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Research ArticleBianchi Type-V Bulk Viscous Cosmic String in π(π , π) Gravitywith Time Varying Deceleration Parameter
B\naya K. Bishi1 and K. L. Mahanta2
1Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur 440010, India2Department of Mathematics, C.V. Raman College of Engineering, Bidya Nagar, Mahura, Janla, Bhubaneswar 752054, India
Correspondence should be addressed to BΔ±Μnaya K. Bishi; [email protected]
Received 20 February 2015; Revised 13 May 2015; Accepted 18 May 2015
Academic Editor: George Siopsis
Copyright Β© 2015 B. K. Bishi and K. L. Mahanta.This is an open access article distributed under theCreativeCommonsAttributionLicense, which permits unrestricted use, distribution, and reproduction in anymedium, provided the originalwork is properly cited.The publication of this article was funded by SCOAP3.
We study the Bianchi type-V string cosmological model with bulk viscosity in π(π , π) theory of gravity by considering a specialform and linearly varying deceleration parameter. This is an extension of the earlier work of Naidu et al., 2013, where they haveconstructed the model by considering a constant deceleration parameter. Here we find that the cosmic strings do not survive inboth models. In addition we study some physical and kinematical properties of both models. We observe that in one of our modelsthese properties are identical to the model obtained by Naidu et al., 2013, and in the other model the behavior of these parametersis different.
1. Introduction
Recent data obtained in observational cosmology [1β5] indi-cate that currently our universe is accelerating. Researchersare trying to describe this late inflation of the universe intwo different ways. Some by modifying Einsteinβs theory ofgravity and others by introducing an exotic type of fluidlike a cosmological constant or quintessential type of scalarfield. But the use of cosmological constant is surroundedwithserious theoretical problems. Therefore various alternativesare used such as K-essence [6], tachyon [7], Phantom [8], andquintom [9]. While all these alternatives have both positiveand negative aspects, a large number of researchers used amatter field, namely, chaplygin gas. In the second approachwhere gravitational theory is modified researchers study theaction described by an arbitrary function of the scalar curva-ture π , which is called π(π ) theory of gravity [10β12]. Carrollet al. [13] pointed out that the late time acceleration of theuniverse can be explained by π(π ) gravity. Sharif and Yousaf[14] analyzed the role of the electromagnetic field and a viableπ(π )model on the range of dynamical instability. Sharif andKausar [15] obtained thatπ(π )with constant scalar curvatureplays the role of cosmological constant and slows down thecollapsing process. Sharif and Yousaf [16] performed stability
analysis of an adiabatic anisotropic cylindrical collapsingsystem in metric π(π ) gravity. Sharif and Yousaf [17] studiedthe effects of polynomial π(π ) model on the stability ofhomogeneous energy density in self-gravitating sphericalstellar object and found that shear, pressure, dissipativeparameters, and π(π ) terms affect the existence of inho-mogeneous energy density. Sharif and Yousaf [18] obtainedthat in addition to other fluid variables higher order π(π )corrections, relaxation processes, and electromagnetic fieldaffect the energy density in homogeneity of spherical stars.Furthermore Sharif and Yousaf [19] explored the effects of thethree-parametric π(π ) model on the stability of the regularenergy density of planar fluid configurations with the Palatiniπ(π ) formalism.
There exist other interesting classes of modified gravitytheories such as π(πΊ) gravity, π(π , πΊ) gravity, π(π) gravity,and π(π , π) gravity. In π(π , π) gravity the gravitationalLagrangian is given by an arbitrary function of theRicci scalar(π ) and the trace (π) of the stress energy tensor and alsoit depends on a source term representing the variation ofthe matter stress energy tensor with respect to the metric.The source term expression is obtained as a function matterLagrangian (πΏ
π); as a result for each choice of πΏ
π, it would
generate a specific set of field equations.The field equation of
Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2015, Article ID 491403, 8 pageshttp://dx.doi.org/10.1155/2015/491403
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π(π , π) gravity is obtained by Harko et al. [20] from Hilbert-Einstein type variational principle.The action for this gravityis given by
π = β«ββπ(1
16ππΊπ (π , π) + πΏ
π)π4π₯, (1)
where π(π , π) is an arbitrary function of the Ricci scalar(π ) and trace (π) of energy tensor of the matter π
ππand πΏ
π
represents matter Lagrangian density. They have derived thefield equations of π(π , π) gravity by varying the action πof the gravitational field with respect to the metric tensorcomponents π
ππ. For the choice of π(π , π) = π + 2π(π), the
field equation takes the form
πΊππ= [8π+ 2π (π)] π
ππ+ [2ππ (π) +π (π)] π
ππ, (2)
where the overhead prime indicates derivative with respect tothe argument and π
ππis given by
πππ= (π +π) π’
ππ’π+ππππ, (3)
whereπ andπ stand for energy density and isotropic pressure,respectively.
Studies of string cosmological models are crucial as theyplay an important role in structure formation of the earlystages of evolution of the universe. Cosmic strings are one-dimensional topological defects, which may be formed dur-ing symmetry breaking phase transition in the early universealong with other defects like domain walls and monopoles.The density perturbation arising out of them plays animportant role in cosmology since they are considered asthe route for galaxy formation. After Stachel [21] and Hendiand Momeni [22] many researchers [23β32] have workedon string cosmological models in the context of Einsteinβstheory or modified theories of gravity. Pradhan et al. [33]studied the LRS Bianchi type-II massive string cosmologicalmodel in general relativity. Five-dimensional Bianchi type-Istring cosmological models in Lyra manifold are analysed bySamanta and Debata [34]. Rikhvitsky et al. [35] discussed themagnetic Bianchi type-II string cosmological model in loopquantum cosmology.
Bulk viscosity is useful for the study of early stages ofevolution of the universe. Bulk viscosity driven inflation isprimarily due to the negative bulk viscous pressure givingrise to a total negative effective pressure whichmay overcomethe pressure due to the usual gravity of matter distribution inthe universe and provides an impetus for rapid expansion ofthe universe. Thus many researchers have shown interest tostudy bulk viscous string cosmological model in the contextof Einstein theory or modified theories of gravity. Severalauthors [36β43] studied bulk viscous cosmological modelsin general relativity. Anisotropic bulk viscous cosmologicalmodels with particle creation have been investigated by Singhand Kale [44]. Rao and Sireesha [45] studied the Bianchitypes II, VIII, and IX string cosmological models with bulkviscosity in Brans-Dicke theory of gravitation. Numerousresearchers, Samanta and Dhal [46]; Chaubey and Shukla[47]; Adhav [48]; Reddy et al. [49], have studied cosmologicalmodel of the universe in π(π , π) theory of gravity in different
contexts. Recently Naidu et al. [50], Kiran and Reddy [51],and Reddy et al. [52] discussed the Bianchi type-V, Bianchitype-III, Kaluza-Klein space time with cosmic strings, andbulk viscosity in π(π , π) gravity, respectively. Mahanta [53]investigated the bulk viscous cosmological models in π(π , π)theory of gravity.
Motivated by the above discussion, here we have con-structed Bianchi type-V bulk viscous string cosmologicalmodel in π(π , π) gravity with special form of decelerationparameter and linearly varying deceleration parameter. Themain reason to explore the Bianchi type-V model is that thestandard FLRW models are contained as special cases of theBianchi models. The Bianchi type-V model generalizes theopen (π = β1) Friedmann model and represents a modelin which the fluid flow is not necessarily orthogonal to thethree surfaces of homogeneity. At early stage of evolution,the universe was not so smooth as it looks in presenttime. Therefore anisotropic cosmological models have takenconsiderable interest of researchers.
In this work we investigate the role of variable decel-eration parameter in Bianchi type-V space time with bulkviscous cosmic string and π(π , π) gravity.
2. Metric and Field Equations
The spatially homogeneous and anisotropic Bianchi type-Vspace time is given by
ππ 2 = βππ‘2 +π΄2ππ₯2 + π2πΌπ₯ (π΅2ππ¦2 +πΆ2ππ§2) , (4)
where πΌ is a constant andπ΄, π΅, andπΆ are functions of cosmictime π‘ only.
The energy momentum tensor for a bulk viscous fluidcontaining one-dimensional cosmic strings is considered as
πππ= (π +π) π’
ππ’π+ππππβππ₯ππ₯π, (5)
π = πβ 3ππ», (6)
where π(π‘) is the rest energy density of the system, π» is theHubble parameter, π is the pressure, π(π‘) is the coefficient ofbulk viscosity,π(π‘) is the string tensiondensity,π’π = (0, 0, 0, 1)is the four-velocity vector of the fluid, and π₯π is the directionof the string. Also π’π and π₯π satisfy the relation
ππππ’ππ’π= π₯ππ₯π= β 1,
π’ππ₯π= 0.
(7)
The field equation (2) for the metric (4), with the help of (5)along with π(π) = ππ (π is a constant quantity), is given asfollows:
οΏ½ΜοΏ½
π΅+οΏ½ΜοΏ½
πΆ+οΏ½ΜοΏ½οΏ½ΜοΏ½
π΅πΆβπΌ2
π΄2
= βπ (8π+ 7π) + π (8π+ 3π) + ππ,
οΏ½ΜοΏ½
πΆ+οΏ½ΜοΏ½
π΄+οΏ½ΜοΏ½οΏ½ΜοΏ½
π΄πΆβπΌ2
π΄2= βπ (8π+ 7π) + (π + π) π,
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οΏ½ΜοΏ½
π΄+οΏ½ΜοΏ½
π΅+οΏ½ΜοΏ½οΏ½ΜοΏ½
π΄π΅βπΌ2
π΄2= βπ (8π+ 7π) + (π + π) π,
οΏ½ΜοΏ½οΏ½ΜοΏ½
π΄π΅+οΏ½ΜοΏ½οΏ½ΜοΏ½
π΅πΆ+οΏ½ΜοΏ½οΏ½ΜοΏ½
π΄πΆβ3πΌ2
π΄2= βπ (8π+ 7π) β 5ππ+ππ,
2 οΏ½ΜοΏ½π΄βοΏ½ΜοΏ½
π΅βοΏ½ΜοΏ½
πΆ= 0,
(8)
where an overhead dot denotes differentiation with respect toπ‘.
3. Solution of the Field Equations
The field equations (8) reduce to the following independentequations:
οΏ½ΜοΏ½
π΅βοΏ½ΜοΏ½
π΄+οΏ½ΜοΏ½οΏ½ΜοΏ½
π΅πΆβοΏ½ΜοΏ½οΏ½ΜοΏ½
π΄πΆ= π (8π+π) , (9)
οΏ½ΜοΏ½οΏ½ΜοΏ½
π΄π΅+οΏ½ΜοΏ½οΏ½ΜοΏ½
π΅πΆ+οΏ½ΜοΏ½οΏ½ΜοΏ½
π΄πΆβ3πΌ2
π΄2= βπ (8π+ 7π) β 5ππ+ππ, (10)
οΏ½ΜοΏ½
π΅βοΏ½ΜοΏ½
πΆ+οΏ½ΜοΏ½οΏ½ΜοΏ½
π΄π΅βοΏ½ΜοΏ½οΏ½ΜοΏ½
π΄πΆ= 0, (11)
2 οΏ½ΜοΏ½π΄βοΏ½ΜοΏ½
π΅βοΏ½ΜοΏ½
πΆ= 0. (12)
Here there are four equations involving six unknowns. Sincethe field equations are highly nonlinear for the completedeterminacy, we need extra conditions among the variables.We consider these conditions in the form SET-I and SET-IIas defined below.
(i) The shear (π) is proportional to the expansion (π)[54]. Collins et al. [55] pointed out that, for spatiallyhomogeneous metric, the normal congruence to thehomogeneous hyper surface satisfies the condition[π/π = Const.] which yields
π΅ = πΆπ. (13)
(ii) The combined effect of the proper pressure and thebulk viscous pressure for barotropic fluid [50] can bewritten as
π = πβ 3ππ» = (π0 β πΎ) π, 0 β€ π0 β€ 1, π = π0π, (14)
where π0 and πΎ are constants. The SET-I consists of (i), (ii),and a special form of deceleration parameter [56]
π = β 1+π½
1 + π π½, (15)
where π½(> 0) is a constant and π is scale factor of themetric. The SET-II consists of (i), (ii), and a linearly varyingdeceleration parameter [57]
π = β ππ‘ + π β 1, (16)
where π(β₯ 0) and π(β₯ 0) are constants.
3.1. Solution of the Field Equations along with Conditions inSET-I. In this section, we have discussed the solution of thefield equations by considering the extra conditions as in SET-I. The Hubble parameterπ» is defined as π» = οΏ½ΜοΏ½/π and from(15) we obtained
π» =οΏ½ΜοΏ½
π = π΄1 (1+π
βπ½) , (17)
where π΄1 is a constant of integration. Integrating (17) andusing the initial conditions π = 0 at π‘ = 0 we have found
π = (ππ΄1π½π‘ β 1)1/π½
. (18)
The scale factor of metric (4) is defined as
π 3 = π΄π΅πΆ. (19)
With the help of (12), (13), and (19) we have found
π΄ = (ππ΄1π½π‘ β 1)1/π½ (20)
π΅ = (ππ΄1π½π‘ β 1)2π/π½(π+1) (21)
πΆ = (ππ΄1π½π‘ β 1)2/π½(π+1)
. (22)
From (9), with the help of (20)β(22) we obtained the stringtension density (π) as
π = βπ΄11ππ΄1π½π‘ (π½ β 3ππ΄1π½π‘)
(ππ΄1π½π‘ β 1)2, (23)
whereπ΄11 = π΄21(πβ1)/(π+1)(8π+2π). From (10), with the
help of (14) and (22), we obtained the rest energy density (π)as
π =1
8π + π (3 β 5π0 + 5πΎ)[(π΄13 (π½ β 3π
π΄1π½π‘) + π΄12) ππ΄1π½π‘
(ππ΄1π½π‘ β 1)2
β3πΌ2
(ππ΄1π½π‘ β 1)2/π½] ,
(24)
where π΄12 = 2π΄21(π
2 + 4π+ 1)/(π + 1)2, π΄13 = ππ΄11. From(14), with the help of (24), we have obtained the total pressure
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(π), proper pressure (π), and the coefficient of bulk viscosity(π) as follows:
π =π0 β πΎ
8π + π (3 β 5π0 + 5πΎ)[(π΄13 (π½ β 3π
π΄1π½π‘) + π΄12) ππ΄1π½π‘
(ππ΄1π½π‘ β 1)2
β3πΌ2
(ππ΄1π½π‘ β 1)2/π½] ,
π =π0
8π + π (3 β 5π0 + 5πΎ)[(π΄13 (π½ β 3π
π΄1π½π‘) + π΄12) ππ΄1π½π‘
(ππ΄1π½π‘ β 1)2
β3πΌ2
(ππ΄1π½π‘ β 1)2/π½] ,
π =πΎ
3π΄1 [8π + π (3 β 5π0 + 5πΎ)][π΄13 (π½ β 3π
π΄1π½π‘) + π΄12
(ππ΄1π½π‘ β 1)
β3πΌ2πβπ΄1π½π‘
(ππ΄1π½π‘ β 1)2/π½β1] .
(25)
Using (13), (20), and (22) in (11) we get
(πβ 1) [οΏ½ΜοΏ½πΆ+(
οΏ½ΜοΏ½
πΆ)
2
+οΏ½ΜοΏ½οΏ½ΜοΏ½
π΄πΆ] = 0, (26)
which provideπ = 1 as
[οΏ½ΜοΏ½
πΆ+(
οΏ½ΜοΏ½
πΆ)
2
+οΏ½ΜοΏ½οΏ½ΜοΏ½
π΄πΆ]
=2π΄21ππ΄1π½π‘
(π + 1)2 (ππ΄1π½π‘ β 1)2
Γ [π½ (π+ 1) β (π+ 5) ππ΄1π½π‘] ΜΈ= 0.
(27)
Thus, forπ = 1, the expressions in (17)β(24) take the form
π΄ = π΅ = πΆ = (ππ΄1π½π‘ β 1)1/π½
,
π = 0,
π =3
8π + π (3 β 5π0 + 5πΎ)[
π΄21ππ΄1π½π‘
(ππ΄1π½π‘ β 1)2
βπΌ2
(ππ΄1π½π‘ β 1)2/π½] ,
π =3 (π0 β πΎ)
8π + π (3 β 5π0 + 5πΎ)[
π΄21ππ΄1π½π‘
(ππ΄1π½π‘ β 1)2
βπΌ2
(ππ΄1π½π‘ β 1)2/π½] ,
π =3π0
8π + π (3 β 5π0 + 5πΎ)[
π΄21ππ΄1π½π‘
(ππ΄1π½π‘ β 1)2
βπΌ2
(ππ΄1π½π‘ β 1)2/π½] ,
π =3πΎ
3π΄1 [8π + π (3 β 5π0 + 5πΎ)][
π΄21(ππ΄1π½π‘ β 1)
βπΌ2πβπ΄1π½π‘
(ππ΄1π½π‘ β 1)2/π½β1] .
(28)
3.1.1. Some Physical Properties of the Model. Some physicalproperties of the model are given below, which are crucial inthe discussion of cosmology:
(i) The spatial volume π = (ππ΄1π½π‘ β 1)1/π½.
(ii) The scalar of expansion π = 3π΄1ππ΄1π½π‘/(ππ΄1π½π‘ β 1).
(iii) ThemeanHubble parameterπ» = π΄1ππ΄1π½π‘/(ππ΄1π½π‘β1).
(iv) The mean anisotropy parameter is defined as 3π΄β=
β3π=1((π»π β π»)/π»)
2 and obtained as π΄β= 0.
(v) The shear scalar π2 = 0.
In this model, we observed that at initial epoch the values ofenergy density (π), proper pressure (π), total pressure (π),coefficient of bulk viscosity (π), and Hubble parameter (π»)are very high and these values gradually decrease with theevolution of time; that is, π, π, π, π, π» tend to zero as π‘ ββ (see Figures 1β5). Spatial volume (π) increases with theevolution of time; that is, π β β as π‘ β β (see Figure 6).Thismodel is isotropic as the average anisotropy parameter iszero and it is also shear free. In this model cosmic string doesnot survive.
3.2. Solution of the Field Equations along with Conditions inSET-II. In this section, we have discussed the solution of thefield equations by considering the conditions as in SET-II.From (16) we have Akarsu and Dereli [57]:
π =
{{{{{{{{{{{{{{{
π 1 exp[[
[
2
βπ2 β 2π1πarctanh( ππ‘ β π
βπ2 β 2π1π)]]
]
, for π > 0, π β₯ 0
π 2 (ππ‘ + π2)1/π
, for π = 0, π > 0
π 3ππ3π‘, for π = 0, π = 0,
(29)
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0 5 10 15 20 25 30 35β1
0
1
2
3
4
5
6
t
π obtained from SET-Iπ obtained from SET-II
π
Figure 1: Density π versus time π‘ for π = 0.5, π0 = 1/3, πΎ = 1,π½ = 1.5, πΌ = 0.1, π΄1 = 1, π 1 = 1, π = 1.6, and π = 0.097.
0 5 10 15 20 25 30 35β0.2
00.20.40.60.8
11.21.41.61.8
p
t
p obtained from SET-Ip obtained from SET-II
Figure 2: Proper pressure π versus time π‘ for π = 0.5, π0 = 1/3,πΎ = 1, π½ = 1.5, πΌ = 0.1, π΄1 = 1, π 1 = 1, π = 1.6, and π = 0.097.
whereπ 1,π 2,π 3, π1, π2, and π3 are constants of integration.Thelast two values of π give the constant deceleration parameter.So we neglect these values of π as π = constant is studiedby Naidu et al. [50]. Thus we focus on the first value of scalefactor. The scale factor π can also be expressed as follows forπ > 0 and π > 1 as [57]:
π = π 1 exp [2πarctanh(ππ‘ β π
π)] . (30)
With the help of (12), (13), and (29) we have obtained
π΄ = π 1 exp [2πarctanh(ππ‘ β π
π)] ,
π΅ = (π 1)2π/(π+1) exp [ 4π
π (π + 1)arctanh(ππ‘ β π
π)] ,
πΆ = (π 1)2/(π+1) exp [ 4
π (π + 1)arctanh(ππ‘ β π
π)] .
(31)
0 5 10 15 20 25 30 35β0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
t
p obtained from SET-Ip obtained from SET-II
p
Figure 3: Total pressure π versus time π‘ for π = 0.5, π0 = 1/3, πΎ =1/4, π½ = 1.5, πΌ = 0.1, π΄1 = 1, π 1 = 1, π = 1.6, and π = 0.097.
0 5 10 15 20 25 30 35β0.05
00.05
0.10.15
0.20.25
0.30.35
0.40.45
t
π obtained from SET-Iπ obtained from SET-II
π
Figure 4: Coefficient of bulk viscosity π versus time π‘ for π = 0.5,π0 = 1/3, πΎ = 1/4, π½ = 1.5, πΌ = 0.1, π΄1 = 1, π 1 = 1, π = 1.6, andπ = 0.097.
From (9), with the help of (31), we found the string tensiondensity (π) as
π =2 (π β 1) (ππ‘ β π + 3)
(π + 1) (4π + π) π‘2 (2π β ππ‘)2. (32)
From (10) with the help of (14) and (32), we found the restenergy density (π):
π =2
8π + π (3 β 5π0 + 5πΎ)[
π΄14
π‘ (2π β ππ‘)2
+π΄15
π‘2 (2π β ππ‘)2
β3πΌ2
2π 21 exp [(4/π) arctanh ((ππ‘ β π) /π)]] ,
(33)
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0 5 10 15 20 25 30 350
10203040506070
H
H obtained from special form of qH obtained from linear varying q
t
Figure 5: Hubble parameter π» versus time π‘ for π½ = 1.5, π΄1 = 1,π = 1.6, and π = 0.097.
0 5 10 15 20 25 30 350
0.51
1.52
t
0 5 10 15 20 25 30 350
1020304050
t
V
V
V obtained from SET-II
V obtained from SET-I
Γ1014
Figure 6: Spatial volumeπ versus time π‘ forπ½ = 1.5,π΄1 = 1, π = 1.6,π = 0.097, and π 1 = 1.
where
π΄14 =ππ (1 β π)
(π + 1) (4π + π),
π΄15
=(16π + π + ππ)π2 + 16 (4π + π)π + 16π + 7π β ππ
(π + 1)2 (4π + π).
(34)
From (14), with the help of (33), we obtained the total pressure(π), proper pressure (π), and the coefficient of bulk viscosity(π) as follows:
π =2 (π0 β πΎ)
8π + π (3 β 5π0 + 5πΎ)[
π΄14
π‘ (2π β ππ‘)2
+π΄15
π‘2 (2π β ππ‘)2
β3πΌ2
2π 21 exp [(4/π) arctanh ((ππ‘ β π) /π)]] ,
(35)
π =2π0
8π + π (3 β 5π0 + 5πΎ)[
π΄14
π‘ (2π β ππ‘)2
+π΄15
π‘2 (2π β ππ‘)2
β3πΌ2
2π 21 exp [(4/π) arctanh ((ππ‘ β π) /π)]] ,
(36)
π =πΎ
3 [8π + π (3 β 5π0 + 5πΎ)][
π΄142π β ππ‘
+π΄15
π‘ (2π β ππ‘)
β3πΌ2π‘ (2π β ππ‘)
2π 21 exp [(4/π) arctanh ((ππ‘ β π) /π)]] .
(37)
Similar to that of (26), here alsowe getπ = 1.Thus, forπ = 1,the expressions in (31)β(37) take the form
π΄ = π΅ = πΆ = π 1 exp [2πarctanh(ππ‘ β π
π)] ,
π = 0,
π =3
8π + π (3 β 5π0 + 5πΎ)[
4π‘2 (2π β ππ‘)2
βπΌ2
2π 21 exp [(4/π) arctanh ((ππ‘ β π) /π)]] ,
π =3 (π0 β πΎ)
8π + π (3 β 5π0 + 5πΎ)[
4π‘2 (2π β ππ‘)2
βπΌ2
2π 21 exp [(4/π) arctanh ((ππ‘ β π) /π)]] ,
π =3π0
8π + π (3 β 5π0 + 5πΎ)[
4π‘2 (2π β ππ‘)2
βπΌ2
2π 21 exp [(4/π) arctanh ((ππ‘ β π) /π)]] ,
π =πΎ
2 [8π + π (3 β 5π0 + 5πΎ)][
4π‘ (2π β ππ‘)
βπΌ2π‘ (2π β ππ‘)
2π 21 exp [(4/π) arctanh ((ππ‘ β π) /π)]] .
(38)
3.2.1. Some Physical Properties of the Model. Here somephysical properties of the model are given below:
(i) The spatial volume π = π 1 exp[(2/π) arctanh((ππ‘ βπ)/π)].
(ii) The scalar of expansion π = 6/π‘(2π β ππ‘).(iii) The mean Hubble parameterπ» = 2/π‘(2π β ππ‘).(iv) The mean anisotropy parameter π΄
β= 0.
(v) The shear scalar π2 = 0.
In this model, the energy density (π), proper pressure (π),total pressure (π), coefficient of bulk viscosity (π), andHubble
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parameter (π») gradually decrease with the evolution of timebut when π‘ =βΌ 33, all of them diverge (see Figures 1β5).Spatial volume (π) increases with the evolution of time up toπ‘ =βΌ 33; after that it also diverges (see Figure 6). This modelis isotropic as the average anisotropy parameter is zero and itis also shear-free. In this model also cosmic string does notsurvive.
4. Graphical Results
See Figures 1, 2, 3, 4, 5, and 6.
5. Concluding Remarks
In this work we have studied the Bianchi type-V bulk viscousstring cosmological model in π(π , π) theory of gravity withvariable deceleration parameters.This work is an extension ofthe earlier work of Naidu et al. [50], in which they solve thefield equations with constant deceleration parameter alongwith two other conditions. Here we found that in one ofour models the behavior of the physical and kinematicalparameters is same as that of Naidu et al. [50] and theseparameters in our other model behave differently. In both themodels cosmic strings do not survive. We observed that thetype of time variations of deceleration parameter consideredhere does not affect the nonexistence of cosmic string inthis model. Hence the consideration of variable decelerationparameter does not contribute towards the existence ofcosmic strings in this theory for Bianchi type-V space time.We have also investigated the stability of our system by theuse of speed of sound in viscous fluid. The condition for thestability of the theory is πΆ2
π = ππ/ππ β₯ 0. In both of our
models we obtained thatπΆ2π = π0βπΎ. Here the theory is stable
if π0 β πΎ β₯ 0; that is, π0 β₯ πΎ.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
Acknowledgment
The authors are grateful to the referee for his valuable sugges-tions and comments which led to the quality of presentationof the paper.
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