Timothy Clifton - The Problem of Averaging in Cosmology

AIMS, South Africa21st February 2011

Timothy Clifton (CERN & Oxford)

The FLRW model of the Universe

• Motivated by

• The isotropy of the CMB (~1 part in 105)

• The homogeneity of large-scale structure

• The homogeneity of the early universe

• Its many successes

The FLRW model of the Universe

• Motivated by

• The isotropy of the CMB (~1 part in 105)

• 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

Why is averaging a problem?

• Foliation invariance gives ambiguity


• Maintaining general covariance



[see, e.g., ]


• Non-commutativity with evolution under Einstein’s equations



Define:

where

[see, e.g., ]


• Averaged and observing do not commute, in general


and .

and redshifts are given by .



• Foliation invariance gives ambiguity




How to deal with averaging?

• Exact solutions

• Applying averaging to FRW

• Solutions without averaging


• Exact solutions



• Fully non-linear

• Usually requires symmetry

• Often involves a fluid


• Exact solutions






• Allows for asymmetry

• Already assumes FRW

• Requires a fluid


• Exact solutions






• Allows for asymmetry

• Already assumes FRW

• Requires a fluid

• Assumes nothing

• Not many known solutions

• Not very realistic

Cosmology without averaging


The goal is to construct a space-time filled with many

regularly spaced discrete objects:


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?

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)].

McVittie model

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

where

McVittie model


where

>

McVittie model


where

>

Einstein static universe is a different size after averaging.

McVittie model


where

>

Einstein static universe is a different size after averaging.

Also, objects can be added to de Sitter space without changing expansion.

Einstein-Strauss model

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

Observables in ES model

[TC & J. Zuntz , Mon. Not. Roy. Ast. Soc. 400, 2185 (2009)].

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.


• Dynamics governed by the Friedmann equation:


• Dynamics governed by the Friedmann equation:

• Scale of expansion approaches FRW as N→∞:

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.


The frequency of a photon is measured as , so that the

redshift is given by:

Numerical integration then gives redshifts that look like:


For a typical trajectory, the low z luminosity distance is given by:


(c.f. ).

For a typical trajectory, the low z luminosity distance is given by:


Comparison with the 115 Sne of Astier et al gives:


Comparison with the 115 Sne of Astier et al gives:

and

Perturbative Lattice model

[TC, arXiv:1005.0788 (2010)].


Consider the metric ansatz:

and a boundary at .


Consider the metric ansatz:

and a boundary at .

The field equations then give , and the boundary conditions give

and .


Compare with FLRW:


Compare with FLRW:

Evolution as given by the Friedmann eq. with

i.e. no back-reaction when masses are regularly arranged!

Conclusions

Conclusions

• One can investigate the emergence of FLRW space-time, without

performing any averaging procedures on the space-time.

Conclusions



• It can be seen that back-reaction is a small effect when matter is

clumped into regularly arranged masses.

Conclusions





• 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.

Conclusions








• Investigations of this type allow one to test the validity of various

different averaging procedures.

Conclusions








• 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.

Timothy Clifton - The Problem of Averaging in Cosmology

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