Walker) cosmology, in such a way that models without a cos-

mological constant seem to be eŒectively ruled out (Refs. 12,13, and refer-

ences therein) . Any model of the universe should yield a lifet ime greater

than that of the oldest objects in it. The ages of the oldest stars in glob-

ular clusters, the oldest known objects in the universe, are of the order of

16 billion years [14]. Even allowing for the uncertainty in the determina-

tion of Hubble’ s constant (H0 ), it is di� cult for the frw models without a

cosmological constant to lead to an age of the universe greater than that

of these stars [13,15,16].

Apart from the age problem, such models also suŒer from other prob-

lems. There is the discrepancy in the local value of H0 as compared to

its global value (see references in Ref. 15) . Dynamical estimates of the

mass density on the scale of galact ic clusters, the largest gravit at ionally

bound systems, lead to a density parameter that is much smaller than

that predicted by in¯ ation [6,17]. The standard cold dark matter mod-

els of structure formation do not provide a good ® t to the shape of the

observed power spectrum, using linear perturbat ion theory and index of

power spectrum n = 1 [13,18± 21]. Even taking mixed dark matter models

and relaxing the requirement n = 1 does not help much [13].

The role played by viscosity and the consequent dissipat ive mecha-

nism in cosmology have been discussed by many authors (Ref. 22, p. 594,

Refs. 23± 25) . The heat represented by the large entropy per baryon in the

microwave background provides a useful clue to the early universe, and a

possible explanat ion for this huge entropy per baryon is that it was gener-

ated by physical dissipat ive processes acting at the begining of evolut ion.

B u lk V is c ou s C osm olog ic a l M od e ls w it h V a r ia b le G a n d L 5 7 5

These dissipat ive processes may indeed be responsible for the smoothing

out of init ial anisot ropies [26].

Ever since Dirac ® rst considered the possibility of a variable G [27],

there have been numerous modi® cations of general relat ivity to allow for a

variable G [28]. These theories have not gained wide acceptance. However,

recently [29± 32] a modi® cation linking the variat ion of G with that of Lhas been considered within the framework of general relat ivity. However,

cosmological models with t ime-dependent G and L and the solut ions L ~R - 2 ~ t - 2 were ® rst obtained by Bertolami [33,34]. Some frw models with

variable G and L have been discussed by Abdussat tar and Viswakarma

[35]. It is found that the solut ions of the ® eld equat ions, which are based

on the variat ion of L as R - 2 but with diŒerent constants of proport ionality,

could describe signi® cantly diŒerent cosmological scenarios.

The modi® cat ion mentioned above is appealing since it leaves the form

of Einstein’ s equat ions formally unchanged by allowing a variat ion of G

to be accompanied by a change in L. However we make some remarks on

the present approach. It is non-covariant , and the ® eld equat ions cannot

be derived from a Lagrangian density. The propagat ion equat ions for the

scalar ® elds are not contained in the ® eld equat ions. Barraco [36] and

Lau and Prokhovnik [37] const ructed full scalar-tensor theories similar to

Brans± Dicke theory, but the simplicity of the present approach was largely

lost . It could be the limit of a more viable fully covariant theory such as

the scale covariant theory [38] or a ® ve-dimensional theory of gravity of

the Kaluza± Klein type [39]. Advantages of the present approach are that

the problems which the in¯ ationary universe scenario solves can be solved,

as well as the cosmological constant problem [2], and the init ial singularity

can be averted [40].

Recent ly in the same formalism, Arbab [41] has discussed a viscous

model with variable cosmological and gravitational ª constantsº , claiming

that energy is conserved. Several solut ions were presented and some have

been shown to be equivalent to those of Berman [31] and Kalligas et al.

[29]. However, a closer scrut iny reveals that energy is not conserved and

that the conservat ion equat ion used by Arbab seems to be an ad hoc as-

sumption.

In the present work we will re-examine the viscous ¯ at frw universe

within a framework where the gravitational parameter G and the cosmo-

logical parameter L vary, while retaining the usual conservat ion law.


2. BASIC EQUATIONS

We consider the Friedmann± Robertson± Walker space-t ime metric

ds2

= dt2

± R2(t)

dr 2

1 ± kr 2+ r

2(dh

2+ sin

2hd F 2

) (1)

where k is the curvature index. Einstein’ s ® eld equat ions with time de-

pendent cosmological and gravit ational ª constantsº are

Ra b ± 12 Rga b = 8pGTa b + Lga b . (2)

Equat ions (1), (2) and the perfect ¯ uid energy-momentum tensor

Ta b = (r + p)u a u b ± pga b (3)

yield the two independent equat ions

3ÈR

R= ± 4pG r + 3p ±

L

8pG, (4)

3ÇR2

R2+ 3

k

R2= 8pG r +

L

8pG. (5)

Eliminat ion of ÈR from eqs. (4) and (5) gives

3(p + r) ÇR = ± rÇG

G+ Çr +

ÇL

8pGR, (6)

where an overdot denotes diŒerentiat ion with respect to time t.

To include the eŒect of bulk viscosity, one has to modify the perfect

¯ uid pressure p by p ± 3gH , where p represents the equilibrium pressure,

g is the bulk viscosity coe� cient and H = ÇR / R is the Hubble parameter.

Then we get from eq. (6)

Çr + 3H (p + r) ± 9gH2

= ±ÇG

Gr +

ÇL

8pG. (7)

The conservat ion of energy and momentum (u a Ta b;b = 0) yields

Çr + 3H (p + r) ± 9gH2

= 0, (8)


which, from eq. (7) leads to

ÇG

Gr +

ÇL

8pG= 0 . (9)

To solve the system of equat ions, we assume the usual barot ropic

equat ion of state relat ing the equilibrium pressure to the energy density

p = (c ± 1)r, (10)

where c is a constant. We also assume the usual power law form for the

bulk viscosity coe� cient

g = g0 rn

, g0 ³ 0, n constant. (11)

Following Arbab [41], we take the variat ion of L to be of the form

L = 3bH2, b constant. (12)

3. THE MODEL

Now considering k = 0 which represents a ¯ at frw model, we obtain

from eqs. (5) and (12)

8pGr = 3(1 ± b)H2 . (13)

Using eqs. (12) and (13) , eq. (9) can be written as

2

(1 ± b)

ÇH

H=

Çr

r. (14)

Integrat ing eq. (14) , we obtain

H = C1r1 / d

, (15)

where C1 is a constant of integrat ion and d = 2/ (1 ± b). Making use of

eqs. (10) , (11) and (15) in eq. (8), we obtain

Çrr - n - ( 2 / d )+ 3C1 c r

1 - n - ( 1 / d )= 9g0C1

2, (16)

which yields a part icular solut ion for n = (1 + b)/ 2 as

r(t) =1

(a0 t)d

, (17)


where a0 = (3c C1 ± 9g0 C12)d - 1 and the constant of integrat ion has been

taken to be zero for simplicity. For realist ic models for which r is decreas-

ing, we require 0 < b < 1.

To ® nd R , we use eqs. (15) and (17) , and ® nd that

R (t) = C2 [t]1 / [( 3 c - 9g0 C 1 ) ( 1 - n ) ]

(18)

where C2 is a constant of integrat ion. Hence eqs. (11) and (13) yield

g(t) =g0

(a0 t)n d

(19)

and

G (t) =6C1

2

8pd[a0 t]2 - d . (20)

It should be noted that eqs. (17) ± (20) are the same as eqs. (23) ± (26) of

[41], the only diŒerence being that c has to be replaced by c ± 3g0 C1 .

The Hubble parameter is

H (t) =C1

a0 t, (21)

and from eq. (12) , the cosmological parameter becomes

L(t) =3bC1

2

a20 t2

. (22)

For the scale factor R (t) given by eq. (18) , the decelerat ion parameter

q = ± (R ÈR) / ÇR2 is given by

q =a0

C1± 1. (23)

This shows that the decelerat ion parameter is constant.

The horizon distance is given by

dH = R (t)

t 0

t

dt 9

R (t 9 ).

Using expression for R (t) from eq. (18) , we ® nd that horizons exit if

(3c ± 9g0 C1 )(n ± 1) ± 1

(3c ± 9g0C1 )(n ± 1)> 0

and do not exist if

(3c ± 9g0 C1 )(n ± 1) ± 1

(3c ± 9g0C1 )(n ± 1)< 0

or

(3c ± 9g0C1 ) (n ± 1) = 1.


4. DISCUSSION

In this paper we have reinvest igated Einstein’ s equat ions for the vis-

cous ¯ at frw universe within the framework of general relat ivity, where the

gravitational parameter G and the cosmological parameter L are variable,

while still retaining the usual energy conservat ion law. We have obtained

expressions for r, R , g, G and L as funct ions of t. Arbab [41] has discussed

some classes of solut ions for speci® c values of n and c . From our solut ion

(18) , we see that it is possible to generate all the solut ions that Arbab has

obtained.

From our result s we can conclude that r and g are decreasing with

time. From eq. (20) , we can express G in terms of n using d = 2/ (1 ± b)

and b = 2n ± 1. It turns out that G µ t( 2n - 1) / ( 1 - n ) . Since we require

that 0 < b < 1, it follows that12 < n < 1. Thus the gravitational pa-

rameter G turns out to be an increasing function of t ime. Our solut ions

represent power law behaviour as suggested by Berman [42]. It is also

found that the cosmological parameter L varies as the inverse square of

time, which matches its natural unit s. This support s the views in favour

of the dependence L~ t - 2 ® rst expressed by Bertolami [33,34] and later

by other authors [42± 44]. Beesham [43] has shown that only the require-

ment of a constant decelerat ion parameter is su� cient to ensure L~ t - 2 .

Berman, Som and Gomide [44] found this relat ion in Brans± Dicke static

models, Berman [42] and Bertolami [33,34] found it in Brans± Dicke the-

ory as modi® ed by Endo± Fukui. Kalligas, Wesson and Everitt [29] have

point ed out that if L varies as t - 2 , there is then no dimensional constant

associat ed with the L term in the ® eld equat ions. It is also compat ible

with Dirac’ s cosmology with varying cosmological constant [45].

Arbab [41] has claimed that G and L are allowed to vary in such a

way that energy is conserved. However, a closer inspect ion reveals that

energy is not really being conserved. To have energy conservat ion, we

must have Ta b;b = 0, where Ta b is the total energy-momentum tensor, i.e.,

Ta b = (r + Åp)u a ub ± Åpga b where Åp = p ± 3gH is the total pressure including

the bulk viscous stress. Arbab has taken Åp to be the equilibrium pressure

only. Hence energy is being transferred from the ¯ uid to the G and Lscalar ® elds. This does not appear to be physical. On the other hand, our

approach represents the least possible departure from Einstein’ s general

relat ivity while allowing for a possible variat ion of G and L.

ACKNOWLEDGEMENTS

The authors are grateful to the FRD (South Africa) for ® nancial sup-

port . TS would also like to thank the authorit ies of Shri RKNEC, Nagpur,


India for grant ing leave and the University of Zululand for facilit ies. AB

is grateful to Roy Maartens of Portsmouth University for interest ing dis-

cussions. The authors would like to express their grat itude to the referees

for their comments which led to an improvement in the manuscript and to

the correction of minor errors.

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Bulk Viscous Cosmological Models with Variable G and Λ

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