Density perturbation gauge transformation
= Density perturbation gauge transformation
{title2=$\widetilde{\delta\rho}=\delta\rho-\bar\rho'T$}
Density is a <scalar field>. Under a first-order time shift $\widetilde\tau=\tau+T$, its background changes by $\bar\rho'T$, so its perturbation changes by the negative of that amount. Spatial shifts act only at second order on a homogeneous background density.