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Ehlers–Geren–Sachs theorem

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The inhomogeneities in the temperature of the cosmic background radiation recorded in this image from the satellite probe WMAP amount to no more than Kelvin.

The models of modern cosmology are based on the Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes of general relativity. Robertson-Walker universes are perfectly isotropic and homogeneous – but how do we know that this is a good approximative description of our actual universe? While observations of the large-scale distribution of galaxies provides valuable evidence, the most stringent limit comes from another data set: the properties of the cosmic background radiation, and "electromagnetic echo" produced around 400,000 years after the big bang.[1]

The Ehlers-Geren-Sachs theorem, published in 1968 by Jürgen Ehlers, P. Geren and Rainer Sachs, shows that if, in a given universe, all freely falling observers measure the cosmic background radiation to have exactly the same properties in all directions (that is, they measure the background radiation to be isotropic), then that universe is an isotropic and homogeneous FLRW spacetime. Using the fact that, as measured from Earth, the cosmic microwave background is indeed highly isotropic – the temperature characterizing this thermal radiation varies only by tenth of thousandth of a Kelvin with the direction of observations –, and making the Copernican assumption that Earth does not occupy a privileged cosmic position, this constitutes the strongest available evidence for our own universe's homogeneity and isotropy, and hence for the foundation of current standard cosmological models. Strictly speaking, this conclusion has a potential flaw. While the Ehlers-Geren-Sachs theorem concerns only exactly isotropic measurements, it is known that the background radiation does have minute irregularities. This was addressed by a generalization published in 1995 by W. R. Stoeger, Roy Maartens and George Ellis, which shows that an analogous result holds for observers who measure a nearly isotropic background radiation, and can justly infer to live in a nearly FLRW universe.[2]

  1. ^ For galaxy observations, cf. Peebles, P.J.E.; Schramm, D.N.; Turner, E.L.; Kron, R.G. (1991), "The case for the relativistic hot Big Bang cosmology", Nature, 352: 769–776, doi:10.1038/352769a0. For a basic description of the cosmic background radiation, cf. chapter 11 in Bergström, Lars; Goobar, Ariel (2003), Cosmology and Particle Astrophysics (2nd ed.), Wiley & Sons, ISBN 3-540-43128-4.
  2. ^ See pp. 351ff. in Hawking, Stephen W.; Ellis, George F. R. (1973), The large scale structure of spacetime, Cambridge University Press, ISBN 0-521-09906-4. The original work is Ehlers, J., Geren, P., Sachs, R.K.: Isotropic solutions of Einstein-Liouville equations. J. Math. Phys. 9, 1344 (1968). For the generalization, see Stoeger, W. R.; Maartens, R; Ellis, George (2007), "Proving Almost-Homogeneity of the Universe: An Almost Ehlers-Geren-Sachs Theorem", Ap. J., 39: 1–5, doi:10.1086/175496.