Track relative composition as an independent perturbation

For separately conserved fluids define ζi=−ψ−ℋδρi/ρi′ when each background derivative is nonzero. A common time shift cancels in each component combination. Adiabaticity means the ratios δρi/ρi′ agree, so the component curvatures agree. Equal fractional density contrasts are generally not the adiabatic condition because different equations of state dilute at different rates. A matter component and radiation provide the simplest concrete example.

Let c denote pressureless matter and γ photons, with no energy exchange in this illustration. Their relative entropy is Scγ=3(ζc−ζγ)=δc−3δγ/4. The ψ terms cancel, leaving a gauge-invariant composition disturbance. A compensated perturbation can have zero total density disturbance in a chosen flat slice while retaining nonzero relative entropy. That cancellation at one time does not imply the relative composition is unphysical or that total curvature will remain zero through later evolution.

The total uniform-density curvature is a weighted sum of component curvatures with weights proportional to their background density derivatives. For separately conserved components these are enthalpy weights, proportional to ρi+pi. Weighting simply by density would be wrong when radiation pressure matters. Even if the individual barotropic ζi remain constant on regular large scales, their evolving weights can change total ζ when they are unequal. This is one explicit way composition sources curvature.

During two-field inflation, perturbations can be resolved along the background field trajectory and perpendicular to it. A turn can couple the perpendicular entropy displacement to curvature outside the Hubble scale. Whether the entropy mode is light, decays, or survives into relative species abundances depends on the potential, kinetic geometry, and reheating couplings. A heavy field can sometimes be integrated out, but only within a controlled effective regime; the mere field count neither guarantees observable isocurvature nor justifies the canonical single-clock tensor ratio.

A schematic large-scale transfer matrix can summarize linear evolution after defining an entropy variable and its normalization. Its curvature-source coefficient measures conversion; it is not fixed universally by the horizon-exit amplitude. Correlated initial modes require their cross power when computing final curvature power. Observational species entropy can also vanish after conversion even though entropy affected the final curvature. Thus absence of a surviving isocurvature component would not by itself establish that the primordial evolution involved only one dynamical field.

The matrix lab illustrates coupling without an area change. Reset and set diagonal scale 1,shear 0; predict final-matrix determinant 1 and singular values (1,1). Reset and set scale 1,shear 1; predict determinant 1 and singular values ((√5+1)/2,(√5−1)/2). At reset the displayed geometry is identity; Play or Step interpolates it, while these readouts always describe selected final A. This positive triangular 2D family is not a multifield perturbation solver or a physical entropy-transfer calculation.