Anisotropic Cosmological Model of Cosmic String with Bulk Viscosity in Lyra Geometry

Anisotropic Cosmological Model of Cosmic

String with Bulk Viscosity in Lyra Geometry

R.N.Patra

P.G. Department of Mathematics, Berhampur University, Odisha(India)

Abstract

A four dimensional spherically symmetric cosmological model with cosmic string and bulk viscosity in Lyra Geometry has been considered. The solution for cloud string model𝜌 + 𝜆𝑠= 0yields that, the spatial volume

decreases with increase of time, i.e. the universe is contracting with a constant deceleration and shows anisotropy throughout the evolution. Themodel is non-rotating, not a recurrent space and also not a space of constant curvature in generalized curved space.

Keywords: Spherically Symmetric, Lyra Geometry, Cosmic String, Bulk Viscosity, Recurrent Space, Space of Constant Curvature, Deceleration Parameter.

1. Introduction:

Einstein’s idea of geometrizing gravitation in general theory of relativity motivated others to

geometrize other physical fields. Lyra [1] proposed a modification of Riemannian geometry in which he

introduced a gauge function to remove the non-integrability of lengh of a vector under paralled transport.

Various Cosmological models in Lyra manifold has been constructed by Rahaman[2], singh [3], Mohantry [4]

and Reddy [5].

The study of string theory is important in the early stage of evolution of the universe before the particle

creation. Cosmic strings are considered as two lines of concentrated energy forming a tangled web permitting

the entire universe with closed loops. Kibble [6] Vilenkin [7] believed that strings may be one source of density

for large scale structures of the universe. The desirable thing for the string models is that, either the strings fade

away at a certain epoch of cosmic evolution or it has a particle dominated future asymptote with barely visible

strings. This fact attracted many researchers for further studies.

2. Field Equation and the Cosmological Model:

Here we have considered the four dimensional spherically symmetric metric in the form

2 2

2 2 2 2 2 2

sin

d

e

dt

r

d

r

dr

e

ds



















 (1)

Where



and



are functions of ‘t’ only. Assuming the coordinates to be co-moving i.e.

u

1



u

2



u

3



0

and

u

₄2



g

₄₄ and the displacement vector in the form

𝜙𝑖= (𝛽, 0 ,0, 0) (2)

(where



is a constant), the energy momentum tensor

T

_ij for cosmic string with bulk viscosity is

𝑇𝑖𝑗 = 𝜌 𝑢𝑖𝑢𝑗− 𝜆𝑠𝑥𝑖𝑥𝑗− 𝜉𝜃(𝑢𝑖𝑢𝑗 − 𝑔𝑖𝑗) , (3)

𝜆𝑠→String tenson density

𝜉 → Bulk viscous coefficient



i

u

Four velocity vector



ij

g

covariant fundamental tensor

𝑥𝑖 _{→direction of anisotropy of cosmic string}

satisfying

u

u





x

i

x

1



1

i

i and



0

i i

x

u

. (4)

The Einstein’s field equation based on Lyra’s manifold is given by

ij m

m ij j

i ij

ij

Rg

g

T

R





















4

3

2

3

2

1

, (5)

where

R



Rici

Scalar.

Using equations (2),(3) and(4) the explicit form of field equation (5) for the line element (1) are obtained as

−1₄𝑒𝜆−𝜇_𝜆 4 2₋𝑒𝜆 −𝜇

8 𝜆4𝜇4− 3 2𝛽

2₊3 4𝑒

𝜆_𝛽2_{= −𝜒𝜆}

𝑠+ 𝜒𝜉𝜃 𝑒𝜆(6)

2 4

2

3

_



_



r

(7)

𝑟2 2𝜆44𝑒

−𝜇 ₊𝑟2 2 𝜆4

2_𝑒−𝜇₋𝑟2 8𝜆4𝜇4𝑒

−𝜇₋3 2𝛽

2₊3 4𝑟

2_𝛽2_{= 𝜒𝜉𝜃𝑟}2_{. (8)}

−3₂𝛽2_{𝑐𝑜𝑠𝑒𝑐}2_{𝜃 +}3

4𝑟2𝛽2= 𝜒𝜉𝜃𝑟2.(9)

−1₈𝜆4𝜇4𝑒−𝜇 +𝑒

−𝜆

𝑟2 −

1 𝑟2−

9 4𝛽

2_{= −𝜒𝜌. (10)}

3. Cosmological Solution:

From equation (7), we get the value of ‘𝜆’ as

c

rt







2

2

3

_



, (11)

where ‘c’ is the constant of integration.

Adding equation (6) and (10), we get

−1₄𝜆42𝑒𝜆−𝜇−𝑒

𝜆 −𝜇

8 𝜆4𝜇4− 1 8𝜆4𝜇4𝑒

−𝜇₊𝑒−𝜆 𝑟2 −

15

4𝛽

2₋ 1

𝑟2+

3 4𝛽

2_𝑒𝜆 _{= −𝜒 𝜆}

𝑠+ 𝜌 + 𝜒𝜉𝜃𝑒𝜆 (12)

Special Case:

Cloud String Model









_s



0



:

−1₄𝜆42𝑒−𝜇 − 1 8𝜆4𝜇4𝑒

−𝜇₋1 8𝜆4𝜇4𝑒

−𝜇−𝜆 ₊𝑒−2𝜆 𝑟2 −

15

4 𝛽

2₊ 1

𝑟2 𝑒−𝜆+

3 4𝛽

2_{= 𝜒𝜉𝜃 (13)}

Multiplying equation (12) with ‘𝑟2_{’ and equating its LHS of equation (9) and using}

‘𝜆44= 0’ from equation (11), we get

3 4𝜆4

2_𝑒−𝜇₊𝑟2 8 𝜆4𝜇4𝑒

−𝜇−𝜆_{= −} 15 4 𝑟

2_𝛽2_{+ 1 𝑒}−𝜆_{+ 𝑒}−2𝜆₊3 2𝛽

2_{. (14)}

Putting the value of ‘𝜆’ from equation (12) in equation (14), we get

−27₁₆𝑟2_𝛽4_𝑒−𝜇₋ 3 16𝑟

3_𝛽2_𝜇 4𝑒 −𝜇+

3

2𝛽2 𝑟 𝑡−𝑐 = − 15 𝑟2𝛽2+4

4 𝑒

3

2𝛽2 𝑟 𝑡−𝑐 + 𝑒 3 𝛽2 𝑟 𝑡−2𝑐 +3

2𝛽 2 ₍₁₅₎

To avoid the complicacy in finding the solution,

let us take, 𝜇4= 0 or 𝜇 = 𝐾(constant) (16)

for a particular case𝑡 = ln

27

16𝑟2𝛽4𝑒−𝐾 − 3 2𝛽2 −15 𝑟2𝛽 2+4

4

+ 𝑐 _{3𝑟 𝛽}2₂ .

Using equation (12) and (16) the metric (1) takes the form

𝑑𝑠2_{= −𝑒} −3₂𝑟2 𝛽 𝑡+𝑐 _𝑑𝑟2_{− 𝑟}2 _𝑑𝜃2_{− 𝑟}2_𝑠𝑖𝑛2_{𝜃 𝑑𝜙}2_{+ 𝑒}𝐾 _𝑑𝑡2_{. (17)}

4. Physical and Geometrical Properties:

1. Taking



= constant, at initial epoch



t



0



, the metric (11) becomes flat. As ‘t’ increases, the first dimension contracts, but the others don’t change.

2. Volume 𝑉 = −𝑔

1 2= 𝑒 −

3 4𝛽2 𝑟 𝑡+

𝐾′

2 _𝑟2 _{𝑠𝑖𝑛𝜃, where 𝐾}′ _{= 𝑐 + 𝐾 =constant.}

Here we observe that, the spatial volume of the universe decreases with the increase ofcosmic time and may

collapse shortly which has shown in the



V

~

t



graph.

3. The mean anisotropy parameter 𝐴 =1₂ ΔH_𝐻i 2 2

𝑖=1 =1₂ i.e.𝐴 ≠ 0. Hence the model show

anisotropy through out the evolution.

0 0.05 0.1 0.15 0.2

1 2 3 4 5 6 7 8 9 10

V

o

lu

m

e

4. The model of the universe is non rotating since the verticity tensor

 

w

_ij



0

5. It is found that

R

_ij

Rg

_ij

4

1



. Hence the space of the string model is not a Einstein’s space.

6. the space of constant curvature is given by



hj ik hk ij



hijk

k

g

g

g

g

R





(18)

where

















_

















_j ik_j

k hj k

h ij j

i hk hijk

rx

rx

g

x

x

g

r

rx

x

g

rx

rx

g

r

R

2 2

2 2

2

2

1



b



hj a ik ab b hk a ij

ab

g

g













= Riemannian curvature tensor.

K

is a constant known as the curvature of Riemannian manifold.

We found that, the value of

K

after comparing the existing values of L.H.S. and R.H.S. of equation (18) is not an unique constant. So our string space time in Lyra’s manifold is not a space of constant curvature.

7. Recurrent Space:

hijk e hijk

l

K

R

R

,



For our matric, the covariant derivatives of the existing components of Riemannian curvature tensor vanishes.

Which implies that, ultimately the corresponding values of

K

_l are also zero,



₁

,

₂

,

₃

,

₄

 



0

,

0

,

0

,

0





k

k

k

k

K

_l .

Hence the matrics is not a recurrent space.

8.The Hubbleparameter (H) = 1₂ 𝐻1+ 𝐻2 =₂1 𝜆_𝜆4+𝜇_μ4 = 𝛽

2 _𝑟

2 𝛽2 _{𝑟 𝑡−}2 3𝑐

, which is a function of ‘𝑡’. The 𝐻~𝑡

graph shows the necessary variation.

0 0.5 1 1.5 2

1 2 3 4 5 6 7 8 9 10

H

u

b

b

le

p

ara

m

e

te

r

9. The scalar expansion 𝜃 = 2𝐻 = 𝛽

2 _𝑟

𝛽2 _{𝑟 𝑡−}2 3𝑐

, which means that the volume expansion depend on the

cosmic time.

10. The deceleration parameter ‘𝑞’ is defined as

𝑞 =_𝑑𝑡𝑑 _𝐻1 − 1 = 1, as 1 > 0,

the model of the universe is a decelerating one, where the spatial volume decreases with the increase in cosmic time.

5.Conclusion:-We havestudied various physical and geometrical properties of a fourdimensional cosmological model of cosmic string with bulk viscosity in Lyra geometry. We found the exact cosmological solution for the

value of ‘𝜆’ but using cloud string model ‘𝜇’ is found to be independent of cosmic time. The verticity tensor,

0



ij

w

indicates the non-rotating nature of the universe. The deceleration parameter (q) is grater than zero

shows that the spatial volume decreases uniformly with ‘𝑡’ and ultimately the universe may collapse at some

time. The mean anisotropy parameter ‘𝐴’ is not equal to zero shows the anisotropy nature of the model. Also it

is observed that the universe is not a space of constant curvature, neither it is an Einstein’s space nor a recurrent

space.

References:

1. G.Lyra, Math. Z., 54,(1951), 52.

2. F.Rahaman, S.Chakrabarty, N.Begum, M.Hossain and M.Kalam, FIZIKAB, 11(2002), 59. 3. G.P.Singh, R.V. Deshpande and T.Singh, Pramana. J. Phys, 63, (2004), 937.

4. G.Mohanty, G.c. Samanta and K.L.Mahanta, Communication in Physics, 17(2007), 213. 5. D.R.K.Reddy, Astrophys, Space Sci., 300(2005), 381.