The perturbed motion of cold dark matter particles (pressure-free, $P=0$ ) in an expanding universe can be parametrized by the trajectories

$$\mathbf{r}(\mathbf{q}, t)=a(t)[\mathbf{q}+\boldsymbol{\psi}(\mathbf{q}, t)]$$

where $a(t)$ is the scale factor of the universe, $\mathbf{q}$ is the unperturbed comoving trajectory and $\boldsymbol{\psi}$ is the comoving displacement. The particle equation of motion is $\ddot{\mathbf{r}}=-\nabla \Phi$, where the Newtonian potential satisfies the Poisson equation $\nabla^{2} \Phi=4 \pi G \rho$ with mass density $\rho(\mathbf{r}, t)$.

(a) Discuss how matter conservation in a small volume $d^{3} \mathbf{r}$ ensures that the perturbed density $\rho(\mathbf{r}, t)$ and the unperturbed background density $\bar{\rho}(t)$ are related by

$$\rho(\mathbf{r}, t) d^{3} \mathbf{r}=\bar{\rho}(t) a^{3}(t) d^{3} \mathbf{q}$$

By changing co-ordinates with the Jacobian

$$\left|\partial r_{i} / \partial q_{j}\right|^{-1}=\left|a \delta_{i j}+a \partial \psi_{i} / \partial q_{j}\right|^{-1} \approx a^{-3}\left(1-\nabla_{q} \cdot \psi\right),$$

show that the fractional density perturbation $\delta(\mathbf{q}, t)$ can be written to leading order as

$$\delta \equiv \frac{\rho-\bar{\rho}}{\bar{\rho}}=-\nabla_{q} \cdot \psi,$$

where $\nabla_{q} \cdot \boldsymbol{\psi}=\sum_{i} \partial \psi_{i} / \partial q_{i}$.

Use this result to integrate the Poisson equation once. Hence, express the particle equation of motion in terms of the comoving displacement as

$$\ddot{\boldsymbol{\psi}}+2 \frac{\dot{a}}{a} \dot{\boldsymbol{\psi}}-4 \pi G \bar{\rho} \boldsymbol{\psi}=0$$

Infer that the density perturbation evolution equation is

$$\ddot{\delta}+2 \frac{\dot{a}}{a} \dot{\delta}-4 \pi G \bar{\rho} \delta=0$$

[Hint: You may assume that the integral of $\nabla^{2} \Phi=4 \pi G \bar{\rho}$ is $\nabla \Phi=-4 \pi G \bar{\rho} \mathbf{r} / 3$. Note also that the Raychaudhuri equation (for $P=0$ ) is $\ddot{a} / a=-4 \pi G \bar{\rho} / 3 .$.]

(b) Find the general solution of equation $(*)$ in a flat $(k=0)$ universe dominated by cold dark matter $(P=0)$. Discuss the effect of late-time $\Lambda$ or dark energy domination on the growth of density perturbations.