Advanced Cosmology : Primordial non-Gaussianities

with amplitude |f(k)| = √ ak + b 2 k. Reality of f(x) imposes ak = a−k , bk = −b−k. In words, for a given configuration in real space f(x), a set of real numbers (ak, bk), parameterized by the vector k, completely (and uniquely) describe it. Different configurations of f(x) are described by different set of numbers. Suppose now we want to randomly generate a field configuration f(x), then one way to do it is to prescribe a probability distribution function for the set (ak, bk). If, furthermore, we want f(x) to be a Random Gaussian Field configuration, then this put a tight constraint on what is the PDF for (ak, bk). Let’s begin by studying the PDF for one particular mode k. Definition : A Gaussian distribution for a mode is one such that ak and bk is drawn from the following Gaussian distribution in k space with zero mean, and k-dependent variance σk