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Adiabatic initial conditions perturb every species by the same local time shift, so their relative number-density ratios are initially unperturbed. Photon acoustic modes then begin predominantly as displacement modes with cosine phase.

Equivalently, $\delta {\rho }_i/{\dot{\bar{\rho }}}_i$ is independent of the component $i$. The non-adiabatic pressure perturbation then vanishes, and the comoving curvature perturbation is conserved on superhorizon scales when anisotropic stress and gradient terms can be neglected.

The squared adiabatic sound speed is the ratio of background pressure and energy-density time derivatives. For a barotropic equation of state it is $dP/d\rho$. The non-adiabatic pressure perturbation is the difference $\delta P-c_a^2\delta \rho$; in a general field or multicomponent model this background quantity need not equal the rest-frame propagation speed.

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 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.

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.

With the convention $\zeta =\Phi -\delta \rho /[3(\bar{\rho }+\bar{P})]$, energy conservation gives ${\zeta }^{\prime }=\mathcal{H}\delta P_{\mathrm{nad}}/(\bar{\rho }+\bar{P})+(\nabla \cdot \mathbf{v})/3$. The non-adiabatic pressure perturbation vanishes for adiabatic cosmological perturbations, and regular velocity gradients are suppressed on superhorizon scales. Curvature is then conserved to leading gradient order, allowing primordial initial conditions to be propagated between cosmological eras.