1. SIGRAV and INFN School GGI, Firenze: 2009
DESCRIBING INHOMOGENEOUS COSMOLOGIES: TOOLS TO USE IN THE INVESTIGATION George F R Ellis University of Cape Town
SIGRAV and INFN School
GGI, Firenze: 2009

2. Importance again Validity of weak field methods: Ehlers estimates:
. 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]

3. Importance again
. Small scale non-linear inhomogeneities plus dynamical back reactions can in principle give observations that mimic FLRW universe with dark energy
BUT
Considerable controversy as to whether this is important or not
Is it significant, or completely negligible?

4. . Genuinely GR effects can occur: Wiltshire: importance of voids
Importance again
. Genuinely GR effects can occur:
Wiltshire: importance of voids

5. 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

6. 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

7. Lindquist and Wheeler

8. Exact equations and theorems
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

9. Exact equations and theorems
. Warning example:
Zero shear results
not given by Newtonian limit

10. 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

11. 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?

12. Fitting a smoothed model
Fit background globally;
determine pointwise difference
This fixes a best-fitting gauge

13. 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

14. 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)

15. 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

16. LTB (Lemaitre-Tolman Bondi models
Metric: In comoving coordinates,
ds2 = -dt2 + B2(r,t) + A2(r,t)(dΘ2+sin2 Θ dΦ2)
where
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).

17. Mustapha, Hellaby, & Ellis
Alnes, Amarzguioui, and Gron astro-ph/
Mustapha, Hellaby, & Ellis
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.
NOTE THAT THESE ARE STRONGLY FAVOURED over FL IF WE ASSUME THERES NO DE!

18. 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: v2 [gr-qc]

19. 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: ].
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

20. scales probed by different observations: different distances

21. 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:

22. can be made to match the WMAP observations.”
Ishak et al
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.
NOTE THAT THESE ARE STRONGLY FAVOURED over FL IF WE ASSUME THERES NO DE!
“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.”

23. 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.

24. 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

25. Improbability “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 for Λ in a FRW universe.
Also: there is no proof universe is probable.
May be improbable!! Indeed, it is!!

26. Improbability 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

27. 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: ,1995: arXiv:hep-th/ v1

28. Improbability
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?

29. 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.

30. Biswas, Monsouri and Notari, astro-ph/0606703
Spherical Symmetry
Biswas, Monsouri and Notari, astro-ph/
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.
NOTE THAT THESE ARE STRONGLY FAVOURED over FL IF WE ASSUME THERES NO DE!
given that we can always find
can we distinguish between the two?

31. [Goodman 1995; Caldwell & Stebbins 2007]
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]

32. 1: Consistency test of LTB
Must not have observational cusp at origin – implies singularity there
Vanderveld, Flangan and Wasserman astro-ph/
“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

33. Distance Measurements
two effects on distance measurements:
curvature bends null geodesics
expansion
eg, positive curvature increases angular sizes
These are coupled in FLRW, decoupled in LTB

34. 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/ ]

35. 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

36. 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

37. 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]

38. 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/ ]

39. 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: ].

40. 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

41. 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

43. G F R Ellis, H van Elst "Cosmological models"
References:  
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).