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In Synchronous gauge in cosmology with coordinates comoving with cold dark matter, the linearized Einstein field equations and matter conservation give the displayed density equation. Primes denote conformal time derivatives. The radiation term contains its pressure contribution and cannot be dropped on superhorizon scales merely because radiation oscillates after cosmological horizon entry.

For scalar perturbations of a generic fluid in cosmic time, momentum conservation in synchronous gauge in cosmology gives

Take $w$ constant and $w\ne -1$, with a barotropic perturbation $\delta P=w\delta \rho$. Define the density contrast $\delta =\delta \rho /\bar{\rho }$ and use the scalar velocity potential convention $v_i(\mathbf{k})=i{k}_i\theta (\mathbf{k})$, so ${\partial }_i{v}^i=-k^2\theta$. This $\theta$ is a potential, not the sometimes-used velocity-divergence variable. In synchronous gauge in cosmology, the inverse metric is $g^{00}=-a^{-2}$ and $g^{ij}=a^{-2}({\delta }^{ij}-h^{ij})$ to first order. Discarding products of perturbations in the perfect fluid in general relativity givesFor the linearization, each $h_{ij}$ must be small: the PDF's determinant condition alone is insufficient, since $h=\mathrm{diag}(L,0,0)$ has zero determinant even for large $L$.

Write $\mathcal{H}=a^{\prime }/a$. The time component of covariant conservation of stress-energy isHere ${\Gamma }_{\mu 0}^{\mu }=4\mathcal{H}+h^{\prime }/2$. To first order, ${\Gamma }_{ij}^0{T}^{ij}=a^{-2}w\bar{\rho }[3\mathcal{H}(1+\delta )+h^{\prime }/2]$: the two metric-$h$ contributions cancel. Multiplying the conservation equation by $a^2$, including ${\Gamma }_{00}^0{T}^{00}$, givesThe background cosmological perfect-fluid continuity equation, ${\bar{\rho }}^{\prime }=-3\mathcal{H}(1+w)\bar{\rho }$, cancels the first and third terms. ThusThe $h^{\prime }$ term accounts for perturbation of the spatial volume expansion in synchronous gauge in cosmology. The PDF's correct connection ${\Gamma }_{0j}^i=\mathcal{H}{\delta }_j^i+h^{\prime i}{}_j/2$ is garbled in the TeX aid.

Use a dot for the coordinate-time derivative and let $\bar{N}$ be the homogeneous background lapse function. The passive linear transformation is $\delta {\tilde{g}}_{\mu \nu }=\delta g_{\mu \nu }-{\mathcal{L}}_{\xi }{\bar{g}}_{\mu \nu }$. Its time-time and time-space components yield the scalar gauge transformations with a background lapse:For example, ${\bar{g}}_{00}=-{\bar{N}}^2$ gives $-({\mathcal{L}}_{\xi }\bar{g}{)}_{00}=2\bar{N}\dot{\bar{N}}{\xi }^0+2{\bar{N}}^2{\dot{\xi }}^0$; ${\bar{g}}_{0i}=0$ gives $-({\mathcal{L}}_{\xi }\bar{g}{)}_{0i}={\bar{N}}^2{\partial }_i{\xi }^0-a^2{\partial }_i\dot{\lambda }$. These establish the signs directly.

To preserve synchronous gauge in cosmology, both transformed quantities must remain zero. HenceIntegrating gives the residual synchronous-gauge freedomThe integration lower limit can be absorbed into $D$. Spatially homogeneous additions to $\lambda$ generate no spatial displacement, so they are physically irrelevant to this scalar parametrization.

The spatial transformation also gives $\tilde{\Phi }=\Phi +(\dot{a}/a){\xi }^0$ and $\tilde{E}=E+\lambda$. Thus fixing the lapse and shift perturbations does not fix the entire coordinate system; the arbitrary functions $C,D$ can still change the remaining scalar cosmological perturbations.

In synchronous gauge in cosmology, $h_{0\mu }=0$, so the explicit metric terms vanish. Symmetry of the covariant energy-momentum tensor and the background metric implyHence the two expansion terms in part (c) combine to $3H\delta T^0{}_j$. Substituting the scalar velocity potential and scalar anisotropic stress definitions gives

Yes. Synchronous gauge in cosmology uses freely falling time lines and sets the lapse and shift perturbations to zero, so no gravitational-potential force appears explicitly in this component of momentum conservation. Metric perturbations still affect the other field equations, the evolution of matter variables, and the relation between coordinate-dependent variables and gauge-invariant observables; synchronous gauge also retains residual gauge modes.

In Newtonian gauge in cosmology, the time-time metric perturbation is a Newtonian gravitational potential. Its spatial gradient therefore appears explicitly as a force term in the Euler equation. The difference is a coordinate representation of the same covariant conservation law.

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.

For the FRW metric in conformal time, a displacement ${\xi }^{\mu }=(\alpha ,{\xi }^i)$ changes a raw metric perturbation by $q_{\mu \nu }\mapsto q_{\mu \nu }-{\mathcal{L}}_{\xi }{\bar{g}}_{\mu \nu }$. The displayed first-order time equations set the lapse and shift perturbations to zero. They can be integrated locally for smooth perturbations, demonstrating accessibility of synchronous gauge in cosmology. Their homogeneous solutions leave residual spatial functions, so the gauge is not fully fixed. A globally regular synchronous chart need not survive caustics of its freely falling congruence.

For constant $P=w\rho$ and velocity potential $v_i=i{k}_i\theta$, stress-energy conservation gives the displayed density equation in synchronous gauge in cosmology. Here $h$ is the trace of the spatial metric perturbation, and primes denote conformal time. In particular cold baryons obey ${\delta }_b^{\prime }=k^2{\theta }_b-h^{\prime }/2$. A positive sign in front of $k^2{\theta }_b$ on the left contradicts this potential convention.