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Simultaneously varying a metric tensor and a scalar field by the Lie derivative along a compactly supported vector field leaves a covariant matter action invariant. Integration by parts gives the displayed Noether identity. It implies stress-energy conservation when the scalar equation holds without needing to impose the metric field equation.

For constant $P=w\rho$, the perfect fluid stress tensor and inverse perturbed spatial metric give $T^{ij}=a^{-2}w\bar{\rho }[(1+\delta ){\delta }^{ij}-h^{ij}]$ at first order. Normalizing the velocity makes $u^0=a^{-1}$ at this order. Stress-energy conservation yields the synchronous perfect-fluid density equation and Cosmological Euler equation in synchronous gauge; the pressure projection includes $v_i{\bar{P}}^{\prime }$, which changes the velocity damping from $\mathcal{H}$ to $(1-3w)\mathcal{H}$.

// Target: cosmology.bigb

Assume a metric theory in four dimensions, a Levi-Civita connection, covariant stress-energy conservation and field equations linear in second metric derivatives. The contracted Bianchi identity selects $G_{ab}=R_{ab}-g_{ab}R/2$. Its trace-reversed equation is $R_{ab}=\kappa (T_{ab}-g_{ab}T/2)+\Lambda g_{ab}$. In the convention $R_{00}\simeq -{\nabla }^2\phi$ and signature $(+---)$, nonrelativistic matter gives $R_{00}\simeq \kappa \rho /2$. The Poisson equation ${\nabla }^2\phi =4\pi G\rho$ fixes $\kappa =-8\pi G$. Reversing the curvature convention reverses this displayed coupling sign without changing the physical theory. The strong equivalence principle alone does not establish uniqueness without the stated additional assumptions.

For a matter-dominated fluid with constant $c_s^2=w\ll 1$, combine stress-energy conservation in synchronous gauge with the scalar trace Einstein equation. Discard pressure corrections to the background gravitational and velocity-damping terms while retaining the potentially large pressure-gradient term $w{k}^2\delta$. This produces the displayed leading nonrelativistic equation. It is not an exact equation at finite $w$. At negligible pressure the growing and decaying matter-era solutions are ${\tau }^2$ and ${\tau }^{-3}$.

// Target: cosmology.bigb

The number current is $J^a=n{u}^a$, where $n$ is the rest-frame number density and $u^a$ the unit four-velocity. Its covariant continuity equation is ${\nabla }_a{J}^a=0$, equivalently $u^a{\nabla }_{a}n+n{\nabla }_a{u}^a=0$. Integration over a spacetime volume and the divergence theorem equate the particle numbers crossing its boundary. This conservation law is separate from stress-energy conservation.

The Pound-Rebka experiment measures a small gravitational redshift by comparing nuclear emission and resonant absorption at different heights in Earth's field. The $14.4\mathrm{keV}$ transition of iron-57 supplies the gamma-ray frequency standard. The Mössbauer effect permits recoil-free emission and absorption by nuclei bound in solids, giving a sufficiently narrow resonance. A source and absorber separated vertically by about $22.6\mathrm{m}$ have a predicted fractional frequency difference of order $2.5\times 10^{-15}$. A calibrated Doppler effect from slow source motion measures the shift through its effect on resonant absorption. Interchanging the upper and lower arrangements reverses the gravitational contribution and helps separate it from intrinsic source-absorber offsets; temperature-dependent frequency shifts must also be controlled. The observed change agrees with the weak-field prediction.

The prediction follows either from an accelerating laboratory and the equivalence principle, or directly from a static metric. With $x^0=ct$, write $g_{00}=N^2\simeq 1+2\Phi /c^2$. The time-translation Killing vector field gives a conserved photon energy $E_{\xi }=p_a{\xi }^a$, while a static observer has four-velocity $u^a={\xi }^a/N$ and measures $h\nu =E_{\xi }/N$. ThusFor an upward ray over height $H$, $\Delta \nu /\nu \simeq -gH/c^2$; downward propagation has the opposite sign. The equivalent compensating Doppler speed has magnitude $gH/c$, about $7.4\times 10^{-7}\mathrm{m}{\mathrm{s}}^{-1}$ for this baseline. Identical clocks at rest have $d\tau =Ndt$, so a clock at higher potential runs faster relative to the common static coordinate time. This does not mean that the nuclear transition changes its locally measured frequency with height: the comparison involves transporting a photon between different local proper-time standards.

The weak equivalence principle asserts universal free fall for test bodies. The Einstein equivalence principle adds local Lorentz invariance and local position invariance for nongravitational experiments in a freely falling laboratory. The redshift comparison tests the clock aspect of local position invariance and supports a metric description of gravity. The strong equivalence principle extends these requirements to self-gravitating bodies and local gravitational experiments, including the contribution of gravitational binding energy. Their outcomes should be independent of the external location and motion of a freely falling laboratory when tidal effects are negligible. A nuclear redshift experiment is therefore not a test of the whole strong equivalence principle, and its success does not uniquely establish the gravitational field equations.

The geometrical realization uses the Levi-Civita connection of a spacetime metric. At any event, normal coordinates make $g_{ab}={\eta }_{ab}$ and the Christoffel symbols vanish. Freely falling test particles follow geodesics; local nongravitational equations take their special-relativistic form at that event. The Riemann curvature tensor need not vanish, so finite laboratories can measure tidal geodesic deviation. The strong principle concerns removing a uniform external gravitational field locally, not removing curvature throughout an extended region.

For a perfect fluid, local rest-frame isotropy and the absence of heat flux or viscosity give $T^{\hat{a}\hat{b}}=\mathrm{diag}(\rho ,p,p,p)$, where $\rho$ is rest-frame energy density. With the PDF's signature $(+---)$, $u^a{u}_a=1$ and $h^{ab}=g^{ab}-u^a{u}^b$ projects spatially. Its covariant form isThe pressure sign follows from the negative spatial metric; it gives positive diagonal stresses in the rest frame. Local stress-energy conservation, ${\nabla }_a{T}^{ab}=0$, produces the two projectionsThe first is the energy equation, including pressure work; the second is the relativistic Euler equation. Its rest-frame spatial pressure force is $-\nabla p$.

Further assumptions are needed to derive the Einstein field equations. Take a local generally covariant metric equation, linear in curvature and of second differential order, and demand compatibility with stress-energy conservation. The available rank-two tensors are $R_{ab}$, $R{g}_{ab}$ and a constant multiple of $g_{ab}$. Defining $R_{ab}=R^c{}_{acb}$, the contracted Bianchi identity gives ${\nabla }^a{R}_{ab}=\frac{1}{2}{\nabla }_{b}R$. Hence the combination $R_{ab}+\alpha R{g}_{ab}$ is identically divergence-free only when $\alpha =-\frac{1}{2}$. This determines the Einstein tensor $G_{ab}=R_{ab}-R{g}_{ab}/2$; a cosmological constant term remains allowed.

The coupling sign must be matched to the actual curvature convention. Here $R^a{}_{bcd}$ represents $[{\nabla }_d,{\nabla }_c]$, and a static weak field has $R_{00}\simeq -{\nabla }^2\Phi /c^2$. With negligible pressure and cosmological term, a trial equation $G_{ab}=C{T}_{ab}$ implies $R_{00}\simeq C{\rho }_{\mathrm{mass}}{c}^2/2$. Comparing with the Poisson equation ${\nabla }^2\Phi =4\pi G{\rho }_{\mathrm{mass}}$ fixes $C=-8\pi G/c^4$. Thus the Einstein equation in the sequential-derivative curvature convention isEquivalently, $R_{ab}=\Lambda g_{ab}-(8\pi G/c^4)(T_{ab}-T{g}_{ab}/2)$, with $T=\rho -3p$. Pressure therefore gravitates as well as supplying a material force. The cosmological constant is not fixed by the local Newtonian matching.

Setting $c=1$ for the action normalization, the same metric equation follows from the Einstein-Hilbert action $S_g=(2\kappa {)}^{-1}\int \sqrt{-g}(R-2\Lambda )d^{4}x$, where $\kappa =8\pi G$, together with a matter action obeying $\delta S_m=\frac{1}{2}\int \sqrt{-g}{T}_{ab}\delta g^{ab}{d}^{4}x$. After the metric boundary term is removed or supplied separately, the bulk variation is proportional to $G_{ab}+\Lambda g_{ab}+\kappa T_{ab}$. This also makes the sign convention explicit. General covariance, the field equation and the contracted Bianchi identity are mutually consistent with the fluid conservation laws. The experiment motivates the local metric description; the curvature and conservation assumptions, plus the Newtonian limit, supply the additional route to the field equation.

Let $Q=g^{ab}{\chi }_{,a}{\chi }_{,b}$ and $\mathcal{L}=p(ϵ)+n(Q-{ϵ}^2)/(2ϵ)$. Vary $n,ϵ,\chi$ independently, taking compactly supported variations. The multiplier variation is $\partial \mathcal{L}/\partial n=(Q-{ϵ}^2)/(2ϵ)=0$. With $u_a={\chi }_{,a}/ϵ$, it gives the unit four-velocity condition $u^a{u}_a=1$. The enthalpy variation isUsing $Q={ϵ}^2$ yields $p^{\prime }(ϵ)=n$. Finally,The integration by parts boundary term vanishes. Thus the three matter equations areThe last is particle number conservation in a relativistic fluid. Off shell $n,ϵ$ are independent variational fields; on shell the thermodynamic relation ties them to the specified function $p(ϵ)$.

For the Hilbert stress-energy tensor, keep these matter fields fixed while varying the inverse metric. The volume variation is $\delta \sqrt{-g}=-\frac{1}{2}\sqrt{-g}{g}_{ab}\delta g^{ab}$, and $\delta Q={\chi }_{,a}{\chi }_{,b}\delta g^{ab}$. ThereforeOn shell, $\mathcal{L}=p$ and $nϵ=\rho +p$. With signature $(+---)$,For variation of the covariant metric the same definition reads ${\delta }_{g}S=-\frac{1}{2}\int \sqrt{-g}{T}^{ab}\delta g_{ab}{d}^{4}x$. In a local rest frame the tensor has entries $\mathrm{diag}(\rho ,p,p,p)$, which checks both the energy density and pressure sign.

Define a derivative along the flow by $\dot{F}=u^a{\nabla }_{a}F$, and set $\theta ={\nabla }_a{u}^a$, $a^b=u^a{\nabla }_a{u}^b$. The number equation immediately gives $\dot{n}+n\theta =0$. The specific enthalpy per particle relation and $dp=ndϵ$ giveConsequently the energy equation follows from the number equation: $\dot{\rho }=ϵ\dot{n}=-(\rho +p)\theta$.

The velocity equation follows directly from the potential, rather than requiring an independent assumption about the fluid force. Since $ϵu_a={\nabla }_a\chi$ and the connection has zero torsion, the canonical vorticity of a relativistic fluid vanishes:Contract with $u^a$ and use $u^a{u}_a=1$ and $u^a{\nabla }_b{u}_a=0$. This gives $ϵa_b={\nabla }_bϵ-u_b\dot{ϵ}$. Multiplication by $n$ and use of ${\nabla }_{b}p=n{\nabla }_bϵ$ produces the relativistic Euler equation. The evolution system is thereforeIt is equivalent to stress-energy conservation together with number conservation, subject to the potential-flow restriction. The action describes an isentropic barotropic sector of a perfect fluid in general relativity; it does not provide general vortical or entropy-carrying fluid motions. The constraint on canonical vorticity is stronger than the Euler equation alone.

For the weak static metric, use $c=1$. To the retained order, $\sqrt{-g}=1+\varphi$, $u^0=1-\varphi +O({\varphi }^2,|\mathbf{v}{|}^2)$ and $u^i=v^i$. The correction to $u^0$ is required by normalization, even though its zeroth-order value is one. The relevant Christoffel symbols arewith no spatial connection at this order. The cancellation $\sqrt{-g}{u}^0=1+O({\varphi }^2,|\mathbf{v}{|}^2)$ makes the leading number equation particularly simple:The corresponding leading energy transport equation is ${\partial }_t\rho +\mathbf{v}\cdot \nabla \rho +(\rho +p)\nabla \cdot \mathbf{v}=0$. The spatial Euler equation contains the gravitational acceleration $-\nabla \varphi$ and pressure force $-\nabla p/(\rho +p)$. In a strict first-order expansion, convective velocity acceleration and products of the small quantities are discarded.

To exhibit a fully linear system, perturb a homogeneous rest state: $n=n_0+\delta n$, $\rho ={\rho }_0+\delta \rho$, $p=p_0+\delta p$, and $ϵ={ϵ}_0+\delta ϵ$, with the perturbations, $\mathbf{v}$ and $\varphi$ all first order. Let $w_0={\rho }_0+p_0=n_0{ϵ}_0$. The weak-field slow-motion equations for a relativistic fluid areThe equation of state gives $\delta \rho ={ϵ}_0\delta n$ and $\delta p=n_0\delta ϵ$. Where $p^{\prime \prime }({ϵ}_0)\ne 0$, it also gives the barotropic relativistic sound speedWithout this invertibility, the primitive conservation equations remain valid and this sound-speed parametrization must not be used.

The potential furnishes a further check on the pressure and gravity signs. Write $\chi ={ϵ}_{0}t+\delta \chi$. Its time and spatial components implyTheir compatibility reproduces the linear Euler equation. Pressure gradients oppose gravitational acceleration, and hydrostatic balance is $\nabla \delta p=-w_0\nabla \varphi$. Combining the energy and Euler equations gives the forced acoustic equationFor positive $c_s^2$, disturbances propagate as sound waves; the prescribed potential supplies a static forcing. A nonrelativistic equation of state gives $p\ll \rho \simeq mn$, $ϵ\simeq m$, and reduces the inertia $\rho +p$ to mass density, recovering the ordinary linearized continuity and Euler equations. Slow velocity alone does not justify dropping $p$ from the inertia of a relativistically hot fluid. The given metric supplies an external gravitational potential; the matter action by itself supplies no Poisson equation for it.

During matter domination, the spatially flat background has $a\propto {\tau }^2$ and $\mathcal{H}=2/\tau$. Let ${\theta }_m=\nabla \cdot {\mathbf{v}}_m$ denote the scalar velocity divergence. Linear stress-energy conservation in synchronous gauge givesThe first equation follows from the perturbed volume element $a^3(1+h/2)$; the second states that peculiar dust velocity redshifts as $a^{-1}$. Differentiating the first and substituting the trace of the Einstein field equations, with $w_m=0$ and ${\Omega }_m=1$, yieldsThereforeTrying ${\delta }_m\propto {\tau }^p$ gives $p(p-1)+2p-6=(p-2)(p+3)=0$. The complete solution of this density equation isThese are the matter-era growing and decaying density modes, proportional to $a$ and $a^{-3/2}$. In a dust-comoving synchronous gauge, ${\theta }_m=0$ and $h^{\prime }=-2{\delta }_m^{\prime }$. That rest-frame choice distinguishes physical mode coefficients from changes of density induced by residual synchronous-gauge freedom.

Neglecting the rapidly oscillating radiation source on subhorizon scales, stress-energy conservation and the trace of the Einstein field equations giveEquivalently in cosmic time, ${\delta }_{m,tt}+2H{\delta }_{m,t}-4\pi G{\rho }_m{\delta }_m=0$. Radiation still contributes to the background expansion, even when its perturbation source is averaged away.

Put $\eta =a/a_{\mathrm{eq}}$ and let ${\rho }_{\mathrm{eq}}$ be the density of either component at matter-radiation equality. Then ${\rho }_m={\rho }_{\mathrm{eq}}{\eta }^{-3}$ and ${\rho }_r={\rho }_{\mathrm{eq}}{\eta }^{-4}$. With $C=(8\pi G/3){\rho }_{\mathrm{eq}}{a}_{\mathrm{eq}}^2$, the Friedmann equation givesApply the chain rule to the density equation:Thus the Mészáros equation isSubstituting $D_{+}=1+3\eta /2$ gives $(2+3\eta )(3/2)-3(1+3\eta /2)=0$, sois indeed a solution. It is approximately constant during radiation domination and grows as $a$ during matter domination. The second independent solution iswhich behaves as $\log (4/\eta )-3$ at small $\eta$. Thus smoothing radiation does not eliminate every logarithmic solution; matching through actual horizon entry fixes the combination excited.

Modes with $k\ll k_{\mathrm{eq}}$ enter after equality and retain the $P(k)\propto k$ shape. Modes with $k\gg k_{\mathrm{eq}}$ enter during radiation domination: their horizon-entry spectrum is $P_H\propto k^{-3}$ and they gain little growth until equality. The late matter power spectrum therefore turns over near $k_{\mathrm{eq}}$ and has a large-scale $k$ branch and a small-scale $k^{-3}$ branch, with the more accurate radiation-era matching givingEquivalently the cold-dark-matter transfer function falls approximately as $(k_{\mathrm{eq}}/k{)}^2$, up to the logarithm. The turnover is the consequence of the Mészáros effect; it is not a change of the primordial scale-invariance condition.

Use one Fourier mode, with $\mu =\hat{\mathbf{k}}\cdot \hat{\mathbf{n}}$ and unperturbed ray $\mathbf{x}(\tau )=\hat{\mathbf{n}}({\tau }_0-\tau )$. Put $E(\tau )=e^{ik\mu ({\tau }_0-\tau )}$, and let ${\tau }_{\ast }$ denote last scattering. In the supplied branch $h^{\prime }=h_s^{\prime }$, the scalar decomposition givesSince $E^{\prime \prime }=-k^2{\mu }^{2}E$, two integrations by parts give the exact endpoint decomposition of the metric contribution:For the matter-era growing mode, $h_s$ is quadratic in $\tau$, so the last integral vanishes. With $E^{\prime }=-ik\mu E$, the emission endpoint isand the observer endpoint is $-h_s^{\prime \prime }({\tau }_0)/(2k^2)-i\mu h_s^{\prime }({\tau }_0)/(2k)$. The observer terms are only a angular monopole and angular dipole; they are removed when studying the anisotropy multipoles $\ell \ge 2$.

For an adiabatic growing mode outside the horizon at last scattering, ${\delta }_{\gamma }=4{\delta }_m/3$, ${\delta }_m=-h/2$ after fixing the constant trace convention, and $h\propto {\tau }^2$. Hence the intrinsic emission term is smaller than $h_s^{\prime \prime }/(2k^2)$ by order $(k{\tau }_{\ast }{)}^2$. The derivative emission endpoint is smaller by order $k{\tau }_{\ast }$. Stress-energy conservation also makes the peculiar emission velocity subleading; its leading synchronous contribution cancels in ${\delta }_{\gamma }^{\prime }+2h^{\prime }/3$, with gradients supplying the remaining small term. Keeping the Fourier phase, which need not be small over the observer's distance, givesHere $\Theta =\delta T/T$, and the sum over modes is understood. The derivative is the second derivative of $h_s$, as printed in the original PDF. This approximation assumes matter-era growing adiabatic modes and neglects late evolving potentials; large wavelength alone does not remove the last integral in a general cosmology. The equality $h^{\prime }=h_s^{\prime }$ is used as the supplied synchronous condition, rather than as a general identity for every fluid solely from vanishing anisotropic stress.

The strong equivalence principle extends universality of free fall to bodies with appreciable gravitational binding energy and requires results of sufficiently local experiments, including gravitational experiments, in freely falling laboratories to be independent of their location and velocity. The Einstein equivalence principle makes the corresponding statement for nongravitational experiments; the weak principle concerns test-body free fall without significant self-gravity. In a locally inertial frame the nongravitational laws take their special-relativistic form, while internal gravitational experiments retain the same local gravitational laws and coupling constants. They are not declared to have no gravity.

In a metric description, one may set $g_{ab}={\eta }_{ab}$ and ${\partial }_c{g}_{ab}=0$ at a point by Riemann normal coordinates, so the Levi-Civita connection vanishes there. This removes a locally uniform external gravitational acceleration, not the Riemann curvature tensor. Tidal effects between separated bodies, or accumulated over a finite-duration experiment, remain. The local limit must control both the laboratory's size and duration. Gravitationally self-bound bodies are included in the strong version, so it is stronger than equality of inertial and passive gravitational mass for ordinary test particles.

A heuristic field-equation argument supplements this principle with further assumptions: gravity is described only by a Lorentzian metric, matter couples universally to it, the affine connection is Levi-Civita, and the leading local field equation is a symmetric covariant equation with at most second metric derivatives and is linear in those second derivatives. Matter obeys stress-energy conservation ${\nabla }^a{T}_{ab}=0$. Curvature therefore supplies the natural candidates $R_{ab}$ and $R{g}_{ab}$, with a constant multiple of $g_{ab}$ also allowed. Write the candidate as $R_{ab}+cR{g}_{ab}+\Lambda g_{ab}=\kappa T_{ab}$. The contracted Bianchi identity and metric compatibility giveTo permit varying scalar curvature consistently with matter conservation, take $c=-1/2$. This gives the Einstein tensor $G_{ab}=R_{ab}-g_{ab}R/2$ andThe cosmological constant is constant so that its divergence vanishes. These assumptions motivate the usual Einstein equation; the equivalence principle alone does not exclude theories with extra fields or higher curvature terms.

The sign and normalization are fixed by the supplied Newtonian limit. Use signature $(+---)$, the curvature commutator convention of Q1, and the Ricci contractionThis is the contraction for which $R_{00}\simeq -{\nabla }^2\phi$ in the question. In four dimensions, tracing the candidate equation gives $-R+4\Lambda =\kappa T$, and substituting back givesFor slowly moving pressureless matter, $T_{00}\simeq \rho$, $T\simeq \rho$ and $g_{00}\simeq 1$. On local scales where the cosmological term is negligible, $R_{00}\simeq \kappa \rho /2$. Compare this with $R_{00}\simeq -{\nabla }^2\phi$ and the Poisson equation ${\nabla }^2\phi =4\pi G\rho$. The Newtonian normalization of the Einstein field equations is thereforeUnits with light speed one are used, as in the paper; for conventional physical stress-energy units the coupling is $-8\pi G/c^4$. If the opposite Ricci/curvature sign convention is adopted, the displayed geometric coupling sign changes accordingly. With the cosmological term retained in the same weak-field calculation, the Newtonian equation becomes ${\nabla }^2\phi =4\pi G\rho -\Lambda$, consistent with this sign choice.

Separate background stress-energy conservation gives ${\rho }_C^{\prime }=-3\mathcal{H}{\rho }_C$ and ${\rho }_S^{\prime }=-2\mathcal{H}{\rho }_S$. ThereforeFor consistency, compute the expansion derivative from the Friedmann equation, rather than assume the printed second hint. Differentiating ${\mathcal{H}}^2=(8\pi G/3)a^2{\rho }_{\mathrm{tot}}$ and using total stress-energy conservation yieldsThe PDF's hint has ${\rho }_{\mathrm{tot}}+P_{\mathrm{tot}}$ where ${\rho }_{\mathrm{tot}}+3P_{\mathrm{tot}}$ is required. It is incompatible with the first hint and conservation, and would not give the requested equation. The expression above is the corrected conformal-time acceleration identity.

For a function $D(\eta )={\delta }_C$, the chain rule givesSubstituting in the cold-matter equation and dividing by ${\mathcal{H}}^2{\eta }^2$ producesorThis is cold-matter growth in a matter-coasting-fluid universe; its coefficients depend on the background density ratio, not on $k$.

Let $A=\bar{\rho }+\bar{P}$, and assume $A\ne 0$ so that the density-defined curvature is meaningful. Under the cosmological gauge transformation in part ii,Therefore the two time-shift terms cancel:This is the uniform-density curvature perturbation with the paper's overall sign. On a uniform-density slice it is the spatial curvature potential $\Psi$. Its opposite overall sign is also a common convention, so the sign must be kept consistent rather than imported from another metric convention.

To establish time conservation, write $D_t={\bar{N}}^{-1}{\partial }_t$ and $c_a^2=\dot{\bar{P}}/\dot{\bar{\rho }}$. Background stress-energy conservation impliesFor constant $w$ in the given barotropic equation of state, $c_a^2=w$. Differentiate $\zeta =\Psi -\delta \rho /(3A)$:Insert both perturbed evolution equations from the hint, keeping the spatial Laplacian $\Delta ={\nabla }^2/a^2$:Thus the lapse perturbation $\Phi$ and the trace perturbation $\kappa$ cancel exactly. The remaining pressure term is the non-adiabatic pressure perturbation $\delta P_{\mathrm{nad}}=\delta P-c_a^2\delta \rho$. In the assumed barotropic fluid, the perturbed equation of state gives $\delta P=w\delta \rho$ and $c_a^2=w$, so it vanishes. For a Fourier mode,For regular long-wavelength perturbations, the velocity and shear potentials in this combination do not grow as $k^{-2}$, so the right-hand side is gradient-suppressed when $k\ll aH$. Therefore $\zeta$ is conserved to leading superhorizon order, which is the superhorizon conservation of uniform-density curvature requested.

The statement is not exact for arbitrary finite $k$: the displayed gradient term gives its correction. It also requires adiabatic cosmological perturbations; entropy perturbations in a multicomponent system or a nonbarotropic fluid can give $\delta P_{\mathrm{nad}}\ne 0$ and evolve $\zeta$ even at zero gradient order. At exact $w=-1$, $A=0$ and the question's variable is undefined, rather than a well-defined conserved density-slicing perturbation.

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 $\delta g_{00}=-2{\bar{N}}^2\Phi$, $\delta g_{0i}=a^2{B}_{,i}$ 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 ${\bar{N}}^2{T}_{,i}-a^2{\dot{\lambda }}_{,i}$; and the spatial component changes by $-2a\dot{a}T{\delta }_{ij}-2a^2{\lambda }_{,ij}$. HenceThese are the scalar gauge transformations with a background lapse in the paper's sign convention.

Starting in synchronous gauge, preserving both $\Phi =0$ and $B=0$ requiresThe first equation integrates to $\bar{N}T=C(\mathbf{x})$; substituting in the second and integrating givesThe 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:To reach Newtonian gauge from synchronous quantities, impose $\tilde{E}=\tilde{B}=0$. For the nonhomogeneous scalar modes, chooseUsing background stress-energy conservation, $\dot{\bar{\rho }}=-3\bar{N}H(\bar{\rho }+\bar{P})$, gives the synchronous-to-Newtonian density transformation with a lapse:The expression includes the lapse factors because dots refer to the arbitrary time coordinate, not automatically to proper time.

The numerical fractions describe physically very different components. Use $c=1$ so that pressure and energy density have the same units. The stress-energy conservation equation for a homogeneous component, together with its equation of state $p_i=w_i{\rho }_i$, givesNonrelativistic baryons and cold dark matter have $w\simeq 0$ and dilute as $a^{-3}$; radiation has $w=1/3$ and dilutes as $a^{-4}$; vacuum energy has $w=-1$ and remains constant. The curvature term scales as $a^{-2}$ in the Friedmann equation but is a geometric contribution, not an additional material fluid.

There is consequently a coincidence in the present mixture. Normalize $a_0=1$. ThenAt early times vacuum is negligible, while at late times it dominates overwhelmingly. Observing order-unity matter and vacuum fractions during their comparatively recent crossover motivates the cosmic coincidence problem: why should this crossover occur near our observing epoch?

The absolute scale gives a second puzzle. Restoring SI units, $H_0\simeq 2.27\times 10^{-18}{\mathrm{s}}^{-1}$ yieldsIts energy density is ${\rho }_{\Lambda ,0}{c}^2\simeq 5.8\times 10^{-10}\mathrm{J}{\mathrm{m}}^{-3}$. Quantum vacuum contributions associated with ordinary particle-physics scales can be far larger than this observed value. Explaining the tiny net value, including cancellations among contributions, is the cosmological constant problem. The required negative pressure also differs qualitatively from ordinary dilute matter.

The matter budget supplies another explanatory gap: only $0.045$ is baryonic, while $0.255$ must be nonbaryonic matter in this benchmark. Together with $0.7$ in vacuum energy, most of the cosmic density is in components whose microscopic explanation is not supplied by fitting their fractions. Finally, near-zero curvature raises the flatness problem. For nonzero $k$,During radiation domination $H\propto a^{-2}$, so $|{\Omega }_k|\propto a^2$; during matter domination $H\propto a^{-3/2}$, so $|{\Omega }_k|\propto a$. Near-flatness today therefore requires extremely small early curvature in an unmodified decelerating history. Cosmic inflation offers a mechanism that reduces this curvature fraction through accelerated expansion.

The apparent contrivances are the recent matter-vacuum coincidence, the tiny vacuum-energy scale, the near-flat initial geometry and the unexplained microscopic dark components. Their observationally inferred values impose concrete requirements on a deeper physical account.

Let ${\gamma }_{\alpha \beta }={\delta }_{\alpha \beta }$ be the positive Euclidean spatial background metric. Write a tensor-sector metric perturbation as $h_{00}=h_{0\alpha }=0$, $h_{\alpha \beta }=-2E_{\alpha \beta }$, with ${\gamma }^{\alpha \beta }{E}_{\alpha \beta }=0$ and ${\partial }^{\alpha }{E}_{\alpha \beta }=0$. Spatial indices here are raised with $\gamma$, and $\square ={\partial }_t^2-\Delta$.

Substitute the linearized Levi-Civita connection into the cover's curvature definition. To first order,In the transverse-traceless tensor sector $h=0$ and ${\partial }^c{h}_{ac}=0$, so $\delta C=0$ andThe tensor part of the Linearized Einstein equations therefore givesIn a calculation strictly confined to the tensor sector, the source on this equation means its tensor projection; it is already trace-free there.