Primordial non-Gaussianities from inflation

These notes present a detailed introduction to Maldacena’s calculation [1] of the three-point function generated by the simplest class of infl ationary models: those with a single inflaton field whose potential satisfies the slow-roll conditions and whose quantum fluctuations start in the asymptotic Bunch-Davies vacuu m state. The three-point function should be the most readily observed evidence for non-Gaussianities amongst the primordial fluctuations produced by inflation. In these i nflationary theories the nonGaussianities are predicted to be extremely small, being naturally suppressed by the small slow-roll parameters. ANY cosmological model that seeks a deeper or more complete explanation of the universe must be able to account for the origin of the great variety of structures that appear in it today. Most of this structural complexity in the universe appears to have arisen through processes which are largely understood from material that was once in a much simpler state than it is today. Stars and galaxies, for example, which now stand out so clearly from their much emptier surroundings, could have grown through the slow gravitational collapse of gases which were initially distributed far more uniformly. This basic p icture suggests that the universe in its primeval stages was once extremely homogeneous, a prediction that matches well with the observations made so far. However, this mechanism also requires that, even at the very earliest times, some spatial variation must always have existed. In a perfectly homogeneous and isotropic universe, the process of collapse and growth, and the development of ever more complex structures, would never have begun. The theory of inflation provides a mechanism for generating t he initial spatial variations in the universe. In the inflationary picture, space-time its elf fluctuates quantum mechanically about a background that is expanding at an accelerating rate. This extreme expansion spreads the fluctuations, which begin with a tiny spatial ext ent, throughout a vast region of the universe, where they eventually become small classical fluctuations in the space-time curvature—or equivalently, small spatial variations in th e strength of gravity. Since everything in the universe feels the influence of gravity, these flu ctuations in the gravitational field are transferred to the matter and radiation fields, creating slightly overdense and underdense regions. The resulting matter fluctuations then become the ‘ initial conditions’ that start the process of collapse which forms the stars and galaxies of later epochs. To test whether this picture is correct, it is necessary to de scribe very accurately the properties of the pattern generated by inflation for the orig inal, primordial fluctuations in space-time which can then be compared with what is inferred from observations. Before entering into a detailed calculation of these primordial pe rturbations, it is instructive to explain first from a more general perspective how the fluctuatio ns of a quantum field lead to a classical pattern of perturbations. Since a quantum field i s never measured directly, it is necessary also to determine how and in what form its information is available. So to begin, consider a quantum field, which will be written as ζ(t,~ x). Later this field will be connected with the fluctuations of the background spa ce-time, but for now its exact physical meaning will be left a little vague. Although ζ(t,~ x) is a quantum field, its influence