= Solution
{c}
Write $\mathbf x$ for a <comoving coordinate>, $\mathbf u$ for the <peculiar velocity>, $H=\dot a/a$ for the <Hubble parameter>, and $\delta=\delta\rho/\bar\rho$ for the <density contrast>. For a homogeneous self-gravitating fluid, the linearized <continuity equation>, <Euler equations for an inviscid fluid>, and <cosmological Poisson equation> are
$$
\dot\delta=-\frac1a\nabla\cdot\mathbf u,
\qquad
\dot{\mathbf u}+H\mathbf u=-\frac{c_s^2}{a}\nabla\delta-\frac1a\nabla\phi,
\qquad
\nabla^2\phi=4\pi G a^2\bar\rho\delta.
$$
Here $c_s$ is the physical <adiabatic sound speed>, gradients refer to <comoving coordinates>, and the homogeneous background force has already been subtracted. Differentiating the first equation and eliminating $\mathbf u$ and $\phi$ gives
$$
\ddot\delta+2H\dot\delta-\frac{c_s^2}{a^2}\nabla^2\delta-4\pi G\bar\rho\delta=0.
$$
A <Fourier mode> with comoving <wavenumber> $k$ therefore obeys
$$
\ddot\delta_k+2H\dot\delta_k+
\left(\frac{c_s^2k^2}{a^2}-4\pi G\bar\rho\right)\delta_k=0.
$$
The <Jeans wavenumber> occurs where the coefficient in parentheses vanishes. Since a comoving <wavelength> is $2\pi/k$, the result is
$$
\boxed{k_J=\frac{a\sqrt{4\pi G\bar\rho}}{c_s},\qquad
\lambda_J^{\rm com}=\frac{c_s}{a}\sqrt{\frac{\pi}{G\bar\rho}}.}
$$
Below this <comoving Jeans length>, pressure restores a displaced fluid element: there are <acoustic waves> rather than growing <Jeans instability>. In a static background they oscillate; expansion changes their amplitude and frequency, and dissipation can damp them. Pressure support alone does not erase them. In the pre-recombination <photon-baryon fluid>, <Silk damping> supplies a separate erasure mechanism.
For collisionless matter, replace the <sound speed> by a characteristic <velocity dispersion>. For an isotropic <Maxwell-Boltzmann velocity distribution>, defining $\sigma^2=\langle u_x^2\rangle=\langle|\mathbf u|^2\rangle/3$, the static collisionless marginal-stability calculation gives
$$
\boxed{\lambda_{J,\rm collisionless}^{\rm com}
=\frac{\sigma}{a}\sqrt{\frac{\pi}{G\bar\rho}}
=\frac{v_{\rm rms}}a\sqrt{\frac{\pi}{3G\bar\rho}}.}
$$
The coefficient depends on the distribution and the convention for <velocity dispersion>. Small-scale suppression is now <collisionless free streaming> and <phase mixing>, rather than collisional <acoustic waves>. An instantaneous <collisionless Jeans length> must be distinguished from the accumulated comoving distance $\int v\,dt/a$ traveled since decoupling.
For the requested thermal histories, an <Einstein-de Sitter universe> describes the matter-era limit: it cannot literally also have a radiation-dominated epoch. Interpret the question as a spatially flat universe with negligible <cosmological constant>, passing from <radiation domination>, $a\propto t^{1/2}$, to <matter domination>, $a\propto t^{2/3}$. For the usual schematic dynamical Jeans scale use $\bar\rho_{\rm tot}$ in the estimate, so
$$
\lambda_{\rm dyn}^{\rm com}\propto
\begin{cases}
c_s a,&t\ll t_{\rm eq},\quad\bar\rho_{\rm tot}\propto a^{-4},\\
c_s a^{1/2},&t\gg t_{\rm eq},\quad\bar\rho_{\rm tot}\propto a^{-3}.
\end{cases}
$$
Near the <particle horizon> a relativistic perturbation calculation replaces the Newtonian derivation; the scaling estimate still identifies the relevant sound-crossing or streaming scale.
Before <cosmological recombination>, <adiabatic initial conditions> keep the entropy per <baryon> fixed in the tightly coupled <photon-baryon fluid>. With $R_b=3\bar\rho_b/(4\bar\rho_\gamma)\propto a$, its <photon-baryon sound speed> is
$$
c_s^2=\frac{c^2}{3(1+R_b)}.
$$
For example, this follows from $\delta\rho_\gamma/\bar\rho_\gamma=(4/3)\delta\rho_b/\bar\rho_b$ and $\delta P=c^2\delta\rho_\gamma/3$. In the radiation-dominated limit $R_b\ll1$, $c_s\simeq c/\sqrt3$; in the strongly baryon-loaded matter-era limit $R_b\gg1$, $c_s\propto a^{-1/2}$. Thus the <baryon Jeans length across recombination> has the asymptotic history
$$
\boxed{\lambda_{J,b}^{\rm com}\propto
\begin{cases}
a\propto t^{1/2},&t\ll t_{\rm eq},\\
\text{approximately constant},&t_{\rm eq}\ll t<t_{\rm rec},\\
a^{-1/2}\propto t^{-1/3},&t>t_{\rm rec}\text{ with adiabatic gas cooling}.
\end{cases}}
$$
The transition to the plateau is smooth: retaining $1+R_b$ gives $\lambda_{J,b}^{\rm com}\propto a^{1/2}/\sqrt{1+R_b}$ in <matter domination>. If baryon loading is still small, this intermediate segment instead rises as $a^{1/2}$ until loading becomes important.
At <photon decoupling>, radiation ceases to provide pressure support to the <baryons>, so the <sound speed> drops from the coupled-fluid value to that of a nonrelativistic <ideal gas>, $c_{s,b}^2=\gamma k_BT_b/(\mu m_p)$, with $\gamma=5/3$. The drop in <comoving Jeans length> at fixed density is the same ratio of <sound speeds>, typically several orders of magnitude. Subsequently an <isentropic process> gives $T_b\propto\bar\rho_b^{\gamma-1}\propto a^{-2}$, so $c_{s,b}\propto a^{-1}$ and $\lambda_{J,b}^{\rm com}\propto a^{-1/2}$. Residual energy exchange through <Compton scattering> can initially keep $T_b\propto a^{-1}$, producing a short approximately flat gas segment before the decreasing segment. The idealized sketch assumes immediate adiabatic cooling:
``
log(comoving baryon Jeans scale)
^
| __________________
| / |
| / | loss of photon pressure
| / |
| / +---\
| / \
+-------------------|-----------------|----------> log(time)
t_eq t_rec
slope 1/2 slope ~0 slope -1/3
``
For the <cold-dark-matter kinetic Jeans scale>, take the stated toy history to mean that collisions maintain a particle <temperature> equal to the photon <temperature> until <kinetic decoupling>. This is an assumption about thermal coupling; a negligible collision rate after decoupling makes the subsequent treatment collisionless. Assume the ordering $t_{\rm NR}<t_{\rm dec}<t_{\rm eq}<t_{\rm rec}$. While relativistic, the characteristic particle speed is constant; while nonrelativistic but thermally coupled, $\sigma\propto\sqrt T\propto a^{-1/2}$; after <kinetic decoupling>, particle momentum redshifts as $a^{-1}$ and $\sigma\propto a^{-1}$. Combining these with the background dynamical density gives
$$
\boxed{\lambda_{\rm dyn,CDM}^{\rm com}\propto
\begin{cases}
a\propto t^{1/2},&t<t_{\rm NR},\\
a^{1/2}\propto t^{1/4},&t_{\rm NR}<t<t_{\rm dec},\\
\text{constant},&t_{\rm dec}<t<t_{\rm eq},\\
a^{-1/2}\propto t^{-1/3},&t>t_{\rm eq}.
\end{cases}}
$$
There is no sudden change at <cosmological recombination>: <cold dark matter> has already decoupled. Its small initial <velocity dispersion> makes its suppression scale much smaller than the pre-recombination <baryon> scale.
``
log(comoving CDM dynamical Jeans scale)
^
| _______________
| ___/ \
| ___/ \ no jump
| / \ at t_rec
| / \
+-----------|------------|--------------|----|------> log(time)
t_NR t_dec t_eq t_rec
slope 1/2 slope 1/4 slope 0 slope -1/3
``
The density convention matters. If one instead defines a species' self-gravitating <collisionless Jeans length> using only its own nonrelativistic density $\bar\rho_m\propto a^{-3}$, its thermally coupled segment is constant and its decoupled segment decreases as $a^{-1/2}$ even during <radiation domination>. The constant radiation-era segment in the second sketch is the usual instantaneous background dynamical streaming estimate, not an exact single-species self-gravity threshold. A complete multicomponent calculation uses the separate perturbation equations and their combined <cosmological Poisson equation>.
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