FIVE DIMENSIONAL BIANCHI TYPE-I STRING COSMOLOGICAL MODEL WITH ELECTROMAGNETIC FIELD

In this paper we defined the five dimensional Bianchi type-I string cosmological models with electromagnetic field. Here we considered that the energy momentum tensor is equal to the sum of the energy density, tension density and electromagnetic field. In order to get determinate solutions of the Einstein's field equations, we assume the average scale factor as a connecting function of time. Some physical and kinematical parameters are also discussed.


INTRODUCTION
In various conditions of beginning moment of cosmological problems higher dimensional cosmological models play a significant role. The investigation of higher dimensional space-time gives a knowledge that our universe is much smaller at beginning moment of expansion as defined presently. A. Chodos and S. Detweiler [1], K.D. Krori and M. Barua [2] have investigated about 6600 JAGAT DAIMARY, RAJSHEKHAR ROY BARUAH the higher dimension. Several authors [3,4,5,6] have studied higher-dimensional cosmological model using different type of scalar-tensor theory of gravitation. G. Samanta and S. Debata [7], have investigated five dimensional Bianchi Type-I string cosmological models in lyra manifold.
Several types of research suggest that our universe is expanding with an accelerated rate. Einstein's theory of gravitation is assumed to be a perfect fluid to derive our expanding universe in FRW space-time. Bull et al. [8] have investigated the alternative cosmology and encapsulate the current position of ΛCDM like a physical theory besides the standard model ΛCDM in expansion cosmology. FRW space-time explains only isotropic and uniform universe on a large scale. On the other hand, present conclusions and arguments propose that there survive an anisotropic stage before approaching to an isotropic one throughout the cosmic development of the universe. R. Bali, and N. K. Chandnani [9], G. Mohanty et al. [10] studied five dimensional LRS Bianchi type-I string cosmological model in Saez and Ballester theory. The anisotropic universe has acquired a greater majority than the isotropic model universes in the theoretical view. Several researchers [11,12,13,14,15,16,17,18] studying anisotropic Bianchi Type-I cosmological models and higher dimensional bianchi type-I cosmological model in several aspects. Presently A. Banerjee et al. [19] have defined Bianchi type I cosmological model with a viscous fluid.
Since string play a very important role in explaining the expansion of the beginning period of our universe, hence lots of authors in current period comprehensively investigated the string cosmological models of the universe. The string has ability to explain together the character and theoretical structure of the beginning universe. String theory is the largest sincerely investigated access to quantum gravity, and by applying this theory we can debate the physics of beginning universe. String theory gives us with a single theoretical configuration, in which the entire matters and forces are homogeneous and explain the beginning period of our universe in terms of vibrating strings in place of particles. Kibble [20] had suggested that, cosmic string are steady line like such as topological objects that are developed at some period in the stage transition in immature beginning stage of our universe. Also, many authors [21, 22, 23, 24, and 25] have investigated that there is a similarity collapsing in the time of the stage transition in the beginning period of universe 6601 FIVE DIMENSIONAL BIANCHI TYPE-I STRING later the big-bang and these strings derive during the cosmic temperature decline under some critical temperature. Hence before the investigation about the beginning period of universe, string can take part important role. Moreover, large scale configurations such as galaxies, and cluster of galaxies are developed by massive closed loops of string. All of these thinks attract many authors to investigate about the string cosmological models. Several authors [26, 27, 28, 29, 30, and 31] have studied about the string cosmological models using different scalar-tensor.
In the presence of electromagnetic fields, the cosmological model takes a crucial part in the expansion of universe and the development of huge scale configuration such as galaxies and other stellar bodies. The current period of accelerated extension of the universe is for the existence of a cosmological electromagnetic field developed throughout expansion. J.B. Jimenez and A.L.
Maroto [32] have investigated that the existence of a temporal electromagnetic field on cosmological scales develops a sufficient cosmological constant that is explained for accelerated extension of the universe. Also, many researchers [33, 34, 35, 36, 37, 38 and 39] have studied about the electromagnetic field by taking different aspects.
In this paper we investigated five dimensional Bianchi type-I string cosmological models with electromagnetic field. In section 1, we discussed about the higher dimension, Bianchi Type-I cosmological model, string and electromagnetic field. In section 2, we assumed the Bianchi Type-I metric and the field equation in general relativity are defined. We have defined the solution of the survival field equations in section 3. In section 4, physical and kinematic parameters are discussed. Section 5, we discussed physical and geometrical character of our model with the help of the graph. Finally, we give conclusion in section 6.

METRIC AND FIELD EQUATIONS
Let us assume the five dimensional Bianchi type-I metric as follows Here A, B, C are functions of cosmic time t.
Also we assume the Einstein field equation ( 8 4 = 1) is as given below represents the Ricci scalar and denotes the Ricci tensor.
Here, for cloud string the energy momentum tensor with having the electromagnetic field can be written as follows where, denotes the rest energy density of the cloud strings with particles attached to them, denotes the tension density and = + , denotes the particle density of the string. Here represents the five velocity vector and represents the unit space-like vector with the following condition We consider the Electromagnetic field as follows, Here denotes the electromagnetic field tensor.
When we compute the magnetic field along the x axis then we get that 15 is the only non-zero component of the electromagnetic field tensor . Hence, when we consider infinite electromagnetic conductivity, we get that 12 = 13 = 14 = 23 = 24 = 25 = 34 = 35 = 0.
Therefore from equation (6), we get Again from equation (6), we get Here we define the spatial volume V, Scalar expansion θ, Hubble parameter H, the deceleration parameter q, the shear scalar 2 and the mean isotropy parameter ∆ for the metric (1) are given by as follows, Here an overhead dot denoted differentiation with respect to cosmic time t.

SOLUTIONS OF THE FIELD EQUATIONS
Here we have to assume two relations. Since the field equations (15) Here n is a constant.

PHYSICAL AND KINEMATICAL PARAMETERS
We define the energy density , string tensor density , and electromagnetic field tensor 15 for the models given by equation (30) from equations (15)-(18) as follows, Particle density is The important physical quantities such as spatial volume V, Hubble parameter H, scalar expansion θ, Shear stress 2 , mean anisotropy parameter ∆ are defined by in the following way Spatial volume V is The Hubble parameter H is The scalar expansion θ is given by And the anisotropy parameter ∆ is given by Also, from Equation (37) and (38) we get, Here we plot all the graphs by taking 1 = 1, 2 = 0, = = .5

PHYSICAL AND GEOMETRICAL INTERPRETATION
(i) From Equation (31), (32) and (34) we obtained that the energy density , tension density and particle density are decreases when time is increases and they will approach a constant finite value at t⟶∞. All of them will increase when time is decrease (i.e. when time is passes) and 6609 FIVE DIMENSIONAL BIANCHI TYPE-I STRING diverge at the initial epoch (i.e., = 0) (as shown in fig-1, fig-2 and fig-4). From there we obtain that the particle controls the universe as time decreases and communicates to total constant finite number of particles in the universe. Also, since the string tension density vanishes more quickly than the particle density so that at late time period our model defines a matter controls universe, this helps the present day's information data.
(ii) When time increases the electromagnetic field 15 and the volume V given by equations (33), (35) increases and they become infinity at infinitely large time. Here for 1 ≠ 0 the electromagnetic field does not vanish. At beginning period of expansion of the universe it has appreciable effect in construction up strings. Also, we obtain that the volume is 2 4 at t=0. This shows that at = 0 the universe first begins with finite volume and later it is expand with increasing time. (as shown in fig-3, fig-5).
(iii) The Hubble parameter H, scalar expansion θ given by equations (36), (37) are decreases when time is increases and vanish for ⟶ ∞. They will become infinity at initial epoch (see fig-6, fig-7).
(iv) From equation (23) we obtain that the deceleration parameter q is constant and negative. From there it is define that our assume model universe inflates and expanding with constant acceleration (since cosmic time t=0) [see fig-10]. At t⟶∞ the shear scalar 2 and anisotropy parameter ∆ which are given by equation (38), (39) are vanish and they will decrease when time is increase. At initial epoch they will diverge (see fig-8, fig-9).
Though we get our models as anisotropic but it cannot bring any contradiction to the current day's conclusion which shows that our universe is isotropy. Because in the time of development of expansion, the initial isotropy do not appear after some process and reaches to isotropy and ultimately it derives into a FRW universe as advised by [43].

CONCLUSION
Here we have investigated five dimensional Bianchi type-I string cosmological models with electromagnetic field. In this paper we have obtain the model given by equation (30) where H>0 and q<0. This represents that the universe displays power law-expansion after the big-bang influence which begin with some finite volume at initial epoch and increases with an acceleration.
Here the tension density vanishes more quickly than particle density. Hence the particle density controls the universe when time increases. For the model given by equation (30), the universe may have some expectation to be anisotropy during the expansion from beginning to late time period.
Current conclusion gives us that there is a distinction in measuring strength of microwaves approaching from various directions of the sky. This motivated us to investigate the universe with anisotropic Bianchi type-I metric as follows to explain our universe in many sensible positions.
Again, many CMB anomalies such as incompatible of temperature anisotropy in the CMB with perfect homogeneous and isotropic FRW model defined by WMAP satellites, systematics and import topologies are an indication that we stay in a globally anisotropic universe. At the stage of inflation, shear decrease and ultimately it becomes into an isotropic step with unacceptable shear.
Hence currently, we need to promote anisotropy in space-time to develop any considerable quantity of shear.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.