# Discussion

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## Applications of Flexible Electronics

### V. Discussion

Graphs are plotted for particular values of the physical parameters and other integration constants From Fig.1 and Fig. 2, it is initially observed that expansion scalar and Hubble Parameter decreases as time increases.

As T , expansion scalar as well as Hubble Parameter will be zero. From Fig. 3 it is observed that Spatial Volume increases as increase in time as V  as T . From Fig. 4, it is observed that as T ,

### 

 i.e. shape of the universe is changing with increasing time.

Fig. 1 Plot of Scalar expansion Vs. Time for k1=k6=0.1 Fig. 2 Plot of Hubble parameter Vs. Time for k1=k6=0.1

Fig. 3 Plot of Spatial volume vs. Time for Fig. 4 Plot of Shear Scalar vs. Time

k1=k6=0.1 for k1=k6=0.1 Conclusion

In this paper, we have obtained Bianchi Type VI0 cosmological model with strange quark matter attached to the string cloud in scalar tensor theory of gravitation proposed by Saez-Ballester (1985). For solving the field equations relation between metric coefficient C and A is used, obtain model is expanding, shearing, non-rotating and do not approach isotropy for large value of time. The rest energy density for the cloud of string with particles attached to them

###   

, string tension density

###   

s and particle energy density tends to zero as

### T

. Also the kinematical variable

and

### 

tends to zero as T becomes large.

References

[1] Saez, D., Ballester, V. J.: Phys. Lett.A113, 467(1985).

[2] Brans, C. and Dicke, R.H.: Phys.Rev.124, 925. (1961).

[3] Norvedt, K. Jr.: Astrophys. J.161, 1069, (1970).

[4] Reddy, D.R.K., Venkateswara Rao, N.: Astrophys. Space Sci. 277, 461 (2001).

[5] Yavuz, I., Yilmaz, I., Baysal, H.: Int. J. Mod. Phys. D 14 1365-1372 (2005).

[6] Yilmaz, I.: Gen. Rel. Grav. 38, 1397-1406 (2006).

[7] Yilmaz, I.: Phys. Rev. D 71, 103503 (2005).

[8] Adhav, K.S., Nimkar, A.S., Naidu, R.L.: Astrophys. Space Sci.312, 165-169 (2007).

[9] Ugale, M. R.: International Journal of Mathematical Archive 5(3), 280-282(2014).

[10] Pund, A. M., Nimkar, A.S.: International Journal of Mathematical Archive 6(9), 18-21(2015).

[11] Reddy, D.R.K. Rao, M.V.S.: Astrophys. Space Sci. 302, 157 (2006).

[12] Adhav, K.S., Nimkar, A.S., Naidu, R.L.: Astrophys. Space Sci.312, 165-169 (2007).

[13] Adhav, K.S., Mete, V. G., Pund, A. M., Raut, R.B.: Journal of vectorial Relativity 5(4), 26-33(2010).

[14] Katore, S. D., Adhav, K. S., Sancheti, M. M.: Prespacetime Journal, 2(1), (2011).

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[16] Mak and Harko: International Journal of Modern Physics D 13 (01), 149-156 (2004)

Aayushi International Interdisciplinary Research Journal (ISSN 2349-638x) (Special Issue No.66)

Impact Factor 6.293 Peer Reviewed Journal www.aiirjournal.com Mob. 8999250451 121

FRW Cosmological Model with Electromagnetic Field in F(R, T) Theory

Apurav R. Gupta1, Milan M. Sancheti2, Suchita A. Mohta3

1Department of Mathematics, L.P.G. Arts and Science Mahavidyalaya Shirpur (Jain) Dist. Washim 444504(M.S), India

2Department of Mathematics, R. A. Arts, M. K. Commerce and S. R. Rathi Science College Dist. Washim 444505(M.S), India

3Research Scholar Dist. Akola 444001(M.S), India

Abstract:

In this communication, we have investigate the FRWcosmological model with electromagnetic field in f(R, T) modified theory of gravity as proposed by Harko et al. ( Phy. Rev. D. 024020), where gravitational Lagrangian function is replaced by an arbitrary function of Ricci Scalar R and trace T stress energy tensor. But we used T =ETij + STij of stress energy tensor where ETij is the energy momentum tensor for electromagnetic field and STij is the energy momentum tensor for string cloud. Stability analysis of the solutions through cosmological perturbation is performed and it is concluded that the expanding solution is stable against the perturbation with respect to anisotropic spatial direction. We have also studied the string cosmological model by considering the time dependent deceleration parameter with the presence of electromagnetic field. Some cosmological parameters are also discussed and studied.

Keywords: FRWcosmology, f(R, T) theory of gravity and Electromagnetic field. Introduction:

The recent scenario of early inflation and late time accelerated expansions of the universe (Rises et al.

1998; Perlmutter et al. 1999) is not explained by general theory of relativity. Hence to incorporate the above desirable features there have been several modifications of general relativity. Significant among them are scalar-tensor theories of gravitation formulated by Brans and Dicke (1961), and Saez and Ballester (1986) and modified theories of gravity like f (R) theory of gravity formulated by Nojiri and Odinstov (2003) and f (R,T ) theory of gravity proposed by Harko et al. (2011). In recent years there has been an immense interest in constructing cosmological models of the universe to study the origin, physics and ultimate fate of the universe.

In particular, cosmological models of Brans-Dicke and Saez- Ballester scalar-tensor theories of gravitation are attraction more and more attention of scientists. Brans-Dicke (1961) theory is a well known competitor of Einstein‟s theory of gravitation. It is the simplest example of a scalar-tensor theory in which the gravitational interaction is mediated by a scalar field φ as well as the tensor field gij of the Einstein‟s theory. In this theory, the scalar field φ has the dimension of the universal gravitational constant. Subsequently, Saez and Ballester (1986) developed a scalar-tensor theory in which the metric is coupled with a dimensionless scalar field φ in a simple manner. In spite of the dimensionless character of the scalar field an anti gravity regime appears in this theory. Also, this theory gives satisfactory description of the weak fields and suggests a possible way to solve

„missing matter‟ problem in non-flat FRW cosmologies. Friedmann-Robertson-Walker (FRW) models, being spatially homogeneous and isotropic in nature, are best suited for the representation of large scale structure of the present universe. However, it is believed that the early universe may not have been exactly uniform in its expansion phase. Thus, the models with anisotropic background seemed the most suitable to describe the early stages of the universe.

Motivated by the above investigations and discussions we focus our attention, in this paper, to explore the five dimensional FRW Cosmological models with electromagnetic field in f(R, T). The paper is organized as follows. In Sect. 2, we derive the field equations of f(R, T) theory for a five dimensional FRW space-time in electromagnetic field. In Sect. 3, some cosmological models representing stiff fluid, radiation, dust and inflation are obtained. A consolidated physical behavior of the models is studied in Sect. 4. The last section contains some conclusions.

f(R,T)gravity theory:

In modified f(R,T) gravity theory models, the field equations are obtained from the Hilbert-Einstein type variational principle. The action for the modified f (R, T ) gravity is

𝑆 =16𝜋1 −𝑔(𝑓 𝑅, 𝑇 + 𝐿𝑚)𝑑4𝑥 (1)

where f (R, T ) is an arbitrary function of the Ricci scalar R and of the trace T of the energy momentum tensor Tij of the matter. Lm is the matter Lagrangian. The energy momentum tensor of the matter is defined as

𝑇𝑖𝑗 = − 2

−𝑔

𝛿( −𝑔𝐿𝑚)

𝛿𝑔𝑖𝑗 (2)

If we assume that Lm of matter depends only on the metric tensor gij , and not on its derivatives, we obtain 𝑇𝑖𝑗 = 𝑔𝑖𝑗𝐿𝑚 − 2𝜕𝐿𝜕𝑔𝑚𝑖𝑗 (3)

By varying the action S of the gravitational field with respect to the metric tensor gij , we obtain 𝑓𝑅 𝑅, 𝑇 𝑅𝑖𝑗 −1

2𝑓 𝑅, 𝑇 𝑔𝑖𝑗 + 𝑔𝑖𝑗 ⊡ −𝛻𝑖𝛻𝑗 𝑓𝑅 𝑅, 𝑇 = 8𝜋𝑇𝑖𝑗 − 𝑓𝑟 𝑅, 𝑇 𝑇𝑖𝑗 − 𝑓𝑟 𝑅, 𝑇 𝜃𝑖𝑗(4)

Where

𝜃𝑖𝑗 = −2𝑇𝑖𝑗 + 𝑔𝑖𝑗𝐿𝑚− 2𝑔𝛼𝛽𝜕𝑔𝜕𝑖𝑗2𝜕𝑔𝐿𝑚𝛼𝛽 (5) 𝑤𝑕𝑒𝑟𝑒, ⊡= ∇𝑖𝑖, 𝑓𝑅 𝑅, 𝑇 =𝜕𝑓(𝑅, 𝑇)

𝜕𝑅 , 𝑓𝑇 𝑅, 𝑇 =𝜕𝑓(𝑅, 𝑇)

And ∇i denotes the covariant derivative and Tij is the standard matter energy momentum tensor derived 𝜕𝑇 from Lagrangian 𝐿𝑚

A contraction of (4) gives

𝑓𝑅 𝑅, 𝑇 𝑅 + 3 ⊡ 𝑓𝑅 𝑅, 𝑇 − 2𝑓 𝑅, 𝑇 = 8𝜋 − 𝑓𝑟 𝑅, 𝑇 𝑇 − 𝑓𝑟 𝑅, 𝑇 𝜃 (6)

where θ =𝜃𝑖𝑖.Equation (6)gives a relationship between R and T . Using matter Lagrangian Lm, the energy momentum tensor of matter is given by

Tij = + p)uiuj pgij (7)

Here ρ is the energy density and p the pressure of the matter, and ui is the four velocity vector𝑢𝑖𝑢𝑖 = 1.The matter Lagrangian can be taken as Lm = −p since there is no unique definition of the matter Lagrangian. Then with the use of (7),we obtain the variation of the stress energy of a perfect fluid the expression for θij as

θij = −2Tij pgij (8)

Generally, the field equations also depend through the tensor θij on the physical nature of the matter field. Hence, several theoretical models corresponding to different matter sources in f (R, T ) gravity can be obtained. Harko et al. (2007) gave three classes of these models.

𝑓 𝑅, 𝑇 = 𝑅 + 2𝑓 𝑇

= 𝑓1 𝑅 + 𝑓2(𝑇) (9) = 𝑓1 𝑅 + 𝑓2(𝑅)𝑓3(𝑇)

Much attention has been focused on the first class. We have studied the cosmological consequences of the class for which f (R, T ) = f1(R) + f2(T ). Here we use this model to obtain the exact solutions of the field equations for the Bianchi type V metric in the presence of massive strings for the class of model, the gravitational field equation (4) becomes

𝑅 𝑅𝑖𝑗 −1

2𝑓1 𝑅 𝑔𝑖𝑗 + 𝑔𝑖𝑗 ⊡ −∇𝑖𝑗 𝑓1 𝑅 =

8𝜋𝑇𝑖𝑗 + 𝑓2 𝑇 𝑇𝑖𝑗 + (𝑓2 𝑇 𝑝 +12𝑓2 𝑇 )𝑔𝑖𝑗 (10) Where a prime denotes differentiation with respect to the argument,The equation for standard f ( R )

gravity can be recovered for p=0 and 𝑓2 𝑇 = 0 . Here we consider the particular form of the functions 𝑓1 𝑅 = 𝜆1 𝑅 𝑎𝑛𝑑 𝑓2 𝑇 = 𝜆2 𝑇 .we further assume that 𝜆1≠ 𝜆2

So that 𝑓 𝑅, 𝑇 = 𝜆 𝑅, 𝑇 .Equation (10 )can be rearranged as

Aayushi International Interdisciplinary Research Journal (ISSN 2349-638x) (Special Issue No.66)

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𝜆1𝑅𝑖𝑗12𝜆1 𝑅 + 𝑇 𝑔𝑖𝑗 + 𝑔𝑖𝑗 − ∇𝑖𝑗 𝜆 = 8𝜋𝑇𝑖𝑗 − 𝜆2𝑇𝑖𝑗 + 2𝜆2𝑇𝑖𝑗 + 𝜆1𝑝𝑔𝑖𝑗 (11) Assuming 𝑔𝑖𝑗 − ∇𝑖𝑗 𝜆 = 0, we obtain

𝑅𝑖𝑗12𝑅𝑔𝑖𝑗 = 8𝜋+ 𝜆𝜆 2

1 𝑇𝑖𝑗 + (𝑝 +12𝑇)𝑔𝑖𝑗 (12)

FRW Space Time Field equations:

We consider the five dimensional FRW space time in the form

𝑑𝑠2= −𝑑𝑡2+ 𝐵2 𝑡 1−𝑘𝑟𝑑𝑟22+ 𝑟2𝑑𝜃2+ 𝑟2𝑠𝑖𝑛2𝜃𝑑𝜑2 + 𝐴2(𝑡)𝑑𝑥2 (13)

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