Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 71 / 5

Put $\mathcal{H}=a^{\prime }/a$ and choose the residual synchronous frame comoving with cold dark matter, so ${\mathbf{v}}_C=0$. Its Euler equation preserves this choice. The continuity equation then gives ${\delta }_C^{\prime }=h^{\prime }/2$. Neglect background baryon pressure and all unenhanced corrections of order $c_s^2$, but retain $c_s^2{k}^2$ because a short wavelength can make it important. The trace equation becomesUsing $h^{\prime }=2{\delta }_C^{\prime }$ provesFor baryons let ${\theta }_B=i\mathbf{k}\cdot {\mathbf{v}}_B$. The retained continuity and Euler terms are ${\delta }_B^{\prime }=-{\theta }_B+h^{\prime }/2$ and ${\theta }_B^{\prime }+\mathcal{H}{\theta }_B-c_s^2{k}^2{\delta }_B=0$. Differentiate the first and substitute the second:HenceThe pressure sign is stabilizing. The drag terms contain first derivatives of the density perturbations, as printed in the PDF; those primes are lost in the converted TeX. The approximation assumes slowly varying nonrelativistic background baryon pressure and the growing physical mode, not a residual synchronous gauge mode.

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 $c_s\sim c$.