= Solution
In the <tight-coupling approximation>, the Thomson-scattering rate $\Gamma_T=-\tau'$ is much faster than both spatial streaming and cosmological evolution: $k/\Gamma_T\ll1$ and $\mathcal H/\Gamma_T\ll1$. Frequent scattering locks the photon and baryon velocities and suppresses higher angular multipoles. The <photon dipole> is leading order, while the <photon quadrupole> is first order in the mean-free-time expansion.
Use the quadrupole equation. Its time derivative is smaller than its collision term by an additional expansion parameter, and $\Theta_3$ starts one order beyond $\Theta_2$. The leading balance is therefore
$$
-\frac{2k}3\Theta_1=\frac9{10}\tau'\Theta_2.
$$
Thus \b[for the temperature-only collision operator supplied here],
$$
\boxed{\Theta_2=-\frac{20}{27}\frac k{\tau'}\Theta_1
=\frac{20}{27}\frac k{\Gamma_T}\Theta_1.}
$$
The <temperature-only tight-coupling quadrupole> is not numerically interchangeable with a hierarchy that includes polarization feedback. The latter changes the leading collision coefficient and gives the existing <photon quadrupole in tight coupling with polarization> instead. Here the stated temperature-only approximation is the one being expanded.
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