1. Finsler引力研究现状
重庆大学 李昕
2016年8月 合肥

2. Outline Why Finsler geometry? Basic concepts of Finsler geometry
Finslerian cosmology

3. I. Why Finsler geometry? Components of modern cosmology
Cosmological principle
GR and particle standard model
Inflation
->(primordial scalar perturbation: adiabatic, Gaussian, nearly scale-invariant )

4. Signs of space anisotropy
Hemispherical asymmetry in CMB
Spatial variation of fine structure constant

5. Hemispherical asymmetry in CMB
Planck satellite

6. Dipole modulation

8. Spatial variation of fine structure constant

9. What if
Riemann Geometry
Finsler Geometry ？

10. II.Basic concepts of Finsler geometry Finslerian length
S.S.Chern: Finsler geometry is just Riemann geometry without the quadratic restriction
Length element: F(x,ay)=aF(x,y)
Metric tensor:
Cartan tensor:

11. Three types of Finsler spacetime
Randers spacetime
Bogoslovsky spacetime(VSR)
Quartic root

12. Symmetry of Finsler spacetime
Isometric group
Maximum symmetry: n(n-1)/2+1（subgroup of Poincare group）
4 dimensional constant curvature space: 6
4 dimensional Bogoslovsky spacetime: 8
DISIM(2) group

13. Geodesic and Ricci scalar
Geodesic equation (preserve F)
Ricci scalar (geometrical invariant)

14. Gravitational field equation in Finsler spacetime--methods
Second Bianchi identity
Action principle
C. Pfeifer and M. N. R. Wohlfarth, Phys. Rev. D 85, (2012)
->
Pirani’s analogy

15. Geodesic deviation 真空
场方程

16. Geodesic deviation
真空场方程

17. Vacuum field equation
Finslerian geodesic deviation
Vacuum field equation

18. Schwarzschild-like solution
Ric=0
Properties
Non-reversible
Two close geodesics
Busemann-Hausdorff volumm
Volume equals to 4\pi
One independent Killing vector

19. III.Finslerian cosmology
SnIa Hubble diagram and fine structure constant
Anisotropic inflation

20. Dipoles of α and SnIa Hubble diagram
Back ground spacetime
Translational and x-y rotational symmetry are preserved
Gravitational field eq:
where

21. Spatial variation of α
Dipoles of Hubble diagram

22. Dataset( variation ):293+10 measurements
A. M. M. Pinho, C. J. A. P. Martins, Phys. Lett. B 756, 121 (2016)
Best fit
Dipole amplitude
Milky Way limit
SnIa Union 2.1 dataset

23. Anisotropic inflation
Background spacetime
Scalar perturbed spacetime
Primordial power spectrum

24. Dipolar modulation (2<l<600)
S. Aiola, B. Wang, A. Kosowsky, T. Kahniashvili and H. Firouzjahi, Phys. Rev. D. 92, (2015)
CMB correlation coefficients
Anisotropic effect only appears in CMB correlation coefficients if l’=l+1

25. Conclusions and Remarks
Finsler spacetime naturally describes the anisotropic spacetime and Lorentz violation
Exist Schwarzschild-like solutions
Finslerian anisotropic inflation: anisotropic effect only appears in CMB correlation coefficients if l’=l+1

26. Riemann spacetime
Gravitational waves
Finsler spacetime

27. Thanks!