Cosmic string

A cosmic string is a hypothetical one-dimensional topological defect formed by symmetry breaking in the early universe. A cosmic string network consists of many such defects and requires additional assumptions to infer an averaged equation of state.

For $d{s}^2=-(d{x}^0{)}^2+d{r}^2+f(r{)}^{2}d{\varphi }^2+d{z}^2$, the smooth bulk mass density is $-c^2{f}^{\prime \prime }/(8\pi Gf)$. Integrate using proper transverse area $2\pi fdr$ to obtain the boundary term. With no shell at the core edge, $f^{\prime }(R^{-})$ equals the exterior conical slope. A regular axis requires $f(0)=0$ and $f^{\prime }(0)=1$ for angular period $2\pi$. A different printed inner derivative gives a different bulk integral; any singular-axis matter must be specified separately. This prevents silently applying the standard cosmic-string mass-deficit relation to nonregular cores.

A network whose characteristic long-string separation grows with the horizon scale can lose energy into loops and radiation. Its density can scale as $\mu /t^2$ in natural units. This is distinct from the $a^{-2}$ dilution of an ideal frozen noninteracting string network.