Density perturbation gauge transformation ($\widetilde{\delta \rho }=\delta \rho -{\bar{\rho }}^{\prime }T$)

Density is a scalar field. Under a first-order time shift $\tilde{\tau }=\tau +T$, its background changes by ${\bar{\rho }}^{\prime }T$, so its perturbation changes by the negative of that amount. Spatial shifts act only at second order on a homogeneous background density.