Perturbations and linearization stability of closed Friedmann universes

We consider perturbations of closed Friedmann universes. Perturbation modes of two lowest wave numbers (L = 0 and 1) are generally known to be fictitious, but here we show that both are physical. The issue is more subtle in Einstein static universes where closed background space has a timelike Killing vector with the consequent occurrence of linearization instability. Proper solutions of the linearized equation need to satisfy the Taub constraint on a quadratic combination of first-order variables. We evaluate the Taub constraint in the two available fundamental gauge conditions, and show that in both gauges the L >= 1 modes should accompany the L = 0 (homogeneous) mode for vanishing sound speed, c(s). For c(s)(2) > 1/5 (a scalar field supported Einstein static model belongs to this case with c(s)(2) = 1), the L >= 2 modes are known to be stable. In order to have a stable Einstein static evolutionary stage in the early universe, before inflation and without singularity, although the Taub constraint does not forbid it, we need to find a mechanism to suppress the unstable L = 0 and L =1 modes.