The fluctuation equation

The terms $\frac{1}{2}{G}_{\mathrm{lin}}[h_{\mathrm{cl}}]+\frac{\kappa }{2}(G_{\mathrm{lin}}[k_{\mathrm{cl}}]+{\mathcal{G}}_2[h_{\mathrm{cl}},h_{\mathrm{cl}}])$ of (eq:fluct) are the Einstein tensor density of the off-shell background. The background-field prescription of Definition def:content drops them. The remaining equation, with $u=\hat{h}+\kappa \hat{k}$ and $P_{\mathrm{gr}}\hat{h}=0$, reads at order $\kappa$

whose right side, with $\varphi \to \hat{\varphi }$, Wick ordered, and switched by ${\sigma }_{\epsilon }$ on the gravitational terms, is $\mathsf{S}$ of (eq:source). The definition $\hat{k}={{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}\mathsf{S}$ is therefore the first-order retarded Møller correction of the metric slot, as X-E's $k_1$ was its matter part; $k_1={{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}(\frac{1}{2}g\Theta )$ is the first term of $\hat{k}$.