Nonsingular big bounces and the evolution of linear fluctuations

We consider the evolutions of linear fluctuations as the background Friedmann world model goes from contracting to expanding phases through smooth and nonsingular bouncing phases. As long as gravity dominates over the pressure gradient in the perturbation equation, the growing mode in the expanding phase is characterized by a conserved amplitude; we call this a C mode. In spherical geometry with a pressureless medium, we show that there exists a special gauge-invariant combination $\Phi$ which stays constant throughout the evolution from the big bang to the big crunch, with the same value even after the bounce: it characterizes the coefficient of the C mode. We show this result by using a bounce model where the pressure gradient term is negligible during the bounce; this requires the additional presence of exotic matter. In such a bounce, even in more general situations for the equation of state before and after the bounce, the C mode in the expanding phase is affected only by the C mode in the contracting phase; thus the growing mode in the contracting phase decays away as the world model enters the expanding phase. When the background curvature plays a significant role during the bounce, the pressure gradient term becomes important and we cannot trace the C mode in the expanding phase to the one before the bounce. In such situations, perturbations in a fluid bounce model show exponential instability, whereas perturbations in a scalar field bounce model show oscillatory behavior.