I've been studying clustering dark energy when I came across a paper named "A Short Review on clustering dark energy" by Ronaldo Batista. there are 2 equations in this paper (eq.8 and eq.9) which I'm trying to derive:
$$ \begin{gather*} \delta^{\prime}+3 \mathcal{H}(\delta p / \delta \rho-w) \delta+(1+w)\left(\theta-3 \Phi^{\prime}\right)=0 \tag{8}\\ \end{gather*} $$
$$ \begin{gather*} \theta^{\prime}+\mathcal{H}\left(1-3 c_{a}^{2}\right) \theta=k^{2} \Phi+\frac{\delta p / \delta \rho k^{2} \delta}{1+w} \tag{9} \end{gather*} $$
I started with the conservation of stress-energy tensor (like what the paper suggested):
$$\nabla^{v} \delta T_{v}^{\mu}=0$$
but I don't know how to perturb velocity terms inside the stress-energy tensor (I've tried to perturb gamma but it looked messy). I know how to derive these equations for different types of matter (dark matter, baryons,...) but I don't know how can I derive this equation which can hold for anything with an equation of state $w$.
any help is appreciated.
