Conserve curvature only when its hypotheses hold

Use ζ=−ψ−ℋδρ/ρ′ with the preceding metric signs. A passive time shift to uniform density removes δρ while changing ψ; its negative is precisely ζ. At first order the spatial metric on that slice has the scalar conformal factor one plus twice ζ after a suitable spatial threading choice. Equivalently, its long-wavelength local scale factor is a times expζ. Positive ζ in this convention means a locally enlarged spatial scale, not a positive Newtonian lapse by definition.

The background continuity equation gives ρ′=−3ℋ(ρ+p) for a conserved total stress tensor. Thus a barotropic fluid with constant w has ζ=−ψ+δ/[3(1+w)], where δ is its density contrast in the original slicing. This rewrite assumes the background evolves and ρ+p is nonzero. A pure cosmological constant has no usable uniform-density clock of this kind, so dividing by its vanishing density derivative cannot define an independent propagating scalar mode.

Define the adiabatic derivative ca²=p′/ρ′ and split the pressure perturbation into ca²δρ plus a remainder δpnad. This remainder is gauge invariant because the background time-shift pieces cancel. It measures failure of the perturbed pressure to follow the background pressure-density relation; ca² is not automatically the physical propagation speed of every field or mixture. Multiple components can create such a remainder even if each component separately has a simple equation of state.

Rewriting the perturbed energy-conservation equation yields ζ′=−ℋδpnad/(ρ+p) plus spatial-gradient terms. On scales k/ℋ much smaller than one, those terms can be neglected only for regular perturbations whose velocity/shear variables do not compensate the small gradients. With negligible nonadiabatic pressure and that regular large-scale limit, ζ is approximately constant. This energy-conservation argument is broader than one specific gravitational field equation, but still presupposes a meaningful background-plus-perturbation expansion.

The phrase superhorizon is commonly used here for k≪aH, although the Hubble radius is not identical to every particle or event horizon. A comoving curvature variable can coincide with ζ at leading large-scale order for regular adiabatic solutions under the relevant constraints; its sign depends on how that variable was defined. A non-attractor scalar background can make a comoving-curvature mode evolve, so one must not replace the stated conservation hypotheses with the phrase single field alone.

The heat lab illustrates the importance of a rate hierarchy rather than primordial dynamics. Reset and set mode n=1 and diffusivity κ=0.1, then Step twice to time 1; predict amplitude exp(−0.1). Reset and set n=1,κ=1, then Step twice; predict exp(−1). A slowly varying mode can look nearly constant over a short interval without an exact conservation law. This analytic periodic diffusion model has no expansion, density slicing, entropy pressure, or inflationary curvature equation.