Evolution of linear perturbations through a bouncing world model: Is the near Harrison-Zel'dovich spectrum possible via a bounce?

We present a detailed numerical study of the evolutions of cosmological linear perturbations through a simple bouncing world model based on two scalar fields. We properly identify the relatively growing and decaying solutions in expanding and collapsing phases. Using a decomposition based on the large-scale limit exact solution of curvature (adiabatic) perturbations with two independent modes, we assign the relatively growing/decaying one in an expanding phase as the C/d-mode. In the collapsing phase, the roles are reversed, and the C/d-mode is relatively decaying/growing. The analytic solution shows that, as long as the large scale and the adiabatic conditions are met, the C- and d-modes preserve their nature throughout the bounce. Here, by using a concrete nonsingular bouncing world model based on two scalar fields, we numerically follow the evolutions of the correctly identified C- and d-modes which preserve their nature through the bounce, thus confirming our previous anticipation based on the analytic solution. Thus, while the large-scale condition is satisfied and the adiabatic condition is met during the bounce, we conclude that it is not possible to obtain the near Harrison-Zel'dovich scale-invariant density spectrum through a bouncing world model as long as the seed fluctuations were generated from quantum fluctuations of the curvature perturbation in the collapsing phase.