On Cosmological Correlators In The $\alpha$-Vacua

This thesis, based largely on , deals with new techniques for computing cosmological correlation functions of scalar fields with respect to a class of vacuum states known as the -vacua. By extending the Mellin space formalism for computing such correlators in the Bunch-Davies vacuum to the case of non-zero , we show that to all orders in perturbation theory, late-time de Sitter boundary correlators can be written as a linear combination of their counterparts in the Bunch-Davies vacuum. The constituent perturbative Bunch-Davies contributions feature points in both the expanding and the contracting Poincare patches. In turn, this reformulation allows us to relate these perturbative de Sitter correlators to Witten diagrams in Euclidean Anti-de Sitter space. In particular, we show that any de Sitter diagram can be written as a linear combination of EAdS Witten diagrams with analytically continued momenta, each dressed with -dependent coefficients. In addition, we use our de Sitter results to compute the inflationary two- and three-point functions of inflaton perturbations at leading order in slow-roll, for arbitrary spacetime dimension and arbitrary choice of -vacuum.