Lemma: Fluctuation equation and the order-κ kinetic correction

With $q=\bar{q}+\kappa u+{\kappa }^2{k}_{\mathrm{cl}}$, $\bar{q}=\eta +\kappa h_{\mathrm{cl}}$, and $S_{\mathrm{q}}:=S_{\mathrm{gr}}[q]+S_{\mathrm{gf}}[u;\bar{q}]+{S^{\mathrm{sc}}}_g[q,\varphi ]$,

and (eq:Q-def) holds. For each smooth $w$, ${\mathcal{Q}}^{(1)}[w]$ is a linear differential operator of order $\le 2$ in $u$, with coefficients linear in $(w,\partial w,{\partial }^{2}w)$; in particular $\mathrm{supp}({\mathcal{Q}}^{(1)}[w]u)\subset \mathrm{supp}w$. Explicitly, ${\mathcal{F}}^{(1)}[h_{\mathrm{cl}}]u=\delta \int {{\mathfrak{L}}_{\mathrm{gf}}}^{(1)}/\delta u$ with

$C^{\mu }=C^{\mu }[u]$, $\bar{u}=\mathcal{P}u$, $H_{\mathrm{cl}}={\mathrm{tr}}_{\eta }{h}_{\mathrm{cl}}$, ${\Gamma }_1$ X-E's linearized Christoffel symbol.

Proof. $\delta S_{\mathrm{gr}}/\delta u=\frac{1}{2\kappa }\mathcal{E}[q]$ by the chain rule from xe:einstein. With $\kappa h=\kappa (h_{\mathrm{cl}}+u)$, ${\kappa }^{2}k={\kappa }^2{k}_{\mathrm{cl}}$, $\mathcal{E}=\kappa G_{\mathrm{lin}}[h_{\mathrm{cl}}+u]+{\kappa }^2(G_{\mathrm{lin}}[k_{\mathrm{cl}}]+{\mathcal{G}}_2[h_{\mathrm{cl}}+u,h_{\mathrm{cl}}+u])+O({\kappa }^3)$; divide by $2\kappa$ and polarize ${\mathcal{G}}_2$. The matter term is X-E's $\delta {S^{\mathrm{sc}}}_g/\delta q=-\frac{1}{2}g\sqrt{-q}T[q,\varphi ]$ times $\kappa$. The gauge-fixing term is (eq:F-expansion). Grouping by powers of $\kappa$ and using (eq:F0) gives (eq:fluct) and (eq:Q1); the $u$-linear part of the right side of (eq:fluct) is by construction $\mathcal{Q}[\bar{q}]u$, which proves (eq:Q-def). For (eq:Lgf1): $\sqrt{-\bar{q}}=1+\kappa b_1[h_{\mathrm{cl}}]+O({\kappa }^2)$, ${\bar{q}}^{\mu \nu }={\eta }^{\mu \nu }-\kappa {h_{\mathrm{cl}}}^{\mu \nu }+O({\kappa }^2)$, so ${\overline{u^{\rho \nu }}}_{\bar{q}}={\bar{u}}^{\rho \nu }+\kappa {\beta }^{\rho \nu }+O({\kappa }^2)$; ${\bar{\nabla }}_{\rho }{\bar{u}}^{\rho \nu }={\partial }_{\rho }{\bar{u}}^{\rho \nu }+{\bar{\Gamma }}_{\rho \lambda }^{\rho }{\bar{u}}^{\lambda \nu }+{\bar{\Gamma }}_{\rho \lambda }^{\nu }{\bar{u}}^{\rho \lambda }$ with $\bar{\Gamma }=\kappa {\Gamma }_1[h_{\mathrm{cl}}]+O({\kappa }^2)$ and ${\Gamma }_1{}_{\rho \lambda }^{\rho }=\frac{1}{2}{\partial }_{\lambda }{H}_{\mathrm{cl}}$. Insert into $\frac{1}{4}\sqrt{-\bar{q}}{\bar{q}}_{\mu \nu }{C^{\mu }}_{\bar{q}}{C^{\nu }}_{\bar{q}}$. ${\mathcal{G}}_2(h_{\mathrm{cl}},u)$ is the polarization of a bidifferential expression of total order two with constant coefficients; ${{\mathfrak{L}}_{\mathrm{gf}}}^{(1)}$ is quadratic in $(u,\partial u)$ with coefficients linear in $(h_{\mathrm{cl}},\partial h_{\mathrm{cl}})$, so its Euler derivative is of order $\le 2$ in $u$ with coefficients linear in $(h_{\mathrm{cl}},\partial h_{\mathrm{cl}},{\partial }^2{h}_{\mathrm{cl}})$.