A generalized FRW model of space-time is studied, taking into consideration the anisotropic structure of fields which are depended on the position and the direction (velocity).The Raychaudhouri and Friedman-like equations are investigated assuming the Finslerian character of space-time.A long range vector field of cosmological origin is considered in relation to a physical geometry where the Cartan connection has a fundamental role.The Friedman equations are produced including extra anisotropic terms.The variation of anisotropy zt is expressed in terms of the Cartan torsion tensor of the Finslerian manifold.A physical generalization of the Hubble and other cosmological parameters arises as a direct consequence of the equations of motion.

General Very Special Relativity (GVSR) is the curved space-time of Very Special Relativity (VSR) proposed by Cohen and Glashow. The geometry of GVSR possesses a line element of Finsler Geometry introduced by Bogoslovsky. We calculate the Einstein field equations and derive a modified FRW cosmology for an osculating Riemannian space. The Friedman equation of motion leads to an explanation of the cosmological acceleration in terms of an alternative non-Lorentz invariant theory. A first order approach for a primordial spurionic vector field introduced into the metric gives back an estimation of the energy evolution and inflation

We construct a cosmological toy model based on a Finslerian structure of space-time. In particular, we are interested in a specific Finslerian Lorentz violating theory based on a curved version of Cohen and Glashow's Very Special Relativity. The osculation of a Finslerian manifold to a Riemannian leads to the limit of Relativistic Cosmology, for a specified observer. A modified flat FRW cosmology is produced.The analogue of a zero energy particle unfolds some special properties of the dynamics. The kinematical equations of motion are affected by local anisotropies. Seeds of Lorentz Violations may trigger density inhomogeneities to the cosmological fluid.

The similarity between Finsler and Riemann geometry is an intriguing starting point to extend general relativity. The lack of quadratic restriction over the line element (color) naturally generalize the Riemannian case and breaks the local symmetries of general relativity. In addition, the Finsler manifold is enriched with new geometric entities and all the classical identities are suitably extended. We investigate the covariant kinematics of a medium formed by a time-like congruence. After a brief view in the general case we impose particular geometric restrictions to get some analytic insight. Central role to our analysis plays the Lie derivative where even in case of irrotational Killing vectors the bundle still deforms. We demonstrate an example of an isotropic and exponentially expanding cross-section that finally deflates or forms a caustic. Furthermore, using the 1+3 covariant formalism we investigate the expansion dynamics of the congruence. For certain geometric restrictions we retrieve the Raychaudhuri equation where a color-curvature coupling is revealed. The condition to prevent the focusing of neighboring particles is given and is more likely to fulfilled in highly curved regions. Then, we introduce the Levi-Civita connection for the osculating Riemannian metric and develop a (spatially) isotropic and homogeneous dust-like model with a non-singular bounce.

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