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Fitting and Averaging in cosmology: Dynamical and observational aspects George F R Ellis University of Cape Town. SIGRAV and INFN School GGI, Firenze: 2009. The Acceleration of the universe. The explanation of dark energy is a central pre-occupation of present day cosmology.

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Fitting and Averaging in cosmology:Dynamical and observational aspectsGeorge F R EllisUniversity of Cape Town

SIGRAV and INFN School

GGI, Firenze: 2009

the acceleration of the universe

The Acceleration of the universe

The explanation of dark energy is a central pre-occupation of present day cosmology.

Its presence is indicated by the recent speeding up of the expansion of the universe indicated by supernova observations

confirmed by other observations such as those of the cosmic background radiation anisotropies and large scale clustering

Its nature (whether constant, or varying) is a major problem for theoretical physics

inhomogeneity and the acceleration of the universe

Inhomogeneity and the Acceleration of the universe

. The deduction of the existence of dark energy is based on the assumption that the universe has a Robertson-Walker geometry

- spatially homogeneous and isotropic on a large scale.

The observations can at least in principle be accounted for without the presence of any dark energy, if we consider the possibility of inhomogeneity

This can happen in two ways:

locally via backreaction associated with averaging, plus associated observational effects (this talk)

by large scale inhomogeneity (next talk)

1 the hidden averaging scal e

1: The hidden averaging scale

Any mathematical description of a physical system depends on an averaging scale characterizing the nature of the

envisaged model. This averaging scale is usually hidden from view: it is taken to be understood.

Thus, when a fluid is described as a continuum, this assumes one is using an averaging scale large enough that the size of individual molecules is negligible. If the averaging scale is close to molecular scale, small changes in the position or size of the averaging volume lead to large changes in the measured density and velocity of the matter, as individual molecules are included or excluded from the reference volume. Then the fluid approximation is no longer applicable.

the hidden averaging scal e

The hidden averaging scale

.Each variable definition hides an

averaging scale: e.g. density of gas

Averaging scale


Usual work referring to the fluid density and velocity assumes a medium--size averaging scale: not so small that molecular effects matter, but not so large that spatial gradients in the properties of the fluid are significant.

The actual averaging scale, or rather the acceptable range of averaging scales, is not explicitly stated but is in fact a key--feature underlying the description used, and hence the effective macroscopic dynamical laws investigated.

Indeed, different types of physics (particle physics, atomic

physics, molecular physics, macroscopic physics, astrophysics) correspond to different assumed averaging scales. Thus, instead of referring to a density function ρ, one should really refer to a function ρ(L): the density averaged over volumes characterized by scale length L.


The key--point about the fluid approximation is that, provided this length scale is in the appropriate domain, then its actual value does not matter; i.e. when it is in this range, then changing L by a factor of 10, 100, or even much more makes no difference: the measured density and average velocity will not change.

But if you change L by a very large amount until outside this range, this is no longer true.

Hence, there is a range of validity L1 < L < L2 where the fluid approximation holds and explicit mention of the associated averaging scale may be omitted.

averaging scal e relations

Averaging scale relations

Relations between scales: lower level relations underlie higher ones, but

There is a non-commutativity of averaging with dynamics and observations

Averaging leads to extra terms in effective higher level equations

Cosmology: contribution to dark energy?

local inhomogeneity description

Local inhomogeneity:description

Multiple scales of representation of same system

Implicit averaging scale

Stars, clusters, galaxies, universe




In electromagnetic theory,

polarization effects result from a large--scale field being applied to a medium with many microscopic charges. The macroscopic field E differs from the point--to—point microscopic field which acts on the individual charges, because of a fluctuating internal field Ei , the total internal field at each point being D = E + Ei

Spatially averaging, one regains the average field because the internal field cancels out: E = <D>, indeed this is how the macroscopic field is defined (implying invariance of the background field under averaging: E = <E> ).

On a microscopic scale, however, the detailed field D is the effective physical quantity, and so is the field ``measured'' by electrons and protons at that scale. Thus, the way different test objects respond to the field crucially depends on their scale. A macroscopic device will measure the averaged field.


Exactly the same issue arises with regard to the gravitational field. The solar system tests of general relativity theory are at solar system scales. We apply gravitational theory, however, at many other scales: to star clusters, galaxies, clusters of galaxies, and cosmology.

Cosmology utilizes the largest scale averaging envisaged in astrophysics: a representative scale is assumed that is a significant fraction of the Hubble scale, and the cosmological velocity and density functions are defined by averaging on such scales.


The General Relativistic cosmological perturbation solutions used to study structure formation embody two interacting levels: the background (zero--order) model, almost always a Robertson—Walker metric, and the perturbed (first--order) model representing the growth of inhomogeneities, represented by a perturbed Robertson--Walker metric.

Both are quite different than the solar system scale where the EFE are tested.

The question then is how do models on two or more different scales relate to each other in Einstein's gravitational theory. This is a difficult issue both because of the non--linearity of Einstein's equations, and because of the lack of a fixed background spacetime -- one of the core features of Einstein's theory. This causes major problems in defining suitable averaging processes as needed in studying these processes.

2 local inhomogeneity dynamic effects

2. Local inhomogeneity:dynamic effects

Averaging and calculating the field equations do not commute

G. F. R. Ellis: ``Relativistic cosmology: its nature, aims and problems". In General Relativity and Gravitation, Ed B Bertotti et al (Reidel, 1984), 215.

Has implications for cosmology (Kolb, Mataresse, Buchert, Wiltshire, et al.)

local inhomogeneity dynamic effects

Local inhomogeneity:dynamic effects

Averaging and calculating the field equations

do not commute

g1ab R1ab G1ab = T1abScale 1


g3ab R3ab G3ab= T3ab Scale 3

averaging process

averaging gives different answer

averaging effects

Averaging effects

Metric tensor: gabĝab =‹gab›

Inverse Metric tensor: gabĝab =‹gab›

but not necessarily inverse …

need correction terms to make it the inverse

Connection: Γabc‹Γabc› + Cabc

new is average plus correction terms

Curvature tensor plus correction terms

Ricci tensor plus correction terms

Field equations G ab =Tab + Pab

problem of covariant averaging

Problem of covariant averaging

The problem with such averaging procedures is that they are not covariant. Can’t average tensor fields in covariant way (coordinate dependent results).

They can be defined in terms of the background unperturbed space, usually either flat spacetime or a Robertson--Walker geometry, and so will be adequate for linearized calculations where the perturbed quantities can be averaged in the background spacetime.

But the procedure is inadequate for non--linear cases, where the integral needs to be done over a generic lumpy (non--linearly perturbed) spacetime that are not ``perturbations'' of a high--symmetry background. However, it is precisely in these cases that the most interesting effects will occur.

problem of covariant averaging1

Problem of covariant averaging

Can’t average tensor fields in covariant way (coordinate dependent results)

Can use bitensors (Synge) for curvature and matter, but not for metric itself: and leads to complex equations

R Zalaletdinov“The Averaging Problem in Cosmology and Macroscopic Gravity” Int. J. Mod. Phys. A 23: 1173 (2008) [arXiv:0801.3256]

Scalars: can be done (Buchert),

But: usually incomplete, so hides effects


Polarisation Form (flat background)

Peter Szekeres developed a polarization formulation for a

gravitational field acting in a medium, in analogy to electromagnetic polarization. He showed that the linearized Bianchi identities for an almost flat spacetime may be expressed in a form that is suggestive of Maxwell's equations with magnetic monopoles.

Assuming the medium to be molecular in structure, it is shown how, on performing an averaging process on the field

quantities, the Bianchi identities must be modified by the inclusion of polarization terms resulting from the induction of quadrupole moments on the individual ``molecules''. A model of a medium whose molecules are harmonic oscillators is discussed and constitutive equations are derived.


This results in the form:

G ab =Tab + Pab . , Pab = Qabcd;cd

that is Pab is expressed as the double divergence of an effective quadrupole gravitational polarization tensor with suitable symmetries:

Qabcd = Q[ab][cd] = Qcdab

Gravitational waves are demonstrated to slow down in such a medium. Thus the large scale effective equations include polarisation terms, as in the case of electromagnetism

P Szekeres: “Linearised gravitational theory in macroscopic media” Ann Phys 64: 599 (1971)

the averaging problem in cosmology

The averaging problem in cosmology

Buchert equations for scalars gives modified Friedmann equation

T Buchert “Dark energy from structure: a status report”. GRG Journal 40: 467 (2008) [arXiv:0707.2153].


Expansion and averaging do not commute:

in any domain D, for any field Ψ

∂t<Ψ> - <∂tΨ> = <θΨ> - <θ><Ψ>

the averaging problem in cosmology1

The averaging problem in cosmology

Buchert equations for scalars gives modified Friedmann and Raychaudhuri equations: e.g.

∂t<Θ>D = Λ - 4πGρD + 2 <II>D - <I>D2

where II = Θ2/3 - σ2 and I = Θ.

This in principle allows acceleration terms to arise from the averaging process

local inhomogeneity dynamic effects1

Local inhomogeneity:dynamic effects

Claim: weak field approximation is adequate and shows effect is negligible (Peebles)

Counter claim: it certainly matters

Kolb, Mattarrese, others

NB one can check if it can explain dark energy issue fully

But if not it might still upset the cosmic concordance: it might show spatial sections are not actually flat

local inhomogeneity and the observations

Local inhomogeneity:and the observations

Fully explain it? Maybe:

B.M. Leith, S.C.C. Ng and D.L. Wiltshire

"Gravitational energy as dark energy: Concordance of cosmological tests" Astrophys. J. 672, L91 (2008) [arXiv:0709:2535].

T. Mattsson “Dark energy as a mirage” (2007) [arXiv:0711.4264]


But others disagree:

S. Rasanen: “Evaluating backreaction with the peak model of structure formation”arxiv:0801.2692 (2008).

But then it still can alter basic relations: density to curvature

3 local inhomogeneity observational effects

3: Local inhomogeneity:observational effects

Ricci focusing and Weyl focusing

B. Bertotti “The Luminosity of Distant Galaxies” Proc Royal Soc London. A294, 195 (1966).

dθ/dv = -RabKaKb - 2σ2 – θ2

d σmn/dv = - Emn

Θ = expansion

σ = shear

Rab = Ricci tensor, determined pointwise by matter

Eab = Weyl tensor, determined non-locally by matter


Robertson-Walker observations:

zero Weyl tensor and non-zero Ricci tensor.

dθ/dv = -RabKaKb

d σmn/dv = 0

Actual observations are best described by zero Ricci tensor and non-zero Weyl tensor

dθ/dv = - 2σ2 – θ2

d σmn/dv = - Emn

This averages out to FRW equations when averaged over whole sky But supernova observations are preferentially where there is no matter

observations and averaging

Observations and averaging

Dyer Roeder equations take matter into account but not shear: allows a fraction of the uniform density

C. C Dyer. & R C Roeder, “Observations in Locally Inhomogeneous Cosmological Models” Astrophysical Journal, Vol. 189: 167 (1974)

NB: must take shear and caustics into account

Claim: averaging over whole sky leads to standard FRW form

Not obvious! It does not follow from energy conservation (Weinberg):

- depends on how area distances average out


Swiss-Cheese models: FRW regions joined to vacuum regions Exact inhomogeneous solutions

The null geodesic equations can be exactly integrated in each domain and matched. Includes shear effects.

Ron Kantowski has obtained analytic expressions for distance--redshift relations that have been corrected for the effects of inhomogeneities in the density.

The values of the density parameter and cosmological constant inferred from a given set of observations depends on the fractional amount of matter in inhomogeneities and can significantly differ from those obtained by using the Mattig relations for the FLRW universes.


As an example, “a determination of Ω0 made by

applying the homogeneous distance--redshift relation to SN 1997ap at z = 0.83 could be as much as 50% lower than its true value.”

R. Kantowski “The Effects of Inhomogeneities on Evaluating the mass parameter Ωm and the cosmological constant Λ” (1998) [astro-ph/9802208]

It could be that the apparent acceleration term detected is at least partly due to this optical effect: focusing of null geodesics is different in a lumpy universe than in a smooth one. Debatable if enough to account for apparent acceleration; probably is enough to significantly influence concordance model values.

fitting problem and gauge invariance

Fitting Problem and gauge invariance

What is the best background model?

Gauge freedom with change of background model,

And of fitting of perturbed model to background model

The variables and their dynamical equations depend on this fitting, which will take presumably place by some process of averaging

Change of background model changes effects!

local inhomogeneity the fitting problem

Local inhomogeneity:the fitting problem

Which background model to use?

G.F.R. Ellis, W.R. Stoeger: ``The fitting problem in cosmology", Class. Quant. Grav. 4 (1987) 1697.



fitting problem and averaging

Fitting Problem and averaging

An obvious approach to fitting is some kind of averaging

But what quantities should we average?

Over what surfaces/volumes should we average?

Spacelike of null averages?

Explicit or implicit (via observational relations as in usual FLRW fitting)?

fitting problem and averaging1

Fitting Problem and averaging


Without insufficient care, adding in a statistical distribution of density perturbations can alter the average.

e.g. in lensing models

Should use compensated inhomogeneities where each overdensity is surrounded by an equal underdensity

Else changes the average density of the universe contemplated

The original background does not give a best fit any more

fitting problem and gauge invariance1

Fitting Problem and gauge invariance

Combine approaches:

Use 1+3 covariant and gauge invariant variables to describe the inhomogeneous geometry and dynamics (Dunsby)

This gives exact and perturbed equations to any desired order, at the detailed scale

2. Choose a best fitting of background to real universe when doing the averaging; use this best fitting to fix the gauge and coordinates



3.Averaging via bitensors (Zalaletdinov)

4. Use only scalars: introduce a sufficient number of scalars to completely determine the spacetime geometry

The run into the equivalence problem of GR: how do you use scalars to completely characterise a solution?

issue of significance does backreaction matter

Issue of significance: does backreaction matter?

Two views:

Peebles, Wald, et al; negligible

Buchert, Kolb, Mattarese, Wiltshire, et al: important

Note issue of voids in universe (Wiltshire)

Not like FRW at all!

local inhomogeneity observational and dynamic effects

Local inhomogeneity:observational and dynamic effects

Counter claim: as there are major voids in the expanding universe a weak-field kind of approximation is not adequate

You have to model (quasi-static) voids and junction to expanding external universe

D.L. Wiltshire "Cosmic clocks, cosmic variance and cosmic averages" New J. Phys.9, 377 (2007) [arXiv:gr-qc/0702082].



  :   G.F.R. Ellis, T. Buchert: "The universe seen at different scales"

       G F R Ellis and W R Stoeger: ``The Fitting Problem in Cosmology". CQG 4, 1679-1690 (1987).