The relationship between continuum homogeneity and statistical homogeneity in cosmology

Although the standard Friedmann–Lemaitre–Robertson–Walker (FLRW) Universe models are based on the concept that the Universe is spatially homogeneous, up to the present time no definition of this concept has been proposed that could in principle be tested by observation. We propose such a definition, based on a simple spatial averaging procedure, which relates observable properties of the Universe to the continuum homogeneity idea that underlies the FLRW models. It turns out that the statistical homogeneity often used to describe the distribution of matter on a large scale does not imply spatial homogeneity according to this definition, and so cannot be simply related to a FLRW Universe model. We propose values for the homogeneity parameter ε and length scale Lc of homogeneity of the Universe.