Having discussed the main elements of the theory of cosmological structure formation, we now turn to the statistical treatment of data - which is how theory and observation will be confronted. The density perturbation field, , inhabits a universe that is isotropic and homogeneous in its large-scale properties, suggesting that the statistical properties of should also be homogeneous. This statement sounds contradictory, and yet it makes perfect sense if there exists an ensemble of universes. The concept of an ensemble is used every time we apply probability theory to an event such as tossing a coin: we imagine an infinite sequence of repeated trials, half of which result in heads, half in tails. The analogy of coin tossing in cosmology is that the density at a given point in space will have different values in each member of the ensemble, with some overall variance <2> between members of the ensemble. Statistical homogeneity of the field then means that this variance must be independent of position. The actual field found in a given member of the ensemble is a realization of the statistical process.
There are two problems with this line of argument: (i) we have no evidence that the ensemble exists; (ii) in any case, we only get to observe one realization, so how is the variance <2> to be measured? The first objection applies to coin tossing, and may be evaded if we understand the physics that generates the statistical process - we only need to imagine tossing the coin many times, and we do not actually need to perform the exercise. The best that can be done in answering the second objection is to look at widely separated parts of space, since the fields there should be causally unconnected; this is therefore as good as taking measurements from two different member of the ensemble. In other words, if we measure the variance <2> by averaging over a sufficiently large volume, the results would be expected to approach the true ensemble variance, and the averaging operator < ... > is often used without being specific about which kind of average is intended. Fields that satisfy this property, whereby
are termed ergodic. Giving a formal proof of ergodicity for a random process is not always easy (Adler 1981); in cosmology it is perhaps best regarded as a common-sense axiom.