= Solution
Preserving <synchronous gauge> means keeping both $\delta g_{00}$ and $\delta g_{0i}$ zero. Applying the metric transformation to the background gives
$$
\delta\widetilde g_{00}=2a^2[(\xi^0)'+\mathcal H\xi^0],\qquad
\delta\widetilde g_{0i}=a^2[\partial_i\xi^0-(\xi^i)'].
$$
The first equation requires $(a\xi^0)'=0$, so $\xi^0=C(\mathbf x)/a$. For the scalar displacement $\xi^i=\partial^i\lambda$, the second requires $\lambda'=\xi^0$, up to a spatially homogeneous coordinate displacement that does not affect scalar perturbations. Integrating gives
$$
\boxed{\xi^0=\frac{C(\mathbf x)}a,\qquad\lambda=C(\mathbf x)\int^\tau\frac{d\tau'}{a(\tau')}+D(\mathbf x).}
$$
This 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 displacement
$$
\widetilde\delta=\delta-\frac{\bar\rho'}{\bar\rho}\xi^0.
$$
For a fluid with constant equation of state, background conservation gives $\bar\rho'/\bar\rho=-3(1+w)\mathcal H$, hence
$$
\boxed{\Delta\delta=3(1+w)\frac{\mathcal HC(\mathbf x)}a.}
$$
The <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: $h_{ij}=\operatorname{diag}(L,0,0)$ has zero determinant even for large $L$. This is <small determinant does not imply a small metric perturbation>.
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