= Solution
\b[Growing and decaying modes.] Neglect <pressure> and insert the Einstein-de Sitter background. A trial $\Delta\propto t^p$ gives
$$
p(p-1)+\frac43p-\frac23=0,\qquad p=\frac23,-1.
$$
Therefore
$$
\boxed{\Delta=A t^{2/3}+B t^{-1},\qquad
\Delta_{\rm growing}\propto R\propto(1+z)^{-1}}.
$$
The quoted growth law selects the growing mode; arbitrary initial data can also contain the decaying one. In a static homogeneous Jeans model, $\ddot\Delta=(4\pi G\bar\rho-k^2c_s^2)\Delta$, so an unstable mode grows exponentially, $e^{\sqrt{4\pi G\bar\rho-k^2c_s^2}\,t}$. Expansion introduces the $2H\dot\Delta$ damping term and a falling background <density>, yielding power-law growth instead. If the <wavevector> in the supplied <pressure> term is comoving, that term would need $R^{-2}$; the pressureless result does not depend on this convention.
\b[Amplitude problem.] Using the stated <recombination> epoch, matter-era growth from then until today is only about $1501$. A baryonic <density> perturbation of order $10^{-5}$ would reach merely $1.5\times10^{-2}$, far below the order-unity regime needed for nonlinear collapse. <photon> coupling also prevents <baryons> from freely collapsing before <recombination>. The observed microwave <temperature> contrast is not an exact direct measurement of <baryon> <density contrast> at every scale: acoustic transfer and gravitational <redshift> matter. The estimate expresses the insufficient growth of tiny photon-coupled baryonic seeds, not an exact conversion between the two observables.
\b[Coupled gravitational response.] Both pressureless components respond to the same perturbed potential. The equations are
$$
\boxed{\begin{aligned}
\ddot\Delta_B+2H\dot\Delta_B&=4\pi G(\bar\rho_B\Delta_B+\bar\rho_D\Delta_D),\\
\ddot\Delta_D+2H\dot\Delta_D&=4\pi G(\bar\rho_B\Delta_B+\bar\rho_D\Delta_D).
\end{aligned}}
$$
Their difference obeys $\ddot D+2H\dot D=0$, where $D=\Delta_D-\Delta_B$. Thus $\dot D\propto R^{-2}$ and, in Einstein-de Sitter, $D=C_0+C_1t^{-1/3}=C_0+\widetilde C_1R^{-1/2}$.
\b[Catch-up branch and its condition.] With <dark matter> dominating the source and in its growing mode, write $\Delta_D=AR$. The full <baryon> solution is
$$
\boxed{\Delta_B=AR-C_0-\widetilde C_1R^{-1/2}}.
$$
The requested <baryon catch-up approximation after recombination> selects the branch with negligible relative decaying mode, $\widetilde C_1=0$. Defining $C_0=AR(z_0)$ then gives
$$
\boxed{\frac{\Delta_B}{\Delta_D}=1-\frac{R(z_0)}{R(z)}
=1-\frac{1+z}{1+z_0}}.
$$
This branch applies at and after $z_0$ and has $\Delta_B(z_0)=0$ and matched <baryon>/dark-matter growth velocities at that epoch. It does not follow uniquely from vanishing initial <baryon> <density contrast> alone. If both $\Delta_B$ and its initial time <derivative> vanish at $R=R(z_0)$, the <zero-initial-baryon-velocity catch-up transient> instead gives
$$
\frac{\Delta_B}{\Delta_D}=1-3x^{-1}+2x^{-3/2},\qquad x=R/R(z_0).
$$
Both approach unity for $x\gg1$. The displayed target therefore needs its stated no-relative-decaying-mode assumption; a finite <baryon> fraction modifies the transient further.
\b[Physical consequence.] <cold dark matter> is not held to the radiation by <Thomson scattering>, and can build gravitational wells before <baryon> decoupling, with appreciable matter-era growth after equality. Its growing perturbations then attract <baryons> once <pressure> and radiation drag become unimportant. The relative offset stays constant or decays while the common mode grows, so <baryons> catch up. \b[Nonlinear structure need not grow solely from the tiny baryonic perturbation inferred at decoupling.] <cold dark matter> growth during <radiation domination> is slower than the matter-era law, and the stated formula is neither a radiation-era nor a nonlinear-collapse solution.
\Image[/past-exam-of-the-formation-of-structure-in-the-universe-course-of-the-university-of-cambridge/2013/paper-1-disk-stability-baryon-catchup.png]
{title=Original local gas-disk stability curves and comparison of constant-offset and zero-initial-velocity baryon catch-up branches}
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