3 cosmology, our Universe is very flat now. We may imag- ine that the Universe is a flat Riemannian manifold with flat Friedmann metric The light traveling along the radial direction satisfies the geodesic equation Then, we obtain The equations (9) and (17) can be solved. Supposing the Universe is matter dominated (no more cosmological constant and dynamical dark energy), we obtain where W$_{i}$ = h$_{ij}$W j and λ = 1 − h ( W, W ). One should notice that there is a map between the Riemannian space which influence by the “wind” and the Randers- Finsler space[29]. It means the effect of “wind” already accounted in the gravitational field equation in Finsler space. The theorem[26] of the classification supplies an interesting case where the Randers metric F has con- stant flag curvature K : for K = − 1 $_{16}$σ 2 < 0 and h is flat. And σ satisfies the constraint L $_{W}$h = − σh , L de- notes Lie differentiation. We set the vector field to be radial W = ϵ ( t )( x$_{1}$∂x$_{1}$ + x$_{2}$∂x$_{2}$ + x$_{3}$∂x$_{3}$ ). Then the flag curvature of Randers metric is K = − 1 $_{4}$ϵ$^{2}$. After doing coordinate change and taking spherical coordinate, we get the Randers metric as where H$_{0}$ is the Hubble constant and Ω$_{m}$ is the density parameter for matter and satisfies Substituting the equation (18) into (15), we obtain the relation between luminosity distant and redshift in the navigation cosmological model and the cosmic scale factor changes as R ( t ) = 2 R$_{h}$ ( t ) ϵ ( t ) . Taking the spatial part of the Finsler structure (5) to be of the form (13), we get metric of the Universe in the framework of Finsler geometry, and the space curvature of the Universe is − 1. One can deduce directly from such metric that the relation between the redshift z and the scale factor R ( t ) is the same with the Riemannian case Here, we present numerical result for the relation be- tween luminosity distant and redshift in the framework of the navigation cosmological model. The best fit curve is shown in Fig.1 with the matter density parameter taken as Ω (0) m = 0 . 92. And the average value of Chisquare is 1 . 077588. Thus, our prediction could account for exper- iment data given by Riess et al .[2]. It should be noticed that the Hubble constant H$_{0}$ we took is different from the Hubble constant H$_{h}$$_{0}$ measured in flat space. The relation for H and H$_{h}$ is where the subscript zero denotes the quantities given at the present epoch. Since the non-radial part of the metric (13) is the same with FRW metric, the luminosity distant in the navigation cosmological model is given as d$_{L}$ = R$_{0}$r (1 + z ). However, the proper distant is not the case, and it is given as By making use of the value of the Hubble constant mea- sured in flat space H$_{h}$$_{0}$ = 73 ± 0 . 3 km · s − 1 · Mpc − $^{1}$[31], we have This geometrical parameter H$_{ϵ}$$_{0}$ represents an “acceler- ated” effect provided by the vector field W . ds 2 = dt 2 − R 2 $_{h}$( t )( dx 2 + dy 2 + dz $^{2}$) , (10) ¯ F 2 = dt $^{2}$− R $^{2}$( t ) ( 1 √$_{1 + 4}$$_{r}$ 2 − 2 r 1 + 4 r 2 ) 2 dr 2 = 0 . (16) d$_{p}$ = 1 R$_{0}$ ∫ 1 (1+ z ) − 1 da a ˙ a (17) and it influenced by the “wind” W . The relation between the Riemannian manifold ( M, h ) and the Randers metric F is F = 1 λ ( √ λh$_{ij}$y $^{i}$yj + ( W$_{i}$y $^{i}$) 2 − W$_{i}$y i ) , (11) d$_{p}$ = log ( √ 1 + Ω (0) m z − √ 1 − Ω (0) m )(1 + √ 1 − Ω (0) m ) ( 1 + Ω (0) m z + 1 − Ω (0) m )(1 − 1 − Ω (0) m ) , (18) 1 − Ω (0) m = 1 H 2 $_{0}$R 2 0 . (19) H$_{0}$d$_{L}$ = 1 + z 2 √ 1 − Ω (0) m | e 2 d$_{p}$ − 1 | e d$_{p}$ √$_{2}$ e 2 $_{d$_{p}$}$. (20) F = √ dr 2 1 + ϵ $^{2}$r2 + r $^{2}$( dθ 2 + sin $^{2}$θdφ$^{2}$) − ϵrdr 1 + ϵ $^{2}$r$_{2}$. (12) F = √ dr 2 1 + 4 r 2 + r $^{2}$( dθ 2 + sin $^{2}$θdφ$^{2}$) − 2 rdr 1 + 4 r $_{2}$, (13) After scale change r → 2 r/ϵ , the metric F changes as NUMERICAL RESULT 1 + z = R$_{0}$ R ( t ) = 1 a ( t ) , (14) H = H$_{h}$ − ˙ ϵ ϵ . (21) H$_{ϵ}$$_{0}$ ≡ ˙ ϵ$_{0}$ ϵ$_{0}$ = 5 . 42 km · s − 1 · Mpc − 1 (22) d$_{p}$ = ∫ r 0 ( 1 √$_{1 + 4}$$_{r}$ 2 − 2 r 1 + 4 r 2 ) dr = sinh − $^{1}$2 r 2 − 1 4 ln(1 + 4 r $^{2}$) . (15)