The time dependence of each perturbation is well defined, being determined by laws of physics. Viewed instead as a function of position at fixed time, the perturbations have random distributions. It is the statistical properties of these distribution that we wish to uncover via observation, and relate to fundamental physics models for the origin of perturbations. Those are usually referred to as stochastic properties.

The inherent randomness means that one shouldn't aim to predict things like the precise location of particular galaxies. Questions should refer to stochastic properties only. Don't ask ‘how far is it to the nearest large galaxy?’; instead ask ‘what is the typical separation between large galaxies?’. This randomness echoes simple quantum mechanics, e.g. one shouldn't hope to predict the precise position of a single particle in a closed box, but could compute the typical distance of the particle from its centre averaged over many such boxes. Indeed, we will see that in the inflationary cosmology the randomness of cosmological perturbations does have its origin in quantum uncertainty.

To describe the stochastic properties of the perturbations, one invokes the mathematical concept of a random field. In this chapter we describe the relevant aspects of that concept, without tying ourselves at this stage to any particular perturbation.

Random fields

Consider just one perturbation, evaluated at some instant, which we denote by g(x). We take g(x) to be associated with what is called a random field.