Linear Perturbation Growth

Cosmology · 1946

Equation: δ̈ + 2H·δ̇ = 4πG·ρ̄·δ
Field: Cosmology
Named After: Evgeny Lifshitz
Year: 1946

Symbols

δ = δρ/ρ̄ the density contrast, how much denser a patch is than average · δ̇, δ̈ its time derivatives · H the Hubble parameter · ρ̄ the mean matter density · G gravitational constant

Significance

How a nearly smooth universe turns lumpy. Gravity on the right amplifies any overdensity; the 2Hδ̇ term on the left is Hubble friction, expansion dragging against collapse. The contest is why structure grows as a power law (δ ∝ a in matter domination) rather than running away exponentially as it would in static space. It also forces dark matter: the microwave background shows δ ≈ 10⁻⁵ at recombination, and ordinary matter starting from there cannot reach today's galaxies in the time available — something that had been growing since long before, unimpeded by radiation pressure, must have dug the wells first.