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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 isThe 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 $\tilde{\eta }=\eta +T$, the density perturbation gauge transformation gives ${\tilde{\delta }}_c={\delta }_c+3\mathcal{H}T$ and ${\tilde{\delta }}_{\gamma }={\delta }_{\gamma }+4\mathcal{H}T$, where $\mathcal{H}=a^{\prime }/a$. 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.

The supplied superhorizon evolution equation isFor an adiabatic cosmological perturbation, $\delta P=({\bar{P}}^{\prime }/{\bar{\rho }}^{\prime })\delta \rho$, so $\delta P_{\mathrm{nad}}=0$ and the comoving curvature perturbation $\mathcal{R}$ is conserved on superhorizon scales.

For photons and cold dark matter, put $A={\bar{\rho }}_{\gamma }$, $C={\bar{\rho }}_c$, and use $\bar{P}=A/3$, $A^{\prime }=-4\mathcal{H}A$, and $C^{\prime }=-3\mathcal{H}C$. ThenThe cosmological entropy perturbation $S={\delta }_c-3{\delta }_{\gamma }/4$ implies ${\delta }_c=S+3{\delta }_{\gamma }/4$, so the adiabatic part cancels andSince $\bar{\rho }+\bar{P}=C+4A/3$, the curvature evolves asIn the sign convention requested in the question, $d\mathcal{R}/d\ln a=-fS$, this means