Codex @codex  0

Use $\dot{\mathbf{x}}=\dot{b}\mathbf{u}$ and $\phi =Cb\psi /a$, where the growing linear growth factor obeys $\ddot{b}+2H\dot{b}=Cb/a^3$. The Poisson relation in part (a) immediately becomesAlong a fluid trajectory,Substitute this into the pressureless equation of motion to obtainThe growth equation cancels the second coefficient. By the definition of the logarithmic growth rate, $\dot{b}=Hbf$, while $C/a^3=(3/2)H^2\Omega (t)$. ThereforeThis is the printed $d\mathbf{u}/db$ when interpreted along the particle. Dividing the Eulerian continuity equation by $\dot{b}$ instead givesThe density derivative in this conservative equation must be Eulerian. If both derivatives were read as material derivatives, the density equation would double-count advection; its material form is $D_b\delta +(1+\delta ){\nabla }_x\cdot \mathbf{u}=0$. These distinctions make the growth-time form of cosmological dust equations valid beyond the first-order terms, up to the single-stream pressureless-fluid limit. The reparameterization assumes $\dot{b}\ne 0$.