Lemma: Cubic coefficient on the pure-trace sector

For smooth $\psi$ and $\gamma =\psi \eta$,

Consequently, for the polarized source,

Proof. $q={\Omega }^2\eta$ with ${\Omega }^2=1+\kappa \psi$ is conformally flat. For $\tilde{g}={\Omega }^{2}g$ in four dimensions with $g$ flat and $w=\ln \Omega$, the standard conformal transformation of the Ricci tensor gives

hence ${\tilde{G}}_{\mu \nu }=-2{\partial }_{\mu }{\partial }_{\nu }w+2{\eta }_{\mu \nu }\square w+2{\partial }_{\mu }w{\partial }_{\nu }w+{\eta }_{\mu \nu }{\partial }_{\rho }w{\partial }^{\rho }w$. Since $q^{\mu \nu }={\Omega }^{-2}{\eta }^{\mu \nu }$ and $\sqrt{-q}={\Omega }^4$, $\sqrt{-q}{G}^{\mu \nu }={\eta }^{\mu \alpha }{\eta }^{\nu \beta }{\tilde{G}}_{\alpha \beta }$ with no further $\Omega$ factor. Expand $w=\frac{1}{2}\ln (1+\kappa \psi )=\frac{\kappa }{2}\psi -\frac{{\kappa }^2}{4}{\psi }^2+O({\kappa }^3)$, so that ${\partial }_{\mu }w=\frac{\kappa }{2}{\partial }_{\mu }\psi -\frac{{\kappa }^2}{2}\psi {\partial }_{\mu }\psi$ and ${\partial }_{\mu }{\partial }_{\nu }w=\frac{\kappa }{2}{\partial }_{\mu }{\partial }_{\nu }\psi -\frac{{\kappa }^2}{2}({\partial }_{\mu }\psi {\partial }_{\nu }\psi +\psi {\partial }_{\mu }{\partial }_{\nu }\psi )$, up to $O({\kappa }^3)$. The order-$\kappa$ coefficient is (eq:H1trace); the order-${\kappa }^2$ coefficient is $(\partial \psi \partial \psi +\psi \partial \partial \psi )-\eta (|\partial \psi {|}^2+\psi \square \psi )+\frac{1}{2}\partial \psi \partial \psi +\frac{1}{4}\eta |\partial \psi {|}^2$, which is (eq:H2trace). Polarizing $2{\mathcal{H}}_2$ gives (eq:J3trace).

Two checks. First, by (eq:algebra), $\mathcal{P}(\psi \eta )=-\psi \eta$, $D^{\mu }(\psi \eta )=-{\partial }^{\mu }\psi$, $D^{\ast }D(\psi \eta )=\partial \partial \psi -\frac{1}{2}\eta \square \psi$, so $A(\psi \eta )=\square \psi \eta -2\partial \partial \psi +\eta \square \psi =2{\mathcal{H}}_1[\psi \eta ]$, as required by $A\gamma =2{\mathcal{H}}_1$. Second, XI-F's witness $\gamma =t\eta$ gives $q=(1+\kappa t)\eta$, a spatially flat Robertson–Walker metric in conformal time with $a^2=1+\kappa t$, for which $G_{00}=3(a^{\prime }/a{)}^2$; its ${\kappa }^2$ coefficient is $\frac{3}{4}$, and (eq:H2trace) gives ${{\mathcal{H}}_2}^{00}[t\eta ]=\frac{3}{2}-\frac{3}{4}=\frac{3}{4}$.