Should the linearized field equations of GR with cosmological constant be gauge-invariant? Say I have a solution to Einstein's field equations (EFE) with cosmological constant (CC)
$$
G_{a b}[g] + \Lambda g_{a b}=T_{ab}[g,\Phi]
$$
and want to find a perturbative solution $g_{a b} + \delta g_{a b}$ given a perturbation on the energy-momentum tensor (e.g., we could take the background metric $g_{a b}$ as De Sitter's). Then, I would have to solve the linearized EFE
$$
\delta G_{a b}[g,\delta g] + \Lambda \delta g_{a b} = \delta T_{a b}[g,\delta g]
$$
However, if we do an infinitesimal coordinate transformation $x^{a} \to x^{a} + \xi^{a}$, and taking the background as fixed, we get that the perturbation should change as $\delta g_{a b} \to \delta g_{a b} + \nabla_{a}\xi_{b} + \nabla_{b} \xi_{a}$, a so called gauge transformation. If we substitute this into the linearized EFE, the CC term would have an explicit dependence on $\xi_{a}$ that would not cancel out (I'm actually not even sure if it cancels out in $\delta G_{a b}$ either, although it does on a flat background). Is this a problem? Should the linearized EFE, after a gauge transformation like that, not depend on the $\xi_{a}$? If it is a problem, is it possible to solve it without relying on ad hoc statements like "the CC is small, so you can neglect the CC term" or equivalents?
 A: $G_{ab}$ and $T_{ab}$, just like the metric, are two-tensors. They have the same transformation behavior that the metric has. In particular, for an infinitesimal change of coordinates given by $\xi^a$,
\begin{align}
\delta g_{ab} & = \xi^{c} \partial_c g_{ab} + g_{bc} \partial_a \xi^c + g_{ac} \partial_b \xi^c = \nabla_a \xi_b + \nabla_b \xi_a\\
\delta G_{ab} & = \xi^{c} \partial_c G_{ab} + G_{bc} \partial_a \xi^c + G_{ac} \partial_b \xi^c \\
\delta T_{ab} & = \xi^c \partial_c T_{ab} + T_{bc} \partial_a \xi^c + T_{ac} \partial_b \xi^c \\
\end{align}
and it is simple to see that $g + \delta g$ is a solution to the EFE with energy-momentum tensor $T + \delta T$. Since the only thing that was done was a change of coordinates (and this is a gauge transformation), the physics is invariant.
In your situation, $\delta T_{ab}$ is given and is an actual physical change, not a change of coordinates. It is not the case that $\delta T_{ab} = \xi^c \partial_c T_{ab} + T_{bc} \partial_a \xi^c + T_{ac} \partial_b \xi^c $ and, accordingly, the $\delta g_{ab}$ given by the change of variables will not be a solution: a simple change of coordinates of the metric will not suffice. Instead, one has to solve the linearized EFE having $\delta T_{ab}$ as the source term.
