Seminar given by Timothy Clifton at the African Institute of Mathematical Sciences (AIMS) on the 21st of February 2011.

1. AIMS, South Africa
21st February 2011
Timothy Clifton (CERN & Oxford)

3. The FLRW model of the Universe
• Motivated by
• The isotropy of the CMB (~1 part in 10 5)
• The homogeneity of large-scale structure
• The homogeneity of the early universe
• Its many successes

4. The FLRW model of the Universe
• Motivated by
• The isotropy of the CMB (~1 part in 10 5)
• The homogeneity of large-scale structure
• The homogeneity of the early universe
• Its many successes
• Implicit assumptions
• Isotropy of observables implies isotropy of space-time
• We are typical observers (the Copernican principle)
• General Relativity is correct on cosmological scales
• Averaging is a well defined and viable process in GR

5. Why is averaging a problem?
• Foliation invariance gives ambiguity

6. Why is averaging a problem?
• Foliation invariance gives ambiguity

7. Why is averaging a problem?
• Foliation invariance gives ambiguity

8. Why is averaging a problem?
• Maintaining general covariance

9. Why is averaging a problem?
• Maintaining general covariance
[see, e.g., ]

10. Why is averaging a problem?
• Non-commutativity with evolution under Einstein’s equations

11. Why is averaging a problem?
• Non-commutativity with evolution under Einstein’s equations
Define:
where
[see, e.g., ]

12. Why is averaging a problem?
• Averaged and observing do not commute, in general

13. Why is averaging a problem?
• Averaged and observing do not commute, in general
and .
and redshifts are given by .

14. Why is averaging a problem?
• Foliation invariance gives ambiguity
• Maintaining general covariance
• Non-commutativity with evolution under Einstein’s equations
• Averaged and observing do not commute, in general

15. How to deal with averaging?
• Exact solutions
• Applying averaging to FRW
• Solutions without averaging

16. How to deal with averaging?
• Exact solutions • Fully non-linear
• Usually requires symmetry
• Often involves a fluid
• Applying averaging to FRW
• Solutions without averaging

17. How to deal with averaging?
• Exact solutions • Fully non-linear
• Usually requires symmetry
• Often involves a fluid
• Applying averaging to FRW • Allows for asymmetry
• Already assumes FRW
• Requires a fluid
• Solutions without averaging

18. How to deal with averaging?
• Exact solutions • Fully non-linear
• Usually requires symmetry
• Often involves a fluid
• Applying averaging to FRW • Allows for asymmetry
• Already assumes FRW
• Requires a fluid
• Solutions without averaging • Assumes nothing
• Not many known solutions
• Not very realistic

19. Cosmology without averaging

20. Cosmology without averaging
The goal is to construct a space-time filled with many
regularly spaced discrete objects:

21. Cosmology without averaging
The goal is to construct a space-time filled with many
regularly spaced discrete objects:

22. Cosmology without averaging
The goal is to construct a space-time filled with many
regularly spaced discrete objects:
We can then ask: Does the smoothed geometry have the same
properties as the unsmoothed space-time?

23. Models with no averaging
• McVittie-type n-body models.
[G. C. McVittie, MNRAS 91, 274 (1931)].
• Einstein-Strauss “Swiss cheese” models.
[A. Einstein & E. G. Strauss, Rev. Mod. Phys. 17, 120 (1945)].
• Lindquist-Wheeler lattice models.
[R. W. Lindquist & J. A. Wheeler, Rev. Mod. Phys. 29, 432 (1957)].
• Perturbative lattice models.
[T. Clifton, arXiv:1005.0788 (2010)].

24. McVittie model
[G. C. McVittie, MNRAS 91, 274 (1931)].
where

25. McVittie model
[G. C. McVittie, MNRAS 91, 274 (1931)].
where
>

26. McVittie model
[G. C. McVittie, MNRAS 91, 274 (1931)].
where
>
Einstein static universe is a different size after averaging.

27. McVittie model
[G. C. McVittie, MNRAS 91, 274 (1931)].
where
>
Einstein static universe is a different size after averaging.
Also, objects can be added to de Sitter space without changing expansion.

28. Einstein-Strauss model
[A. Einstein & E. G. Strauss, Rev. Mod. Phys. 17, 120 (1945)].

29. Einstein-Strauss model
[A. Einstein & E. G. Strauss, Rev. Mod. Phys. 17, 120 (1945)].

30. Einstein-Strauss model
[A. Einstein & E. G. Strauss, Rev. Mod. Phys. 17, 120 (1945)].

31. Observables in ES model
[TC & J. Zuntz , Mon. Not. Roy. Ast. Soc. 400, 2185 (2009)].

32. Lindquist-Wheeler model
[R. W. Lindquist & J. A. Wheeler, Rev. Mod. Phys. 29, 432 (1957)].
Lattice model of the Universe, inspired by Wigner-Seitz construction.

33. Lindquist-Wheeler model
[R. W. Lindquist & J. A. Wheeler, Rev. Mod. Phys. 29, 432 (1957)].
Lattice model of the Universe, inspired by Wigner-Seitz construction.

34. Lindquist-Wheeler model
[R. W. Lindquist & J. A. Wheeler, Rev. Mod. Phys. 29, 432 (1957)].
Lattice model of the Universe, inspired by Wigner-Seitz construction.

35. Lindquist-Wheeler model
• Dynamics governed by the Friedmann equation:

36. Lindquist-Wheeler model
• Dynamics governed by the Friedmann equation:
• Scale of expansion approaches FRW as N→∞:

37. Observables in the LW model
To determine observables in a Lindquist-Wheeler model we need
to understand the optical properties of the large-scale space-time.

38. Observables in the LW model
The frequency of a photon is measured as , so that the
redshift is given by:
Numerical integration then gives redshifts that look like:

39. Observables in the LW model
For a typical trajectory, the low z luminosity distance is given by:

40. Observables in the LW model
For a typical trajectory, the low z luminosity distance is given by:

41. Observables in the LW model
For a typical trajectory, the low z luminosity distance is given by:
(c.f. ).

42. Observables in the LW model
Comparison with the 115 Sne of Astier et al gives:

43. Observables in the LW model
Comparison with the 115 Sne of Astier et al gives:
and

44. Perturbative Lattice model
[TC, arXiv:1005.0788 (2010)].

45. Perturbative Lattice model
[TC, arXiv:1005.0788 (2010)].

46. Perturbative Lattice model
[TC, arXiv:1005.0788 (2010)].

47. Perturbative Lattice model
Consider the metric ansatz:
and a boundary at .

48. Perturbative Lattice model
Consider the metric ansatz:
and a boundary at .
The field equations then give , and the boundary conditions give
and .

49. Perturbative Lattice model
Compare with FLRW:

50. Perturbative Lattice model
Compare with FLRW:
Evolution as given by the Friedmann eq. with
i.e. no back-reaction when masses are regularly arranged!

51. Conclusions

52. Conclusions
• One can investigate the emergence of FLRW space-time, without
performing any averaging procedures on the space-time.

53. Conclusions
• One can investigate the emergence of FLRW space-time, without
performing any averaging procedures on the space-time.
• It can be seen that back-reaction is a small effect when matter is
clumped into regularly arranged masses.

54. Conclusions
• One can investigate the emergence of FLRW space-time, without
performing any averaging procedures on the space-time.
• It can be seen that back-reaction is a small effect when matter is
clumped into regularly arranged masses.
• Observables in these models can be calculated, and can themselves
be averaged. The results of this are not necessarily those of FLRW,
even if the global dynamics are well modelled by that space-time.

55. Conclusions
• One can investigate the emergence of FLRW space-time, without
performing any averaging procedures on the space-time.
• It can be seen that back-reaction is a small effect when matter is
clumped into regularly arranged masses.
• Observables in these models can be calculated, and can themselves
be averaged. The results of this are not necessarily those of FLRW,
even if the global dynamics are well modelled by that space-time.
• Investigations of this type allow one to test the validity of various
different averaging procedures.

56. Conclusions
• One can investigate the emergence of FLRW space-time, without
performing any averaging procedures on the space-time.
• It can be seen that back-reaction is a small effect when matter is
clumped into regularly arranged masses.
• Observables in these models can be calculated, and can themselves
be averaged. The results of this are not necessarily those of FLRW,
even if the global dynamics are well modelled by that space-time.
• Investigations of this type allow one to test the validity of various
different averaging procedures.
• Extending the investigations outlined here to more general situations
will allow more realistic situations to be considered.