In the Zel'dovich approximation, particle trajectories in a flat expanding universe are described by $\mathbf{r}(\mathbf{q}, t)=a(t)[\mathbf{q}+\mathbf{\Psi}(\mathbf{q}, t)]$, 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

$$\ddot{\mathbf{r}}=-\nabla \Phi-\frac{1}{\rho} \nabla P$$

where $\rho$ is the mass density, $P$ is the pressure $\left(P \ll \rho c^{2}\right)$ and $\Phi$ is the Newtonian potential which satisfies the Poisson equation $\nabla^{2} \Phi=4 \pi G \rho$.

(i) Show that the fractional density perturbation and the pressure gradient are given by

$$\delta \equiv \frac{\rho-\bar{\rho}}{\bar{\rho}} \approx-\nabla_{\mathbf{q}} \cdot \boldsymbol{\Psi}, \quad \nabla P \approx-\bar{\rho} \frac{c_{s}^{2}}{a} \nabla_{\mathbf{q}}^{2} \boldsymbol{\Psi}$$

where $\nabla_{\mathbf{q}}$ has components $\partial / \partial q_{i}, \bar{\rho}=\bar{\rho}(t)$ is the homogeneous background density and $c_{s}^{2} \equiv \partial P / \partial \rho$ is the sound speed. [You may assume that the Jacobian $\left|\partial r_{i} / \partial q_{j}\right|^{-1}=$ $\left|a \delta_{i j}+a \partial \psi_{i} / \partial q_{j}\right|^{-1} \approx a^{-3}\left(1-\nabla_{\mathbf{q}} \cdot \boldsymbol{\Psi}\right)$ for $\left.|\boldsymbol{\Psi}| \ll|\mathbf{q}| .\right]$

Use this result to integrate the Poisson equation once and obtain then the evolution equation for the comoving displacement:

$$\ddot{\boldsymbol{\Psi}}+2 \frac{\dot{a}}{a} \dot{\boldsymbol{\Psi}}-4 \pi G \bar{\rho} \boldsymbol{\Psi}-\frac{c_{s}^{2}}{a^{2}} \nabla_{\mathbf{q}}^{2} \boldsymbol{\Psi}=0$$

[You may assume that the integral of $\nabla^{2} \Phi=4 \pi G \bar{\rho}$ is $\nabla \Phi=4 \pi G \bar{\rho} \mathbf{r} / 3$, that $\boldsymbol{\Psi}$ is irrotational and that the Raychaudhuri equation is $\ddot{a} / a \approx-4 \pi G \bar{\rho} / 3$ for $P \ll \rho c^{2}$.]

Consider the Fourier expansion $\delta(\mathbf{x}, t)=\sum_{\mathbf{k}} \delta_{\mathbf{k}} \exp (i \mathbf{k} \cdot \mathbf{x})$ of the density perturbation using the comoving wavenumber $\mathbf{k}(k=|\mathbf{k}|)$ and obtain the evolution equation for the mode $\delta_{\mathbf{k}}$ :

$$\ddot{\delta}_{\mathbf{k}}+2 \frac{\dot{a}}{a} \dot{\delta}_{\mathbf{k}}-\left(4 \pi G \bar{\rho}-c_{s}^{2} k^{2} / a^{2}\right) \delta_{\mathbf{k}}=0$$

(ii) Consider a flat matter-dominated universe with $a(t)=\left(t / t_{0}\right)^{2 / 3}$ (background density $\left.\bar{\rho}=1 /\left(6 \pi G t^{2}\right)\right)$ and with an equation of state $P=\beta \rho^{4 / 3}$ to show that $(*)$ becomes

$$\ddot{\delta}_{\mathbf{k}}+\frac{4}{3 t} \dot{\delta}_{\mathbf{k}}-\frac{1}{t^{2}}\left(\frac{2}{3}-\bar{v}_{s}^{2} k^{2}\right) \delta_{\mathbf{k}}=0$$

where the constant $\bar{v}_{s}^{2} \equiv(4 \beta / 3)(6 \pi G)^{-1 / 3} t_{0}^{4 / 3}$. Seek power law solutions of the form $\delta_{\mathbf{k}} \propto t^{\alpha}$ to find the growing and decaying modes

$$\delta_{\mathbf{k}}=A_{\mathbf{k}} t^{n+}+B_{\mathbf{k}} t^{n-} \quad \text { where } \quad n_{\pm}=-\frac{1}{6} \pm\left[\left(\frac{5}{6}\right)^{2}-\bar{v}_{s}^{2} k^{2}\right]^{1 / 2} .$$