Cosmic fluctuations from quantum effective action

Abstract

Does the observable spectrum of cosmic fluctuations depend on detailed initial conditions? This addresses the question if the general inflationary paradigm is sufficient to predict within a given model the spectrum and amplitude of cosmic fluctuations, or if additional particular assumptions about the initial conditions are needed. The answer depends on the number of e-foldings $N_{in}$ between the beginning of inflation and horizon crossing of the observable fluctuations. We discuss an interacting inflaton field in an arbitrary homogeneous and isotropic geometry, employing the quantum effective action $\Gamma$. An exact time evolution equation for the correlation function involves the second functional derivative $\Gamma^{(2)}$. The operator formalism and quantum vacua for interacting fields are not needed. Use of the effective action also allows one to address the change of frames by field transformations (field relativity). Within the approximation of a derivative expansion for the effective action we find the most general solution for the correlation function, including mixed quantum states. For not too large $N_{in}$ the memory of the initial conditions is preserved. In this case the cosmic microwave background cannot disentangle between the initial spectrum and its processing at horizon crossing. The inflaton potential cannot be reconstructed without assumptions about the initial state of the universe. We argue that for very large $N_{in}$ a universal scaling form of the correlation function is reached for the range of observable modes. This can be due to symmetrization and equilibration effects, not yet contained in our approximation, which drive the short distance tail of the correlation function towards the Lorentz invariant propagator in flat space.