[1104.1300] Is the Universe homogeneous?
Authors:  Roy Maartens (Western Cape, ICG, Portsmouth) 
Abstract:  The standard model of cosmology is based on the existence of homogeneous surfaces as the background arena for structure formation. Homogeneity underpins both general relativistic and modified gravity models and is central to the way in which we interpret observations of the CMB and the galaxy distribution. However, homogeneity cannot be directly observed in the galaxy distribution or CMB, even with perfect observations, since we observe on the past lightcone and not on spatial surfaces. We can directly observe and test for isotropy, but to link this to homogeneity, we need to assume the Copernican Principle. First, we discuss the link between isotropic observations on the past lightcone and isotropic spacetime geometry: what observations do we need to be isotropic in order to deduce spacetime isotropy? Second, we discuss what we can say with the Copernican assumption. The most powerful result is based on the CMB: the vanishing of the dipole, quadrupole and octupole of the CMB is sufficient to impose homogeneity. Real observations lead to nearisotropy on large scales  does this lead to nearhomogeneity? There are important partial results, and we discuss why this remains a difficult open question. Thus we are currently unable to prove homogeneity of the Universe on largescales, even with the Copernican Principle. However we can use observations of galaxies and clusters to test the Copernican Principle itself. 
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[1104.1300] Is the Universe homogeneous?
This interesting paper examines perhaps the most fundamental assumption in cosmology: homogeneity. This seemingly obvious assumption is actually quite difficult to test properly since we can only observe on our past , light cone, not on spatial slices of spacetime. The author examines in detail what observations can test for isotropy, and argues that is far from trivial to prove that the isotropy of the CMB for all observers implies homogeneity.
He also discusses tests of the Copernican Principle, which is the key to linking isotropy we observe to homogeneity. These tests include consistency of luminosity distances (testable by SNIa) and comparison of the radial and transverse BAO signals, as well as tests based on the SZ effect in clusters. In principle these tests can be done well with upcoming dark energy experiments, which I think will be a valuable complement to the traditional dark energy measurements.
He also discusses tests of the Copernican Principle, which is the key to linking isotropy we observe to homogeneity. These tests include consistency of luminosity distances (testable by SNIa) and comparison of the radial and transverse BAO signals, as well as tests based on the SZ effect in clusters. In principle these tests can be done well with upcoming dark energy experiments, which I think will be a valuable complement to the traditional dark energy measurements.

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 Affiliation: University of Helsinki
Re: [1104.1300] Is the Universe homogeneous?
These are not really tests of the Copernican Principle i.e. the statement that our position is not special. They are tests of the FRW approximation, i.e. to which degree the FRW model describes the real universe. Even if your position is typical, it may be that your observations are not described by a FRW model because the universe is not exactly homogeneous and isotropic.

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[1104.1300] Is the Universe homogeneous?
True, these tests are all framed as "If we observe such and such, then our universe is nonFLRW." But the idea is to combine them with independent tests of isotropy as discussed in the first half of the paper. Since isotropy+copernican principle > FLRW, then a universe that fails the FLRW tests but passes the isotropy tests violates the copernican principle.

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Re: [1104.1300] Is the Universe homogeneous?
Observing exact homogeneity and isotropy would prove that the universe is exactly FRW. But this is not the kind of a universe we live in.
Observing statistical homogeneity and isotropy on large scales does not prove that the universe is close to FRW. The FRW model describes an exactly homogeneous and isotropic universe, not one that is statistically homogeneous and isotropic.
Observing statistical homogeneity and isotropy on large scales does not prove that the universe is close to FRW. The FRW model describes an exactly homogeneous and isotropic universe, not one that is statistically homogeneous and isotropic.

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Re: [1104.1300] Is the Universe homogeneous?
Since that may be too abstract for some people, here is a more concrete way of saying that. The Earth exists, with an overdensity of about [tex]10^{30}[/tex]. Virialised clusters of galaxies exist, with overdensities above 200. These are uncontroversial observational results. Hence, we do not live in an exactly FLRW universe.Syksy Rasanen wrote:Observing exact homogeneity and isotropy would prove that the universe is exactly FRW. But this is not the kind of a universe we live in.
In fact, the FLRW model describes an exactly locally homogeneous and locally isotropic universe. Some FLRW universes are locally homogeneous and locally isotropic, but globally inhomogeneous and/or globally anisotropic, e.g. an E_2 FLRW model better fits WMAP data than an R^3 FLRW model with the same (concordance model) metric: 1009.5880.The FRW model describes an exactly homogeneous and isotropic universe, not one that is statistically homogeneous and isotropic.