= Solution
The <density contrast> is $\delta(\mathbf x,t)=[\rho(\mathbf x,t)-\rho_0(t)]/\rho_0(t)$, and $\delta_k(t)$ is one spatial <Fourier mode> with comoving wavevector $\mathbf k$. The <scale factor> is $a(t)$, $H=\dot a/a$ is the <Hubble parameter>, $c_s$ is the physical <sound speed>, $\rho_0$ the mean matter density, and $G$ the <gravitational constant>. For each fixed $k$ the displayed equation is an ordinary differential equation in cosmic time. It compares <pressure> restoration, self-gravity and expansion-induced damping:
$$
\ddot\delta_k+2H\dot\delta_k+\left(\frac{c_s^2k^2}{a^2}-4\pi G\rho_0\right)\delta_k=0.
$$
The <pressure> term resists compression; self-gravity enhances an overdensity; the $2H\dot\delta$ term slows the motion relative to a static background. The Newtonian treatment is appropriate to small perturbations on subhorizon scales within the specified fluid model, rather than an arbitrary gauge-independent treatment of superhorizon perturbations.
The physical <Jeans length> follows by equating <pressure> and gravity:
$$
\boxed{k_{J,\rm phys}=\frac{\sqrt{4\pi G\rho_0}}{c_s},\qquad
\lambda_{J,\rm phys}=\frac{2\pi}{k_{J,\rm phys}}=c_s\sqrt{\frac\pi{G\rho_0}}.}
$$
The comoving Jeans wavenumber is $a k_{J,\rm phys}$. Perturbations well below the physical Jeans length are <pressure>-supported acoustic oscillations whose amplitude is modified by expansion; those above it admit gravitational growth when the background evolution permits. Pressureless matter in a matter-dominated universe is the example below. Merely having a large wavelength is not sufficient to apply this Newtonian equation outside the horizon.
For a gas of particle <mass> $m$ and <temperature> $T$, write $c_s^2=\Gamma k_BT/m$, with $\Gamma=1$ for the isothermal sound-speed convention or the appropriate adiabatic heat-capacity ratio otherwise. Then
$$
\boxed{\lambda_{J,\rm phys}=\left(\frac{\pi\Gamma k_BT}{G\rho_0m}\right)^{1/2}.}
$$
At fixed density it is larger for hotter or lighter particles, and it decreases with increasing density. Numerical coefficients differ between sound-crossing and self-gravitating-sphere conventions; the expression here uses the Fourier instability boundary of the stated equation.
In an <Einstein-de Sitter universe>, the spatially flat <Friedmann-Lemaître-Robertson-Walker metric> is $ds^2=-c^2dt^2+a^2(t)d\mathbf x^2$. Conservation of pressureless matter gives $\rho_0a^3=\mathrm{constant}$; the <Friedmann equation> then gives $a\propto t^{2/3}$, $H=2/(3t)$ and $\rho_0=1/(6\pi Gt^2)$. With $c_s=0$,
$$
\ddot\delta+\frac4{3t}\dot\delta-\frac2{3t^2}\delta=0.
$$
Trying $\delta=t^p$ gives $p^2+p/3-2/3=(p-2/3)(p+1)=0$. Hence the <Einstein-de Sitter density-growth modes> are
$$
\boxed{\delta=C_+t^{2/3}+C_-t^{-1}=D_+a+D_-a^{-3/2}.}
$$
The growing mode increases only as a power law because the expansion dilutes and slows the collapsing matter.
<Conformal time> is defined by $d\eta=dt/a(t)$, so the metric becomes $ds^2=a^2(-c^2d\eta^2+d\mathbf x^2)$. Choosing its origin at the big bang gives $\eta\propto t^{1/3}$ and $a\propto\eta^2$. Therefore
$$
\boxed{\delta=A_+\eta^2+A_-\eta^{-3}.}
$$
Equivalently the equation is $\delta''+2\delta'/\eta-6\delta/\eta^2=0$, where primes denote conformal-time derivatives.
Before and through <cosmological recombination>, <photons> diffuse between hotter and cooler parts of the tightly coupled <photon>–baryon fluid. Radiative viscosity and heat transport erase short-wavelength acoustic perturbations. As free <electrons> disappear, the <photon> mean free path grows, enhancing this diffusion before decoupling. <Silk damping> can be represented by a factor approximately $e^{-k^2/k_D^2}$ in the linear acoustic amplitude. The comoving Silk wavelength may be defined as $\lambda_D=2\pi/k_D$; the associated diffusion-length convention is $r_D=1/k_D$.
A random-walk estimate is $r_D^2\sim\int c\lambda_\gamma\,dt/a^2$ up to transport and baryon-loading factors, with physical $\lambda_\gamma=(n_e\sigma_T)^{-1}$. Scales much shorter than the damping wavelength are strongly suppressed. This is <photon> diffusion, not the Jeans boundary and not erasure of every cold-dark-matter perturbation. After decoupling, <photon> free streaming is a distinct part of the evolution.
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