Litterature on Statistical Homogeneity and Isotropy

[center o bass]

When dealing with cosmological perturbations, there are a lot of different notions that are thrown around in the literature like statistical homogeneity and isotropy. However, these terms are often not motivated and clearly defined.

Could anyone recommend any good references where these notions are motivated and clearly defined?
It would also be good if they could be contrasted to other notions of homogeneity and isotropy.

 

[Chalnoth]

When it comes to cosmological perturbations, one way to say it is that the probability distribution that determines the perturbations is independent of angle. That is to say, in a spherical coordinate system, the derivatives of the PDF with respect to the angles $\theta$ and $\phi$ are zero. In terms of spherical harmonics, this statement translates into each $a_{\ell m}$ for the same $\ell$ being drawn from the same probability distribution function.

The statement of homogeneity can be represented by stating it is possible to change the origin of the coordinate system without having any impact on the probability distribution function (If I recall, it's possible to prove homogeneity if there are three non-colinear points that are in causal contact view an isotropic universe).