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Preserving synchronous gauge means keeping both $\delta g_{00}$ and $\delta g_{0i}$ zero. Applying the metric transformation to the background givesThe first equation requires $(a{\xi }^0{)}^{\prime }=0$, so ${\xi }^0=C(\mathbf{x})/a$. For the scalar displacement ${\xi }^i={\partial }^i\lambda$, the second requires ${\lambda }^{\prime }={\xi }^0$, up to a spatially homogeneous coordinate displacement that does not affect scalar perturbations. Integrating givesThis is the residual gauge freedom in synchronous gauge. The lapse and shift conditions do not uniquely choose a slicing and spatial labeling.

A background density is a scalar, so under this remaining time displacementFor a fluid with constant equation of state, background conservation gives ${\bar{\rho }}^{\prime }/\bar{\rho }=-3(1+w)\mathcal{H}$, henceThe synchronous density gauge mode scales as ${\tau }^{-2}$ in radiation domination and as ${\tau }^{-3}$ in matter domination. It can mimic a changing density perturbation without changing physical observables. A density time dependence alone therefore does not identify a physical growing or decaying mode. Choosing coordinates comoving with cold matter fixes the residual time freedom, and a remaining spatial relabeling fixes the initial metric-trace integration convention. Gauge-invariant curvature perturbations provide another way to specify physical initial conditions.

For consistency, linear metric theory requires all background-orthonormal components of $h_{ij}$ to be small. The PDF's small-determinant condition by itself is insufficient: $h_{ij}=\mathrm{diag}(L,0,0)$ has zero determinant even for large $L$. This is small determinant does not imply a small metric perturbation.