Understanding Statistical Homogeneity in Cosmology

[latentcorpse]

For a random field, $f$, the two point correlator is defined as
$\xi(x,y)=\langle f(x) f(y) \rangle = \int \mathcal{D}f \text{Pr}[f] f(x) f(y)$
where $\text{Pr}[f]$ is the probability of realising some field configuration.
Statistical homogeneity means that $\text{Pr}[f(x)]=Pr[f(x-a)]$


Apparently this means that the two point correlator satisfies $\xi(x,y)=\xi(x-a,y-a) \forall a \Rightarrow \xi(x,y)=\xi(x-y)$
I do not understand that last line. Why does it suddenly just become dependent on the seperation?

Thanks.

 

[NateD]

The last line follows from the definition of statistical homogeneity. Since the probability distribution of the field is invariant under any translation, it follows that the two point correlator must also be invariant under any translation. That is, \xi(x,y)=\xi(x-a,y-a) for any translation a. This in turn implies that the two point correlator is only a function of the separation between x and y (i.e. x-y), and not of the individual points x and y separately. Hence, \xi(x,y)=\xi(x-y).