Adiabatic initial conditions

Adiabatic initial conditions perturb all relevant local physical quantities along the same background time trajectory. For quantities with nonzero background time derivatives, their perturbations satisfy $\delta X/{\bar{X}}^{\prime }=\delta Y/{\bar{Y}}^{\prime }$. For separately conserved barotropic fluids, cosmological energy conservation then givesFor example, pressureless matter and radiation in cosmology obey ${\delta }_m=3{\delta }_{\gamma }/4$. Their relative particle-number ratio is unperturbed at leading order.

The density condition tests relative entropy between components. In a general nonbarotropic component it does not by itself eliminate intrinsic entropy, measured by $\delta P_i-({\bar{P}}_i^{\prime }/{\bar{\rho }}_i^{\prime })\delta {\rho }_i$. The stronger statement that every physical variable follows the same time shift includes the pressure and any independent field-velocity data. A single scalar species alone does not make the density-ratio test sufficient, since that test is then vacuous.

For vanishing total non-adiabatic pressure perturbation and negligible regular gradient terms, uniform-density curvature is conserved on superhorizon scales.^[ref] Relating this to conserved comoving curvature perturbation also requires the appropriate regular adiabatic mode; a non-attractor phase such as ultra-slow-roll inflation needs separate analysis. The ordinary growing adiabatic photon mode selects coherent acoustic phases, explaining the adiabatic acoustic-peak spacing.

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.