= 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> is
$$
\boxed{\nabla_x^2\varphi=4\pi Ga^2\bar\rho\,\delta=\frac Ca\delta.}
$$
For the comoving velocity field $\mathbf w=\dot{\mathbf x}$, local <mass conservation> gives
$$
\boxed{\partial_t\delta+\nabla_x\cdot[(1+\delta)\mathbf w]=0.}
$$
One 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 $a\mathbf w$.
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