In a flat expanding universe with scale factor $a(t)$, average mass density $\bar{\rho}$ and average pressure $\bar{P} \ll \bar{\rho} c^{2}$, the fractional density perturbations $\delta_{k}(t)$ at co-moving wavenumber $k$ satisfy the equation

$$\ddot{\delta}_{k}=-2\left(\frac{\dot{a}}{a}\right) \dot{\delta}_{k}+4 \pi G \bar{\rho} \delta_{k}-\frac{c_{s}^{2} k^{2}}{a^{2}} \delta_{k}$$

Discuss briefly the meaning of each term on the right hand side of this equation. What is the Jeans length $\lambda_{J}$, and what is its significance? How is it related to the Jeans mass?

How does the equation $(*)$ simplify at $\lambda \gg \lambda_{J}$ in a flat universe? Use your result to show that density perturbations can grow. For a growing density perturbation, how does $\dot{\delta} / \delta$ compare to the inverse Hubble time?

Explain qualitatively why structure only forms after decoupling, and why cold dark matter is needed for structure formation.