Relations between Correlation Functions in Minkowski and de Sitter Spacetime and Recursion Relations

In this thesis, we study the relation between the n-point correlation functions for a scalar field on Minkowski space and the curvature perturbation in de Sitter space. With the Schwinger-Keldysh formalism, we can write down the perturbation expansion for the correlation functions, and each term in the expansion can be represented by a Feynman diagram. We consider a scalar field theory on Minkowski space with direct couplings only, and for the curvature perturbation we consider an effective field theory which contains derivative couplings. For any given Feynman diagram, we found that the values of that diagram for the two theories mentioned above are related by a differential operator which depends only on the form of the couplings. We developed a systematic way for constructing these operators. With these operators, formulae for correlation functions in Minkowski space can be translated into formulae for correlation functions in de Sitter space. In particular, in this work, we will derive recursion relations with this method.