1. Accelerating Expansion from Inhomogeneities ? Je-An Gu ( 顧哲安 ) National Taiwan University IoPAS 2006/03/17 Collaborators: Chia-Hsun Chuang ( 莊家勛 ) W-Y. P. Hwang ( 黃偉彥 ) (astro-ph/0512651)

2. Acceleration 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 ? 

4. Einstein equations satisfy Einstein equations BUT in general DO NOT.

5. Supernova data  ?  Cosmic Acceleration Cosmic Acceleration requires Dark Energy ? Questions

6. 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

7. 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.

8. What is Accelerating Expansion ? (I) Line Acceleration What is Accelerating Expansion ? (II) Domain Acceleration Lemaitre-Tolman-Bondi (LTB) Solution (exact solution in GR) (spherically symmetric dust fluid)

9. 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.

10. 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

11. 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)

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

13. 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

14. 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  0.9  Observations  q ~  1 (based on FRW cosmology) Acceleration

15. Examples of Line (Radial) Acceleration : q L < 0 nknk khkh rkrk 00 rLrL t qLqL qDqD 2010.7111  0.9  k(r) = 0 at r k = 0.7 Over-densityUnder-density

16. Examples of Line (Radial) Acceleration : q L < 0 Deceleration Acceleration nknk khkh rkrk 00 rLrL t qLqL qDqD 2010.7111  0.9  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 larger n k larger inhomogeneity 1 khkh rKrK k(r)k(r) r 0 Easy to generate n k =3

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. Domain Acceleration : q D < 0  t b (r) = 0 : NO acceleration  k(r)k(r)  parameters : (n k, k h, r k ), (n t, t bh, r t ),  0, r D, t [Nambu and Tanimoto: incorrect example.] arbitrary functions of r : k(r),  0 (r), t b (r) tb(r)tb(r)

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 k(r) = 0 at r = 0.82

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.  Against the common intuition and consensus : normal matter  attractive gravity  deceleration, Counter-examples for both the Line and the Domain Acceleration are found.  These examples support : Inhomogeneity  Acceleration  These examples raise two issues : (next slide)

36. How to understand the examples ? (GR issue) Can Inhomog. explain “Cosmic Acceleration”? (Cosmology issue)

37. IF YES Can Inhomog. explain “Cosmic Acceleration”? SN Ia DataCosmic Acceleration Inhomogeneities ? ? Mathematically, possible. In Reality ?? ? Can Inhomogeneities explain SN Ia Data? Do these Inhomog. Indicate Cosmic Acceleration?

38. source earth LTB (Each circle represents a LTB region.) over-density under-density Can Inhomogeneities explain SN Ia Data ?

39. energy density  (x) x light Can Inhomogeneities explain SN Ia Data ?  The effects of inhomogeneities on the cosmic evolution should be restudied. (No matter whether inhomogeneities can solely explain SN Ia data, …)

40. 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.

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