 Solution

The distinction between the evolution equation and the constraint is essential. For strings alone, the exact equations reduce toLinearizing the acceleration equation gives $\delta a=A+Bt$. At the exactly tuned static $C_s$, however, the full Friedmann equation requires ${\dot{a}}^2=0$, so $B=0$ for an admissible exact perturbation.

A small nonzero velocity $v$ can be admissible if its initial density also changes:Arbitrarily small $v$ and the corresponding second-order density change therefore produce a secular departure. There is no Lyapunov stability against all nearby constrained initial data for the static closed cosmic-string universe. Neutral radius shifts at fixed normalization and drift when the normalization changes are different perturbation classes.