Ehlers–Geren–Sachs Theorem
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The Ehlers–Geren–Sachs theorem, published in 1968 by
Jürgen Ehlers Jürgen Ehlers (; 29 December 1929 – 20 May 2008) was a German physicist who contributed to the understanding of Albert Einstein's theory of general relativity. From graduate and postgraduate work in Pascual Jordan's relativity research group ...
, P. Geren and Rainer K. Sachs, shows that if, in a given universe, all freely falling observers measure the
cosmic background radiation Cosmic background radiation is electromagnetic radiation that fills all space. The origin of this radiation depends on the region of the spectrum that is observed. One component is the cosmic microwave background. This component is redshifted ...
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, if the one uses a kinetic picture and the collision term vanishes, i.e. in the so-called Vlasov case or if there is a so-called detailed balance. This result was later extended to the full Boltzmann case by R. Treciokas and G.F.R. Ellis. Using the fact that, as measured from
Earth Earth is the third planet from the Sun and the only astronomical object known to Planetary habitability, harbor life. This is enabled by Earth being an ocean world, the only one in the Solar System sustaining liquid surface water. Almost all ...
, the cosmic microwave background is indeed highly isotropic—the temperature characterizing this
thermal radiation Thermal radiation is electromagnetic radiation emitted by the thermal motion of particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation. The emission of energy arises from a combination of electro ...
varies only by tenth of thousandth of a
kelvin The kelvin (symbol: K) is the base unit for temperature in the International System of Units (SI). The Kelvin scale is an absolute temperature scale that starts at the lowest possible temperature (absolute zero), taken to be 0 K. By de ...
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. However the paper by Stoeger et al. assumes that derivatives of the cosmic background temperature multipoles are bounded in terms of the multipoles themselves. The derivatives of the multipoles are not directly accessible to us and would require observations over time and space intervals on cosmological scales. In 1999 John Wainwright, M. J. Hancock and Claes Uggla show a counterexample in the non-tilted perfect fluid case. Thus an almost isotropic cosmic microwave temperature does not imply an almost isotropic universe. Using the methods of Wainwright et al. Ho Lee, Ernesto Nungesser and John Stalker could show that they can be applied to Vlasov as well, which was the original matter model of the EGS-theorem.


References

Coordinate charts in general relativity Cosmic background radiation {{physical-cosmology-stub