Derivation of the prescribed response equation

Consider formally

Its fluctuation Euler derivative through first order is

This follows directly from Lemma lem:action, polarization, and (eq:backgroundgauge).

The background-field prescription removes the terms independent of the fluctuation from its equation: equivalently, it removes constant and linear fluctuation terms from the local action expansion. It does not remove the linear kinetic equation. The classical backgrounds $h,k$ are not solved for.

Now write the fluctuation as $r+\kappa k_q$, and multiply the order-$\kappa$ gravitational sources, after Euler differentiation, by $\sigma$. The resulting definition is

For smooth inputs this is the unique zero-past solution of ${\mathsf{P}}_{\mathrm{gr}}{k}_q=\mathrm{Src}$.

Every source in (eq:response) is past-compact: its intersection with the causal past of a compact set is compact. This follows from its vanishing for $t<0$ and flat finite propagation. No spatial decay is needed BGP. Without onset multiplication the self source of a general free field is not past-compact, and the retarded Green operator cannot be applied on that basis.