ABSTRACT We present a new Cosmological solution for a Bianchi type-I Cosmological model filled with viscous fluid in a modified Brans-Dicke theory in which the variable cosmological term is an explicit function of a scalar field. The physical and geometrical properties of this model have been discussed. Finally, this model has been transform to the original form (1961) of Bras-Dicke theory

G M Wali Ullah

Iftekhar Yousuf

Md Sayeed

Mohammed Ashraful Islam

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                                        VOL. 9, NO. 9, SEPTEMBER 2014                                                                                                               ISSN 1819-6608            
ARPN Journal of Engineering and Applied Sciences 
 
©2006-2014 Asian Research Publishing Network (ARPN). All rights reserved. 
 
www.arpnjournals.com 
 
 
1615
ANISOTROPIC BIANCHI TYPE-I COSMOLOGICAL MODEL FOR 
VISCOUS FLUID IN A MODIFIED BRANS-DICKE COSMOLOGY 
 
Mohammed Ashraful Islam1, G. M. Wali Ullah1 and Md. Sayeed Iftekhar Yousuf2 
1Department of Mathematics, University of Chittagong, Bangladesh 
2Department of Computer Science and Mathematics, Bangladesh Agricultural University, Mymensingh, Bangladesh 
E-Mail: islamashraful38@yahoo.com  
 
ABSTRACT 
We present a new Cosmological solution for a Bianchi type-I Cosmological model filled with viscous fluid in a 
modified Brans-Dicke theory in which the variable cosmological term is an explicit function of a scalar field. The physical 
and geometrical properties of this model have been discussed. Finally, this model has been transform to the original form 
(1961) of Bras-Dicke theory. 
 
Keywords: spatially homogeneous, anisotropic, bianchi type-I, cosmological model, brans-dicke theory. 
 
1. INTRODUCTION 
Einstein succeeded in geometrizing gravitation by 
expressing gravitational potential in terms of metric 
tensor. Various observations indicate that spatially 
homogeneous space-time is Bianchi type’s model and 
interpreted as cosmological models. Cosmological models 
with a cosmological constant are currently serious 
candidates to describe the dynamics of the universe. The 
significance of cosmological constant was studied by 
various cosmologists but no satisfactory results of its 
meaning have been supported as yet. Zel’dovich [1] has 
tried to visualize the meaning of this term from theory of 
elementary particles. Linde [2] argued that the 
cosmological term arises from spontaneous symmetry 
breaking and suggested that the term is not a constant but a 
function of temperature. In cosmology, the term may be 
understood by incorporation with Mach’s principle that 
suggests the acceptance of Brans-Dicke Lagrangian as 
realistic one [3]. Endo and Fukui [4] have also studied the 
variable cosmological term in  
Brans-Dicke theory [3] and elementary particle 
physics. Also Rai, Rai andSingh [5] have studied the 
variable cosmological term for viscous fluid in modified 
Brans-Dicke theory by imposing Petrov type -D condition. 
Astronomical observation for large scale 
distribution of galaxies indicates that the distribution of 
matter can be satisfactorily described by a perfect fluid. 
Also at the early stage of the universe when galaxies were 
formed, the matter distribution behaved like a viscous 
fluid [6]. So, in this paper we consider Bianchi type-I 
cosmological model for viscous fluid in a modified Brans-
Dicke theory in which the variable cosmological term  Q 
in an explicit function of a scalar field φ  proposed by 
Bergmann[7] and Wagonar [8], detailed discussed by both 
Endo and Fukui [4] and Rai, Rai and Singh [5].    
The Brans-Dicke field equations with 
cosmological term Q [5, 9] are:  
 
( )ααφφφφφφφφ
ω
φ
π ,
;;
,
,2
1
2
18
ijji
k
kijjiijijij ggTQgG −+⎟
⎠
⎞
⎜
⎝
⎛ −+=+ ,                                                             (1) 
 
( )32
8,
; +
=
ω
πµφ αα
T
,                    (2) 
 
( ) ( ) ( )TQ
φ
µπ
φ
φ
µ
µω αα
4
181
4
32 ,; −=−+=                          (3) 
 
where the constant µ   shows how much this theory 
including )(φQ deviates from that of Brans and Dicke  
and as usual ω  is coupling constant and ijT  is the energy-
momentum tensor for a viscous fluid 
distribution[9].Covariant derivative  with respect to the 
metric ijg  is denoted by semicolons and partial 
differentiation with respect to the coordinate 
ix  is 
denoted by comas. Then under the conformal 
transformation: 
 
ijijij ggg φ=→                     (4) 
 
Equations (1) – (3) becomes (5) 
 
( ) ⎟
⎠
⎞
⎜
⎝
⎛ ΛΛ−ΛΛ++=+ kkijjiijijij gTQgG
,
,,, 2
132
2
18 ωπ ,    (5) 
 
( ) ,log,32
8,
; φω
πµα
α =Λ+
=Λ
T
                   (6) 
 
                                        VOL. 9, NO. 9, SEPTEMBER 2014                                                                                                               ISSN 1819-6608            
ARPN Journal of Engineering and Applied Sciences 
 
©2006-2014 Asian Research Publishing Network (ARPN). All rights reserved. 
 
www.arpnjournals.com 
 
 
1616
( ) ( ) ( )TQ
φ
µπ
µ
µω α
α 4
181
4
32 ,
;
−
=Λ
−+
=
   
                  (7) 
 
where all barred and unbarred quantities are defined in 
terms of metric ijg  and ijg  respectively. 
In this paper, in sect. 2 we considered Bianchi 
type-I metric and energy- momentum tensor for viscous 
fluid [5, 9]. In sect.3 we obtained the solution of the field 
equations for the Bianchi type I metric. In sect.4 we 
obtained expressions for pressure, density and 
cosmological term for spatially homogeneous and 
anisotropic cosmological model. Finally in sect.5 we have 
transformed this model to the 1961 form of Brans-Dicke 
theory. 
 
2. FIELD EQUATIONS  
The Bianchi type-I metric is considered as 
 
22222222 dzCdyBdxAdtds +++−=          (8) 
 where ,,, CBA are functions of tx =4  only. The 
energy-momentum tensor ijT  
for a viscous fluid distribution is given by [5,9] 
 
( ) ( )
( )kikiaa
k
a
a
iai
akk
i
k
i
k
i
k
i
k
i
vvgv
vvvvvvvvgpvvpT
+⎟
⎠
⎞
⎜
⎝
⎛ −−
+++−++=
;
;;;
;
3
2ηξ
ηε
      (9) 
 
along with 
 
1−=jiij vvg                    (10) 
 
where p is the isotropic pressure, ε the density, η and ξ  
the two coefficient of viscosity and semicolons defined 
covariant differentiation. iv is the components of the fluid 
four-velocity satisfying equation (10). We assume that the 
coordinate to be co-moving i.e. 0321 === vvv  
and 14 =v . The scalar field Λ  is also taken to be a 
function of t only. The field equations (5) and (6) for the 
line element (8) turn into 
 
( ) 2
; 4
32
3
228 Λ++
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
⎟
⎠
⎞
⎜
⎝
⎛ −−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−=+⎥
⎦
⎤
⎢
⎣
⎡
++ &
&&&&&&& ωηξηπ aavA
ApQ
BC
CB
C
C
B
B                                                     (11) 
 
( ) 2
; 4
32
3
228 Λ++
⎭
⎬
⎫
⎩
⎨
⎧
⎟
⎠
⎞
⎜
⎝
⎛ −−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−=+⎥
⎦
⎤
⎢
⎣
⎡
++ &
&&&&&&& ωηξηπ aavB
BpQ
AC
CA
C
C
A
A                                                     (12) 
 
( ) 2
; 4
32
3
228 Λ++
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
⎟
⎠
⎞
⎜
⎝
⎛ −−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−=+⎥
⎦
⎤
⎢
⎣
⎡
++ &
&&&&&&& ωηξηπ aavC
CpQ
AB
BA
B
B
A
A                                                     (13) 
 
( ) 2
4
328 Λ+−−=+⎥
⎦
⎤
⎢
⎣
⎡
++ &
&&&&&& ωεπQ
BC
CB
AC
CA
AB
BA                                                                                           (14) 
 
( ) ( ) ( )⎥⎦
⎤
⎢⎣
⎡ −−
+
−=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
++Λ+Λ ABC
dt
dp
C
C
B
B
A
A log33
32
8 ξε
ω
πµ&&&&&&                                                    (15) 
 
where dot represent differentiation with respect to t. The 
coefficients of viscosity η and ξ are taken as constants. 
For complete determinacy of the system one extra 
condition is needed, which we will impose later. 
 
3. SOLUTION OF FIELD EQUATIONS 
The equation (11)-(15) are five equations in six 
unknowns pCBA ,,,, ε  andΛ . 
 Let we set L=πη16 . 
 Equation (11)-(12) yields    
 
( ) LteCBABAm =− &&  , =m constant                 (16) 
Equation (12)-(13) yields 
 
( ) keABCCBk Lt ,=− && =constant                  (17) 
 
Equation (11)-(13) yields     
 
( ) αα ,LteBACCA =− && =constant                (18) 
 
To avoid the complicacy, we assume 
 
ttt ethCetgBetfA ρνµ )(,)(,)( ===   
 
                                        VOL. 9, NO. 9, SEPTEMBER 2014                                                                                                               ISSN 1819-6608            
ARPN Journal of Engineering and Applied Sciences 
 
©2006-2014 Asian Research Publishing Network (ARPN). All rights reserved. 
 
www.arpnjournals.com 
 
 
1617
By using above, equation (16) gives two conditions 
 
s
f
f
g
g
mfgh
=+−=−
&&1µν
 
(say) (i), L=++ ρνµ       (ii) 
 
where sm,  are constants. 
Similarly equation (17) and (18) give extra 
conditions except (ii) are respectively 
 
b
rfghh
h
g
g
=+−=−
1&&νρ ,                    (iii) 
 
and 
 
c
f
f
h
h
fgh
=+−=−
&&
α
µρ 1 ,                    (iv) 
 
where cb, are constants. 
 
Consider ∫ = )(log
1 tKdt
fgh
. Then equation (i), (iii) and 
(iv) yield respectively 
 
stm gefK =
1
(v), btkr hegK =
1
 (vi) and 
cthefK =α
1
  (vii) 
 
Using the above (v), (vi) and (vii), we obtained 
the relation 
 
( )tscmkrkrbt heKgKhe −
−+
== α
1111
. 
 
Comparing with (vi) we will get  
 
1111 =−+
αmk
,⇒
mkkm
mk
−+
=α  (viii) and scb −=       (ix) 
Again from (v) and (vi), we obtained 
 
( )tscmk
mk
tsbmk
mk
efghKgorefhKg 23)(2 −
−
−
−
== , 
(by using (ix))                                              (19) 
 
Now, we impose the condition 
 
( )γγ
1
)( lttK += , which gives ltfgh += γ .           (x) 
 
Equation (19) and (x) gives 
 
( )
( ) ( )tsc
km
mmk
eltg 3
2
3
1 −−+
+= γ
γ
γ                      (20) 
 
Using (20), equation (v) and (vii) give, respectively 
 
( )
( ) ( )tsc
km
mmk
eltf 33
2 +−−
+= γ
γ
γ                 (21) 
 
and 
 
( )
( ) ( )tcs
km
mmk
elth 3
2
3
21 −++
+= γ
γ
γ                 (22) 
 
Again, using equations (i), (ii), (iii), (iv) and (ix), 
we obtained 
 
sLccLsscL −+
=
−+
=
−−
=
2,
3
2,
3
ρνµ
  
(xi) 
 
where πη16=L  
Hence, the required solution is 
 
 
( )
( )
( )
( )
( )
( )
23
32
3
412
23
32
3
212
23
32
3
222
22
dzelt
dyeltdxeltdtds
t
km
mmk
t
km
mmkt
km
mmk
πη
γ
γ
πη
γ
γπη
γ
γ
γ
γγ
++
−+−−
++
++++−=
                  (23) 
 
By the following transformation of coordinate ( ) tlt →+γ  we get 
 
( ) ( ) ( )
⎭
⎬
⎫
⎩
⎨
⎧
+++−=
++−+−−
23
412
23
212
23
222
3
32
22 dztdytdxtedtds km
mmk
km
mmk
km
mmktπη
                  (24) 
 
                                        VOL. 9, NO. 9, SEPTEMBER 2014                                                                                                               ISSN 1819-6608            
ARPN Journal of Engineering and Applied Sciences 
 
©2006-2014 Asian Research Publishing Network (ARPN). All rights reserved. 
 
www.arpnjournals.com 
 
 
1618
4. SOME PHYSICAL AND GEOMETRICAL 
PROPERTIES 
The pressure and density in the model (24) are 
given by 
 
( )
( )
( )
( ) Qt
km
kmmkkm
tkm
kmmkmk
ttkm
kmmkkm
t
p
+
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
+
−−−−−−
−
⎟
⎠
⎞
⎜
⎝
⎛ ++
−−−
++=
β
ω
πηπξπηηππ
log
323
sec
3
1168
99
128
9
10248
22
2222
2
222
222
222
222222
  (25) 
 
( ) ( )
( ) ( )
Q
t
t
km
kmmkkm
tkm
kmmkkm
−−−
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
+
−−−−−−
−=
3
32
3
256
log
323
sec
9
8
22
22
2222
2
222
2222
πηηπ
β
ω
επ   (26) 
 
and 
 
( ) ( )
( ) ( )
( )( ) ( )
( ) ( )
( )
4
324116
3
70
3
40
log
323
tan
3
32
log
323
sec
124
11
22
22
2222
22
2222
22
2222
2
222
2222
ξπξπηηππη
β
ω
ω
µ
η
β
ωπµ
−
⎟
⎠
⎞
⎜
⎝
⎛ +−−−
+
−−−+−−−
−
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
+
−−−−−−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−=
tt
t
km
kmmkkm
km
kmmkkm
t
t
km
kmmkkm
tkm
kmmkkmQ   (27) 
 
Also the scalar field Λ is given by 
 
( )
( ) ( )⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
+
−−−
=Λ t
km
kmmkkm β
ω
log
323
seclog 22
2222
2    (28) 
 
The model will be real if the conditions 
pp 3,0,0 fff εε hold when 
 
( ) 1,0,
2
3,02222 ≥−−−− µω ppp Qkmmkkm  (29) 
 
The average scale factor )(tR is given by 
 
( ) 31ABCR = .                   (30) 
 
The Hubble parameter H  is given 
by RRH &= .Volume expansionθ , deceleration 
parameter q and shear σ for the metric (1.1) can be 
written as 
 
t
tH
3
116 +
=
πη ,                                  (31) 
 
showing that Hubble parameter is decreasing during time 
evolution and become constant when 
→∝t and 14 −≤≤− q .  
 
R
RH
&
33 ==θ =
t
116 +πη ,    (32) 
 
2
1
tdt
d
−=
θ ,                  32(a) 
showing that the rate of volume expansion decreases 
during time evolution. 
The non-zero components of shear tensor ijσ  are 
 
)116(
3
2
3
22
3
1
3
2
3
1
3
22
3
1
4
4
3
3
2
2
1
1
t
mkt
mk
B
B
A
A
C
C
mkt
mk
C
C
A
A
B
B
mkt
km
C
C
B
B
A
A
+−=
+
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−−=
−
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−−=
+
−=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−−=
πησ
σ
σ
σ
&&&
&&&
&&&
            (33) 
 
and the shear σ  is 
 
( ) ( ) ( ) ( ){ }
( ) 2222222
4
4
23
3
22
2
21
1
2
)116(
9
2
3
2
2
1
t
kkmm
tkm
++++=
+++=
πη
σσσσσ
        (34) 
 
showing that the shearσ decreases during the time 
evolution. 
The deceleration parameter is 
 
( ) ⎭
⎬
⎫
⎩
⎨
⎧
+
+−=−= 22 116
31
tRH
Rq
πη
&&
                (35) 
 
showing that deceleration parameter increases during time 
evolution and become -1 when →∝t  and 14 −≤≤− q  
 
5. TRANSFORMATIONS OF SOLUTIONS AND  
     DISCUSSIONS 
Under the transformation given by [5]: 
 
⎪
⎪
⎪
⎭
⎪⎪
⎪
⎬
⎫
=→=→
=→=→
=→=→
=→=→
Λ
iii
ijijijijijij
vvvQQQ
e
pppTTT
TTTggg
21
2
22
,
,
,
,1
φφ
φφεφεε
φφ
φ
φ
                (36) 
 
The field equations (5)-(7) are changed into (1)-
(3). 
 
( )
( ) ( )⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
+
−−−
= t
km
kmmkkm β
ω
φ log
323
sec 22
2222
2
     
37(a) 
 
                                        VOL. 9, NO. 9, SEPTEMBER 2014                                                                                                               ISSN 1819-6608            
ARPN Journal of Engineering and Applied Sciences 
 
©2006-2014 Asian Research Publishing Network (ARPN). All rights reserved. 
 
www.arpnjournals.com 
 
 
1619
( )
( ) ( ) ijij gtkm
kmmkkmg
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
+
−−−
= β
ω
log
323
cos 22
2222
2     37(b) 
i.e. 
( )
( ) ( )
( )
3
32
3
22
22
2222
2
11 log323
cos
t
mk
kmmk
ett
km
kmmkkmg
πη
β
ω
−−
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
+
−−−
=  
 
( )
( ) ( )
3
32
3
)(2
22
2222
2
22 log323
cos
t
mk
mkmk
ett
km
kmmkkmg
πη
β
ω
+−
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
+
−−−
=   
 
( )
( ) ( )
( )
3
32
3
22
22
2222
2
33 log323
cos
t
mk
mmkk
ett
km
kmmkkmg
πη
β
ω
++
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
+
−−−
=  
 
( )
( ) ( )⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
+
−−−
−= t
km
kmmkkmg β
ω
log
323
cos 22
2222
2
44  
 
( ) ( )
( ) ( )
( )( )
( ) ( ) )13.4(log323sec48
112
log
323
tan
3
)32)((2
4
3116
99
64
9
8log
323
sec8
22
2222
6
422
2222
22
2222
22
2222
22
222222
22
2222
4
ct
km
kmmkkm
tkm
kmmkkm
t
km
kmmkkm
km
kmmkkm
tt
km
kmmkkm
t
t
km
kmmkkmp
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
+
−−−−−−+
−
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
+
−−−+−−
−⎟
⎠
⎞
⎜
⎝
⎛ +⎟
⎠
⎞
⎜
⎝
⎛ ++
⎢
⎣
⎡ −−−
++
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
+
−−−
=
β
ωµπ
πµ
β
ω
ω
µ
ηπξξπη
ηππηβ
ω
π
 
 
( )( ) ( )
( ) ( )
( ) ( )
( ) ( ) ⎥⎦
⎤
⎟
⎠
⎞
⎜
⎝
⎛ −⎟
⎠
⎞
⎜
⎝
⎛ +++−
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
+
−−−
×
+−−−
⎢
⎣
⎡
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
+
−−−
+
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
+
−−−−−−−
−=
ξπξπηπηµπβ
ω
ω
µ
ηβ
ω
β
ωµπ
πµεπ
4
36116
3
8
3
64log
323
tan
3
)32)((log
323
sec
log
323
sec
48
1208
22
22
222
22
2222
22
2222
4
22
2222
6
222
2222
tt
t
km
kmmkkm
km
kmmkkm
t
t
km
kmmkkm
t
km
kmmkkm
tkm
kmmkkm
                              37(d) 
 
( ) ( )
( ) ( )
( ) ( )
( )( )
( )
( ) ( ) ⎥⎦
⎤
⎟
⎠
⎞
⎜
⎝
⎛ −⎟
⎠
⎞
⎜
⎝
⎛ −−+−
+
−−−
×
+−−−
⎢
⎣
⎡
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
+
−−
−
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
+
−−−−−−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −
=
ξπξπηηππηβ
ω
ω
µ
ηβ
ω
β
ωπµ
πµ
4
36116
3
640
3
40log
323
tan
3
32log
323
sec
log
323
sec
316
14
22
22
2222
22
2222
22
2222
2
22
2222
4
222
2222
tt
t
km
kmmkkm
km
kmmkkm
t
t
km
kmmkkm
t
km
kmmkkm
tkm
kmmkkmQ
                               
37(e) 
 
                                        VOL. 9, NO. 9, SEPTEMBER 2014                                                                                                               ISSN 1819-6608            
ARPN Journal of Engineering and Applied Sciences 
 
©2006-2014 Asian Research Publishing Network (ARPN). All rights reserved. 
 
www.arpnjournals.com 
 
 
1620
,0321 === vvv ( )( ) ( )⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
+
−−−
= t
km
kmmkkmv β
ω
log
323
sec 22
2222
4
                         
                           37(f) 
The reality conditions (29) may be imposed on 
the solutions in 37(a)-37(f). The solution obtained in this 
paper is new. Both pressure p and density ε  are positive 
and decreases during the time evolution. The cosmological 
term Q  is negative which differ from Rai,Rai and Singh 
[5] solution. The viscosity prevents the free gravitational 
field. Equation (32) indicates that the effect of viscosity is 
to retard the expansion of the model. 
 
REFERENCES 
 
[1] Zel’dovich, Ya. B.: Sov. phys. Usp.11, 381 (1968). 
 
[2] Linde, A.D.: JETP Lett. 19, 183 (1974). 
 
[3] Brans, C.H., Dicke, R.H.: Phys. Rev. 124, 925 (1961). 
 
[4] Endo, M., Fukui, T: Gen. Relativ. Gravit. 8, 
833(1977). 
 
[5] Rai, P., Rai, L. N., Singh, V. K.: Int. Theor. Phys.51, 
2127(2012). 
 
[6] Ellis, G.F. R.: In: Sachs, R.K. (ed.) General Relativity 
and Cosmology. Academic Press, New York and 
London (1971). 
 
[7] Bergmann, P.G.: Int.J.Theor.Phys.1, 25 (1968). 
 
[8] Wagoner, R.V.: Phys. Rev. D 1, 3209 (1970). 
 
[9] Landau, L.D., Lifshitz, E.M.: Fluid Mechanics. 
Pergamon, Oxford (1963). 


ANISOTROPIC BIANCHI TYPE-I COSMOLOGICAL MODEL FOR VISCOUS FLUID IN A MODIFIED BRANS-DICKE COSMOLOGY

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ANISOTROPIC BIANCHI TYPE-I COSMOLOGICAL MODEL FOR VISCOUS FLUID IN A MODIFIED BRANS-DICKE COSMOLOGY

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