Cosmological entropy perturbation

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.