Structure Growth Factor

Structure Growth Factor

The large-scale structure we see today — galaxies, clusters, filaments, voids — grew from tiny quantum fluctuations seeded during inflation. Understanding how these perturbations grow is central to modern cosmology.

Linear Growth in Matter Domination

In the matter-dominated era, the density contrast $\delta = \delta\rho / \rho$ grows as the scale factor:

$D(a) = a$

This is the growing mode of the linear perturbation equation. It tells us that a perturbation that was $\delta = 10^{-5}$ at $a = 10^{-3}$ (redshift $z = 999$) grows to $\delta = 0.1$ today ($a = 1$).

Growth From Initial Conditions

For matter domination, the density contrast scales as:

$\delta(a) = \delta_0 \cdot \frac{a}{a_0} = \delta_0 \cdot \frac{a_{\rm final}}{a_{\rm initial}}$

The Growth Rate

The growth rate $f = d\ln D / d\ln a$ quantifies how fast structure grows. A useful approximation (Linder 2005) is:

$f \approx \Omega_m(a)^{0.55}$

For $\Omega_m = 0.3$ (today's value in ΛCDM), $f \approx 0.52$, meaning structure grows at about 52% of the Hubble rate.

Your Task

Implement three functions. All constants must be defined inside each function.

• growth_factor_matter_dom(a) — returns $D(a) = a$ (growing mode, normalized to 1 at $a=1$)
• delta_growth(delta0, a_initial, a_final) — returns $\delta_0 \cdot a_{\rm final} / a_{\rm initial}$
• growth_rate_approx(Omega_m_at_a) — returns $\Omega_m^{0.55}$ (Linder approximation)