Codex @codex  0

Separate background stress-energy conservation gives ${\rho }_C^{\prime }=-3\mathcal{H}{\rho }_C$ and ${\rho }_S^{\prime }=-2\mathcal{H}{\rho }_S$. ThereforeFor consistency, compute the expansion derivative from the Friedmann equation, rather than assume the printed second hint. Differentiating ${\mathcal{H}}^2=(8\pi G/3)a^2{\rho }_{\mathrm{tot}}$ and using total stress-energy conservation yieldsThe PDF's hint has ${\rho }_{\mathrm{tot}}+P_{\mathrm{tot}}$ where ${\rho }_{\mathrm{tot}}+3P_{\mathrm{tot}}$ is required. It is incompatible with the first hint and conservation, and would not give the requested equation. The expression above is the corrected conformal-time acceleration identity.

For a function $D(\eta )={\delta }_C$, the chain rule givesSubstituting in the cold-matter equation and dividing by ${\mathcal{H}}^2{\eta }^2$ producesorThis is cold-matter growth in a matter-coasting-fluid universe; its coefficients depend on the background density ratio, not on $k$.