# Ergodic Hypothesis in cosmology

I'm studying primordial fluctuations of the Universe from a statistical point of view and I'am aware of the following problem:

A fundamental limitation arises in cosmology – because there is only one universe to observe, i.e. there is only one realization of the stochastic process that generates the fluctuations whose consequences we observe. Therefore we cannot measure ensemble averages or expectation values, as we would in a repeatable laboratory experiment. What we can do when observing a fluctuation on a given scale $$\lambda$$ is to average over many distinct regions of size $$\sim\lambda$$. An ergodic-type hypothesis allows us to replace the ensemble average by a spatial average over these regions. (see pag 307 from Relativistic cosmology-George F.R. Ellis, R. Maartens, M.A.H MacCallum; Cambridge University Press (2012))

So theoretically if i wanted to get the average of a stochastic field or of a function, we say $$\delta(\vec x)$$ or $$f[\delta(\vec x)]$$, in a point $$\vec x_1$$ I should do the average over the ensemble of all the universes (it is obviously impossible because we have only one Universe). I have imagined this thing thinking about a stack of planes representing the different universes of the ensemble and drawing the point $$\vec x_1$$ in each plane. Now i can take the value of $$\delta(\vec x_1)$$ in the first universe, in the second, and so on and finally make the average of these values; if I repeat this procedure for all the points I'll get the average of $$\delta (\vec x)$$, i.e. $$\langle\delta\rangle$$ over the ensemble of the universes. Ok it's fine...but it is not possible because I have only one Universe, so I have to appeal to Ergodic Hypothesis (really a spatial version of the Ergodic Hypothesis).

From here on I didn't understand how things work... Can I do a spatial average over the entire existing Universe? Someone talks about a division of the Universe in "fair samples" considering these regions as different realisations of our Universe.

In particular I want to understand:

An ergodic-type hypothesis allows us to replace the ensemble average by a spatial average over these regions

It seems that I have to divide the Universe in regions (we can imagine squares for simplicity in two dimensions, I thought the Universe like a chessboard) and I can replace the ensemble averege with a spatial average over any single region. Any square of the chessboard is a fair sample of the Universe, so I have many squares that can "reproduce" the Universe (I divided my sky in many regions and I can consider these squares like the realisations of the "Universe experiment", the outcomes of the experiment). Now if all this is correct(?) and I want to get the average of a gaussian stochastic field (fluctuations density field for example) or of a function of this field how I can do?

Ok I divided my sky into squares (any region has a reference frame), if I want to know $$\langle f[\delta(\vec x)]\rangle$$ in a point, say $$\vec x_1$$, I have to evaluate $$f[\delta(\vec x_1)]$$ in the first square, in the second, in the third and so on and then take the averege over this values or I have to average $$f[\delta(\vec x)]$$ over the area of the firt squares, of the second, of the third and so on and then to divide for the total area?