1. Accelerating Expansion from Inhomogeneities ? Je-An Gu (National Taiwan University) Collaborators: Chia-Hsun Chuang (astro-ph/0512651) IRGAC2006, 2006/07/14

2. Accelerating Expansion Based on FRW Cosmology (homogeneous & isotropic)

3. Based on FRW Cosmology (homogeneous & isotropic) Supernova data  ?  Cosmic Acceleration However, apparently, our universe is NOT homogeneous & isotropic.  At large scales, after averaging, the universe IS homogeneous & isotropic. But, averaging !? Is it legal ? Does it make sense ?   Existence of cosmic acceleration  Dark energy as a necessity of understanding acceleration

4. Einstein equations For which satisfy Einstein equations, in general DO NOT.

5. Effects of Inhomogeneities through averaging Einstein equations Toy Model: ds 2 = dt 2  a 2 (1 + h coskx cosky coskz) (dx 2 + dy 2 + dz 2 ) Einstein equations after averaging in space : (perturb: h << 1)   p eff =   eff / 3  eff  p eff Dark Geometry

6. Supernova data  ?  Cosmic Acceleration Cosmic Acceleration requires Dark Energy ? Questions (or Inhomogeneity-induced Acceleration ?)

7. Cosmic Acceleration requires Dark Energy ? Normal matter  attractive gravity  slow down the expansion Need something abnormal : e.g. cosmological constant, dark energy -- providing anti-gravity (repulsive gravity) Is This True ? Common Intuition / Consensus

8. Is This True ? Intuitively, YES ! (of course !!) Normal matter  attractive gravity  slow down the expansion Common Intuition / Consensus ** Kolb, Matarrese, and Riotto (astro-ph/0506534) : Inhomogeneities of the universe might induce acceleration. Mission Impossible ? or Mission Difficult ? Two directions: 1.Prove NO-GO theorem. 2.Find counter-examples. This is what we did. We found counter-examples for a dust universe of spherical symmetry, described by the Lemaitre-Tolman-Bondi (LTB) solution.

9. Lemaitre-Tolman-Bondi (LTB) Solution (exact solution in GR) (unit: c = 8  G = 1) Dust Fluid + Spherical Symmetry k(r) = const.,  0 (r) = const., a(t,r) = a(t)  FRW cosmology Solution (parametric form with the help of ) arbitrary functions of r : k(r),  0 (r), t b (r)

10. Line (Radial) Acceleration ( q L < 0 ) Radial : Inhomogeneity  Acceleration Angular : No Inhomogeneity  No Acceleration

11. What is Accelerating Expansion ? (I) Line Acceleration L homogeneous & isotropic universe: RW metric: We found examples of q L < 0 (acceleration) in a dust universe described by the LTB solution.

12. Line (Radial) Acceleration : q L < 0 Inhomogeneity  the less smoother, the better arbitrary functions of r : k(r),  0 (r), t b (r)  parameters : (n k, k h, r k ),  0, r L, t 1 khkh rkrk k(r)k(r) r 0

13. Examples of Line (Radial) Acceleration : q L < 0 arbitrary functions of r : k(r),  0 (r), t b (r) parameters : (n k, k h, r k ),  0, r L, t 1 khkh rkrk k(r)k(r) r 0 nknk khkh rkrk 00 rLrL t qLqL qDqD 2010.7111  0.8  Observations  q ~  1 (based on FRW cosmology) Acceleration

14. Examples of Line (Radial) Acceleration : q L < 0 nknk khkh rkrk 00 rLrL t qLqL qDqD 2010.7111  0.8  k(r) = 0 at r k = 0.7 Over-densityUnder-density

15. Examples of Line (Radial) Acceleration : q L < 0 nknk khkh rkrk 00 rLrL t qLqL qDqD 2010.7111  0.8  k(r) = 0 at r k = 0.7 characterizing the accel/deceleration status of the radial line elements

16. Examples of Line (Radial) Acceleration : q L < 0 Deceleration Acceleration nknk khkh rkrk 00 rLrL t qLqL qDqD 2010.7111  0.8  k(r) = 0 at r k = 0.7

17. Examples of Line (Radial) Acceleration : q L < 0

18. Acceleration Inhomogeneity Examples of Line (Radial) Acceleration : q L < 0

19. nknk khkh rkrk 00 rLrL t (20) 10.7111 Deceleration Acceleration Easy to generate n k =5 larger n k larger inhomogeneity 1 khkh rKrK k(r)k(r) r 0

20. Examples of Line (Radial) Acceleration : q L < 0 nknk khkh rkrk 00 rLrL t 2010.711 (1) Deceleration Acceleration

21. Domain Acceleration ( q D < 0 ) spherical domain r = 0 r = r D

22. What is Accelerating Expansion ? (II) Volume V D a large domain D (e.g. size ~ H 0  1 ) NO-GO q D  0 > 0 (deceleration) in a dust universe (see, e.g., Giovannini, hep-th/0505222) We found examples of q D < 0 (acceleration) in a dust universe described by the LTB solution. [Nambu and Tanimoto (gr-qc/0507057) : incorrect example.] Domain Acceleration

23. Examples of Domain Acceleration : q D < 0 parameters : (n k, k h, r k ), (n t, t bh, r t ),  0, r D, t arbitrary functions of r : k(r),  0 (r), t b (r) nknk khkh rkrk ntnt t bh rtrt 00 rDrD t qDqD 40 0.940100.910 5 1.10.1 11 Acceleration tb(r)tb(r) k(r)k(r)

24. Examples of Domain Acceleration : q D < 0 nknk khkh rkrk ntnt t bh rtrt 00 rDrD t qDqD 40 0.940100.910 5 1.10.1 11 k(r) = 0 at r = 0.82 Over-densityUnder-density

25. Examples of Domain Acceleration : q D < 0 Deceleration Acceleration nknk khkh rkrk ntnt t bh rtrt 00 rDrD t qDqD 40 0.940100.910 5 1.10.1 11 characterizing the accel/deceleration status of the radial line elements

26. Examples of Domain Acceleration : q D < 0 nknk khkh rkrk ntnt t bh rtrt 00 rDrD t (40) 400.940100.910 5 1.10.1 Acceleration

27. Examples of Domain Acceleration : q D < 0 nknk khkh rkrk ntnt t bh rtrt 00 rDrD t 40 (40) 0.940100.910 5 1.10.1 Deceleration Acceleration

28. Examples of Domain Acceleration : q D < 0 nknk khkh rkrk ntnt t bh rtrt 00 rDrD t 40 (0.9) 4010 (0.9) 10 5 1.10.1 Deceleration Acceleration

29. Examples of Domain Acceleration : q D < 0 Deceleration Acceleration larger n t larger inhomogeneity tb(r)tb(r) nknk khkh rkrk ntnt t bh rtrt 00 rDrD t 40 0.9 (40) 100.910 5 1.10.1

30. Examples of Domain Acceleration : q D < 0 Acceleration nknk khkh rkrk ntnt t bh rtrt 00 rDrD t 40 0.940 (10) 0.910 5 1.10.1 Deceleration

31. Examples of Domain Acceleration : q D < 0 Acceleration nknk khkh rkrk ntnt t bh rtrt 00 rDrD t 40 0.940100.9 (10 5 ) 1.10.1

32. Examples of Domain Acceleration : q D < 0 Deceleration Acceleration nknk khkh rkrk ntnt t bh rtrt 00 rDrD t 40 0.940100.910 5 (1.1) 0.1

33. Examples of Domain Acceleration : q D < 0 Deceleration Acceleration nknk khkh rkrk ntnt t bh rtrt 00 rDrD t 40 0.940100.910 5 1.1 (0.1)

34. Summary and Discussions

35. Model: Inhomogeneous Universe (Reality?)  Against the common intuition and consensus : normal matter  attractive gravity  deceleration, Counter-examples (acceleration) are found.  These examples support : Inhomogeneity  Acceleration  Toy Model: ds 2 = dt 2  a 2 (1 + h coskx cosky coskz) (dx 2 + dy 2 + dz 2 )   p eff =   eff / 3  These examples raise two issues : ? Can inhomogeneities explain cosmic acceleration ? (cosmology issue) ? How to understand these counter-intuitive examples ? (GR issue)

36. Can Inhomog. explain “Cosmic Acceleration”? SN Ia DataCosmic Acceleration Inhomogeneities ? ? Mathematically, possible. In Reality ?? ? Can Inhomogeneities explain SN Ia Data? IF YES Does Cosmic Acceleration exist?

37. How to understand the examples ? Normal matter  attractive gravity  slow down the expansion Common Intuition / Consensus Intuition for GR ? NO !?  (x) (x) (valid only for … ?) Newton? NO. GR? YES. Intuition from Newtonian gravity, not from GR.

38. Summary and Discussions GR is still not fully understood after 90 years !!

40. Line (Radial) Acceleration : q L < 0 Sufficient and Necessary Condition: 1 khkh rKrK k(r)k(r) r 0  Sharp enough change in k h (r)  Tuning/choosing the boundary condition ( For constant  0 )