 Solution

Let $\delta =(\rho -\bar{\rho })/\bar{\rho }$ be the density contrast, let $\mathbf{x}$ be a comoving coordinate, and use $a_0=1$. Define $C=3H_0^2{\Omega }_0/2$, so $4\pi G\bar{\rho }=C/a^3$ for the pressureless background. The perturbation Poisson equation isFor the comoving velocity field $\mathbf{w}=\dot{\mathbf{x}}$, local mass conservation givesOne obtains this by subtracting $\dot{\bar{\rho }}+3H\bar{\rho }=0$ from ${\partial }_t\rho +3H\rho +{\nabla }_x\cdot (\rho \mathbf{w})=0$. In the linear regime it reduces to ${\partial }_t\delta =-{\nabla }_x\cdot \mathbf{w}$, the linearized cosmological continuity equation. The spatial derivatives are with respect to $\mathbf{x}$, and the time derivative in the conservative continuity equation is at fixed $\mathbf{x}$. In contrast the acceleration of an individual fluid element is a material derivative. The physical peculiar velocity is $a\mathbf{w}$.