Adiabatic matter-string density relation (${\delta }_s=\frac{2}{3}{\delta }_c$)

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 $S^{\prime \prime }+2\mathcal{H}{S}^{\prime }=0$ 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.