Residual synchronous-gauge freedom (${\xi }^0=C/\bar{N},\lambda =C\int \bar{N}{a}^{-2}dt+D$)

Preserving $\Psi =B=0$ under the scalar gauge transformations with a background lapse requires $\bar{N}{\xi }^0=C(\mathbf{x})$ and $\dot{\lambda }=C(\mathbf{x})\bar{N}/a^2$. Thus arbitrary spatial functions $C,D$ remain after imposing synchronous gauge in cosmology. This freedom can generate changes in the remaining scalar metric perturbations.

A residual synchronous time displacement ${\xi }^0=C(\mathbf{x})/a$ changes a fluid density contrast by $-{\bar{\rho }}^{\prime }{\xi }^0/\bar{\rho }$. For constant equation of state this gives the displayed mode: it scales as ${\tau }^{-2}$ during radiation domination and as ${\tau }^{-3}$ during matter domination. Such density changes alone do not indicate physical structure growth. Choosing the cold-matter rest frame fixes the residual time displacement, while a spatial relabeling fixes the remaining integration convention for the metric trace.