= Adiabatic matter-string density relation
{title2=$\delta_s=\tfrac23\delta_c$}
For separately conserved barotropic components, an <adiabatic cosmological perturbation> has equal $\delta_N/(1+w_N)$. Pressureless matter and a $w=-1/3$ string component therefore have $\delta_s=2\delta_c/3$ on <superhorizon scales>. In the supplied perfect-fluid synchronous equations, neglecting spatial gradients gives $S''+2\mathcal H S'=0$ for $S=\delta_s-2\delta_c/3$. Both $S$ and its independent decaying-mode amplitude must vanish to select the growing adiabatic solution. At finite wavenumber, pressure gradients generate departures from this leading long-wavelength relation.
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