Adiabatic initial conditions

Adiabatic initial conditions perturb every species by the same local time shift, so their relative number-density ratios are initially unperturbed. Photon acoustic modes then begin predominantly as displacement modes with cosine phase.

Equivalently, $\delta {\rho }_i/{\dot{\bar{\rho }}}_i$ is independent of the component $i$. The non-adiabatic pressure perturbation then vanishes, and the comoving curvature perturbation is conserved on superhorizon scales when anisotropic stress and gradient terms can be neglected.

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.