 Solution

First perturb the radius consistently with the conserved string normalization. If $a_E\mapsto a_E(1+ϵ)$ while $C_s={\rho }_s{a}^2$ stays at its static value, the density becomesThe Friedmann equation still has an exact cancellation between the string and curvature terms, and the Friedmann acceleration equation still vanishes. ThusThe homogeneous stability of a static cosmic-string universe is neutral in this restricted sense: there is no restoring force and no exponentially growing displacement. Holding the density fixed while changing the radius would instead change $C_s$ and would not be this pure displacement perturbation.