String Cosmology in Anisotropic Bianchi-II Space-time

0.23 MB·English

by  Suresh Kumar

String Cosmology in Anisotropic Bianchi-II Space-time
Suresh Kumar
Department of Applied Mathematics,
Delhi Technological University (Formerly Delhi College of Engineering),
Bawana Road, Delhi-110 042, India. E-mail: [email protected]
Abstract
The present study deals with a spatially homogeneous and anisotropic Bianchi-II cosmological model
1 representingmassivestrings. Theenergy-momentumtensor,asformulated byLetelier(1983), hasbeenused
1 to construct a massive string cosmological model for which the expansion scalar is proportional to one of
0
the components of shear tensor. The Einstein’s field equations have been solved by applying a variation
2
law for generalized Hubble’s parameter that yields a constant value of deceleration parameter in Bianchi-II

n space-time. A comparative study of accelerating and decelerating modes of the evolution of universe has
a been carried out in the presence of string scenario. The study reveals that massive strings dominate the
J
early Universe. The strings eventually disappear from the Universe for sufficiently large times, which is in

4 agreement with the current astronomical observations.
1
Keywords: Massive string, Bianchi-II model, Accelerating universe

] PACS number: 98.80.Cq, 04.20.-q,04.20.Jb
c
q
-
r 1 Introduction
g
[
In recentyears,there hasbeen considerableinterestinstring cosmology. Cosmic stringsaretopologicallystable

3
objects which might be found during a phase transition in the early universe (Kibble [1]). Cosmic strings play
v
an important role in the study of the early Universe. These arise during the phase transition after the big bang
1
8 explosion as the temperature goes down below some critical temperature as predicted by grand unified theories
6 (Zel’dovichetal. [2];Kibble[1,3];Everett[4];Vilenkin[5]). Itisbelievedthatcosmicstringsgiverisetodensity
0 perturbationswhichleadtotheformationofgalaxies(Zel’dovich[6]). However,recentobservationssuggestthat
.
0 cosmicstringscannotbewhollyresponsibleforeithertheCMBfluctuationsortheobservedclusteringofgalaxies
1 [7, 8]. The cosmic strings have stress-energy, and couple to the gravitational field. Therefore, it is interesting
0 to study the gravitationaleffects that arise from strings. The pioneering work in the formulationof the energy-
1
momentum tensor for classical massive strings was done by Letelier [9] who considered the massive strings to
:
v be formed by geometric strings with particle attached along its extension. Letelier [10] first used this idea in
i obtainingcosmologicalsolutionsinBianchi-IandKantowski-Sachsspace-times. Stachel[11]hasstudiedmassive
X
string.
r
a The present day Universe is satisfactorily described by homogeneous and isotropic models given by the
FRWspace-time. Butatsmallerscales,the Universeisneitherhomogeneousandisotropicnordoweexpectthe
Universeinitsearlystagestohavetheseproperties. Homogeneousandanisotropiccosmologicalmodelshavebeen
widelystudiedintheframeworkofgeneralrelativityinthesearchofarealisticpictureoftheUniverseinitsearly
stages. Althoughthese aremorerestrictedthan the inhomogeneousmodels whichexplainanumber ofobserved
phenomena quite satisfactorily. A spatially homogeneous Bianchi model necessarily has a three-dimensional
group,whichactssimplytransitivelyonspace-likethree-dimensionalorbits. Hereweconfineourselvestomodels
of Bianchi-II.Asseo andSol [12] emphasizedthe importance of Bianchitype-II Universe. Bianchitype-II space-
time has a fundamental role in constructing cosmological models suitable for describing the early stages of
evolution of Universe.
Roy and Banerjee [13] have dealt with locally rotationally symmetric (LRS) cosmologicalmodels of Bianchi
type-IIrepresentingcloudsofgeometricalaswellasmassivestrings. Wang[14]studiedtheLeteliermodelinthe
context of LRS Bianchi type-II space-time. Recently, Pradhan et al. [15, 16] and Amirhashchi and Zainuddin
[17]obtainedLRSBianchitype II cosmologicalmodels with perfectfluid distributionofmatter and stringdust,
respectively. Belinchon[18,19]studiedBianchitype-II space-timeinconnectionwithmassivecosmicstringand
perfectfluidmodelswithtimevaryingconstantsunderthe self-similarityapproachrespectively. Recently,Tyagi
and Sharma [20] have investigated string cosmologicalmodels in Bianchi type-II space-time.
Motivated by the above discussions, in this paper, we have investigated a new class of Bianchi type-II
cosmologicalmodelsforacloudofstringsbyusingthelawofvariationforgeneralizedmeanHubble’sparameter.
1
This approach is different from what the other authors have adapted. The paper is organized as follows. The
metric and the field equations are presented in Section 2. Section 3 deals with exact solutions of the field
equations with cloud of strings. Physical behavior of the derived model is elaborated in detail. Finally, in
Section 4, concluding remarks are given.
2 The metric and field equations
We consider totally anisotropic Bianchi type-II line element, given by
ds2 =−dt2+A2(dx−zdy)2+B2dy2+C2dz2, (1)
where the metric potentials A, B and C are functions of t alone. This ensures that the model is spatially
homogeneous.
The Einstein’s field equations ( in gravitational units c=1,8πG=1) read as
Gi =−Ti, (2)
j j
where Gi = Ri − 1Rgi is the Einstein tensor. The energy-momentum tensor Ti for a cloud of massive strings
j j 2 j j
and perfect fluid distribution is taken as
Ti =(ρ+p)viv +pgi −λxix , (3)
j j j j
where p is the isotropic pressure; ρ is the proper energy density for a cloud strings with particles attached to
them;λisthestringtensiondensity;vi =(0,0,0,1)isthefour-velocityoftheparticles,andxi isaunitspace-like
vector representing the direction of string. The vectors vi and xi satisfy the conditions
v vi =−x xi =−1, vix =0. (4)
i i i
Choosing xi parallel to ∂/∂z, we have
xi =(0,0,C−1,0). (5)
Here the cosmic string has been directed along z-direction in order to satisfy the condition T1 = T2. As a
1 2
result, the off-diagonalcomponentof Einsteintensor,viz., G1 =z(T2−T1) vanishes. A detailedanalysis about
2 2 1
the choice of energy-momentum tensor for Bianchi type-II models is given by Saha [21].
If the particle density of the configuration is denoted by ρ , then
p
ρ=ρ +λ. (6)
p
TheEinstein’sfieldequations(2)forlineelement(1)withenergy-momentumtensor(3),leadtothefollowing
set of independent differential equations:
B¨ C¨ B˙C˙ 3 A2
+ + − =−p, (7)
B C BC 4B2C2
C¨ A¨ C˙A˙ 1 A2
+ + + =−p, (8)
C A CA 4B2C2
A¨ B¨ A˙B˙ 1 A2
+ + + =−p+λ, (9)
A B AB 4B2C2
A˙B˙ B˙C˙ C˙A˙ 1 A2
+ + − =ρ. (10)
AB BC CA 4B2C2
Here, and in what follows, an over dot indicates ordinary differentiation with respect to t. The energy
conservation equation Tij =0, leads to the following expression:
;j
A˙ B˙ C˙ C˙
ρ˙+(ρ+p) + + −λ =0, (11)
A B C C
!
which is a consequence of the field equations (7)-(10).
2
3 Solutions of the Field Equations
Equations(7)-(10)arefourequationsinsixunknownparametersA,B,C,p,ρandλ. Twoadditionalconstraints
relating these parameters are required to obtain explicit solutions of the system.
First, we utilize the special law of variation for the Hubble’s parameter given by Berman [22], which yields
a constant value of deceleration parameter. Here, the law reads as
−n
H =ℓ(ABC) 3, (12)
where ℓ > 0 and n ≥ 0 are constants. Such type of relations have already been considered by Berman and
Gomide [23] for solving FRW models. Later on, many authors (see, Kumar and Singh [24], Akarsu and Kilinc
[25]andreferencestherein)havestudiedflatFRWandBianchitypemodelsbyusingthespeciallawforHubble’s
parameter that yields constant value of deceleration parameter.
Considering(ABC)13 astheaveragescalefactoroftheanisotropicBianchi-IIspace-time,theaverageHubble’s
parameter may be written as
1 A˙ B˙ C˙
H = + + . (13)
3 A B C
!
Equating the right hand sides of (12) and (13), and integrating, we obtain
ABC =(nℓt+c1)n3, (n6=0) (14)
ABC =c3e3ℓt, (n=0) (15)
2
where c and c are constants of integration. Thus, the law (12) provides power-law (14) and exponential-law
1 2
(15) of expansion of the Universe.
Following Pradhan and Chouhan [26], we assume that the component σ1 of the shear tensor σj is propor-
1 i
tional to the expansion scalar (θ). This condition leads to the following relation between the metric potentials:
A=(BC)m, (16)
where m is a positive constant.
Now, subtracting (8) from (7), and taking integral of the resulting equation two times, we get
B 1 A3 1
=c exp dt dt+c dt , (17)
4 3
A ABC BC ABC
(cid:20)Z (cid:26) Z (cid:27) Z (cid:21)
where c and c are constants of integration. In the following subsections, we discuss the string cosmology
3 4
using the power-law (14) and exponential-law (15) of expansion of the Universe.
3.1 String Cosmology with Power-law
Solving the equations (14), (16) and (17), we obtain the metric functions as
3m
A(t)=(nℓt+c1)n(m+1) , (18)
B(t)=c4(nℓt+c1)n(m3m+1) exp (m2ℓ+2M1)2(nℓt+c1)n6((mm−+11))+2+ ℓ(nc−3 3)(nℓt+c1)n−n3 , (19)
(cid:20) (cid:21)
C(t)=c−41(nℓt+c1)n3((1m−+m1)) exp −(m2ℓ+2M1)2(nℓt+c1)n6((mm−+11))+2− ℓ(nc−3 3)(nℓt+c1)n−n3 , (20)
(cid:20) (cid:21)
where M =(9m−3+mn+n)(3m−3+mn+n) and n6=3.
In the special case n=3, we have
m
A(t)=(3ℓt+c1)m+1 , (21)
3mℓ+c3(m+1) (m+1)2 4m
B(t)=c4(3ℓt+c1) 3ℓ(m+1) exp 144ℓ2m2(3ℓt+c1)m+1 , (22)
(cid:20) (cid:21)
C(t)=c−41(3ℓt+c1)3(1−m3ℓ)(ℓm−+c31()m+1) exp −(1m44+ℓ2m1)22(3ℓt+c1)m4m+1 , (23)
(cid:20) (cid:21)
3
Thus, the metric (1) is completely determined.
The expressions for the isotropic pressure (p), the proper energy density (ρ), the string tension (λ) and the
particle density (ρ ) for the above model are obtained as
p
p = 3ℓ2[n(m+1()m−+3(1m)22−m+1)](nℓt+c1)−2+ 4(93m(m−+31+)(mn+n+1)n)(nℓt+c1)−n6((1m−+m1))
+3c3ℓm(1+−12m)(nℓt+c1)−n3−1− ℓ2(9m−(m3++1m)2n+n)2(nℓt+c1)−[1n2((m1−+m1))−2]
−ℓ(9m2−c33(m++m1n)+n)(nℓt+c1)−[n3((3m−+m1))−1]−c23(nℓt+c1)−n6, (24)
ρ = 9ℓ(2mm(+2−1)2m)(nℓt+c1)−2+ 41(59−m3−3m3+−mmnn+−nn)(nℓt+c1)−n6((1m−+m1))
+3c3ℓm(1+−12m)(nℓt+c1)−n3−1− ℓ2(9m−(m3++1m)2n+n)2(nℓt+c1)−[1n2((m1−+m1))−2]
−ℓ(9m2−c33(m++m1n)+n)(nℓt+c1)−[n3((3m−+m1))−1]−c23(nℓt+c1)−n6, (25)
λ= 3ℓ2(2m−1)(3−n)(nℓt+c1)−2+2(nℓt+c1)−n6((1m−+m1)) , (26)
m+1
ρp = 3ℓ2[3m(2−m))+(m(3+−1n))2(1−2m)(m+1)](nℓt+c1)−2+ 3(143(9−m3−5m3+−m3mnn+−n3)n)(nℓt+c1)−n6((1m−+m1))
+3c3ℓm(1+−12m)(nℓt+c1)−n3−1− ℓ2(9m−(m3++1m)2n+n)2(nℓt+c1)−[1n2((m1−+m1))−2]
−ℓ(9m2−c33(m++m1n)+n)(nℓt+c1)−[n3((3m−+m1))−1]−c23(nℓt+c1)−n6. (27)
The above solutions satisfy the energy conservation equation (11) identically, as expected.
2500
2000
A(t)
B(t)
1500
C(t)
1000
500
0
0 1 2 3 4 5
t
Figure 1: Scale factors vs time with n=0.5, m=0.4, ℓ=2, c =c =1 and c =2.
1 3 4
We observe that all the parameters diverge at t = −c /nℓ. Therefore, the model has a singularity at
1
t = −c /nℓ, which can be shifted to t = 0 by choosing c = 0. This singularity is of Point Type as all the
1 1
scale factors vanish at t = −c /nℓ. The cosmological evolution of Bianchi-II space-time is expansionary since
1
all the scale factors monotonically increase with time (see, Fig.1). So, the Universe starts expanding with a big
bang singularity in the derived model. The parameters p, ρ, ρ and λ start off with extremely large values. In
p
particular, the large values of ρ and λ in the beginning suggest that strings dominate the early Universe. For
p
4
sufficientlylargetimes,ρ andλbecomenegligible. Therefore,thestringsdisappearfromtheUniverseforlarger
p
times. That is why, the strings are not observable in the present Universe.
The rates of expansion in the direction of x, y and z are given by
A˙ 3mℓ
H = = (nℓt+c )−1 , (28)
x 1
A m+1
Hy = BB˙ = m3m+ℓ1(nℓt+c1)−1+ ℓ(9m−m3++m1n+n)(nℓt+c1)−n6((1m−+m1))+1+c3(nℓt+c1)−n3 , (29)
Hz = CC˙ = 3(m1−+m1)ℓ(nℓt+c1)−1− ℓ(9m−m3++m1 n+n)(nℓt+c1)−n6((1m−+m1))+1−c3(nℓt+c1)−n3. (30)
The average Hubble’s parameter, expansion scalar and shear of the model are, respectively given by
1
H = (H +H +H )=ℓ(nℓt+c )−1, (31)
x y z 1
3
θ =3H =3ℓ(nℓt+c )−1, (32)
1
1
σ2 = (H −H )2+(H −H )2+(H −H )2
x y y z z x
6
= 3ℓ(cid:2)2(m(2m+−1)21)2(nℓt+c1)−2− 41(59−m3−3m3+−mmnn(cid:3)+−nn)(nℓt+c1)−n6((1m−+m1))
−3c3ℓm(1+−12m)(nℓt+c1)−n3−1+ ℓ2(9m−(m3++1m)2n+n)2(nℓt+c1)−[1n2((m1−+m1))−2]
+ℓ(9m2−c3(3m++m1n)+n)(nℓt+c1)−[n3((3m−+m1))−1]+c23(nℓt+c1)−n6. (33)
The spatial volume (V) and anisotropy parameter (A¯) are found to be
V =ABC =(nℓt+c1)n3, (34)
A¯ = 32Hσ22 = 2((m2m+−1)12)2 − 6ℓ125(9−m33−m3−+mmnn−+nn)(nℓt+c1)−n6((1m−+m1))+2
−2cℓ3((m1−+21m) )(nℓt+c1)−n3+1+ 3ℓ4(9m2−(m3++1m)2n+n)2(nℓt+c1)−[1n2((m1−+m1))−4]
+3ℓ3(9m4c−3(m3++m1)n+n)(nℓt+c1)−[n3((3m−+m1))−3]+ 23cℓ232(nℓt+c1)−n6+2. (35)
The value of DP (q) is found to be
H˙
q =−1+ =n−1, (36)
H2
which is a constant. A positive sign of q, i.e., n > 1 corresponds to the standard decelerating model whereas
the negative sign of q, i.e., 0 < n < 1 indicates acceleration. The expansion of the Universe at a constant rate
corresponds to n=1, i.e., q =0. Also, recent observations of SN Ia [27]-[34] reveal that the present Universe is
accelerating and value of DP lies somewhere in the range −1<q <0. It follows that in the derived model, one
can choose the values of DP consistent with the observations.
From the above results, it can be seen that the spatial volume is zero at t = −c /nℓ, and it increases with
1
the cosmic time t. The parameters H , H , H , H, θ and σ diverge at the initial singularity. These parameters
x y z
decrease with the evolution of Universe, and finally drop to zero at late times provided m < 1. The mean
anisotropy parameter asymptotically approaches to 2(2m−1)2 for n < 3 and m < 1. Thus, the dynamics of the
(m+1)2
mean anisotropy parameterdepends on the values of n and m. The model does not approachisotropy provided
m6=0.5 as may be observed from Fig. 1. In case m=0.5, at late times, the directional scale factors vary as
A(t)≈(nℓt+c1)n1, B(t)≈(nℓt+c1)n1, C(t)≈(nℓt+c1)n1.
5
Therefore,isotropyis achievedinthe derivedmodelform=0.5. Since the present-dayUniverseis isotropic,
we consider m=0.5 in the remaining discussion of the model.
For n < 1, the model is accelerating whereas for n > 1 it goes to decelerating phase. In what follows, we
compare the two modes of evolution through graphical analysis of various parameters. We have chosen n = 2,
i.e.,q =1todescribethedeceleratingphasewhiletheacceleratingmodehasbeenaccountedbychoosingn=0.5,
i.e., q =−0.5. The other constants are chosen as ℓ=6, c =1, c =1, m=0.5.
1 3
40
20
0
p−20
q =−0.5
−40 q =1
−60
−80
0 1 2 3 4 5
t
Figure 2: Pressure vs time.
80
60
q =−0.5
ρ40 q =1
20
0
0 1 2 3 4 5
t
Figure 3: Rest energy density vs time.
Fig. 2depictsthevariationofpressureversustimeinthetwomodesofevolutionoftheUniverse. Weobserve
thatthepressureispositiveinthedeceleratingUniversewhichdecreaseswiththeevolutionoftheUniverse. But
in the accelerating phase, negative pressure dominates the Universe, as expected. In both cases, the pressure
becomes negligible at late times.
The rest energy density has been graphed versus time in Fig. 3. It is evident that the rest energy density
remains positive in both modes of evolution. However, it decreases more sharply with the cosmic time in the
decelerating Universe.
Fig. 4 and Fig. 5 show the behavior of particle energy density and string tension versus time in the
decelerating and accelerating modes, respectively. We see that ρ >λ, i.e., the particle energy density remains
p
larger than the string tension density during the cosmic expansion , especially in early Universe. This shows
that massive strings dominate the early Universe (see, Refs. [1, 19]). Further, it is observedthat for sufficiently
large times, ρ and λ tend to zero. Therefore, the strings disappear from the Universe at late times.
p
According to Ref. [19], since there is no direct evidence of strings in the present-day Universe, we are in
general, interested in constructing models of a Universe that evolves purely from the era dominated by either
6
80
60
ρ
40 p
λ
20
0
0 1 2 3 4 5
t
Figure 4: Particle energy density and string tension vs time for q =1.
80
60
ρ
40 p
λ
20
0
0 1 2 3 4 5
t
Figure 5: Particle energy density and string tension vs time for q =−0.5.
7
geometricstringormassivestringsandendsupinaparticledominatederawithorwithoutremnantsofstrings.
Therefore,theabovemodeldescribestheevolutionoftheUniverseconsistentwiththepresent-dayobservations.
80
60
40 ρ+p
ρ−p
20
0
0 1 2 3 4 5
t
Figure 6: ρ+p and ρ−p vs time for q =−0.5.
From Fig. 3 to Fig. 6, we observe the following:
(i) ρ≥0
(ii) ρ ≥0
p
(iii) ρ+p≥0
(iv) ρ−p≥0.
This shows that the weak and dominant energy conditions are satisfied in the derived model.
3.2 String Cosmology with Exponential-law
Solving the equations (15), (16) and (17), we obtain the metric functions as
3m 3mℓ
A(t)=cm+1 exp t , (37)
2 m+1
(cid:18) (cid:19)
6(m−1)
B(t)=c4c2m3m+1 expm3m+ℓ1t+ 18ℓc22(m3+m1−(m1)+(m1)−21)e6ℓm(m+−11)t− 3cℓ3c3e−3ℓt, (38)
2
 
6(m−1)
C(t)=c−1c3(m1−+m1) exp 3(1−m)ℓt− c2m+1 (m+1)2 e6ℓm(m+−11)t+ c3 e−3ℓt . (39)
4 2  m+1 18ℓ2(3m−1)(m−1) 3ℓc3 
2
 
The expressions for the isotropic pressure, the proper energy density, the string tension and the particle
density for the derived model are obtained as
6(m−1) 3(m−3)
p = −9ℓ2(m2−m+1) + c2m+1 (m+1)e6ℓm(m+−11)t− 2c3c2m+1 (m+1)e3ℓm(m+−13)t
(m+1)2 4(3m−1) 3ℓ(3m−1)
12(m−1)
+3c3(1−2m)e−3ℓt− c2 m+1 (m+1)2e12ℓm(m+−11)t− c23e−6ℓt, (40)
c3(m+1) 9ℓ2(3m−1)2 c6
2 2
8
6(m−1) 3(m−3)
ρ = 9ℓ2m(2−m) + c2m+1 (5−11m)e6ℓm(m+−11)t− 2c3c2m+1 (m+1)e3ℓm(m+−13)t
(m+1)2 4(3m−1) 3ℓ(3m−1)
12(m−1)
+3c3(1−2m)e−3ℓt− c2 m+1 (m+1)2e12ℓm(m+−11)t− c23e−6ℓt (41)
c3(m+1) 9ℓ2(3m−1)2 c6
2 2
λ= 9ℓ2(2m−1) +2c6(mm+−11)e6ℓm(m+−11)t, (42)
m+1 2
6(m−1) 3(m−3)
ρp = 9ℓ2(1(m++m1−)23m2) + c2m+41(3(m13−−12)5m)e6ℓm(m+−11)t− 2c3c32ℓm(3+m1 −(m1)+1)e3ℓm(m+−13)t
12(m−1)
+3c3(1−2m)e−3ℓt− c2 m+1 (m+1)2e12ℓm(m+−11)t− c23e−6ℓt. (43)
c3(m+1) 9ℓ2(3m−1)2 c6
2 2
The energy conservation equation (11) is satisfied identically by the above solutions, as expected.
The directional Hubble’s parameters are given by
3mℓ
H = , (44)
x
m+1
6(m−1)
Hy = m3m+ℓ1 + c23mℓ+(13m(m−+1)1)e6ℓm(m+−11)t+ cc33e−3ℓt, (45)
2
6(m−1)
Hz = 3ℓm(1+−1m) − c23mℓ+(13m(m−+1)1)e6ℓm(m+−11)t− cc33e−3ℓt. (46)
2
Hence the averagegeneralized Hubble’s parameter is given by
H =ℓ. (47)
Fromequations(44)-(47), weobservethat the directionalHubble’s parametersaretime dependentwhile the
average Hubble’s parameter is constant.
The expressions for kinematical parameters, i.e., the scalar of expansion, shear scalar, the spatial volume,
average anisotropy parameter and deceleration parameter for the derived model are given by
θ =3ℓ, (48)
6(m−1) 3(m−3)
σ2 = 3ℓ2(2m−1)2 − c2m+1 (5−11m)e6ℓm(m+−11)t+ 2c3c2m+1 (m+1)e3ℓm(m+−13)t
(m+1)2 4(3m−1) 3ℓ(3m−1)
12(m−1)
−3c3(1−2m)e−3ℓt+ c2 m+1 (m+1)2e12ℓm(m+−11)t+ c23e−6ℓt, (49)
c3(m+1) 9ℓ2(3m−1)2 c6
2 2
V =c3e3ℓt, (50)
2
6(m−1) 3(m−3)
A¯ = 2(2m−1)2 − c2m+1 (5−11m)e6ℓm(m+−11)t+ 4c3c2m+1 (m+1)e3ℓm(m+−13)t
(m+1)2 6ℓ2(3m−1) 9ℓ3(3m−1)
12(m−1)
−2c3(1−2m)e−3ℓt+ 2c2 m+1 (m+1)2e12ℓm(m+−11)t+ 2c23 e−6ℓt, (51)
ℓ2c3(m+1) 27ℓ4(3m−1)2 3ℓ2c6
2 2
q =−1. (52)
RecentobservationsofSNIa[27]-[34]suggestthattheUniverseisacceleratinginitspresentstateofevolution.
It is believedthat the way Universeis acceleratingpresently; itwill expandatthe fastest possible ratein future
9
and forever. For n = 0, we get q = −1 ; incidentally this value of DP leads to dH/dt = 0, which implies the
greatest value of Hubble’s parameter and the fastest rate of expansion of the Universe. Therefore, the derived
model can be utilized to describe the dynamics of the late time evolution of the actual Universe. So, in what
follows, we emphasize upon the late time behavior of the derived model. At late times, we find
9ℓ2(m2−m+1)
p≈− , (53)
(m+1)2
9ℓ2m(2−m)
ρ≈ , (54)
(m+1)2
9ℓ2(2m−1)
λ≈ , (55)
m+1
9ℓ2(1+m−3m2)
ρ ≈ , (56)
p (m+1)2
3ℓ2(2m−1)2
σ2 ≈ , (57)
(m+1)2
2(2m−1)2
A¯≈ . (58)
(m+1)2
In particular, for m= 1, we have
2
p=−ρ,
λ≈0,
ρ ≈3ℓ2,
p
σ2 ≈0,
A¯≈0.
This shows that vacuum energy dominates the Universe at late times, which is consistent with the ob-
servations. Strings disappear and the Universe evolves with constant particle energy density. The shear and
anisotropy parameter become negligible. So the Universe becomes isotropic.
4 Concluding Remarks
In this paper, a spatially homogeneous and anisotropic Bianchi-II space-time representing massive strings in
general relativity has been studied. The main features of the work are as follows:
• The models are based on exact solutions of the Einstein’s field equations for the anisotropic Bianchi-II
space-time filled with massive strings.
• Thesingularmodel(n6=0)seemstodescribethedynamicsofUniversefrombigbangtothepresentepoch
while the non-singular model (n=0) seems reasonable to project dynamics of future Universe.
• In the derived models, m= 1 turns out to be a condition of isotropy.
2
• In the present models, the weak and dominant energy conditions are satisfied, which in turn imply that
the derived models are physically realistic.
• Thesingularmodelpresentsthedynamicsofstringsintheacceleratinganddeceleratingmodesofevolution
oftheUniverse. IthasbeenfoundthatmassivestringsdominatetheUniverse,whicheventuallydisappear
from the Universe for sufficiently large times. This is in agreement with the astronomical observations.
• The non-singular model for m = 1 predicts a Universe dominated by vacuum energy, which is consistent
2
with the predictions of current observations.
Acknowledgements:
The author is thankful to the anonymous referee for valuable comments on this manuscript.
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