Stable initial conditions and analytical investigations of cosmological perturbations in a modified loop quantum cosmology

Abstract

In this paper, we study cosmological perturbations in a modified theory of loop quantum cosmologies, the so-called mLQC-I model. Our purposes are two-fold: First, using a method developed by Birrell and Davies, we identify an initial state in the remote contracting phase, which turns out to be stable, minimize particle creations and diagonalize the Hamiltonian, despite the fact that at this time some modes may be still outside of the Hubble horizon and not in their adiabatic states. Second, using the uniform asymptotic approximation method, we obtain the first-order approximate solutions of the mode function in terms of either the Airy functions, or the first or second kind of cylindrical functions, depending on the values of the wavenumber. In each case, the mode function contains two integration constants, which are uniquely determined by the initial state.