= Adiabatic initial conditions
= Adiabatic cosmological perturbation
{synonym}
<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 $\delta X/\bar X'=\delta Y/\bar Y'$. For separately conserved <barotropic fluids>, <cosmological energy conservation> then gives
$$
\frac{\delta\rho_i}{\bar\rho_i'}=\frac{\delta\rho_j}{\bar\rho_j'}
\quad\Longleftrightarrow\quad
\frac{\delta_i}{1+w_i}=\frac{\delta_j}{1+w_j}.
$$
For 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 $\delta P_i-(\bar P_i'/\bar\rho_i')\delta\rho_i$. 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>.<A new approach to the evolution of cosmological perturbations on large scales by David Wands and others>[Equations (18)–(19), (29), and Section V.A]{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>.
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