International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 148
ISSN 2229-5518
H. R. Ghate, Arvind S. Patil
Abstract— Higher dimensional dissipative future universe without Big rip in the context of Ecart formalism has been studied. In this work, we have stud-
A
ied the generalized Chaplygin gas characterized by equation of state
p = − 1
ρ α
as a model for dark energy. We have verified that in higher dimen-
sions when the cosmic dark energy behaves simultaneously like a fluid with equation of state
p = ωρ ; ω < −1
as well as Chaplygin gas then big rip
does not arise and the scale factor is found to be regular for all time. The work of Yadav (2011) has been extended and studied in higher dimensions.
Keywords : Phantom fluid, Big rip, Accelerated universe, Higher dimensions.
PACS: 04.50.-g, 04.25.-g, 98.80.-k
—————————— ——————————
ECENT Ia Supernova observations indicate that the current universe is not only expanding but also accelerating. This behavior of universe is confirmed by various independent
observational data, the large scale structure, the cosmic microwave
without specifying the theory from which EOS parameter ω is derived ( Jaffe et. al. [19]). The astrophysical observations also indicate that the universe media is not a perfect fluid (Brevik et. al.
[5]) and the viscosity is considered in the evolution of the universe
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background (CMB) radiation and so on. There is consensus on
conclusion that the universe has entered a state of accelerating
by (Brevik et. al. [6], Caldwell et. al. [7] and Cataldo, et. al. [8]).
On the other hand, in the standard cosmological model if EOS
expansion at red shift
z ≈ 0.5 . These recent observations strongly
parameter ω is less than -1 the universe shows the future finite
indicate that our universe is spatially flat and there exists an exotic fluid called as dark energy with negative pressure.
Most of the perfect fluids relevant to cosmology obey an equation of
state of the form
p = ωρ ,
where ω is the equation of state parameter.
The three most common examples of cosmological fluids with constant ω are
(i) Dust model (ω = 0) ,
singularity called Big rip (Jackiw [18], Nojiri et. al. [23]).
Cosmologists and particle physicists have considerable interest in
obtaining solutions of Einstein’s equations in higher dimensions in the context of physics of early universe. In its early stage, it is well known that the universe was much small than it is today. Cosmologists are interested in theories with more than four space- time dimensions in which extra dimensions are connected to a very small size beyond our present ability of experimental detection ( Krori and Barua [21]). The experimental detection of time variation of fundamental constants could provide strong evidence for the existence of extra dimensions. (Chodos and Detweller [10],
Marciano [22]) proposed cosmological dimensional reduction
(ii) Radiating model ω = 1
3
) and
process such that the five dimensional universe naturally evolves into four dimensional universe as a consequence of dimensional reduction. A number of authors ( [1], [2], [9], [20], [26] ] have
(iii) Vacuum energy model
( ω = −1 )
which is
studied higher dimensional cosmological models containing variety
mathematically equivalent to cosmological constant ∧ .
of matter fields.
In Roberson Walker cosmology, the EOS for Chaplygin gas is
It is well known that fluids with ω < − 1
3
are usually considered in
given by
the context of dark energy (DE), since they give rise to accelerating expansion. As per Coroll et. al. [11] the fluid with constant EOS parameter with ω < − 1 , particularly in constant of DE there have
p = − A ,
ρ
(1)
3
been proposed various scalar field models that can be described by
where p is pressure and ρ is energy density in co-moving
reference frame with ρ > 0 and A is positive constant.
time dependent ω and that can evolve below
− 1 . E.g.
3
The Chaplygin gas has super symmetry generalization ( Bento et. al.
[3], Gorini et. al. [15]). Betrolami et. al. [4] have found that
quintessence
(−1 ≤ ω ≤ 1) , Phantom
(ω ≤ −1) , quintom that
generalized Chaplygin gas (GCG) is better fit for latest Supernova
evolve across cosmological constant boundary
ω = −1 . However,
data.
we still do not have a perfect understanding of EOS of the DE and some time it is useful to think about the Einstein field equations
For generalized Chaplygin gas, the EOS is given by,
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International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 149
ISSN 2229-5518
p = − A
1
, with 1 ≤ α < ∞ .
a = − 1 ( ρ + 2 p)
ρ α a 6
(9)
In particular for
Chaplygin gas.
(2)
α = 1 , generalized Chaplygin gas reduces to
Here (.) dot represents the differentiation with respect to t. The energy conservation equation is given as,
Zhai et. al. [28] have investigated a viscous generalized Chaplygin
T; j = 0
where
1
T; j =
∂ T ij
j
− g ]+ T jk
Γjk
gas.. Szydlowsk et. al. [25] found that dissipative Chaplygin gas can
give rise to structurally stable evolutional scenario. It is interesting to note that generalized Chaplygin gas itself can behave like a fluid with viscosity in the context of Ecart Formalism. Fabris et al. [14] have investigated the equivalence generalized Chaplygin gas and dust like fluid. Recently Cruz et al. [12] have studied dissipative generalized Chaplygin gas as Phantom dark energy and established the cosmological solutions to generalized Chaplygin gas with bulk viscosity.
− g ∂x
Using Equation (3) and (6) which simplifies to
ρ + 4H( p − ρ ) = 0 ,
where ρ is the differentiation of ρ with respect to t.
Using equations (2), (5), (8), equation (10) can be written as
(10)
In this paper, we consider higher dimensional FRW metric for
homogeneous and isotropic flat universe. The work of Yadav [27] is extended in five dimensional space times.
ρ + 4 =a ρ − A
1
ρ α
− 3ξH = 0 .
(11)
The higher dimensional FRW metric for homogeneous and isotropic flat universe is given by
ds 2 = −dt 2 + a 2 (t) (dx 2 + dy 2 + dz 2 + dv 2 ), (3)
where a(t) is a scale factor and t represents the cosmic time; v is the
In most of the investigations in cosmology, the viscosity is assumed
to be a simple power law function of the energy density (Shri Ram
[24], Yadav [27]) i.e
ξ = ξ 0 ρ n , (12)
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fifth dimension in space. Here the suffixes 0, 1, 2, 3, 4 represent the variables t, x, y, z, v respectively.
where ξ 0 and n are constants.
From equations (11) and (12) we get,
The energy momentum tensor for bulk viscous cosmology in early
universe is given by
dρ da ρ α +1α −
ξ ρ n+1 . (13)
i i i
+ 4
dt dt
1 = 2 0
Tj = ( ρ + p) u u j + pg j
ρ α
with
p = p − 3ξH ,
(4)
(5)
Using the transformation which reduces to,
dρ = f ( ρ ) = ρ n+1
dt
in equation (13)
where ρ → energy density, p → isotropic pressure,
p → effective
1+α
ρ α
ρ 1+α α
4(1+α )
a0 α (1−2ξ0 )
pressure,
ξ → bulk viscous coefficient,
H → Hubble’s parameter,
= A + 0
− A
a
, (14)
g ij →
metric tensor.
ui is the four velocity of fluid which satisfy
where
ρ0 , a0
represent the values of
ρ (t )
and
a(t )
at present
the condition gij ui u j = −1 .
Using the above equations, the matter tensor is given by ,
T i = diag.(−ρ , p, p, p, p)
time t0 respectively.
The dark energy simultaneously behaves like GCG obeying equation (2) as well as fluid with equation of state.
The Einstein’s Field equations are
(6)
p = ωρ , with
ω < −1 .
(15)
i 1 i i
From equations (2) and (15) we get,
R j −
g j R = −Tj
2
(7)
ω(t) = −
A .
ρ 1+α α
Where
Rij → Ricci Tensor,
R → Ricci Scalar,
At t = t 0 equation (16) gives
(16)
Tij →
cosmology.
Energy momentum tensor for bulk viscous
1+α
A = −ω0 ρ 0 α ,
Using the equations (4), (5) and (6) for metric (3), the Einstein’s
Field equations (7) reduce to
2
where ω0 is value of ω = ω(t) at t = t 0 .
(17)
a = ρ = H 2
a 2 6
(8)
Put the value of A from equation (17) in equation (14) we get,
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International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 150
ISSN 2229-5518
a0
=4(1+α ) (1 2 0 )
α
1+α
(25)
Expanding the R. H. S. of (25) and neglecting the higher powers of
ρ = ρ
0 − ω0
+ (1 + ω0
) α − ξ .
a
(1 − ω
) a
4(1+α )
α (1−2ξ )
0 0
0 we get,
(18) In homogeneous model of universe, scalar field φ(t) with potential
ω0
a Ω
a(t)
α (1 − ω0 )
a0
=4(1+α )
α (1−2ξ0 )
V (φ ) has energy density =
0 H 0 ω0 2(1+α ) 1 +
. (26)
ρφ = 1 φ 2 + V (φ )
a 2
(19)
2(1 + α ) ω0
a(t)
and
2
pφ = 1 φ 2 − V (φ ) , (20)
2
Integrating (26) we get,
α Ω0 (t −t0 )
.
where φ is differentiation of φ with respect to t.
a(t) =
a0
α (1−2ξ0 )
(α
+ 2(1 +
α ω0
) e 8H0 ω0 2(1+α ) 2
Adding (19) and (20), we get
φ 2 = ρφ + pφ . (21)
[2(1 + α ) ω0 ] 4(1+α )
(27)
With the help of equations (2) and (17), equation (21) gives
From equation (27), it is clear that as t → ∞ ,
a(t) → ∞ , therefore
1+α
ρ α + ρ
1+α α ω
the present model is free from finite time future singularity.
In this case, the Hubble distance is given by,
φ 2 = 0 0 . (22)
=4(1+α )
1
ρ α H −1 =
2
α .1 −
α (1 − ω0 )
a α (1−2ξ0 )
. .
Substitute the value of ρ from equation (18) in equation (22) we
get,
4(1+α )
Ω 0 H 0 ω0 2(1+α )
2 (1
+ α ) ω0
a(t)
(28)
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(1 + ω )ρ a0
α (1−2ξ0 )
Equation (28) shows the growth of Hubble distance
H −1 with time
φ 2 =
a(t)
4(1+α )
11+α
. (23)
such that
H −1 →
2
Ω H ω
α ≠ 0
+α
as t → ∞ . Thus in
a0
α (1−2ξ0 )
0 0 0 2(1 )
− ω0 + (1 + ω0 )
present higher dimensional case, the galaxies will not disappear as
a(t)
t → ∞ , avoiding big rip singularity. Therefore, one can conclude that if Phantom fluid simultaneously behaves like generalized
From equation (23), it is observed that when
φ 2 > 0 , we get
Chaplygin gas and fluid with
p = ωρ
then the future accelerated
positive kinetic energy for (1 + ω0 ) > 0
negative kinetic energy for (1 + ω0 ) < 0 .
and when φ 2 < 0 , we get
expansion of the universe will free from catastrophic situation like big rip in higher dimensions.
Equation (18) can be written as
Thus (1 + ω0 ) > 0
represents a case of quintessence and
α
4(1+α )
(1 + ω0 ) < 0
represents Phantom fluid dominated universe. Similar
1+α
results are obtained by Hoyle and Narlikar [16-17] in C-field with
ρ = ρ
ω + (
− ω ) a0
α (1−2ξ0 )
. (29)
0 0
1 0
a(t)
negative kinetic energy for steady state theory of universe.
Now from equation (8) and (18) we get,
=4(1+α )
2
α
1+α
From equation (29), it is clear that as
α
t → ∞ ,
a
= 0 H 2 ω
+ (1 − ω
) a0
α (1−2ξ0 )
, (24)
ρ → ρ 0 ω0 1+α
> ρ 0 .
0 0
0
a
where
2
ω0 = −ω0 ,
a(t)
H 0 = 100h km / s mpc
ρ
present value of
3 H 2
Thus one can conclude that energy density increases with time,
contrary to other phantom models having future singularity at t = t s .
In higher dimensional FRW universe, we observed that when the
Hubble parameter and
Ω 0 =
0
ρ cr ,0
with
ρ cr ,0 =
0 .
8πG
cosmic dark energy behaves simultaneously like a fluid with
equation of state p = ωρ ; ω < −1
as well as generalized Chaplygin
Taking square root to both sides of equation (25) gives
α
4(1+α )
gas with equation of state
p = − A
1
ρ α
, it is conclude that for
a
Ω α
(1 − ω )
a0
2(1+α )
α (1−2ξ0 ) 2
= H 0 ω0 2(1+α ) 1 +
. φ
> 0 , we get positive kinetic energy when (1 + ω0 ) > 0
a 2 ω0
a(t)
representing the case of quintessence and for
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φ 2 < 0 , we get
International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 151
ISSN 2229-5518
negative kinetic energy when (1 + ω0 ) < 0
representing phantom
063004, (2005).
fluid dominated universe. The results are analogous to the results obtained by Hoyle and Narlikar in C-field with negative kinetic energy for steady state theory of universe. Further it is clear that as
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H −1 ≠ 0
indicating that in
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present higher dimensional FRW universe, the galaxies will not
disappear as t → ∞ , avoiding big-rip singularity. Therefore, one can conclude that when the cosmic dark energy behaves simultaneously
[28] X. H. Zhai, Y. D. Xu., X. Z. Li., Int. J. Mod. Phys. D, 15,
1151, (2006).
like a fluid with equation of state
p = ωρ ;
ω < −1 as well as
Chaplygin gas then big rip problem does not arise and the scale
factor is found to be regular for all time. Also as
α
t → ∞ ,
ρ → ρ 0 ω0 1+α
> ρ 0 , concluding that energy density increases
with time, contrary to other phantom models having future singularity at t = ts . One should note that the results of Yadav [27] can be obtained from our results by putting appropriate values to the function concerned.
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