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DESCRIBING INHOMOGENEOUS COSMOLOGIES: TOOLS TO USE IN THE INVESTIGATION George F R Ellis University of Cape Town. SIGRAV and INFN School GGI, Firenze: 2009. Importance again. . Perturbation methods: validity of weak field methods and Newtonian limit Validity of weak field methods:

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SIGRAV and INFN School

GGI, Firenze: 2009

importance again

Importance again

. Perturbation methods: validity of weak field methods and Newtonian limit

Validity of weak field methods:

Ehlers estimates:

very large density inhomogeneities but small metric differences?

paradox of inhomogeneity:

In this room δρ/ρ=101/10-30 = 10+30

Metric perturbations in solar system very small

Metric variations not very small [Ehlers]

importance again1

Importance again

. Small scale non-linear inhomogeneities plus dynamical back reactions can in principle give observations that mimic FLRW universe with dark energy


Considerable controversy as to whether this is important or not

Is it significant, or completely negligible?

importance again2

Importance again

. Genuinely GR effects can occur:

Wiltshire: importance of voids


Void dominated universe:

geometry and dynamics in and out of void are

quite different (Einstein/Strauss, Lindquist/Wheeler)

Time runs at different rate in and out of voids:

and the difference is cumulative (Harwitt, Wiltshire)

So a key issue:

How large in space and time is the domain where

quasi-Newtonian coordinates can be used in a realistic model of an expanding universe with voids?

Wald: globally; Wiltshire: not so


Exact solutions and results:

Inhomogeneous models that average out to give FRW observations in large

1: Lindquist and Wheeler

2: LTB: Lemaitre-Tolman-Bondi

3: Swiss cheese:

Kantowski, Dyer

Nb not necessarily associated with averaging: just making exact models that give the same kind of observations

exact equations and theorems

Exact equations and theorems

.Exact equations:

1+3 Covariant equations

Tetrad methods and coordinates to complete them

Note problem:

We want to deal with both timelike and null curves (dynamics and also observations)

Coordinates and tetrad to suit the one will not be well fitting for the other

exact equations and theorems1

Exact equations and theorems

.Warning example:

Zero shear results

not given by Newtonian limit

approximate solutions

Approximate solutions:


Gauge dependence, gauge invariance, and Sachs Theorem

1+3 covariant gauge invariant equations

and variables

Averaging and gauge invariance

1+3 covariant and gauge invariant variables and equations

But non-covariant fitting and then averaging to deal with multiple scales and associated back reaction

averaging and covariance

Averaging and covariance

May need to use non covariant methods

Indeed the are probably necessary almost always when we get down to real experimental detail

Solar system tests, binary pulsar calculations

But then what guiding principle to use?

Calculate invariant or observational quantities

use best-fitting procedure to choose background

What is the correct background model?

How to fit it?

What coordinates to use?

fitting a smoothed model

Fitting a smoothed model

Fit background globally;

determine pointwise difference

This fixes a best-fitting gauge

overall averaging and small scale inhomogeneity

Overall: Averaging and Small Scale inhomogeneity

These scale-related effects occur

Dynamical effects may be small

Observational effects may still be large

and may affect estimates of acceleration

and the cosmological constant

large scale inhomogeneity and the acceleration of the universe

Large Scale 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:

- local via backreaction (so far)

- large scale (final section)

large scale inhomogeneity inhomogeneous geometry

Large scale inhomogeneity:inhomogeneous geometry

Perhaps there is a large scale inhomogeneity of the observable universe

such as that described by the Lemaitre-Tolman-Bondi pressure-free spherically symmetric models.

We are near the centre of a void (comment later)

Marie-Noëlle Célérier

Challenging dark energy with exact inhomogeneous models

ltb lemaitre tolman bondi models

LTB (Lemaitre-Tolman Bondi models

Metric: In comoving coordinates,

ds2 = -dt2 + B2(r,t) + A2(r,t)(dΘ2+sin2 ΘdΦ2)


B2(r,t) = A’(r,t)2 (1-k(r))-1

and the evolution equation is

(Å/A)2 = F(r)/A3 + 8πGρΛ/3 - k(r)/A2

with F’ (A’A2)-1 = 8πGρM.

Two arbitrary functions: k(r) (curvature) and F(r) (matter).


Solving the Observer Metric

Charles Hellaby, Alnadhief H. A. Alfedeel

The analysis of modern cosmological data is becoming an increasingly important task as the amount of data multiplies. An important goal is to extract geometric information, i.e. the metric of the cosmos, from observational data. The observer metric is adapted to the reality of observations: information received along the past null cone, and matter flowing along timelike lines. It provides a potentially very good candidate for a developing general numerical data reduction program. As a basis for this, we elucidate the spherically symmetric solution, for which there is to date single presentation that is complete and correct. With future numerical implementation in mind, we give a clear presentation of the mathematical solution in terms of 4 arbitrary functions, the solution algorithm given observational data on the past null cone, and we argue that the evolution from one null cone to the next necessarily involves integrating down each null cone.

arXiv:0811.1676v2 [gr-qc]


Other observations??

Can also fit cbr observations:

Larger values of r

S. Alexander, T. Biswas, A. Notari, D. Vaid “Local void vs dark energy: confrontation with WMAP and Type IA supernovae” (2007) [arXiv:0712.0370].

Nb: cbr dipole can then (partly) be because we are a bit off-centre

Re-evaluate the great attractor analysis

Quadrupole? Perhaps also (and alignment)

Nucleosynthesis: OK

Baryon acoustic oscillations?

Maybe – more tricky


Typical observationally viable model:

We live roughly centrally (within 10% of the central position) in a large void:

a compensated underdense region stretching to z ≈ 0.08 with δ≈ -0.4 and size 160/h Mpc to 250/h Mpc, a jump in the Hubble constant of about 1.20, and no dark energy or quintessence field

Solving inverse problem with inhomogenoeus universe

Yoo, Kai, Nakao arXiv:0807.0932


Ishak et al 0708.2943

“We find that such a model can easily explain the observed luminosity distance-redshift

relation of supernovae without the need for dark energy, when the inhomogeneity is in

the form of an underdense bubble centered near the observer. With the additional

assumption that the universe outside the bubble is approximately described by a

homogeneous Einstein-de Sitter model, we find that the position of the first CMB peak

can be made to match the WMAP observations.”

large scale inhomogeneity dynamic evolution

Large scale inhomogeneity:dynamic evolution

Can we find dynamics (inflation, HBB) that matches the observations?

Same basic dynamics (FRW evolution along individual world lines) but with distant dependent parameters

If we are allowed usual tricks of fiddling potential and adding in multiple fields then of course we can!

Paul Hunt

Constraints on large scale voids from WMAP-5 and SDSS

Problems if Gaussian – but it may not be Gaussian.

large scale inhomogeneity dynamic evolution1

Large scale inhomogeneity:dynamic evolution

Will inflation prevent it? Depends on the initial data, the amount of inflation, and the details of the unknown inflaton

There is sufficient flexibility that it should certainly be possible

In any case we can be conservative about inflation until it becomes a proper physical theory (i.e. the inflaton and its potential are uniquely identified)

rather than a paradigm for speculative theory creation



“It is improbable we are near the centre”

But there is always improbability in cosmology

Can shift it:

FRW geometry

Inflationary potential

Inflationary initial conditions

Position in inhomogeneous universe

Which universe in multiverse

Competing with probability 10-120 for Λ in a FRW universe.

Also: there is no proof universe is probable.

May be improbable!! Indeed, it is!!



There is only one universe

Concept of probability does not apply to a single object, even though we can make many measurements of that single object

There is no physically realised ensemble to apply that probability to, unless a multiverse exists

– which is not proven: it’s a philosophical assumption

and in any case there is no well-justified measure for any such probability proposal


Do We Live in the Center of the World?

Andrei Linde, Dmitri Linde, Arthur Mezhlumian

We investigate the distribution of energy density in a stationary self-reproducing inflationary universe. We show that the main fraction of volume of the universe in a state with a given density $\rho$ at any given moment of time $t$ in synchronous coordinates is concentrated near the centers of deep exponentially wide spherically symmetric holes in the density distribution. A possible interpretation of this result is that a typical observer should see himself living in the center of the world. Validity of this interpretation depends on the choice of measure in quantum cosmology.

Phys.Lett.B345:203-210,1995: arXiv:hep-th/9411111v1



In any case there is a basis for saying that inflation prefers a void dominated universe, where we are near the centre of the universe

Linde, Linde and Mezhlumian

Whatever theory may say, it must give way to such tests

Can we observationally test the inhomogeneity possibility?

5 direct observational tests

5 Direct Observational tests

It follows that:

direct observational tests of the Copernican (spatial homogeneity) assumption are of considerable importance;

particularly those that are independent of field equations or matter content

This is now the subject of investigation

The following section is with the help of

Chris Clarkson.

spherical symmetry

given that we can always find

  • can we distinguish between the two?

Biswas, Monsouri and Notari, astro-ph/0606703

Spherical Symmetry
observational tests
Observational Tests
  • only previously known direct tests use scattered CMB photons - looking inside past null cone
    • if CMB very anisotropic around distant observers, SZ scattered photons have distorted spectrum
    • but model dependent - good for void models but misses, e.g., conformally stationary spacetimes
  • ideally we need a model-independent ‘forensic’ test ... is FLRW the correct metric?

[Goodman 1995; Caldwell & Stebbins 2007]

1 consistency test of ltb
1: Consistency test of LTB
  • Must not have observational cusp at origin – implies singularity there

Vanderveld, Flangan and Wasserman astro-ph/0602476

“Living in a void: Testing the Copernican Principle with distant supernovae”, T Clifton, P G Ferreira and K Land

  • Distance modulus Δdm(z) ≈ - (5/2)q0z in ΛCDM, but if this were true in void model without Λ this implies singularity

- Observational test will be available from intermediate redshift supernovae in future

distance measurements
Distance Measurements
  • two effects on distance measurements:


curvature bends null geodesics

  • eg, positive curvature increases angular sizes
  • These are coupled in FLRW, decoupled in LTB
measuring curvature in flrw
Measuring Curvature in FLRW
  • in FLRW we can combine Hubble rate and distance data to find curvature
  • independent of all other cosmological parameters, including dark energy model, and theory of gravity
  • can be used at single redshift
  • what else can we learn from this?

[see Clarkson Cortes & Bassett JCAP08(2007)011; arXiv:astro-ph/0702670]

2 generic consistency test of flrw
2: Generic Consistency Test of FLRW
  • since independent of z we can differentiate to get consistency relation
  • depends only on FLRW geometry:
    • independent of curvature, dark energy, theory of gravity
  • consistency test for homogeneity and isotropy
  • should expect in FLRW
testing the copernican assumption
Testing the Copernican Assumption
  • Copernican assumption hard to test ... but in non-FLRW
      • even at center of symmetry
      • two free functions in LTB (even for dust) - H(z) & D(z)
      • FLRW has just H(z) - or w(z) [ignoring inflaton]
  • can also be used to find correct ‘FLRW scale’

- cf averaging problem

Errors may be estimated from a series expansion
  • simplest to measure H(z) from BAO [already a 2-sigma discrepancy...?]
  • time drift of redshifts over many years gives [in FLRW]
  • or relative ages of passively evolving galaxies (eg, LRGs) gives
  • can’t use inverted distance data as it assumes FLRW

deceleration parameter measured from distance measurements

deceleration parameter measured from Hubble measurements

[Percival et al]

[Uzan Clarkson & Ellis]

[Jimemez & Loeb]

it s only as difficult as dark energy
It’s only as difficult as dark energy...
  • measuring w(z) from Hubble uses
    • requires H’(z)
  • and from distances requires second derivatives D’’(z)
  • simplest to begin with via

[see Clarkson Cortes & Bassett JCAP08(2007)011; arXiv:astro-ph/0702670]

3 indirect observational tests

3: Indirect Observational tests

If the standard inverse analysis of the supernova data to determine the required equation of state shows

there is any redshift range where

w := p/ρ < -1,

this may well be a strong indication that one of these geometric explanations is preferable to the Copernican (Robertson-Walker) assumption,

for otherwise the matter model indicated by these observations is non-physical (it has a negative k.e.)

M.P. Lima, S. Vitenti, M.J. Reboucas “Energy conditions bounds and their confrontation with supernovae data” (2008) [arXiv:0802.0706].


The physically most conservative approach is to assume no unusual dark energy but rather that geometry might be responsible for the observed apparent acceleration

This could happen due to small scale inhomogeneity that definitely exists, but may not be sufficiently significant

Or due to large scale inhomogeneity that can probably do the job, but may not exist

Observational tests of the latter possibility is as important as pursuing the dark energy (exotic physics) option in a homogeneous universe

Theoretical prejudices as to the universe’s geometry, and our place in it, must bow to observational tests

5 conclusion

5: Conclusion

The issue of what is testable and what is not testable in cosmology is a key issue

Some dark energy proposals, specifically multiverse advocates, propose weakening the link to observations because we believe we have a good theory

We should stand firm and insist that genuine science is based on observational testing of plausible hypotheses

There is nothing wrong with physically motivated philosophical explanation:

but it must be labelled for what it is

Overall: theory must be subject to observational test

There is good progress in this respect as regards both dark matter and dark energy



G F R Ellis, H van Elst "Cosmological models"

G F R Ellis and D R Matravers: "General Covariance in General Relativity."

Gen Rel Grav 27, 777 (1995).