Static closed cosmic-string universe (${\rho }_s{a}^2=3K/(8\pi G)$)

A closed universe containing only a cosmic string network can remain static. Its Friedmann acceleration equation vanishes identically because $P_s=-{\rho }_s/3$, and its Friedmann equation requires ${\rho }_s{a}^2=3K/(8\pi G)$. Once this conserved normalization is tuned, every positive radius gives a static solution. The usual Hubble-normalized cosmological density parameter is undefined at a static epoch.

At fixed string normalization $C={\rho }_s{a}^2$, scale-factor shifts move between static solutions and are neutral, with no restoring force. The acceleration equation gives $\ddot{a}=0$, but the Friedmann constraint also imposes ${\dot{a}}^2=8\pi GC/3-K$. At the static normalization it excludes a nonzero velocity. A slightly larger $C$ admits linearly expanding or contracting branches; a smaller $C$ admits no real closed solution. This distinguishes permitted perturbations from constraint violations.