# Does the average homogeneity of the universe imply the average validity of the FRW metric

One of the big assumption of cosmology is the isotropic and homogeneous character of the matter distribution in the universe (since homogeneity implies isotropy I'll concentrate on the latter).

Obviously this assumption is wrong, as any local experiment can determine. This only works on the assumption that this homogeneity is only average.

How does one really define this, though? I suppose the simplest method would be to define a ball of radius $r$ such that, for instance, for two points $p$ and $q$,

$$\int_{B(p,r)} T_{00} \sqrt{-g} d^3x = \int_{B(q,r)} T_{00} \sqrt{-g} d^3x + \mathcal{O}(r^{-1})$$

The quantity to integrate can differ of course, as well as the drop off of the difference, but I guess something of that kind might work as a definition. The problem is of course to define a sphere of a certain radius if we do not have the metric tensor already.

Also, once we are given such a definition, is there an actual proof that, given some averaged homogeneity of the stress energy tensor, we have some averaged notion of the metric tensor being homogeneous? From the rare discussions I've seen on the topic, we have for some averaging process $\langle \cdot \rangle$

$$\langle R(\Gamma) \rangle \neq R(\langle\Gamma\rangle)$$

Is there a rigorous justification of the assumption of homogeneity for the FRW metric?

• Why does homogeneity imply isotropy? – MBN Apr 12 '17 at 7:40
• homogeneity and isotropy is not an assumption, it's an observation. And it applies to large scales (that's why it's called cosmology, not astrophysics) – Kosm Apr 12 '17 at 19:42
• homogeneous distribution of matter is described by perfect fluid en.m.wikipedia.org/wiki/Perfect_fluid. Small inhomogeneities are described as density perturbations and gravitational waves (tensor perturbations) – Kosm Apr 12 '17 at 19:44
• Your question is answered (partially) in the monograph by H. Ringström. – Ryan Unger May 16 '17 at 14:46