Perturbations and Linearization Stability of Closed Friedmann Universes

We consider perturbations of closed Friedmann universes. Perturbation modes of two lowest wavenumbers ($L=0$ 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 time-like 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 $L\geq 1$ 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 $L\geq 2$ 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.

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