 Cosmic string network

An isotropic cosmic string network averages the directional stress and energy density of many long cosmic strings. For an ideal smooth string whose tension equals its rest energy per unit length, a network frozen into the expansion has negative averaged pressure,A comoving string segment then has physical length proportional to $a$, while its enclosing volume grows as $a^3$, giving $\rho \propto a^{-2}$. This is the nonmoving, noninteracting limit, not the general evolution law of a mobile network.

In units $c=1$, let $v^2$ be the energy-weighted mean squared transverse string velocity. For an isotropic smooth-string network the averaged equation of state isThe kinetic contribution raises the averaged pressure; for example $v^2=1/2$ gives $w=0$. Motion, reconnection and production of smaller loops therefore matter even when the initial network is statistically isotropic.

The corresponding long-string balance, neglecting friction against matter, can be writtenwhere $\mathcal{Q}\ge 0$ is energy removed from the long-string population into loops or other radiation channels. For $\mathcal{Q}=0$ this agrees with $\dot{\rho }+3H(\rho +P)=0$ and proves the equation-of-state relation. If $v$ is constant and losses vanish, the solution is $\rho \propto a^{-2(1+v^2)}$. The primary velocity-dependent model defines $\rho =\mu /L^2$ and evolves both its characteristic length $L$ and its velocity.^[ref]

A horizon-scaling cosmic string network is consequently distinct from a frozen fluid with fixed $w=-1/3$. The ideal frozen equation of state can be used in a specified cosmological toy model, but it cannot be inferred from isotropy alone.