 Solution

At $ϵ=0$ the metric is the constant Minkowski metric, whose Christoffel symbols and curvature vanish, soInsert $R=ϵR^{(1)}+O({ϵ}^2)$ into the Penrose wave equation. Every curvature-square term is $O({ϵ}^2)$, while the covariant wave operator reduces at first order to the flat d'Alembert operator. ThusThe linearized Riemann curvature operator is unchanged by $h_{\mu \nu }\mapsto h_{\mu \nu }+{\partial }_{\mu }{\xi }_{\nu }+{\partial }_{\nu }{\xi }_{\mu }$, because the resulting third derivatives cancel pairwise. The equation therefore requires no gauge choice for $h_{\mu \nu }$.