 Solution

For small baryon fraction the CDM growing mode is ${\delta }_C\propto a$, and its gravity forcesFor an adiabatic cosmological perturbation to follow the unsuppressed growing mode, pressure must be below the gravitational forcing frequency: $c_s^2{k}^2\lesssim 4\pi G{a}^2{\bar{\rho }}_C$. With physical wavelength $\lambda =2\pi a/k$, this becomesThis is Jeans support versus forced baryon response. The wording “only grow” must refer to unsuppressed gravitational structure growth; it is not a theorem that all shorter-wavelength baryon perturbations are absent or constant. In the pressure-dominated regime a forced response ${\delta }_B\simeq (k_J/k{)}^2{\delta }_C$ remains. In particular, a falling sound speed can make this small response grow even before it catches up to CDM.

Before photon decoupling, the photon-baryon fluid has sound speed of order $c/\sqrt{3}$. Its Jeans scale is roughly a horizon scale, so smaller perturbations oscillate acoustically instead of freely following the growing CDM structures. After decoupling the baryon sound speed drops sharply. Above the new Jeans scale, pressure is negligible and subtraction of the CDM equation gives $({\delta }_B-{\delta }_C{)}^{\prime \prime }+\mathcal{H}({\delta }_B-{\delta }_C{)}^{\prime }=0$. For $a\propto {\tau }^2$, the difference is a constant plus a decaying $1/\tau$ term; its ratio to the growing CDM contrast tends to zero. Baryons therefore catch up. Under the stipulated post-decoupling $c_s\propto T\propto a^{-1}$, the physical Jeans length scales as $a^{1/2}$, the comoving scale as $a^{-1/2}$ and the corresponding Jeans mass as $a^{-3/2}$.

For the requested instantaneous mass estimates, define the baryonic Jeans mass as the baryon content of a sphere of radius ${\lambda }_J/2$:Using the supplied matter-era age gives $t_{\mathrm{dec}}=t_0(1+z_{\mathrm{dec}}{)}^{-3/2}\simeq 9.5\times 10^{13}$ s and ${\bar{\rho }}_C\simeq 1/(6\pi G{t}_{\mathrm{dec}}^2)\simeq 6\times 10^{-23}\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$. ThenThe drop is by $(\sqrt{3}\times 10^{-5}{)}^3$. A definition using the total gravitating mass instead of baryon content gives values about ten times larger. The pre-decoupling number is an acoustic Jeans order estimate, not an accurate application of the nonrelativistic equations from part a when $c_s\sim c$.