= 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>,
$$
w=\frac P\rho=-\frac13.
$$
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 is
$$
w=\frac{2v^2-1}{3}.
$$
The 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 written
$$
\dot\rho+2H(1+v^2)\rho=-\mathcal Q,
$$
where $\mathcal Q\ge0$ 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>.<Extending the Velocity-dependent One-scale String Evolution Model by C. J. A. P. Martins and E. P. S. Shellard>[Section II.A, equations (1)–(3), energy-weighted velocity, loop losses and correlation-length evolution]{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.
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