** Applications of Flexible Electronics**

**Stage 3: Computing the SDG India Index Scores**

**S. C. Wankhade**

**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 k_{1}=k_{6}=0.1 **Fig. 2 **Plot of Hubble parameter Vs. Time for k_{1}=k_{6}=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 *VI*_{0} 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).

[15] Pund, A. M. Avachar, G.R.: Prespacetime Journal, 5(3), 207-211 (2014).

[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)**

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

**Apurav R. Gupta**^{1}**, Milan M. Sancheti**^{2}**, Suchita A. Mohta**^{3}

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*^{ =E}*T*_{ij}* + *^{S}*T*_{ij}* of stress *
*energy tensor where *^{E}*T*_{ij}* is the energy momentum tensor for electromagnetic field and *^{S}*T*_{ij}* 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 =_{16𝜋}^{1} −𝑔(𝑓 𝑅, 𝑇 + 𝐿𝑚)𝑑^{4}𝑥 (1)

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

𝑇_{𝑖𝑗} = − ^{2}

−𝑔

𝛿( −𝑔𝐿𝑚)

𝛿𝑔^{𝑖𝑗} (2)

If we assume that* L**m* of matter depends only on the metric tensor* g**ij* , and not on its derivatives, we obtain
𝑇_{𝑖𝑗} = 𝑔_{𝑖𝑗}𝐿_{𝑚} − 2^{𝜕𝐿}_{𝜕𝑔}^{𝑚}_{𝑖𝑗} (3)

By varying the action S of the gravitational field with respect to the metric tensor* g**ij* , we obtain
𝑓_{𝑅} 𝑅, 𝑇 𝑅_{𝑖𝑗} −1

2𝑓 𝑅, 𝑇 𝑔_{𝑖𝑗} + 𝑔_{𝑖𝑗} ⊡ −𝛻_{𝑖}𝛻_{𝑗} 𝑓_{𝑅} 𝑅, 𝑇 =
8𝜋𝑇_{𝑖𝑗} − 𝑓_{𝑟} 𝑅, 𝑇 𝑇_{𝑖𝑗} − 𝑓_{𝑟} 𝑅, 𝑇 𝜃_{𝑖𝑗}(4)

Where

𝜃_{𝑖𝑗} = −2𝑇_{𝑖𝑗} + 𝑔_{𝑖𝑗}𝐿_{𝑚}− 2𝑔^{𝛼𝛽}_{𝜕𝑔}^{𝜕}_{𝑖𝑗}^{2}_{𝜕𝑔}^{𝐿}^{𝑚}_{𝛼𝛽} (5)
𝑤𝑒𝑟𝑒, ⊡= ∇^{𝑖}∇_{𝑖}, 𝑓_{𝑅} 𝑅, 𝑇 =𝜕𝑓(𝑅, 𝑇)

𝜕𝑅 , 𝑓_{𝑇} 𝑅, 𝑇 =𝜕𝑓(𝑅, 𝑇)

And ∇*i* denotes the covariant derivative and* T**ij* is the standard matter energy momentum tensor derived 𝜕𝑇
from Lagrangian 𝐿_{𝑚}

A contraction of (4) gives

𝑓_{𝑅} 𝑅, 𝑇 𝑅 + 3 ⊡ 𝑓_{𝑅} 𝑅, 𝑇 − 2𝑓 𝑅, 𝑇 = 8𝜋 − 𝑓_{𝑟} 𝑅, 𝑇 𝑇 − 𝑓_{𝑟} 𝑅, 𝑇 𝜃 (6)

where* θ* =𝜃_{𝑖}^{𝑖}.Equation (6)gives a relationship between R and* T* . Using matter Lagrangian* L**m*, the energy
momentum tensor of matter is given by

* T**ij* =* (ρ* +* p)u**i**u**j* −* pg**ij** * * *(7)

Here* ρ* is the energy density and p the pressure of the matter, and* u*^{i} is the four velocity vector𝑢^{𝑖}𝑢_{𝑖} =
1.The matter Lagrangian can be taken as* L**m* = −*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* θ**ij* as

* θ**ij* = −2*T**ij* −* pg**ij* (8)

Generally, the field equations also depend through the tensor *θ**ij *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

= 𝑓_{1} 𝑅 + 𝑓_{2}(𝑇) (9)
= 𝑓_{1} 𝑅 + 𝑓_{2}(𝑅)𝑓_{3}(𝑇)

Much attention has been focused on the first class. We have studied the cosmological consequences
of the class for which* f (R, T )* =* f*1*(R)* +* f*2*(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 𝑅_{𝑖𝑗} −1

2𝑓_{1} 𝑅 𝑔_{𝑖𝑗} + 𝑔_{𝑖𝑗} ⊡ −∇_{𝑖}∇_{𝑗} 𝑓_{1}^{′} 𝑅 =

8𝜋𝑇_{𝑖𝑗} + 𝑓_{2}^{′} 𝑇 𝑇_{𝑖𝑗} + (𝑓_{2}^{′} 𝑇 𝑝 +^{1}_{2}𝑓_{2} 𝑇 )𝑔_{𝑖𝑗} (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 𝑓_{2} 𝑇 = 0 . Here we consider the particular form of the
functions 𝑓_{1} 𝑅 = 𝜆_{1} 𝑅 𝑎𝑛𝑑 𝑓_{2} 𝑇 = 𝜆_{2} 𝑇 .we further assume that 𝜆_{1}≠ 𝜆_{2}

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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𝜆_{1}𝑅_{𝑖𝑗} −^{1}_{2}𝜆_{1} 𝑅 + 𝑇 𝑔_{𝑖𝑗} + 𝑔𝑖𝑗 − ∇_{𝑖}∇_{𝑗} 𝜆 = 8𝜋𝑇_{𝑖𝑗} − 𝜆_{2}𝑇_{𝑖𝑗} + 2𝜆2𝑇_{𝑖𝑗} + 𝜆_{1}𝑝𝑔_{𝑖𝑗} (11)
Assuming 𝑔_{𝑖𝑗} − ∇_{𝑖}∇_{𝑗} 𝜆 = 0, we obtain

𝑅_{𝑖𝑗} −^{1}_{2}𝑅𝑔_{𝑖𝑗} = ^{8𝜋+ 𝜆}_{𝜆} ^{2}

1 𝑇_{𝑖𝑗} + (𝑝 +^{1}_{2}𝑇)𝑔_{𝑖𝑗} (12)

**FRW Space Time Field equations: **

We consider the five dimensional FRW space time in the form

𝑑𝑠^{2}= −𝑑𝑡^{2}+ 𝐵^{2} 𝑡 _{1−𝑘𝑟}^{𝑑𝑟}^{2}_{2}+ 𝑟^{2}𝑑𝜃^{2}+ 𝑟^{2}𝑠𝑖𝑛^{2}𝜃𝑑𝜑^{2} + 𝐴^{2}(𝑡)𝑑𝑥^{2} (13)