= Solution
Put $H=2/(3t)$ and let $f_B=\rho_B/\bar\rho$, $f_D=1-f_B$ be constant background matter fractions. Define the total matter perturbation $S=f_B\Delta_B+f_D\Delta_D$ and the relative mode $D=\Delta_D-\Delta_B$. Adding with the background weights and subtracting the two given equations respectively gives
$$
\ddot S+\frac4{3t}\dot S-\frac2{3t^2}S=0,\qquad\ddot D+\frac4{3t}\dot D=0.
$$
The <total and relative density modes of two pressureless matter fluids> are consequently
$$
\boxed{S=A t^{2/3}+B t^{-1},\qquad D=C+E t^{-1/3},\qquad\Delta_B=S-f_DD,\quad\Delta_D=S+f_BD.}
$$
These are the general consequences of the supplied equations; a single ratio additionally needs initial-mode assumptions.
For the usual dark-matter-dominated growing-mode approximation, take $f_B\ll1$, neglect the decaying modes, and set the relative-velocity mode $E=0$. Then $\Delta_D\propto t^{2/3}\propto R$. With $\Delta_B(t_r)=0$ at recombination, $C=\Delta_D(t_r)$, so the <baryon catch-up approximation after recombination> is
$$
\boxed{\Delta_B(t)=\Delta_D(t)\left[1-\frac{R(t_r)}{R(t)}\right]=\Delta_D(t)\left[1-\frac{1+z}{1+z_0}\right].}
$$
At redshifts $z,z_0\gg1$, replacing $1+z$ by $z$ gives the printed $\Delta_B\simeq\Delta_D(1-z/z_0)$. \b[The printed relation is a high-redshift approximation with a specified relative mode, not an exact identity for arbitrary solutions of the two-fluid equations.]
For example, retaining a finite baryon fraction with a growing total mode and constant relative mode gives, with $x=R/R(t_r)$,
$$
\frac{\Delta_B}{\Delta_D}=\frac{f_D(x-1)}{f_Dx+f_B}.
$$
Alternatively, if initially both $\Delta_B=0$ and $\dot\Delta_B=0$ while the dark-matter growing mode is already present, then in the dark-matter-dominated limit the relative transient is not zero and
$$
\frac{\Delta_B}{\Delta_D}=1-3x^{-1}+2x^{-3/2}.
$$
These different solutions show why the initial-velocity convention matters. \b[The quoted catch-up law is the dark-matter-dominated branch with the relative decaying mode neglected], together with the high-redshift approximation stated above. The general solution retains the transient and a finite baryon fraction, so the additional assumptions needed for that special case remain explicit.
The physical conclusion is robust: after decoupling, baryonic <gas> can fall into pre-existing dark-matter potential wells and its fractional <density> perturbation catches up. The absolute constant offset need not vanish; it becomes small relative to the growing common mode. The pressureless approximation applies on scales above the baryonic Jeans scale and while perturbations remain linear; subsequent collapse requires nonlinear treatment.<Cosmology by David Tong>[Sections3.1–3.3: matter growth, decoupling, Jeans support and dark-matter seeds]{ref}
\Image[/past-exam-of-the-formation-of-structure-in-the-universe-course-of-the-university-of-cambridge/2015/paper-1-baryon-catch-up.png]
{title=Baryons approach the dark-matter density contrast, with transient behaviour set by the initial relative velocity}
{source=paper-1-baryon-catch-up.py}
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