Cosmic Microwave Background (CMB) distortions: a new window into the physics of inflation

The Cosmic Microwave Background (CMB) spectrum in the frequency domain is extremely close to a perfect black-body. However, we know there must be tiny distortions of the CMB spectrum which are within the sensitivity of proposed satellite missions, such as PRISM and PIXIE. These spectral distortions provide a potential powerful source of information about the origin of the primordial density perturbations in the early Universe, i.e. the inflationary paradigm. In order to achieve a more solid comprehension, we will first introduce our theoretical framework, based on the Cosmological Standard Model, and the Theory of cosmological perturbations, which represents the traditional approach in Cosmology. In particular we will see how primordial density perturbations, of order 10-5, around a background solution produced the seeds for the Universe we see today. Thereafter, we will consider various aspects related to CMB spectral distortions: what they consist in, their main properties and why they are so important. Recent works proved that a cross correlation between the so-called µ-type (i.e. chemical potential type) distortions and the CMB temperature anisotropy ΔT/T could constraint the level of primordial non-Gaussianity (NG) via the parameter fNLloc at very small scales so far unexplored, lesser than about a Mpc, and potentially with a precision much higher than the present constraints from CMB anisotropies. The most up-to-date and guaranteed constraints on primordial NG come from Planck data and give fNLloc = 2.5 ± 5.7; however, authors of a recent paper (Pajer & Zaldarriaga, 2012) claimed that a cosmic variance limited experiment could in principle reach Δ fNLloc ~ O(10-3), which is the typical level predicted by the standard single-field models of slow-roll inflation. That is a quite bold claim and in this work we will focus on a computation at second-order in the cosmological perturbations to quantify some non-primordial signals that can act as a source of contamination to the measurement of the level of primordial non-Gaussianity. One could be lead to think that a calculation of such kind should in principle follow the same derivation as the bispectrum formula TTT, but this idea, although legitimate, is actually incorrect. In fact, we will see that asserting this second-order contamination is an highly non trivial task and we will discover a few crucial subtleties hidden in the calculation which must be treated very carefully. The procedure we will follow is surely going to shine a light on the very brief and incomplete derivations and implied assumptions in the research literature. In addition to it, we will develop an analytical formula, initially not intended, which provides the basis for a full numerical calculation once plugged into the Second Order Non Gaussianity (SONG) code. The main goals of this Thesis are therefore to present some remarkable results on recent developments in Cosmology, becoming aware of what we know, and, where possible, to take a step toward what we do not know, clarifying some aspects about CMB spectral distortions that have not been yet explored.