|| Title: Cosmological correlation functions from the Hamilton Jacobi formalism ||

Abstract: A better understanding of how correlations evolve during inflation is crucial if we are to extract information about the early Universe from our late-time observables. Using the Hamilton-Jacobi formalism, we review how the state of the Universe flows along cosmological time, analogous to the familiar Renormalisation Group flow. In multi-field holographic realisations of inflation, fields additional to the inflaton cannot display underdamped oscillations (that is, their wavefunctions contain no oscillatory phases independent of the momenta). We show that this result is exact, independent of the number of fields, the field space geometry and the shape of the inflationary trajectory followed in multi-field space. The absence of underdamped oscillations implies that a detection of “cosmological collider” oscillatory patterns in the non-Gaussian bispectrum would not only rule out single field inflation, but also holographic inflation or any inflationary model based on the Hamilton–Jacobi equations. Hence, future observations have the potential to exclude, at once, an entire class of inflationary theories, regardless of the details involved in their model building. We also find a set of recursion relations which relate the time-dependence of n-point correlators to lower point correlators, exhibiting a factorisation reminiescent of flat space amplitudes. These relations allow boundary correlators (which can be fixed by conformal invariance if asymptotically de Sitter) to now be extended into the bulk. ||