A matter dominated navigation Universe in accordance with the Type Ia supernova data Xin Li 1 , 3 ∗ $^{1}$Institute of Theoretical Physics, Chinese Academy of Sciences, 100190 Beijing, China Zhe Chang 2 , 3 † and Minghua Li 2 , 3 ‡ $^{2}$Institute of High Energy Physics, Chinese Academy of Sciences, 100049 Beijing, China $^{3}$Theoretical Physics Center for Science Facilities, Chinese Academy of Sciences We investigate a matter dominated navigation cosmological model. The influence of a possi- ble drift (wind) in the navigation cosmological model makes the spacetime geometry change from Riemannian to Finslerian. The evolution of the Finslerian Universe is governed by the same gravi- tational field equation with the familiar Friedmann-Robertson-Walker one. However, the change of space geometry from Riemannian to Finslerian supplies us a new relation between the luminosity distant and redshift. It is shown that the Hubble diagram based on this new relation could account for the observations on distant Type Ia supernovae. PACS numbers: 02.40.-k,98.80.-k Einstein’s general relativity enables us to come up with a testable theory of the Universe. Hubble[1] first found that the galaxies are receding from us not long after the birth of general relativity. The Hubble’s observation in- dicates that our Universe is expanding. Over the past decade, two groups [2, 3] observing supernovae reported that the luminosity distance can not be explained by a matter dominated Universe. If one accepts the convinci- ble assumption of homogeneity and isotropy of the Uni- verse which is approximately true on large scales, then the general relativity tells us that we now live in a dark energy dominated Universe and the Universe is acceler- ated expanding. Dark energy, which has the property of negative pressure, is different from the classical matter and the found particles. A great amount of models have been proposed to study the possible candidates of dark energy and their dynamics [4]. The most famous and acceptable candidate is the cosmological constant. How- ever, the magnitude of cosmological constant 10 − $^{3}$eV 4 is much smaller than the energy density of vacuum in quantum field theory. The theory of dark energy [5] dominates the modern cosmology in the past decade. This situation is similar to the raise of the theory of ether, which has been considered as direct evidence of the aberration of starlight, an im- portant astronomical effect known since eighteenth cen- tury. Although the rapid progress in technology makes the astronomical observations more and more accurate, up to now, there is no direct evidence indicate what the dark energy it is. Since the theory of dark energy is con- trived, which requires fine tuning and apparently cannot be tested in the laboratory or solar system, several mod- ified theories of general relativity have been developed as the alternative source of cosmological acceleration [6]. In this Letter, we present a matter dominated navi- gation model. The influence of a possible drift (wind) in the navigation cosmolgical model makes the expand- ing Universe “accelerated”. It is remarkable that the navigation cosmological model is described in the frame- work of Finsler geometry. We find that the predictions Finsler geometry as a more general geometry could provide new sight on modern physics. It is of great interest for physicists to investigated the violation of Lorentz symmetry [7]. An interesting case of Lorentz violation, which was proposed by Cohen and Glashow[8], is the model of Very Special Relativity (VSR) character- ized by a reduced symmetry SIM(2). In fact, Gibbons, Gomis and Pope[9] showed that the Finslerian line ele- ment ds = ( η$_{µν}$dx $^{µ}$dx$^{ν}$) (1 − b ) / $^{2}$( n$_{ρ}$dx $^{ρ}$)b is invariant un- der the transformations of the group DISIM$_{b}$(2). In the framework of Finsler geometry, modified dispersion rela- tion has been discussed[10, 11]. Also, the model based on Finsler geometry could explain the recent astronom- ical observations which Einstein’s gravity could not. A list includes: the flat rotation curves of spiral galaxies can be deduced naturally without invoking dark mat- ter [12]; the anomalous acceleration[13] in solar system observed by Pioneer 10 and 11 spacecrafts should corre- spond to Finsler-Randers space [14]; the secular trend in the astronomical unit[15, 16] and the anomalous secular eccentricity variation of the Moon’s orbit[17] should be correspond to the effect of the length change of unit circle in Finsler geometry[18]. Einstein first used the Riemann geometry to describe the theory of gravitation. In this Letter, we suppose that the spacetime in large scale may be described by other geometry instead of Riemann geometry. To preserve re- deeming feature of general relativity, this geometry must take Riemann geometry as its special case. Fortunately, Paul Finsler proposed a natural generation of Riemann geometry-Finsler geometry. INTRODUCTION arXiv:1001.0066v2 [gr-qc] 7 Apr 2010