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Adiabatic initial conditions perturb all relevant local physical quantities along the same background time trajectory. For quantities with nonzero background time derivatives, their perturbations satisfy $\delta X/{\bar{X}}^{\prime }=\delta Y/{\bar{Y}}^{\prime }$. For separately conserved barotropic fluids, cosmological energy conservation then givesFor example, pressureless matter and radiation in cosmology obey ${\delta }_m=3{\delta }_{\gamma }/4$. Their relative particle-number ratio is unperturbed at leading order.

The density condition tests relative entropy between components. In a general nonbarotropic component it does not by itself eliminate intrinsic entropy, measured by $\delta P_i-({\bar{P}}_i^{\prime }/{\bar{\rho }}_i^{\prime })\delta {\rho }_i$. The stronger statement that every physical variable follows the same time shift includes the pressure and any independent field-velocity data. A single scalar species alone does not make the density-ratio test sufficient, since that test is then vacuous.

For vanishing total non-adiabatic pressure perturbation and negligible regular gradient terms, uniform-density curvature is conserved on superhorizon scales.^[ref] Relating this to conserved comoving curvature perturbation also requires the appropriate regular adiabatic mode; a non-attractor phase such as ultra-slow-roll inflation needs separate analysis. The ordinary growing adiabatic photon mode selects coherent acoustic phases, explaining the adiabatic acoustic-peak spacing.

David H. Lyth helped establish when large-scale cosmological curvature is conserved. With David Wands, Karim A. Malik and Andrew R. Liddle, he showed that negligible non-adiabatic pressure perturbations preserve the uniform-density curvature perturbation when spatial-gradient effects are negligible.^[ref]

The result specifies when primordial fluctuations can be carried forward without solving every intervening gravitational detail.

David Wands and collaborators gave a conservation-based account of long-wavelength fluctuations in linear cosmological perturbation theory. Their 2000 work treated slowly varying regions as locally homogeneous expanding universes and identified the conditions for conserved uniform-density curvature perturbations, without selecting particular gravitational field equations.^[ref]

The important qualifications are negligible spatial gradients and negligible non-adiabatic pressure perturbations; large wavelength alone is insufficient.

Karim A. Malik developed the treatment of gauge-invariant cosmological curvature with David Wands, David H. Lyth and Andrew R. Liddle. Their 2000 analysis derived large-scale evolution from stress-energy conservation, exposing how a non-adiabatic pressure perturbation can change the uniform-density curvature perturbation.^[ref]

The uniform-density curvature perturbation is a dimensionless spatial-curvature amplitude evaluated on hypersurfaces where the density perturbation vanishes. Fix the scalar spatial metric conventionHere primes denote conformal time, $\mathcal{H}=a^{\prime }/a$ and ${\bar{\rho }}^{\prime }\ne 0$. Under the passive cosmological gauge transformation $\tilde{\eta }=\eta +T$, $\tilde{\Phi }=\Phi +\mathcal{H}T$ and $\delta \tilde{\rho }=\delta \rho -{\bar{\rho }}^{\prime }T$, so the two changes cancel in $\zeta$. Choosing $T=\delta \rho /{\bar{\rho }}^{\prime }$ reaches uniform-density slicing and gives $\zeta =-{\Phi }_{\delta \rho =0}$.

Using background energy conservation, the same convention isIt agrees with the uniform-density curvature sign test. For a spatially flat background, the intrinsic three-dimensional Ricci perturbation on these slices is $\delta {}^{(3)}R=-4a^{-2}{\nabla }^2\zeta$: $\zeta$ is its scalar amplitude, rather than the Ricci scalar itself. The full-Hessian $E$ term is a spatial-coordinate contribution and cancels from that Ricci scalar.

An overall reversal of the definition of $\zeta$ is also used, including in the explicitly signed superhorizon conservation of uniform-density curvature equation. Signs must follow the declared metric and curvature definition together. Uniform-density slicing also differs from comoving gauge in cosmology; their regular adiabatic large-scale relation needs additional dynamical assumptions.^[ref]