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

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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. Since we are currently in an expanding phase the observationally relevant one in the expanding phase is the relatively growing C-mode whose nature is preserved throughout the bounce. The spectrum of C-mode generated from quantum fluctuations in the collapsing phase has a quite blue spectrum compared with themore » near Harrison-Zel'dovich scale-invariant one. 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. We also study the tensor-type perturbation. For the tensor-type perturbation, however, both C- and d-modes (of the tensor-type perturbations) in the collapsing phase survive as the relatively growing C-mode in the expanding phase. Because of its growing nature in the collapsing phase, the initial d-mode dominates the surviving C-mode spectrum after the bounce.« less