Most inflationary models of the early Universe predict that the initial density fluctuations in the Universe giving rise to the temperature-fluctuations in the Cosmic Microwave Background (CMB) and in the Large Scale Structure (LSS) are nearly Gaussian distributed. "Gaussian distributed" means that their statistical properties are completely characterized by the two-point correlation function. Indeed, the histogram of the CMB temperature fluctuations nicely follows a Gaussian curve, and the higher-order correlation functions deviate only little from the Gaussian ones. Nevertheless, a new and powerful probe of the origin and evolution of structures in the Universe has emerged and been actively developed over the last decade. In the coming decade, primordial non-Gaussianity (NG), that is the study of primordial NG contributions to the correlations of cosmological fluctuations and other cosmological observables, will become an important probe of both the early and the late universe. It will play a leading role in furthering our understanding of two fundamental aspects of cosmology and astrophysics: the physics of the very early Universe that created the primordial seeds for the LSS and the subsequent growth of structures via gravitational instability at later times. On-going and future observations of fluctuations in the CMB and the LSS will test the level of primordial NG and therefore provide us with crucial information about the early and late evolution of our Universe.
Why is the study of NG so relevant? The answer is that, despite the simplicity of the inflationary paradigm, the mechanism by which cosmological perturbations are generated is not yet established. In the standard slow-roll inflationary scenario associated to one-single field, the inflaton, density perturbations are due to fluctuations of the inflaton itself when it slowly rolls down along its potential. In other scenarios, the cosmological perturbations are sourced by some other light field, like the curvaton. This is what makes a positive detection of NG so relevant: it might help in discriminating among competing scenarios which otherwise might be undistinguishable. While single-field models of inflation generically predict a tiny level of NG, other models for the generation of the perturbation may predict a high level of NG. While detection of large primordial NG would not rule out inflation, it would rule out in a single shot the large class of slow-roll models where inflation is driven by a single scalar field.
However, inflation is not the only mechanism leaving NG signatures in the CMB and the LSS. For instance, non-linear structure growth due to gravitational instability produce NG which tend to overwhelm the primordial NG in the LSS; similarly the CMB carries NG imprints from radiation of the Milky Way and other galaxies, from gravitational lensing, and from the Sunyaev-Zel'dovich effect. It is therefore crucial to understand all of these effects in order to separate them from primordial NG created in the early Universe. But, of course, looking at the NG of these astrophysical effects also teaches us a lot about these effects themselves.