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of the navigation cosmological model could account for the observations of Riess and Perlmutter[2, 3] on distant supernovae. 

Instead of defining an inner product structure over the tangent bundle in Riemann geometry, Finsler geometry is based on the so called Finsler structure F with the property F ( x, λy ) = λF ( x, y ) for all λ > 0, where x represents position and y represents velocity. The Finsler metric is given as[19] 

In Finsler manifold, there exists a unique linear connec- tion - the Chern connection[20]. It is torsion freeness and almost metric-compatibility, 

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where γ µν is the formal Christoffel symbols of the sec- ond kind with the same form of Riemannian connec- tion, N µ ν is defined as N µ ν ≡ γ µ $_{να}$y α − A µ $_{νλ}$γ λ $_{αβ}$y$^{α}$y β and A$_{λµν}$ ≡ F 4 ∂ ∂y λ ∂ ∂y µ ∂ ∂y $_{ν}$( F $^{2}$) is the Cartan tensor (regarded as a measurement of deviation from the Riemannian Manifold). In terms of Chern connection, the curvature of Finsler space is given as 

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where δ δx µ = ∂ ∂x µ − N ν µ ∂ ∂y $_{ν}$. The notion of Ricci ten- sor in Finsler geometry was first introduced by Akbar- Zadeh[21] 

where R = y µ F R κ µ κν y ν F . And the scalar curvature in Finsler geometry is given as S = g $^{µν}$Ric$_{µν}$ . 

In standard cosmology, following the cosmological principle, one gets the Friedmann-Robertson-Walker (FRW) metric[22]. In another word, the spatial part of the Universe is a constant sectional curvature space. Here comes our major assumption, the gravity in large scale should be described in terms of Finsler geometry. In light of the cosmological principle, the Finsler structure of the Universe should be written in such form 

where R ( t ) is scale factor with cosmic time t , the struc- ture F is a constant flag curvature space. By making 

## FORMULISM 

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use of the geometrical terms we mentioned above, one obtains the components of Ricci tensor 

where the dot denotes a derivative with respect to t . 

The gravitational field equation in Finsler space should reduce to the Einstein’s gravitational field equation while the Finsler space reduce to the Riemannian space. Thus, the symmetrical tensor G$_{µν}$ ≡ Ric$_{µν}$ − 1 $_{2}$g$_{µν}$ S should in- volve in the gravitational field equation in Finsler space. In general, Finsler space is an anisotropic space. There- fore, it has less Killing vectors than the Riemannian space, and it breaks the Lorentz symmetry[23]. It means the angular momentum in Finsler space is not a conser- vative quantity. This fact implies that the energy mo- mentum tensor is not symmetrical in Finsler space. For these reasons, the gravitational field equation in Finsler space should be taken in such form 

where A$_{µν}$ is an asymmetrical tensor and T s $_{µν}$, T a µν are symmetrical part and asymmetrical part of energy mo- mentum tensor respectively. This general form is agree with the result of Asanov, its gravitational field equa- tion contains the asymmetrical term in Finsler space of Landsberg type[24]. Dealing with field equation in the anisotropic and inhomogeneous cosmology is not a simple staff, and it is hard to find an exact solution of gravita- tional field equation while its energy momentum tensor involves the non diagonal terms[25]. At first glance, we just deal with the symmetrical part of field equation (8) 

By making use of the equations (6) and (7), the solu- tion of equation (9) is the same with the Einstein’s field equation deduced by FRW metric. Taking the energy- momentum tensor T$_{µν}$ to be the form of perfect fluid, one can see that the evolution of scale factor R ( t ) is the same with the Riemannian case. However, since the lu- minosity distant is related to the spatial geometry of the Universe, one may expect that it is different from the Riemannian luminosity distant. 

Unlike Riemann space, a complete classification of the constant flag curvature spaces remain an unsolved prob- lem. However, by making use of the Zermolo navigation on Riemannian space, Bao et al . [26] gave a complete classification of Finsler-Randers space [27] of constant flag curvature. The Zermelo navigation problem [28] aims to find the paths of shortest travel time in a Rie- mannian manifold ( M, h ) under the influence of a drift (“wind”) represented by a vector field W . In standard 

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