Aspects of cosmological observables

Abstract

One of the goals of early universe cosmology is to understand the physics of inflation. Since we cannot visually access this epoch, we rely on indirect observations, e.g., measuring temperature correlations on the CMB. By analysing various features of these correlation functions, we can infer details of the inflationary epoch. However, the observational data is not yet powerful enough to give a detailed sketch of this period. This is because these correlations are very feeble and hence difficult to measure. So far, we have only measured the two-point function and its dependence on the wavelength. Measuring three- and higher-point functions is very challenging; however, one can still make theoretical progress. This can be achieved by understanding various mathematical structures present in the correlation functions and tracing their precise physical origin. This approach is very well developed in the case of flat space scattering amplitudes. For instance, it is well known that locality is related to the pole structure of the amplitude and unitarity is encoded in the Optical theorem and the Cutkosky cutting rules. In the case of cosmology, these ideas have been developed only very recently. In this thesis, we develop these ideas further by first analysing the tree-level analytic structure of scalar correlation functions in momentum space. In particular, we try to establish a connection between the pole structure and the initial state of inflation. We find that while only excited states can produce folded poles, the absence of such poles does not imply a non-excited initial state, i.e., the vacuum. We then proceed to discuss well-known inflationary soft theorems. Although the proof of soft theorems is very general, it is always good to have explicit checks. By taking specific operators in the Effective Field Theory of Inflation (EFToI), we perform explicit checks of soft theorems for mixed correlation functions (scalar + tensor). Such an explicit check was missing in the case of mixed soft theorems (to the best of our knowledge). We then discuss the boostless bootstrap and apply it to certain higher-dimensional EFToI operators. There has been very little progress in computing cosmological loops because of their mathematical complexity. Most of the literature has been devoted to computing the inflationary power spectrum at one-loop. We take the next step and compute the three-point function at one-loop in the EFToI. We explicitly compute the log running and find the following two structures: (1) log(H/µ) and (2) log(ki/kT). We verify our results using both cut-off and dimensional regularisation. In the case of dim-reg, we notice that the unphysical logarithms of the form log(k/µ), which are ratios of a co-moving scale and a physical scale, cancel only after properly renormalising the results. In the case of cut-off regularisation, these unphysical logs never show up.