1. Higher Order Curvature Gravity in Finsler Geometry N.Mebarki and M.Boudjaada Département de Physique Mathématique et Subatomique Faculty of Science, Mentouri University Constantine, Algeria MG12 July 12-18 2009 Paris France

2. Why Finsler Geometry ? *The observed anisotropy of the microwave cosmic radiation is of a dipole type: Intensity of radiation is maximum at one direction and minimum at opposite one. *It is known that this anisotropy can be explained if we use Robertson-walker metric and take into account the motion of our galaxy with respect to distant galaxies of the universe. * However, a small anisotropy is expected, due to the anisotropic distribution of galaxies in space. In framework of Finsler geometry, the flat rotation curves of spiral galaxies can be deduced naturally without involving dark matter Moreover

3. Goal Get a viable theory and a better understanding Criteria for viability Must have cosmological dynamics Connect with observations No instabilities No ghosts Connect with Newtonian/Post Newtonian limits Well formulated Cauchy problem Investigate more Finsler Geometry

4. Finsler Geometry Connect Riemannian metric structure of the space-time to a physical vector field (which is dependent on the position and direction (velocity)) of cosmological origin (emerge out by a physical source of the universe where it is incorporated into the geometry causing an anisotropic structure ) Variation of anisotropy is expressed in terms of Cartan torsion tensor of the Finslerian manifold Can be considered as a physical geometry on which matter dynamic takes place while Riemannian geometry is the gravitational geometry. May be considered as a generalized Riemanian geometry.

5. So the geometrical anisotropies are caused by internal variables. Solutions of Friedman eqs., CMB temperature estimation, distance luminosity etc... will be affected by presence of this anisotropy In this context

6. A Finsler space a metric space Defined by a norm F (on a tangent bundle instead of defining an inner product structure on it) Norm is a real function F(x,y) of a space-time point and a tangent vector Y belonging to T x M which plays the role of an internal variable The Y dependence characterizes the Finslerian field: combined with concept of anisotropy causing deviation from Riemannian geometry Metric function Mathematical Formalism

7. If we denote by Tangent space at Tangent bundle of Each element ofHas the form Natural projection Given by A Finsler structure of M is a function: With properties

8. Regularity Positive homogeneity Strong convexity For all The nxn Hersian Is positive definite

9. To describe dynamics, one has to introd. a Lagrangian L of Randers-type Where and FRW metric

10. Important information about the anisotropy is encoded into the component, or vector We can have an induced Metric of Finsler space (called osculating Riemannian metric) can be directly calculated from the metric function F as: *Under weak field assumption: we can approximate Finslerian metric as a perturbation of the FRW metric since in general relativity a weak vector field in a space (e.g a magnetic field) can be treated as a first order perturbation of the Riemann metric tensor. So:

11. NB: In the framework of Finsler geometry, the four velocity vector is treated as independent variable

12. Osculating affine connection Where Cartan torsion tensor

13. A Finsler space is a Riemanian space if and only if Curvatures

14. VSR Space-time symmetries are certain proper subgroups of the Poincaré group SIM(2) similitude subgroup Lie algebra (has just 04 generators) Generators of rotations and boosts of Poincaré group where NB Present experimental data are not in contradiction with this symmetry Locally Finsler theory

15. Field equations: Where: and Stress-energy-momentum tensor

16. F( R ) Gravity in Palatini Formalism and Finsler Geometry Field equats. Conservation equat. : Action

17. Take Tr. of eq.(*) Also from field eqts. one has Where and Anisotropy

18. If and Where

19. Using conservation eqs. One can show that: If: and Matter dominated universe

20. Special caseGet exact analytical solution If Gives

21. Then, One can calculate

22. Conclusion1: If : In flat space case, we have found: and Get exact explicit time dependence relation of scale factor

23. Getand calculate distance luminosity Can compare with SNIa supernovae data see deviation from the ordinary case (Riemanian Geometry). More important:

28. Metric f( R ) in Finsler Geometry Field eqs. In flat space Take If

29. Where and If Previous field eqs.

30. If we take a power law formula Then get analytical expression Where

31. Compatible solutions Or Static universe deceleration acceleration

34. Conclusion2: In flat space case : we have found with Get If we consider power law formula Then, to have compatible solutions

35. NB: (Riemanian case) If We have At large Then : behaviors are completly different with Riemanian and Finsler Geometry

36. Dynamical Study of Metric F ( R ) Gravity in Finsler Geometry Goal of study stabilities and unstabilities: Provide additional informations about the solutions even if we cannot find the exact ones, and improve our understanding of the theory For this:Apply dynamical system approach and study phase portrait and some critical points. In vaccumField eqs.:

37. where and If

38. Dynamical variables

39. Get Autonomous set of eqs. With the constraint

40. or

41. Critical points 1) Stability matrix Eigenvalues Deceleration parameter andOpen universe In portrait phase space : Unstable Saddle point

42. Stable nodal sink Unstable Saddle point

43. 2) Stability matrix Eigenvalues Deceleration parameter and flat space deceleration

44. Unstable nodal source In portrait phase space : If Unstable nodal source Unstable Critical point

45. 3) Stability matrix Deceleration parameter If decceleration acceleration

46. Eigenvalues flat space Some examples: For

47. 4) Stability matrix Deceleration parameter example Eigenvalues If For open universe closed universe flat space Unstable nodal source Static universe

48. 5) Deceleration parameteracceleration Stability matrix closed universe

49. Stability matrix 6) Deceleration parameter acceleration open universe

50. Matter dominated universe case Introduce additional dynamical variable Get 04 autonomous set of eqs. Get more dynamical variables (4 non linear first order equations), study is similar.

51. We have made similar study near infinity (asymtotic limit), get similar behaviors Stability at infinity Since the phase portrait is not compact, it is possible that the dynamical system has a non trivial asymptotic structure, thus check existence of fixed points at infinity Physically such points represents regimes in which one or more of the terms in generalized Friedman equations become dominant. e.g if the kinetic part of curvature energy density dominates over potential part.

52. In vaccum case Autonomous set of eqts. We can analyze asymptotic feature of system in same way as we did with finite case. The asymptotic analysis can be performed by compactifying phase space portrait using Poincaré method

53. More Critical points Solutions which may be stable in ordinary (Riemannian) case may not be in Finsler case More dynamical variables, more chaotic system. Conclusion3 : In Finsler Geometry

54. Perspectives More studies on stability problem (linear and non linear perturbations etc…) More contact with observations Equivalence with some generalized scalar-tensor gravity LTB in Finsler Geometry etc…

55. Thank You

56. Nodal sink : Eigenvalues all trajectories or orbits in phase portrait converge to the point Nodal source: Eigenvalues all trajectories or orbits in phase portrait diverge from the point Saddle point: Eigenvalues Star pointall orbits coverge into a star point in straight lines Star point orbits diverge from star point in straight lines