LRS Bianchi Type-V Viscous Fluid Universe with a Time
Dependent Cosmological Term Λ
Anirudh Pradhan a,1, J. P. Shahi b and Chandra Bhan Singh c
aDepartment of Mathematics, Hindu Post-graduate College, Zamania-232 331,
Ghazipur, India
E-Addresses: pradhan@iucaa.ernet.in, acpradhan@yahoo.com
b,c Department of Mathematics, Harish Chandra Post-graduate College,
Varanasi, India
Abstract
An LRS Bianchi type-V cosmological models representing a viscous
fluid distribution with a time dependent cosmological term Λ is inves-
tigated. To get a determinate solution, the viscosity coefficient of bulk
viscous fluid is assumed to be a power function of mass density. It turns
out that the cosmological term Λ(t) is a decreasing function of time, which
is consistent with recent observations of type Ia supernovae. Various phys-
ical and kinematic features of these models have also been explored.
PACS number: 98.80.Es, 98.80.-k
Key words: cosmology, variable cosmological constant, viscous universe
1 Introduction
Cosmological models representing the early stages of the Universe have been
studied by several authors. An LRS (Locally Rotationally Symmetric) Binachi
type-V spatially homogeneous space-time creates more interest due to its richer
structure both physically and geometrically than the standard perfect fluid FRW
models. An LRS Bianchi type-V universe is a simple generalization of the
Robertson-Walker metric with negative curvature. Most cosmological models
assume that the matter in the universe can be described by ’dust’ (a pressure-
less distribution) or at best a perfect fluid. However, bulk viscosity is expected
to play an important role at certain stages of expanding universe [1]−[3]. It has
been shown that bulk viscosity leads to inflationary like solution [4] and acts
like a negative energy field in an expanding universe [5]. Furthermore, there
are several processes which are expected to give rise to viscous effects. These
are the decoupling of neutrinos during the radiation era and the decoupling of
1Corresponding Author
http://arxiv.org/abs/0704.0849v2
radiation and matter during the recombination era. Bulk viscosity is associ-
ated with the Grand Unification Theories (GUT) phase transition and string
creation. Thus, we should consider the presence of a material distribution other
than a perfect fluid to have realistic cosmological models (see Grøn [6] for a
review on cosmological models with bulk viscosity). A number of authors have
discussed cosmological solutions with bulk viscosity in various context [7]−[9].
Models with a relic cosmological constant Λ have received considerable at-
tention recently among researchers for various reasons (see Refs.[10]−[14] and
references therein). Some of the recent discussions on the cosmological constant
“problem” and consequence on cosmology with a time-varying cosmological con-
stant by Ratra and Peebles [15], Dolgov [16]−[18] and Sahni and Starobinsky
[19] have pointed out that in the absence of any interaction with matter or radi-
ation, the cosmological constant remains a “constant”. However, in the presence
of interactions with matter or radiation, a solution of Einstein equations and
the assumed equation of covariant conservation of stress-energy with a time-
varying Λ can be found. For these solutions, conservation of energy requires
decrease in the energy density of the vacuum component to be compensated by
a corresponding increase in the energy density of matter or radiation. Earlier
researchers on this topic, are contained in Zeldovich [20], Weinberg [11] and
Carroll, Press and Turner [21]. Recent observations by Perlmutter et al. [22]
and Riess et al. [23] strongly favour a significant and positive value of Λ. Their
finding arise from the study of more than 50 type Ia supernovae with redshifts
in the range 0.10 ≤ z ≤ 0.83 and these suggest Friedmann models with negative
pressure matter such as a cosmological constant (Λ), domain walls or cosmic
strings (Vilenkin [24], Garnavich et al. [25]) Recently, Carmeli and Kuzmenko
[26] have shown that the cosmological relativistic theory (Behar and Carmeli
[27]) predicts the value for cosmological constant Λ = 1.934 × 10−35s−2. This
value of “Λ” is in excellent agreement with the measurements recently obtained
by the High-Z Supernova Team and Supernova Cosmological Project (Garnavich
et al. [25], Perlmutter et al. [22], Riess et al. [23], Schmidt et al. [28]). The
main conclusion of these observations is that the expansion of the universe is
accelerating.
Several ansätz have been proposed in which the Λ term decays with time
(see Refs. Gasperini [29, 30], Berman [31], Freese et al. [14], Özer and Taha [14],
Peebles and Ratra [32], Chen and Hu [33], Abdussattar and Viswakarma [34],
Gariel and Le Denmat [35], Pradhan et al. [36]). Of the special interest is the
ansätz Λ ∝ S−2 (where S is the scale factor of the Robertson-Walker metric)
by Chen and Wu [33], which has been considered/modified by several authors
( Abdel-Rahaman [37], Carvalho et al. [14], Waga [38], Silveira and Waga [39],
Vishwakarma [40]).
Recently Bali and Yadav [41] obtained an LRS Bianchi type-V viscous fluid
cosmological models in general relativity. Motivated by the situations discussed
above, in this paper, we focus upon the exact solutions of Einstein’s field equa-
tions in presence of a bulk viscous fluid in an expanding universe. We do this
by extending the work of Bali and Yadav [41] by including a time dependent
cosmological term Λ in the field equations. We have also assumed the coefficient
of bulk viscosity to be a power function of mass density. This paper is organized
as follows. The metric and the field equations are presented in section 2. In
section 3 we deal with the solution of the field equations in presence of viscous
fluid. The sections 3.1 and 3.2 contain the two different cases and also con-
tain some physical aspects of these models respectively. Section 4 describe two
models under suitable transformations. Finally in section 5 concluding remarks
have been given.
2 The Metric and Field Euations
We consider LRS Bianchi type-V metric in the form
ds2 = −dt2 +A2dx2 +B2e2x(dy2 + dz2), (1)
where A and B are functions of t alone.
The Einstein’s field equations (in gravitational units c = 1, G = 1) read as
i + Λg
i = −8πT
i , (2)
where R
i is the Ricci tensor; R = g
ijRij is the Ricci scalar; and T
i is the stress
energy-tensor in the presence of bulk stress given by
i = (ρ+ p)viv
j + pg
i − (v
i; + v
;i + v
jvℓvi;ℓ + viv
ξ − 2
vℓ;ℓ(g
i + viv
j). (3)
Here ρ, p, η and ξ are the energy density, isotropic pressure, coefficients of
shear viscosity and bulk viscous coefficient respectively and vi the flow vector
satisfying the relations
ivj = −1. (4)
The semicolon (; ) indicates covariant differentiation. We choose the coordinates
to be comoving, so that vi = δi4.
The Einstein’s field equations (2) for the line element (1) has been set up as
= −8π
p− 2ηA4
ξ − 2
− Λ, (5)
= −8π
p− 2η
− Λ, (6)
2A4B4
= −8πρ− Λ, (7)
= 0. (8)
The suffix 4 after the symbols A, B denotes ordinary differentiation with respect
to t and
θ = vℓ;ℓ
3 Solutions of the Field Eqations
In this section, we have revisited the solutions obtained by Bali and Yadav [41].
Equations (5) - (8) are four independent equations in seven unknowns A, B, p,
ρ, ξ, η and Λ. For complete determinacy of the system, we need three extra
conditions.
Eq. (8), after integration, reduce to
A = Bk, (9)
where k is an integrating constant. Equations (5) and (6) lead to
− A44
− A4B4
= −16πη
. (10)
Using Eq. (9) in (10), we obtain
k + 1
f = −16πη, (11)
where B4 = f(B). Eq. (11) leads to
f = − 16πη
(k + 2)
, (12)
where L is an integrating constant. Eq. (12) again leads to
B = (k + 2)
k1 − k2e−16πηt
k+2 , (13)
where
, (14)
, (15)
N being constant of integration. From Eqs. (9) and (13), we obtain
A = (k + 2)
k1 − k2e−16πηt
k+2 . (16)
Hence the metric (1) reduces to the form
ds2 = −dt2 + (k + 2)
k1 − k2e−16πηt
k+2 dx2
+ e2x(k + 2)
k1 − k2e−16πηt
k+2 (dy2 + dz2). (17)
The pressure and density of the model (17) are obtained as
8πp =
(8π)(16πη)k2e
−16πηt
3(k + 2)2(k1 − k2e−16πηt)2
k1(k + 2)
2(4η + 3ξ)− {k2(4η + 3ξ)
+4k(η+3ξ)+2(5η+6ξ)}k2e−16πηt
[(k + 2)(k1 − k2e−16πηt)]
−Λ, (18)
8πρ = − (2k + 1)
(k + 2)2
(16πη)2k22
e−32πηt
(k1 − k2e−16πηt)2
[(k + 2)(k1 − k2e−16πηt)]
+ Λ. (19)
The expansion θ in the model (17) is obtained as
(16πη)k2e
−16πηt
(k1 − k2e−16πηt)
. (20)
For complete determinacy of the system we have to consider three extra condi-
tions. Firstly we assume that the coefficient of shear viscosity is constant, i.e.,
η = η0 (say). For the specification of Λ(t), we secondly assume that the fluid
obeys an equation of state of the form
p = γρ, (21)
where γ(0 ≤ γ ≤ 1) is a constant.
Thirdly bulk viscosity (ξ) is assumed to be a simple power function of the
energy density [42]−[45].
ξ(t) = ξ0ρ
n, (22)
where ξ0 and n are constants. For small density, n may even be equal to unity
as used in Murphy’s work [46] for simplicity. If n = 1, Eq. (22) may correspond
to a radiative fluid [47]. Near the big bang, 0 ≤ n ≤ 1
is a more appropriate
assumption [48] to obtain realistic models.
For simplicity and realistic models of physical importance, we consider the
following two cases (n = 0, 1):
3.1 Model I: Solution for n = 0
When n = 0, Eq. (22) reduces to ξ = ξ0 = constant. Hence, in this case Eqs.
(18) and (19), with the use of (21), lead to
8π(1 + γ)ρ =
k1(k + 2)
2(4η0 + 3ξ0)− {k2(4η0 + 3ξ0)
+ 4k(η0 + 3ξ0) + 2(5η0 + 6ξ0)}k2e−16πη0t
− (2k + 1)M
. (23)
Eliminating ρ(t) between Eqs. (19) and (23), we obtain
(1 + γ)Λ =
k1(k + 2)
2(4η0 + 3ξ0)− {k2(4η0 + 3ξ0)
+ 4k(η0 + 3ξ0) + 2(5η0 + 6ξ0)}k2e−16πη0t
+ (2k + 1)γ
(1− 3γ)
, (24)
where
M = 16πk2η0e
−16πη0t,
N = (k + 2)(k1 − k2e−16πη0t),
P = 2k2 + 2k + 5,
Q = k2 + 4k + 4. (25)
3.2 Model II: Solution for n = 1
When n = 1, Eq. (22) reduces to ξ = ξ0ρ . Hence, in this case Eqs. (18) and
(19), with the use of (21), leads to
8πρ =
16πM{2k1(k + 2)2η0 − Pk2η0e−16πη0t}
3 [(1 + γ)N2 −M{k1(k + 2)2ξ0 −Qk2ξ0e−16πη0t}]
k+2 − (2k + 1)M2
[(1 + γ)N2 −M{k1(k + 2)2ξ0 −Qk2ξ0e−16πη0t}]
. (26)
Eliminating ρ(t) between Eqs. (19) and (26), we get
Λ = 16πM [2k1(k + 2)
2η0 − Pk2η0e−16πη0t] +
γ(2k + 1)
(1 + γ)
(1− 3γ)
(1 + γ)N
M [k1(k + 2)
2ξ0 −Qk2ξ0e−16πη0t]{4N
k+2 − (2k + 1)M2}
(1 + γ)N2 [(1 + γ)N2 −M{k1(k + 2)2ξ0 −Qk2ξ0e−16πη0t}]
From Eqs. (23) and (26), we note that ρ(t) is a decreasing function of time
and ρ > 0 for all time in both models. The behaviour of the universe in these
models will be determined by the cosmological term Λ; this term has the same
effect as a uniform mass density ρeff = −Λ/4πG, which is constant in space
and time. A positive value of Λ corresponds to a negative effective mass density
(repulsion). Hence, we expect that in the universe with a positive value of Λ,
the expansion will tend to accelerate; whereas in the universe with negative
value of Λ, the expansion will slow down, stop and reverse. From Eqs. (24) and
(27), we observe that the cosmological term Λ in both models is a decreasing
function of time and it approaches a small positive value as time increase more
and more. This is a good agreement with recent observations of supernovae Ia
(Garnavich et al. [25], Perlmutter et al. [22], Riess et al. [23], Schmidt et al.
[28]).
The shear σ in the model (17) is given by
(k − 1)M√
. (28)
The non-vanishing components of conformal curvature tensor are given by
C2323 = −C1414 =
(k − 1)M
[kM − 16πη0k1(k + 2)], (29)
C1313 = −C2424 =
(k − 1)M
[16πη0k1(k + 2)− kM ], (30)
C1212 = −C3434 =
(k − 1)M
[16πη0k1(k + 2)− kM ]. (31)
Equations (20) and (28) lead to
(k − 1)√
3(k + 2)
= constant. (32)
The model (17) is expanding, non-rotating and shearing. Since σ
= conatant,
hence the model does not approach isotropy. The space-time (17) is Petrov type
D in presence of viscosity.
4 Other Models
After using the transformation
k1 − k2e−16πηt = sin (16πητ), k + 2 = 1/16πη, (33)
the metric (17) reduces to
ds2 = −
cos (16πητ)
k1 − sin (16πητ)
dτ2 +
sin (16πητ)
]2(1−32πη)
+ e2x
sin (16πητ)
](32πη)
(dy2 + dz2). (34)
The pressure (p), density (ρ) and the expansion (θ) of the model (34) are ob-
tained as
8πp =
(16πη)2{k1 − sin (16πητ)
3 sin2 (16πητ)
2k1−2(1−48πη+1152π2η2){k1−sin (16πητ)}
(16πη)(8πξ){k1 − sin (16πητ)}
sin (16πητ)
sin (16πητ)
]2(1−32πη)
− Λ, (35)
8πρ =
2(24πη − 1)(16πη)3{k1 − sin (16πητ)}2
sin2 (16πητ)
sin (16πητ)
]2(1−32πη)
(16πη){k1 − sin (16πητ)}
sin (16πητ)
. (37)
4.1 Model I: Solution for n = 0
When n = 0, Eq. (22) reduces to ξ = ξ0 = constant. Hence, in this case Eqs.
(35) and (36), with the use of (21), lead to
8π(1 + γ)ρ =
2(16πη0)
3 sin2(16πη0τ)
[k1 − P1M1] +
(16πη0)(8πξ0)M1
sin(16πη0τ)
+ 4N1 +
2(24πη0 − 1)(16πη0)3M21
sin2(16πη0τ)
. (38)
Eliminating ρ(t) between Eqs. (36) and (38), we obtain
(1 + γ)Λ =
2(16πη0)
3 sin2(16πη0τ)
[k1 − P1M1] +
(16πη0)(8πξ0)M1
sin(16πη0τ)
+ (1− 3γ)N1 +
2γ(24πη0 − 1)(16πη0)3M21
sin2(16πη0τ)
. (39)
4.2 Model II: Solution for n = 1
When n = 1, Eq. (22) reduces to ξ = ξ0ρ . Hence, in this case Eqs. (35) and
(36), with the use of (21), lead to
8πρ =
2(16πη0)
2M1[(k1 − P1M1) + 3(24πη0 − 1)(16πη0)M1]
3 sin(16πη0τ)[(1 + γ) sin(16πη0τ) − 16πη0ξ0M1]
4N1 sin(16πη0τ)
[(1 + γ) sin(16πη0τ) − 16πη0ξ0M1]
. (40)
Eliminating ρ(t) between Eqs. (36) and (40), we obtain
2(16πη0)
2M1(k1 − P1M1)
3 sin(16πη0τ)[(1 + γ) sin(16πη0τ)− 16πη0ξ0M1]
[3(16πη0ξ0)M1 + (1− 3γ) sin(16πη0τ)]
[(1 + γ) sin(16πη0τ)− 16πη0ξ0M1]
2(24πη0 − 1)(16πη0)3M21 [γ(1 + γ) sin(16πη0τ) − (1− γ)(16πη0ξ0)M1]
(1 + γ) sin2(16πη0τ)[(1 + γ) sin(16πη0τ)− (16πη0ξ0)M1]
, (41)
where
M1 = k1 − sin(16πη0τ),
16πη0
sin (16πη0τ)
]2(1−32πη)
P1 = 1− 48πη0 + 1152π2η20 . (42)
The shear (σ) in the model (34) is obtained as
(1− 48πη0)(16πη0)[k1 − sin(16πη0τ)√
3 sin(16πη0τ)
. (43)
The models descibed in cases 4.1 and 4.2 preserve the same properties as in the
cases of 3.1 and 3.2.
5 Conclusions
We have obtained a new class of LRS Bianchi type-V cosmological models of
the universe in presence of a viscous fluid distribution with a time dependent
cosmological term Λ. We have revisited the solutions obtained by Bali and Ya-
dav [41] and obtained new solutions which also generalize their work.
The cosmological constant is a parameter describing the energy density of
the vacuum (empty space), and a potentially important contribution to the dy-
namical history of the universe. The physical interpretation of the cosmological
constant as vacuum energy is supported by the existence of the “zero point”
energy predicted by quantum mechanics. In quantum mechanics, particle and
antiparticle pairs are consistently being created out of the vacuum. Even though
these particles exist for only a short amount of time before annihilating each
other they do give the vacuum a non-zero potential energy. In general relativity,
all forms of energy should gravitate, including the energy of vacuum, hence the
cosmological constant. A negative cosmological constant adds to the attractive
gravity of matter, therefore universes with a negative cosmological constant are
invariably doomed to re-collapse [49]. A positive cosmological constant resists
the attractive gravity of matter due to its negative pressure. For most universes,
the positive cosmological constant eventually dominates over the attraction of
matter and drives the universe to expand exponentially [50].
The cosmological constants in all models given in Sections 3.1 and 3.2 are
decreasing functions of time and they all approach a small and positive value
at late times which are supported by the results from recent type Ia supernova
observations recently obtained by the High-z Supernova Team and Supernova
Cosmological Project (Garnavich et al. [25], Perlmutter et al. [22], Riess et al.
[23], Schmidt et al. [28]). Thus, with our approach, we obtain a physically rele-
vant decay law for the cosmological term unlike other investigators where adhoc
laws were used to arrive at a mathematical expressions for the decaying vacuum
energy. Our derived models provide a good agreement with the observational
results. We have derived value for the cosmological constant Λ and attempted
to formulate a physical interpretation for it.
Acknowledgements
The authors wish to thank the Harish-Chandra Research Institute, Allahabad,
India, for providing facility where part this work was done. We also thank to
Professor Raj Bali for his fruitful suggestions and comments in the first draft of
the paper.
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	Introduction
	The Metric and Field Euations
	Solutions of the Field Eqations
	Model I:       Solution for n = 0
	Model II:       Solution for n = 1
	Other Models
	Model I:       Solution for n = 0
	Model II:       Solution for n = 1
	Conclusions

  An LRS Bianchi type-V cosmological models representing a viscous fluid
distribution with a time dependent cosmological term $\Lambda$ is investigated.
To get a determinate solution, the viscosity coefficient of bulk viscous fluid
is assumed to be a power function of mass density. It turns out that the
cosmological term $\Lambda(t)$ is a decreasing function of time, which is
consistent with recent observations of type Ia supernovae. Various physical and
kinematic features of these models have also been explored.

Introduction
Cosmological models representing the early stages of the Universe have been
studied by several authors. An LRS (Locally Rotationally Symmetric) Binachi
type-V spatially homogeneous space-time creates more interest due to its richer
structure both physically and geometrically than the standard perfect fluid FRW
models. An LRS Bianchi type-V universe is a simple generalization of the
Robertson-Walker metric with negative curvature. Most cosmological models
assume that the matter in the universe can be described by ’dust’ (a pressure-
less distribution) or at best a perfect fluid. However, bulk viscosity is expected
to play an important role at certain stages of expanding universe [1]−[3]. It has
been shown that bulk viscosity leads to inflationary like solution [4] and acts
like a negative energy field in an expanding universe [5]. Furthermore, there
are several processes which are expected to give rise to viscous effects. These
are the decoupling of neutrinos during the radiation era and the decoupling of
1Corresponding Author
http://arxiv.org/abs/0704.0849v2
radiation and matter during the recombination era. Bulk viscosity is associ-
ated with the Grand Unification Theories (GUT) phase transition and string
creation. Thus, we should consider the presence of a material distribution other
than a perfect fluid to have realistic cosmological models (see Grøn [6] for a
review on cosmological models with bulk viscosity). A number of authors have
discussed cosmological solutions with bulk viscosity in various context [7]−[9].
Models with a relic cosmological constant Λ have received considerable at-
tention recently among researchers for various reasons (see Refs.[10]−[14] and
references therein). Some of the recent discussions on the cosmological constant
“problem” and consequence on cosmology with a time-varying cosmological con-
stant by Ratra and Peebles [15], Dolgov [16]−[18] and Sahni and Starobinsky
[19] have pointed out that in the absence of any interaction with matter or radi-
ation, the cosmological constant remains a “constant”. However, in the presence
of interactions with matter or radiation, a solution of Einstein equations and
the assumed equation of covariant conservation of stress-energy with a time-
varying Λ can be found. For these solutions, conservation of energy requires
decrease in the energy density of the vacuum component to be compensated by
a corresponding increase in the energy density of matter or radiation. Earlier
researchers on this topic, are contained in Zeldovich [20], Weinberg [11] and
Carroll, Press and Turner [21]. Recent observations by Perlmutter et al. [22]
and Riess et al. [23] strongly favour a significant and positive value of Λ. Their
finding arise from the study of more than 50 type Ia supernovae with redshifts
in the range 0.10 ≤ z ≤ 0.83 and these suggest Friedmann models with negative
pressure matter such as a cosmological constant (Λ), domain walls or cosmic
strings (Vilenkin [24], Garnavich et al. [25]) Recently, Carmeli and Kuzmenko
[26] have shown that the cosmological relativistic theory (Behar and Carmeli
[27]) predicts the value for cosmological constant Λ = 1.934 × 10−35s−2. This
value of “Λ” is in excellent agreement with the measurements recently obtained
by the High-Z Supernova Team and Supernova Cosmological Project (Garnavich
et al. [25], Perlmutter et al. [22], Riess et al. [23], Schmidt et al. [28]). The
main conclusion of these observations is that the expansion of the universe is
accelerating.
Several ansätz have been proposed in which the Λ term decays with time
(see Refs. Gasperini [29, 30], Berman [31], Freese et al. [14], Özer and Taha [14],
Peebles and Ratra [32], Chen and Hu [33], Abdussattar and Viswakarma [34],
Gariel and Le Denmat [35], Pradhan et al. [36]). Of the special interest is the
ansätz Λ ∝ S−2 (where S is the scale factor of the Robertson-Walker metric)
by Chen and Wu [33], which has been considered/modified by several authors
( Abdel-Rahaman [37], Carvalho et al. [14], Waga [38], Silveira and Waga [39],
Vishwakarma [40]).
Recently Bali and Yadav [41] obtained an LRS Bianchi type-V viscous fluid
cosmological models in general relativity. Motivated by the situations discussed
above, in this paper, we focus upon the exact solutions of Einstein’s field equa-
tions in presence of a bulk viscous fluid in an expanding universe. We do this
by extending the work of Bali and Yadav [41] by including a time dependent
cosmological term Λ in the field equations. We have also assumed the coefficient
of bulk viscosity to be a power function of mass density. This paper is organized
as follows. The metric and the field equations are presented in section 2. In
section 3 we deal with the solution of the field equations in presence of viscous
fluid. The sections 3.1 and 3.2 contain the two different cases and also con-
tain some physical aspects of these models respectively. Section 4 describe two
models under suitable transformations. Finally in section 5 concluding remarks
have been given.
2 The Metric and Field Euations
We consider LRS Bianchi type-V metric in the form
ds2 = −dt2 +A2dx2 +B2e2x(dy2 + dz2), (1)
where A and B are functions of t alone.
The Einstein’s field equations (in gravitational units c = 1, G = 1) read as
i + Λg
i = −8πT
i , (2)
where R
i is the Ricci tensor; R = g
ijRij is the Ricci scalar; and T
i is the stress
energy-tensor in the presence of bulk stress given by
i = (ρ+ p)viv
j + pg
i − (v
i; + v
;i + v
jvℓvi;ℓ + viv
ξ − 2
vℓ;ℓ(g
i + viv
j). (3)
Here ρ, p, η and ξ are the energy density, isotropic pressure, coefficients of
shear viscosity and bulk viscous coefficient respectively and vi the flow vector
satisfying the relations
ivj = −1. (4)
The semicolon (; ) indicates covariant differentiation. We choose the coordinates
to be comoving, so that vi = δi4.
The Einstein’s field equations (2) for the line element (1) has been set up as
= −8π
p− 2ηA4
ξ − 2
− Λ, (5)
= −8π
p− 2η
− Λ, (6)
2A4B4
= −8πρ− Λ, (7)
= 0. (8)
The suffix 4 after the symbols A, B denotes ordinary differentiation with respect
to t and
θ = vℓ;ℓ
3 Solutions of the Field Eqations
In this section, we have revisited the solutions obtained by Bali and Yadav [41].
Equations (5) - (8) are four independent equations in seven unknowns A, B, p,
ρ, ξ, η and Λ. For complete determinacy of the system, we need three extra
conditions.
Eq. (8), after integration, reduce to
A = Bk, (9)
where k is an integrating constant. Equations (5) and (6) lead to
− A44
− A4B4
= −16πη
. (10)
Using Eq. (9) in (10), we obtain
k + 1
f = −16πη, (11)
where B4 = f(B). Eq. (11) leads to
f = − 16πη
(k + 2)
, (12)
where L is an integrating constant. Eq. (12) again leads to
B = (k + 2)
k1 − k2e−16πηt
k+2 , (13)
where
, (14)
, (15)
N being constant of integration. From Eqs. (9) and (13), we obtain
A = (k + 2)
k1 − k2e−16πηt
k+2 . (16)
Hence the metric (1) reduces to the form
ds2 = −dt2 + (k + 2)
k1 − k2e−16πηt
k+2 dx2
+ e2x(k + 2)
k1 − k2e−16πηt
k+2 (dy2 + dz2). (17)
The pressure and density of the model (17) are obtained as
8πp =
(8π)(16πη)k2e
−16πηt
3(k + 2)2(k1 − k2e−16πηt)2
k1(k + 2)
2(4η + 3ξ)− {k2(4η + 3ξ)
+4k(η+3ξ)+2(5η+6ξ)}k2e−16πηt
[(k + 2)(k1 − k2e−16πηt)]
−Λ, (18)
8πρ = − (2k + 1)
(k + 2)2
(16πη)2k22
e−32πηt
(k1 − k2e−16πηt)2
[(k + 2)(k1 − k2e−16πηt)]
+ Λ. (19)
The expansion θ in the model (17) is obtained as
(16πη)k2e
−16πηt
(k1 − k2e−16πηt)
. (20)
For complete determinacy of the system we have to consider three extra condi-
tions. Firstly we assume that the coefficient of shear viscosity is constant, i.e.,
η = η0 (say). For the specification of Λ(t), we secondly assume that the fluid
obeys an equation of state of the form
p = γρ, (21)
where γ(0 ≤ γ ≤ 1) is a constant.
Thirdly bulk viscosity (ξ) is assumed to be a simple power function of the
energy density [42]−[45].
ξ(t) = ξ0ρ
n, (22)
where ξ0 and n are constants. For small density, n may even be equal to unity
as used in Murphy’s work [46] for simplicity. If n = 1, Eq. (22) may correspond
to a radiative fluid [47]. Near the big bang, 0 ≤ n ≤ 1
is a more appropriate
assumption [48] to obtain realistic models.
For simplicity and realistic models of physical importance, we consider the
following two cases (n = 0, 1):
3.1 Model I: Solution for n = 0
When n = 0, Eq. (22) reduces to ξ = ξ0 = constant. Hence, in this case Eqs.
(18) and (19), with the use of (21), lead to
8π(1 + γ)ρ =
k1(k + 2)
2(4η0 + 3ξ0)− {k2(4η0 + 3ξ0)
+ 4k(η0 + 3ξ0) + 2(5η0 + 6ξ0)}k2e−16πη0t
− (2k + 1)M
. (23)
Eliminating ρ(t) between Eqs. (19) and (23), we obtain
(1 + γ)Λ =
k1(k + 2)
2(4η0 + 3ξ0)− {k2(4η0 + 3ξ0)
+ 4k(η0 + 3ξ0) + 2(5η0 + 6ξ0)}k2e−16πη0t
+ (2k + 1)γ
(1− 3γ)
, (24)
where
M = 16πk2η0e
−16πη0t,
N = (k + 2)(k1 − k2e−16πη0t),
P = 2k2 + 2k + 5,
Q = k2 + 4k + 4. (25)
3.2 Model II: Solution for n = 1
When n = 1, Eq. (22) reduces to ξ = ξ0ρ . Hence, in this case Eqs. (18) and
(19), with the use of (21), leads to
8πρ =
16πM{2k1(k + 2)2η0 − Pk2η0e−16πη0t}
3 [(1 + γ)N2 −M{k1(k + 2)2ξ0 −Qk2ξ0e−16πη0t}]
k+2 − (2k + 1)M2
[(1 + γ)N2 −M{k1(k + 2)2ξ0 −Qk2ξ0e−16πη0t}]
. (26)
Eliminating ρ(t) between Eqs. (19) and (26), we get
Λ = 16πM [2k1(k + 2)
2η0 − Pk2η0e−16πη0t] +
γ(2k + 1)
(1 + γ)
(1− 3γ)
(1 + γ)N
M [k1(k + 2)
2ξ0 −Qk2ξ0e−16πη0t]{4N
k+2 − (2k + 1)M2}
(1 + γ)N2 [(1 + γ)N2 −M{k1(k + 2)2ξ0 −Qk2ξ0e−16πη0t}]
From Eqs. (23) and (26), we note that ρ(t) is a decreasing function of time
and ρ > 0 for all time in both models. The behaviour of the universe in these
models will be determined by the cosmological term Λ; this term has the same
effect as a uniform mass density ρeff = −Λ/4πG, which is constant in space
and time. A positive value of Λ corresponds to a negative effective mass density
(repulsion). Hence, we expect that in the universe with a positive value of Λ,
the expansion will tend to accelerate; whereas in the universe with negative
value of Λ, the expansion will slow down, stop and reverse. From Eqs. (24) and
(27), we observe that the cosmological term Λ in both models is a decreasing
function of time and it approaches a small positive value as time increase more
and more. This is a good agreement with recent observations of supernovae Ia
(Garnavich et al. [25], Perlmutter et al. [22], Riess et al. [23], Schmidt et al.
[28]).
The shear σ in the model (17) is given by
(k − 1)M√
. (28)
The non-vanishing components of conformal curvature tensor are given by
C2323 = −C1414 =
(k − 1)M
[kM − 16πη0k1(k + 2)], (29)
C1313 = −C2424 =
(k − 1)M
[16πη0k1(k + 2)− kM ], (30)
C1212 = −C3434 =
(k − 1)M
[16πη0k1(k + 2)− kM ]. (31)
Equations (20) and (28) lead to
(k − 1)√
3(k + 2)
= constant. (32)
The model (17) is expanding, non-rotating and shearing. Since σ
= conatant,
hence the model does not approach isotropy. The space-time (17) is Petrov type
D in presence of viscosity.
4 Other Models
After using the transformation
k1 − k2e−16πηt = sin (16πητ), k + 2 = 1/16πη, (33)
the metric (17) reduces to
ds2 = −
cos (16πητ)
k1 − sin (16πητ)
dτ2 +
sin (16πητ)
]2(1−32πη)
+ e2x
sin (16πητ)
](32πη)
(dy2 + dz2). (34)
The pressure (p), density (ρ) and the expansion (θ) of the model (34) are ob-
tained as
8πp =
(16πη)2{k1 − sin (16πητ)
3 sin2 (16πητ)
2k1−2(1−48πη+1152π2η2){k1−sin (16πητ)}
(16πη)(8πξ){k1 − sin (16πητ)}
sin (16πητ)
sin (16πητ)
]2(1−32πη)
− Λ, (35)
8πρ =
2(24πη − 1)(16πη)3{k1 − sin (16πητ)}2
sin2 (16πητ)
sin (16πητ)
]2(1−32πη)
(16πη){k1 − sin (16πητ)}
sin (16πητ)
. (37)
4.1 Model I: Solution for n = 0
When n = 0, Eq. (22) reduces to ξ = ξ0 = constant. Hence, in this case Eqs.
(35) and (36), with the use of (21), lead to
8π(1 + γ)ρ =
2(16πη0)
3 sin2(16πη0τ)
[k1 − P1M1] +
(16πη0)(8πξ0)M1
sin(16πη0τ)
+ 4N1 +
2(24πη0 − 1)(16πη0)3M21
sin2(16πη0τ)
. (38)
Eliminating ρ(t) between Eqs. (36) and (38), we obtain
(1 + γ)Λ =
2(16πη0)
3 sin2(16πη0τ)
[k1 − P1M1] +
(16πη0)(8πξ0)M1
sin(16πη0τ)
+ (1− 3γ)N1 +
2γ(24πη0 − 1)(16πη0)3M21
sin2(16πη0τ)
. (39)
4.2 Model II: Solution for n = 1
When n = 1, Eq. (22) reduces to ξ = ξ0ρ . Hence, in this case Eqs. (35) and
(36), with the use of (21), lead to
8πρ =
2(16πη0)
2M1[(k1 − P1M1) + 3(24πη0 − 1)(16πη0)M1]
3 sin(16πη0τ)[(1 + γ) sin(16πη0τ) − 16πη0ξ0M1]
4N1 sin(16πη0τ)
[(1 + γ) sin(16πη0τ) − 16πη0ξ0M1]
. (40)
Eliminating ρ(t) between Eqs. (36) and (40), we obtain
2(16πη0)
2M1(k1 − P1M1)
3 sin(16πη0τ)[(1 + γ) sin(16πη0τ)− 16πη0ξ0M1]
[3(16πη0ξ0)M1 + (1− 3γ) sin(16πη0τ)]
[(1 + γ) sin(16πη0τ)− 16πη0ξ0M1]
2(24πη0 − 1)(16πη0)3M21 [γ(1 + γ) sin(16πη0τ) − (1− γ)(16πη0ξ0)M1]
(1 + γ) sin2(16πη0τ)[(1 + γ) sin(16πη0τ)− (16πη0ξ0)M1]
, (41)
where
M1 = k1 − sin(16πη0τ),
16πη0
sin (16πη0τ)
]2(1−32πη)
P1 = 1− 48πη0 + 1152π2η20 . (42)
The shear (σ) in the model (34) is obtained as
(1− 48πη0)(16πη0)[k1 − sin(16πη0τ)√
3 sin(16πη0τ)
. (43)
The models descibed in cases 4.1 and 4.2 preserve the same properties as in the
cases of 3.1 and 3.2.
5 Conclusions
We have obtained a new class of LRS Bianchi type-V cosmological models of
the universe in presence of a viscous fluid distribution with a time dependent
cosmological term Λ. We have revisited the solutions obtained by Bali and Ya-
dav [41] and obtained new solutions which also generalize their work.
The cosmological constant is a parameter describing the energy density of
the vacuum (empty space), and a potentially important contribution to the dy-
namical history of the universe. The physical interpretation of the cosmological
constant as vacuum energy is supported by the existence of the “zero point”
energy predicted by quantum mechanics. In quantum mechanics, particle and
antiparticle pairs are consistently being created out of the vacuum. Even though
these particles exist for only a short amount of time before annihilating each
other they do give the vacuum a non-zero potential energy. In general relativity,
all forms of energy should gravitate, including the energy of vacuum, hence the
cosmological constant. A negative cosmological constant adds to the attractive
gravity of matter, therefore universes with a negative cosmological constant are
invariably doomed to re-collapse [49]. A positive cosmological constant resists
the attractive gravity of matter due to its negative pressure. For most universes,
the positive cosmological constant eventually dominates over the attraction of
matter and drives the universe to expand exponentially [50].
The cosmological constants in all models given in Sections 3.1 and 3.2 are
decreasing functions of time and they all approach a small and positive value
at late times which are supported by the results from recent type Ia supernova
observations recently obtained by the High-z Supernova Team and Supernova
Cosmological Project (Garnavich et al. [25], Perlmutter et al. [22], Riess et al.
[23], Schmidt et al. [28]). Thus, with our approach, we obtain a physically rele-
vant decay law for the cosmological term unlike other investigators where adhoc
laws were used to arrive at a mathematical expressions for the decaying vacuum
energy. Our derived models provide a good agreement with the observational
results. We have derived value for the cosmological constant Λ and attempted
to formulate a physical interpretation for it.
Acknowledgements
The authors wish to thank the Harish-Chandra Research Institute, Allahabad,
India, for providing facility where part this work was done. We also thank to
Professor Raj Bali for his fruitful suggestions and comments in the first draft of
the paper.
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	Introduction
	The Metric and Field Euations
	Solutions of the Field Eqations
	Model I:       Solution for n = 0
	Model II:       Solution for n = 1
	Other Models
	Model I:       Solution for n = 0
	Model II:       Solution for n = 1
	Conclusions
