Gauge Invariance and Frame Independence in Cosmology

In this thesis the mathematical formulation of cosmological perturbations is studied. First we discuss the gauge problem of general relativity: perturbations of the metric and matter fields in an expanding universe are dependent on the choice of coordinate system, i.e. gauge dependent, even though the complete, unperturbed theory is covariant. Secondly, we discuss the frame problem of perturbations: the action for a universe with scalar field matter can be formulated in two different ways, in the Jordan or Einstein frame. These formulations are related by redefinitions of the metric and scalar field, and are therefore physically equivalent. However, perturbations in one frame and another are related by non-linear redefinitions, which makes it complicated to see how to formulate physical correlation functions. In this thesis we show that it is possible to express the action for perturbations in terms of gauge invariant and frame independent perturbations. These perturbations are then used to compute physical correlation functions which are independent of coordinate reparametrizations or field redefinitions. We first compute the second order action for linear perturbations. Later we outline a procedure to compute the action for non-linear perturbations, and apply this to find the gauge invariant action for second order perturbations. We furthermore show that there are different gauge invariant variables related by non-linear redefinitions, but that their actions are only related by boundary terms, such that the evolution of non-Gaussianity is unique. Moreover, as a final application we use the formulation of the action in terms of gauge invariant and frame independent variables in order to readdress the naturalness in Higgs inflation. In this formulation, Higgs inflation does not suffer from unitarity problems