= Cosmological entropy perturbation
= Isocurvature perturbation
{synonym}
A <cosmological entropy perturbation> describes relative composition changes between cosmic components, rather than only a common displacement along the background evolution. For <cold dark matter> and <photons>, define fractional density perturbations $\delta_c=\delta\rho_c/\bar\rho_c$ and $\delta_\gamma=\delta\rho_\gamma/\bar\rho_\gamma$. Their relative entropy variable is
$$
S_{c\gamma}=\delta_c-\frac34\delta_\gamma.
$$
The factor $3/4$ follows from the different background dilution laws $\bar\rho_c\propto a^{-3}$ and $\bar\rho_\gamma\propto a^{-4}$, with <scale factor> $a$. Under a first-order time shift $\widetilde\eta=\eta+T$, the <density perturbation gauge transformation> gives $\widetilde\delta_c=\delta_c+3\mathcal HT$ and $\widetilde\delta_\gamma=\delta_\gamma+4\mathcal HT$, where $\mathcal H=a'/a$. These shifts cancel in $S_{c\gamma}$.
Thus this relative entropy is gauge invariant at first order. The condition $S_{c\gamma}=0$ gives the common adiabatic relation $\delta_c=3\delta_\gamma/4$. A pure <isocurvature perturbation> initial condition is chosen with vanishing initial <comoving curvature perturbation>; a mixed initial state can instead have both nonzero curvature and nonzero $S_{c\gamma}$. Vanishing initial curvature is therefore a condition on a pure initial mode, not a requirement for defining the relative entropy variable.
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