Scalar gauge transformations with a background lapse ($\delta \tilde{g}=\delta g-{\mathcal{L}}_{\xi }\bar{g}$)

For $N=\bar{N}(1+\Psi )$, $g_{0i}=a^2{\partial }_{i}B$ and spatial scalar convention $h_{ij}=a^2[(1-2\Phi ){\delta }_{ij}-2{\partial }_i{\partial }_{j}E]$, a passive displacement $({\xi }^0,{\partial }^i\lambda )$ gives $\tilde{\Psi }=\Psi -{\dot{\xi }}^0-(\dot{\bar{N}}/\bar{N}){\xi }^0$, $\tilde{B}=B+{\bar{N}}^2{\xi }^0/a^2-\dot{\lambda }$, $\tilde{\Phi }=\Phi +(\dot{a}/a){\xi }^0$ and $\tilde{E}=E+\lambda$. These follow from subtracting the background metric Lie derivative.

For spatial scalar convention $g_{ij}=a^2[(1-2\Psi ){\delta }_{ij}-2E_{,ij}]$ and $g_{0i}=a^2{B}_{,i}$, a passive gauge displacement gives $\tilde{E}=E+\lambda$ and $\tilde{B}=B-\dot{\lambda }+{\bar{N}}^2{\xi }^0/a^2$. Moving from synchronous gauge to Newtonian gauge therefore requires $\lambda =-E_S$ and ${\xi }^0=-a^2{\dot{E}}_S/{\bar{N}}^2$. A scalar density transforms by $\delta \tilde{\rho }=\delta \rho -\dot{\bar{\rho }}{\xi }^0$, giving the displayed density contrast relation. Keeping the background lapse function explicit prevents cosmic-time and conformal-time factors from being confused.