Watch density perturbations grow under gravity—see how dark energy slows structure formation
The linear growth factor D(z) describes how small density perturbations in the early universe grow over time to form the large-scale structure we see today—galaxies, clusters, and the cosmic web. In the linear regime (before gravitational collapse), overdensities δρ/ρ grow as D(z), where z is redshift. Understanding this growth is crucial for cosmology because it depends sensitively on the universe's matter and dark energy content.
The growth factor is normalized to D(0) = 1 today. At high redshift (early times), D(z) is smaller—structures were less developed. In a matter-dominated universe, D ∝ a = 1/(1+z), growing linearly with the scale factor. But dark energy changes this: once Λ dominates, expansion accelerates and structure growth slows or even halts. The suppression of growth in late times is a key signature of dark energy.
The growth rate f(z) = d ln D / d ln a measures how fast perturbations are growing at a given epoch. It's related to the matter density by the famous approximation f ≈ Ω_m(z)γ, where γ ≈ 0.55 for ΛCDM. This provides a powerful test: by measuring f(z) from galaxy surveys (via redshift-space distortions), we can test whether gravity follows General Relativity on cosmic scales or if modified gravity is needed.
Structure growth has distinct phases. During radiation domination (z > 3000), gravity can't overcome radiation pressure—growth is suppressed. In matter domination (3000 > z > 0.3), gravity wins and D(z) grows rapidly. Once dark energy dominates (z < 0.3), the accelerating expansion counteracts gravity and growth slows dramatically. This is why most structure formation happened in the past, not today.
In linear perturbation theory, the growth factor obeys a second-order ODE: D'' + (2 + d ln H / d ln a) D' - (3/2) Ωm(a) D = 0, where primes are derivatives with respect to ln a. This couples the growth to the Hubble expansion H(a), which itself depends on Ω_m, ΩΛ, and curvature. Solving this equation numerically gives D(z) for any cosmology. The growth rate is then f = d ln D / d ln a.
ΛCDM Universe: Set Ωm = 0.30, ΩΛ = 0.70 (the standard model). Notice how D(z) flattens at low z (recent times) as dark energy takes over. The growth rate f(z) drops sharply. Structure growth is effectively frozen today—our universe's large-scale structure is mostly set from earlier epochs.
Matter-Only Universe: Set Ωm = 1.0, ΩΛ = 0. Now D(z) ∝ a continues growing forever. The growth rate f ≈ 1 throughout. This was the favored model before 1998. But observations show real structure growth is suppressed in late times, confirming dark energy.
Open Universe: Set Ωm = 0.3, ΩΛ = 0.5 (Ω_total < 1). Negative curvature accelerates expansion even without Λ, suppressing growth. Compare the growth rate to the flat ΛCDM case. Open universes grow structure more slowly in late times.
High Dark Energy: Set Ω_Λ = 0.9, Ω_m = 0.1. Dark energy dominates early, so D(z) flattens at higher redshift. Structure barely grows after z ≈ 1. This extreme case shows how dark energy competes with gravity: too much Λ and you can't form galaxies at all!
Compare Models: Enable "Compare Models" to overlay multiple cosmologies. Notice how they converge at high z (where matter always dominates) but diverge at low z (where Λ matters). This divergence is what surveys like DESI and Euclid measure via galaxy clustering and weak lensing.
Growth vs Expansion: Switch between D(z) and f(z) plots. The growth factor D shows cumulative structure, while f shows the instantaneous rate. f(z) drops sharply when Λ dominates, signaling the transition from gravitational growth to accelerated expansion.
The linear growth factor is central to modern cosmology for several reasons:
Current surveys like DESI, Euclid, and the Vera Rubin Observatory are measuring f(z) to unprecedented precision. Any deviation from ΛCDM predictions would revolutionize cosmology—either revealing the nature of dark energy or forcing us to modify General Relativity itself. The linear growth factor, despite being "linear" and "simple," is one of the most powerful tools we have for understanding the dark universe.