 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,Thus the photon-baryon sound speed isBecause ${\rho }_r\propto a^{-4}$ and ${\rho }_b\propto a^{-3}$,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 isDuring radiation domination $\bar{\rho }\propto a^{-4}$, so ${\lambda }_{\mathrm{com}}\propto a$. During matter domination before recombination, $\bar{\rho }\propto a^{-3}$ and $c_s\propto a^{-1/2}$, so ${\lambda }_{\mathrm{com}}$ is approximately constant. Recombination causes a sharp downward jump in sound speed, after which ${\lambda }_{\mathrm{com}}\propto a^{-1/2}$. 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 $v_{\mathrm{rms}}$:For a thermally produced cold relic, identify four epochs:

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.