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With negligible shear, a minimally coupled scalar field serving as a clock, and $\Pi \ne 0$, the Hamiltonian constraint and momentum constraint imply $\dot{H}=-4\pi GN{\Pi }^2$ and ${\partial }_{i}H=-4\pi G\Pi {\partial }_i\varphi$. Differentiating the local Friedmann relation yields $\dot{\Pi }=-N(3H\Pi +V_{,\varphi })$ and ${\partial }_i\Pi =-(3H+V_{,\varphi }/\Pi ){\partial }_i\varphi$. Inserting these identities into ${\dot{\zeta }}_i$ cancels all terms, including gradients of the lapse function. The conserved covector becomes the gradient of a uniform-density curvature perturbation at linear adiabatic order. Dropping the shear constraint is essential; this is not a proof of conservation in an arbitrary non-attractor phase.

Small primordial fluctuations seed later structure. For adiabatic cosmological perturbations on scales outside the horizon, the uniform-density curvature perturbation is conserved when the non-adiabatic pressure perturbation and the decaying mode can be neglected. A density contrast outside the horizon is gauge-dependent, so its apparent power-law growth should not be confused with directly measured local collapse.

For subhorizon, pressureless matter, the linearized continuity equation, Euler equation and Poisson equation combine intoHubble expansion damps peculiar motion; self-gravity amplifies overdensities. During radiation domination, radiation pressure drives acoustic oscillations and cold dark matter growth after horizon entry is only logarithmic. During matter domination the growing and decaying solutions are $\delta \propto a$ and $\delta \propto a^{-3/2}$. Dark-energy domination later suppresses growth relative to the matter-era law.

For a pressure-supported fluid the Fourier-mode equation also contains $+c_s^2{k}^2\delta /a^2$, giving a Jeans length separating acoustic support from gravitational growth. Before recombination, baryons are tightly coupled to photons and oscillate rather than freely following dark matter. Silk damping from photon diffusion damps small-scale photon-baryon fluctuations; after decoupling, baryons can fall into dark-matter potential wells. free streaming smooths species-dependent scales and distinguishes cold from warm or hot dark matter.

Schematically, the linear matter power spectrum can be written $P_{\delta }(k,a)=A{k}^{n_s}{T}^2(k)D^2(a)$ under the usual primordial-curvature normalization. The cosmological transfer function records horizon entry, matter-radiation equality, acoustic physics and damping; the linear growth factor records later amplification. density contrasts approaching unity require nonlinear collapse and halo formation beyond this approximation. cosmic microwave background anisotropies and galaxy clustering test the same initial perturbations through different transfer histories; galaxy bias, redshift-space distortions and observational selection enter the galaxy comparison.

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.

Use the spatial-potential convention of this paper and assume $\bar{\rho }+\bar{P}\ne 0$. A passive time displacement changes the spatial scalar by $\tilde{\Psi }-\Psi =\bar{N}H{\xi }^0$ and changes the energy density byThe two contributions to the uniform-density curvature perturbation therefore cancel:A spatial scalar displacement changes $E$ but not this combination. Hence $\zeta$ is gauge invariant at linear order. Pure cosmological-constant matter, for which $\bar{\rho }+\bar{P}=0$, does not supply a energy density clock and this formula is not defined for that component alone.

Now take the definite constant-$w$ barotropic equation of state and use physical-time differentiation $\mathcal{D}={\bar{N}}^{-1}{\partial }_t$. Put $A=\bar{\rho }+\bar{P}$. Background stress-energy conservation gives $\mathcal{D}A=-3H(1+w)A$. Differentiating $\zeta =-\Psi +\delta \rho /(3A)$, then substituting the two supplied perturbation equations, givesThus the lapse and expansion perturbations cancel without imposing a gauge. In terms of the non-adiabatic pressure perturbation $\delta P_{\mathrm{nad}}=\delta P-w\delta \rho$,For the stipulated barotropic equation of state, $\delta P=w\delta \rho$, so the first term vanishes. For a Fourier mode, $\Delta =-k^2/a^2$. For regular long-wavelength solutions the remaining gradients are suppressed when $k\ll aH$. Consequently $\dot{\zeta }\simeq 0$ on superhorizon scales, with exact conservation in the leading zero-gradient limit. It is not an exact finite-$k$ identity. Entropy perturbations or a singular gradient expansion can invalidate that conclusion; the supplied adiabatic assumption is what removes the unsuppressed pressure source.

This superhorizon conservation of uniform-density curvature is useful because inflationary fluctuations leave the horizon during cosmic inflation and may re-enter only much later. In an adiabatic single-clock evolution, the conserved uniform-density curvature perturbation transports their initial amplitude through reheating and the later radiation in cosmology and matter eras without solving every short-distance process. It then determines the initial conditions for large-scale structure and Cosmic microwave background anisotropy. Extra fluctuating fields or fluids can introduce isocurvature perturbations and convert them into curvature, so conservation must be checked rather than assumed during such transitions.

Now impose the stipulated negligible shear and work where $\Pi \ne 0$. The Hamiltonian constraint, momentum constraint and local expansion evolution giveTime differentiation of the first equation, followed by $\dot{\varphi }=N\Pi$, gives $\dot{\Pi }=-N(3H\Pi +V_{,\varphi })$. Spatial differentiation of the same equation and use of the momentum constraint givesThese are the temporal and spatial identities needed for the nonlinear curvature covector. Differentiate its definition, keeping the spatially varying lapse function:The terms proportional to ${\partial }_{i}N$ cancel. The terms $-N{\partial }_{i}H$ and $-4\pi GN\Pi {\partial }_i\varphi$ cancel by the momentum constraint, and the remaining two terms cancel by the spatial momentum identity. Thereforeto the retained gradient order. No homogeneous-lapse assumption was needed.

For linear adiabatic cosmological perturbations, expansion of the local scale and scalar field gives ${\zeta }_i={\partial }_i[\Psi +\bar{H}\delta \varphi /\bar{\Pi }]$. For nonzero spatial modes, linearizing the Hamiltonian constraint and the momentum constraint givesThusThis uses the paper's overall sign for the uniform-density curvature perturbation; the opposite overall sign is also common. For an adiabatic single-field attractor, superhorizon conservation of single-field comoving curvature allows the primordial amplitude to be transported through later epochs without following every short-scale process. This proof specifically discarded the shear contribution to the momentum constraint and assumed a usable field clock. It is not a claim that every single-field background conserves every curvature mode: growing curvature perturbation in ultra-slow-roll inflation is a non-attractor counterexample, and $\Pi =0$ makes this particular covector undefined.

Because density is a scalar field, $\tilde{\rho }(\tilde{x})=\rho (x)$. Write both sides as background plus perturbation and expand the background time shift:The coordinate shift acting on an already first-order perturbation contributes only at second order. Thus the density perturbation gauge transformation isThe uniform-density curvature perturbation in the paper's sign convention isThe derivative on ${\bar{\rho }}^{\prime }$ is essential and is present in the PDF; the TeX transcription drops it. The scalar-curvature combination transforms aswhile the density term changes by $-\mathcal{H}T$. Hence the two time-slicing changes cancel:On a uniform-density slice, $\delta \rho =0$ and this variable is the signed spatial-curvature perturbation. The construction assumes ${\bar{\rho }}^{\prime }\ne 0$; a pure constant-density cosmological constant does not define such a time slicing.

In Newtonian gauge in cosmology, $E=B=0$ and $C=-\Phi$. The background cosmological perfect-fluid continuity equation gives ${\bar{\rho }}^{\prime }=-3\mathcal{H}(\bar{\rho }+\bar{P})$, so, writing $D=\bar{\rho }+\bar{P}$,Set the cosmological adiabatic sound speed $c_a^2={\bar{P}}^{\prime }/{\bar{\rho }}^{\prime }$. Then $D^{\prime }=-3\mathcal{H}D(1+c_a^2)$. Differentiate the previous expression and insert the perturbed energy-conservation equation:Consequently the curvature evolution equation in this sign convention isThe last equality defines the total energy-frame velocity by $\mathbf{q}=D\mathbf{v}$. For adiabatic cosmological perturbations, the non-adiabatic pressure perturbation vanishes. On superhorizon scales, the stated suppression of the velocity divergence makes the remaining gradient term negligible. Thus $\zeta$ is conserved to leading order in $k/\mathcal{H}$. The positive sign of the pressure source follows from the paper's definition of $\zeta$ and must not be replaced by a formula using the opposite curvature convention.

Superhorizon conservation of uniform-density curvature lets one carry a primordial perturbation from its inflationary generation through otherwise complicated eras and predict the initial conditions for later Cosmic microwave background anisotropy and structure growth. It relies on adiabaticity: an isocurvature perturbation can source curvature evolution, so the same conclusion is not automatic in an arbitrary multifield model.