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.
