Carry an initial fluctuation through linear matter evolution

For weak scalar potentials, nonrelativistic pressureless matter, negligible velocity dispersion and scalar anisotropic stress, and wavelengths well inside the Hubble radius, combine linear mass conservation, Euler acceleration, and the Poisson equation. In cosmic time the density growth obeys a second-order equation balancing expansion drag against gravity. Here Φ is the lapse potential in the preceding conformal Newtonian metric. With the Fourier sign fixed earlier, an overdensity corresponds to negative Φ in the usual isolated-well convention because −k²Φ is positive.

This Newtonian sub-Hubble density contrast agrees with an appropriate relativistic density variable only within the stated approximation. Near or outside the Hubble scale, use the relativistic constraint and evolution system and specify the slicing; an arbitrary gauge density cannot be inserted into the small-scale Poisson rule. Pressure, radiation driving, modified gravity, or free-streaming components can make growth scale dependent, so the simplest common growth factor should not be imposed before checking those effects.

In an Einstein–de Sitter background, matter dominates with negligible curvature and cosmological constant. The independent density solutions scale as a and a⁻³ᐟ². A pure growing mode has logarithmic growth rate f=1, but an arbitrary initial density value does not select that branch; its initial velocity or time derivative is also needed. A decaying admixture can change an early inferred growth rate without modifying gravity or the primordial spectrum.

For the growing branch, background matter density falls as a⁻³ while δ rises as a. The combination a²ρmδ sourcing the potential is therefore constant at fixed k. A nearly constant Φ is perfectly compatible with growing density contrast. This statement uses matter domination and the relevant mode branch; it need not hold through radiation–matter transition, dark-energy domination, or significant anisotropic stress. Confusing density growth with potential growth can give the wrong sign and time dependence in an observational transfer calculation.

When the physics allows separation, write a late density mode as D(a)T(k) times a defined seed mode. Specify the reference epoch where D=1 and the long-wavelength normalization of T, with the seed chosen consistently. Matter–radiation equality, photon–baryon acoustics, and free streaming leave characteristic scale dependence; they are not generated by the late pressureless growth equation alone. Galaxy counts additionally involve bias, velocities, selection, and projection rather than equaling matter density directly.

The heat lab illustrates scale-dependent attenuation only. Reset and set mode 1,diffusivity0.5, then Step twice to time 1; predict amplitude exp(−0.5). Reset and set mode 2,diffusivity0.5, then Step twice; predict exp(−2). The higher wave number is more strongly suppressed because this model’s rate is κn². It is analytic periodic diffusion, not a radiation-era transfer function, gravitational instability calculation, or matter-power inference. Its lab time is not the cosmological scale factor.