Perturbed Klein-Gordon equation with background lapse ($\delta \Pi =\mathcal{D}\delta \varphi -\bar{\Pi }\Psi$)

Let $\mathcal{D}={\bar{N}}^{-1}{\partial }_t$ and $\bar{\Pi }=\mathcal{D}\bar{\varphi }$. For negative-expansion extrinsic curvature of a spatial hypersurface, its trace perturbation is $\kappa$. With $\delta \Pi =\mathcal{D}\delta \varphi -\bar{\Pi }\Psi$, scalar evolution is $\mathcal{D}\delta \Pi -\Psi \mathcal{D}\bar{\Pi }+3H\delta \Pi -\kappa \bar{\Pi }-\Delta \delta \varphi +V_{,\varphi \varphi }\delta \varphi =0$. Combining background acceleration terms produces the standard lapse source $-\bar{\Pi }(\kappa +\mathcal{D}\Psi -3H\Psi )+2\Psi V_{,\varphi }$.