= Solution
Before recombination, tight coupling makes photons and baryons share an adiabatic perturbation. Since $\delta\rho_b/\rho_b=(3/4)\delta\rho_r/\rho_r$ and only radiation supplies appreciable pressure,
$$
\delta P=\frac{c^2}{3}\delta\rho_r,\qquad
\delta\rho=\delta\rho_r\left(1+\frac{3\rho_b}{4\rho_r}\right).
$$
Thus the <photon-baryon sound speed> is
$$
\boxed{c_s=\frac c{\sqrt3}\left(1+\frac{3\rho_b}{4\rho_r}\right)^{-1/2}}.
$$
Because $\rho_r\propto a^{-4}$ and $\rho_b\propto a^{-3}$,
$$
c_s\propto
\begin{cases}
a^0,&t<t_{\rm eq},\\
a^{-1/2},&t_{\rm eq}<t<t_{\rm rec}.
\end{cases}
$$
After <cosmological recombination>, baryons decouple from radiation; for adiabatically cooling nonrelativistic gas, $T_b\propto a^{-2}$ and $c_s\propto a^{-1}$.
The <Comoving Jeans length> is
$$
\lambda_{\rm com}=\frac{c_s}{a}\sqrt{\frac{\pi}{G\bar\rho}}.
$$
During radiation domination $\bar\rho\propto a^{-4}$, so $\lambda_{\rm com}\propto a$. During matter domination before recombination, $\bar\rho\propto a^{-3}$ and $c_s\propto a^{-1/2}$, so $\lambda_{\rm com}$ is approximately constant. Recombination causes a sharp downward jump in sound speed, after which $\lambda_{\rm com}\propto a^{-1/2}$. The requested log-log sketch therefore rises with slope one, reaches a plateau after equality, drops at recombination, and then declines with slope $-1/2$.
For collisionless particles, replace $c_s$ by their one-dimensional velocity dispersion $v_{\rm rms}$:
$$
\boxed{\lambda_{\rm com}^{\rm coll}
=\frac{v_{\rm rms}}a\sqrt{\frac{\pi}{G\bar\rho}}}.
$$
For a thermally produced cold relic, identify four epochs:
* while coupled and relativistic, $v_{\rm rms}\simeq c$ and $\lambda_{\rm com}\propto a$ in radiation domination;
* after decoupling but while still relativistic, momentum redshifts but speed remains near $c$, so the same $\lambda_{\rm com}\propto a$ scaling continues with collisionless free streaming;
* after becoming nonrelativistic but before equality, $v_{\rm rms}\propto a^{-1}$ and $\bar\rho\propto a^{-4}$, giving $\lambda_{\rm com}\propto a^0$;
* after equality, $v_{\rm rms}\propto a^{-1}$ and $\bar\rho\propto a^{-3}$, giving $\lambda_{\rm com}\propto a^{-1/2}$.
Its plot rises through the two relativistic epochs, turns onto a plateau at the nonrelativistic transition, and falls after equality. Recombination does not dynamically affect collisionless dark matter.
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