Cosmological Consequences with Randers Conformal of Finsler space

In this paper we study an anisotropic model of space – time with Finslerian metric. The observed anisotropy of the microwave background radiation is incorporated in the Finslerian metric of space time.


1-INTRODUCTION:
Recently in literature we have gone one paper published in physical letter respectively, B668(2008)453 and B676(2009)173 under which the aouther has considered that in Finslerian manifold there exists a unique linear connection known as Chern connection. It is torsion firness and metric compatibility, we are not agree with the result of the paper because there are well known result published by H. Rund that in Finsler geometry there exits and infinite number of linear connection defined by the same metric structure and that the Chern and Bernwald connection are not metric compatable.
cosmology and other reference of the same kind [ 1, 2 ,3 ].We are fully agreed with the theory expressed in the above referred publication regarding as it is based on tangent bundle on space time manifold are positively with local Lorentz violations which may be related with dark energy and dark matter models with variable ᴧ in cosmology. Certainly those may be one of the most recent hidden connections between Finsler geometry and cosmology. Now in these days many researchers are constructing suitable cosmological models with variable Lambada term including our own research group [29][30][31][32][33] Here in this paper we will study anisotropic model of General Theory of Relativity based on the frame work of Finsler geometry.

BASIC ASSUMPTIONS & CONSIDERATION:
In 1984, Shibata [7] introduce the concept of change and defined as Where, is a one-form metric and L is any Finsler metric, and obtained very interesting result in his paper. The conformal theory of Finsler spaces has been initiated by M. S. Knebelman [8] in 1929 and has been investigated in detail by many authors [9,10,11,12,13] etc. This change is defined as Where is a function of position only and known as conformal factor.

3-Generalization of result with different assumption:
In the present paper we generalize the above changes and defined a Randers conformal metric as x e L (1) Where, is a one-form and is the conformal factor. If , then this change is reduces to simple Randers metric and for this metric Stavrions and Diakogiannis [16] obtained the relationship between the anisotropic cosmological models of space time and Randers Finslerian metric. The purpose of the present study is to obtain the relationship between the anisotropic cosmological models of space time with above generalized Finslerian metric.
Let us consider an n-dimensional Finsler space (M n , L) and an adaptable 1-form on M n . We shall use a Lagrangian function on M n , given by the equation: The vector i bˆ represents the observed anisotropic of the microwave background radiation.
A coordinate transformation on the total space TM is given by  The Carton torsion coefficients Cijk are given by The torsions and curvatures which we use are given by [4,5,15]: Differentiating equation (3) with respect to y i , the normalized supporting elements From equation (12) and (13) is the angular metric tensor in the Finsler Space F n .

3-Anisotropic cosmological model with Randers conformal of Finsler space:
The Lagrangian function on M n , given by the equation: Where ij a is the Riemannian metric with signature (-, +, +. +).Because of the anisotropy, we must insert an additional term to the Riemannian line element α. This term fulfill the following requirements: It must give absolute maximum contribution for direction of movement parallel to the anisotropy axis. (ii) The new line element must coincide with the Riemannian one for the direction vertical to the anisotropic axis.
(iii) It must not symmetric with respect to replacement We see that a term which satisfies the above conditions is In order for the Finslerian metric to be physically consistent with General theory of relativity, it must have the same signature with the Riemannian metric (-, +, +, +). We have 0 dx ct d dt c cd (17) Where 2 2 1 c v and v: 3-velocity in Riemannian space-time. One possible explanation of the anisotropy axis could be that it represents the resultant of spin densities of the angular momenta of galaxies in a restricted area of space ( bi(x) space like). It is known that the mass is anisotropically distributed for regions of space with radius ≤ 10 8 light years [4].

4-Concluding Remarks:
Particular attention over the last decade has been paid on the so-called Finsler-Randers cosmological model [17].In general metrical extensions of Riemann geometry can provide a Finslerian geometrical structure in a manifold which leads to generalized gravitational field theories. During the last decade there is a rapid development of applications of Finsler geometry in its FR context, mainly in the topics of general relativity, astrophysics and cosmology [18][19][20][21][22][23][24][25][26][27][28]. Keeping this in mind in this paper we have considered the randers conformal metric to the general relativity. From the above section it is clear that If , then this change is reduces to simple Randers metric and GTR is well connected with Randers metric. For this metric, we have also obtained the relationship between the anisotropic cosmological models of space time with above generalized Finslerian metric which is shown in the equations derived in section 3