= Solution
Use $T=\xi^0$ and $\xi^i=\partial^i\lambda$, and define the physical background <Hubble parameter> by $H=\dot a/(\bar N a)$. At first order, the spacetime metric components are $\delta g_{00}=-2\bar N^2\Phi$, $\delta g_{0i}=a^2B_{,i}$ and $\delta g_{ij}=-2a^2(\Psi\delta_{ij}+E_{,ij})$. Apply the given passive <cosmological gauge transformation> component by component. The temporal component changes by $2\bar N\dot{\bar N}T+2\bar N^2\dot T$; the mixed component changes by $\bar N^2T_{,i}-a^2\dot\lambda_{,i}$; and the spatial component changes by $-2a\dot aT\delta_{ij}-2a^2\lambda_{,ij}$. Hence
$$
\boxed{\widetilde\Phi=\Phi-\dot T-\frac{\dot{\bar N}}{\bar N}T,\qquad
\widetilde B=B+\frac{\bar N^2}{a^2}T-\dot\lambda,\qquad
\widetilde\Psi=\Psi+\bar NH T,\qquad
\widetilde E=E+\lambda.}
$$
These are the <scalar gauge transformations with a background lapse> in the paper's sign convention.
Starting in <synchronous gauge>, preserving both $\Phi=0$ and $B=0$ requires
$$
\partial_t(\bar N T)=0,\qquad
\dot\lambda=\frac{\bar N^2}{a^2}T.
$$
The first equation integrates to $\bar NT=C(\mathbf x)$; substituting in the second and integrating gives
$$
\boxed{T=\frac{C(\mathbf x)}{\bar N(t)},\qquad
\lambda=C(\mathbf x)\int^t\frac{\bar N(s)}{a(s)^2}\,ds+D(\mathbf x).}
$$
The arbitrary lower integration limit is absorbed into $D$. Thus specifying synchronous lapse and shift does not fix the origins of the freely falling clocks or all spatial coordinate labels. This is the <residual gauge freedom in synchronous gauge>.
During <cosmic inflation>, choosing these clocks remains arbitrary. For example, the $C$ mode shifts the spatial potential by $HC$ and the density perturbation by $3HC(\bar\rho+\bar P)$. In the <de Sitter approximation>, $H$ varies little, so a nearly constant <metric perturbation> can contain a pure coordinate contribution. An <inflaton> fluctuation or <density contrast> in <synchronous gauge> therefore cannot be declared physical solely from its time behavior. Use a <uniform-density curvature perturbation> or an appropriate field-based gauge-invariant variable, or fix a physical initial slicing. The exact de Sitter case has no evolving homogeneous density clock, which makes density-defined slicing degenerate.
During the standard hot Big Bang, a cold-matter rest frame supplies a convenient additional condition: imposing zero cold-matter velocity removes the nontrivial spatially varying time-shift mode. Before that condition is imposed, the pure-gauge <density contrast> for constant $w$ is $3H(1+w)C$, which decays as $t^{-1}$ in either radiation or <matter domination>. In <conformal time> this is proportional to $\tau^{-2}$ during <radiation domination> and $\tau^{-3}$ during <matter domination>. It can be confused with an independent decaying physical mode unless gauge freedom is fixed. The $D$ mode is a time-independent spatial relabeling and must likewise be fixed by a coordinate or initial-metric convention.
Finally, a scalar density transforms by evaluating the homogeneous scalar at the shifted time:
$$
\delta\widetilde\rho=\delta\rho-\dot{\bar\rho}T.
$$
To reach <Newtonian gauge> from synchronous quantities, impose $\widetilde E=\widetilde B=0$. For the nonhomogeneous scalar modes, choose
$$
\lambda=-E_S,\qquad T=-\frac{a^2}{\bar N^2}\dot E_S.
$$
Using background <stress-energy conservation>, $\dot{\bar\rho}=-3\bar NH(\bar\rho+\bar P)$, gives the <synchronous-to-Newtonian density transformation with a lapse>:
$$
\boxed{\left(\frac{\delta\rho}{\bar\rho}\right)_N
=\left(\frac{\delta\rho}{\bar\rho}\right)_S
+\frac{a^2\dot E_S}{\bar N^2}\frac{\dot{\bar\rho}}{\bar\rho}
=\left(\frac{\delta\rho}{\bar\rho}\right)_S
-3H\left(1+\frac{\bar P}{\bar\rho}\right)\frac{a^2}{\bar N}\dot E_S.}
$$
The expression includes the lapse factors because dots refer to the arbitrary time coordinate, not automatically to proper time.
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