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For an adiabatic cosmological perturbation, the baryon and photon fractional density perturbations obey ${\delta }_b=3{\delta }_{\gamma }/4$. Since the pressure of the photon-baryon fluid is supplied by the photons,The photon-baryon sound speed is therefore

Use ${\bar{\rho }}_b\propto a^{-3}$ and ${\bar{\rho }}_{\gamma }\propto a^{-4}$ in the Comoving Jeans lengthWell before matter-radiation equality, photon inertia dominates, $c_s\simeq c/\sqrt{3}$, andOnce baryon loading dominates while tight coupling still holds, $c_s\propto a^{-1/2}$, so ${\lambda }_{J,\mathrm{com}}$ is approximately constant. At cosmological recombination, photon pressure support disappears and the baryonic Jeans scale drops sharply. For a subsequently adiabatic monatomic gas, $T_b\propto a^{-2}$ and $c_s\propto a^{-1}$, givingin an Einstein-de Sitter universe. The requested graph therefore rises as $t^{1/4}$, flattens before recombination, jumps downward there, and then decreases as $t^{-1/3}$.

For collisionless matter, the same instantaneous estimate uses its one-dimensional velocity dispersion ${\sigma }_v$ instead of $c_s$; more precisely, suppression is described by collisionless free streaming. ThusWhile the particles are relativistic, ${\sigma }_v\simeq c$ and ${\lambda }_{J,\mathrm{com}}\propto a^{1/2}$. Once nonrelativistic but thermally coupled to radiation, $T_{\mathrm{cdm}}\propto a^{-1}$, so ${\sigma }_v\propto a^{-1/2}$ and the scale is constant. After kinetic decoupling, momentum redshifts as $a^{-1}$, so ${\sigma }_v\propto a^{-1}$ and ${\lambda }_{J,\mathrm{com}}\propto a^{-1/2}$. The second graph joins these three power laws at $t_{\mathrm{NR}}$ and $t_{\mathrm{dec}}$; unlike the baryonic graph, its final decline begins at dark-matter kinetic decoupling rather than recombination.

Let $k_{\mathrm{eq}}$ be the horizon scale at matter-radiation equality. At a fixed time soon after recombination, the cold-dark-matter transfer function defined as $T_c={\Delta }_c/\mathcal{R}$ has the schematic behaviorwith a smooth turnover near $k_{\mathrm{eq}}$. Large-scale modes enter the horizon during matter domination and the Poisson equation converts an almost scale-independent primordial potential into a density contrast proportional to $k^2$. Small-scale modes enter during radiation domination; radiation controls the potential and pressure prevents it from clustering, so cold-dark-matter growth is only logarithmic until equality. Subsequent matter-era growth multiplies all these modes by the same linear growth factor.

Equivalently, if the conventional matter transfer function is normalized to one as $k\to 0$, it is constant for $k\ll k_{\mathrm{eq}}$ and falls approximately as $\log (k/k_{\mathrm{eq}})/k^2$ for $k\gg k_{\mathrm{eq}}$. Multiplication by the Poisson factor $k^2$ gives the behavior of the definition used here.