In the <tight-coupling approximation>, finiteness of the photon Euler equation as $\Gamma\to\infty$ requires
$$
v_b=-3\Theta_1+O(\Gamma^{-1}).
$$
Let $S=3\Theta_1+v_b$. Rearranging the baryon Euler equation and substituting the leading relation only on its right-hand side gives
$$
S=-\frac R\Gamma(v_b'+\mathcal Hv_b+k\Psi)
=\frac R\Gamma(3\Theta_1'+3\mathcal H\Theta_1-k\Psi)
+O(\Gamma^{-2}).
$$
Insert this slip into the photon Euler equation:
$$
3(1+R)\Theta_1'+3\mathcal HR\Theta_1
=k\Theta_0+k(1+R)\Psi.
$$
The photon continuity equation gives $\Theta_1=-(\Theta_0'-\Phi')/k$. Eliminating the dipole yields
$$
\boxed{
\Theta_0''+\frac{\mathcal HR}{1+R}\Theta_0'
+c_s^2k^2\Theta_0
=-\frac{k^2}{3}\Psi+\Phi''
+\frac{\mathcal HR}{1+R}\Phi'},
$$
where the <photon-baryon sound speed> is
$$
\boxed{c_s^2=\frac1{3(1+R)}}.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 312 / 4 / b / Solution

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The tight-coupling approximation expands the photon-baryon Boltzmann equations in powers of $k/\Gamma$ and $\mathcal H/\Gamma$, where $\Gamma$ is the Thomson scattering rate.


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 Incoming links: Tight-coupling approximation