= Solution
Insert the result for the <neutrino tensor anisotropic stress> into the tensor wave equation and divide out $m_{ij}e^{i\mathbf k\cdot\mathbf x}/\sqrt2$. Since the <Fourier transform> replaces $-\nabla^2$ by $k^2$, the source coefficient is $-64\pi Ga^2\bar\rho_\nu$. The flat <Friedmann equation> gives
$$
\mathcal H^2=\left(\frac{\dot a}{a}\right)^2
=\frac{8\pi G}{3}a^2\bar\rho_{\rm tot},
$$
so $64\pi Ga^2\bar\rho_\nu=24\mathcal H^2\bar\rho_\nu/\bar\rho_{\rm tot}$. Hence
$$
\boxed{\ddot h^{(\pm2)}+2\mathcal H\dot h^{(\pm2)}+k^2h^{(\pm2)}
=-24\mathcal H^2\frac{\bar\rho_\nu}{\bar\rho_{\rm tot}}
\int_0^\eta d\eta'\,\dot h^{(\pm2)}(\eta')
\frac{j_2(k(\eta-\eta'))}{[k(\eta-\eta')]^2}}.
$$
This memory term transfers coherent tensor motion into directional neutrino perturbations. \b[Free-streaming neutrinos damp the tensor oscillation amplitude after horizon entry], suppressing the gravitational-wave contribution to cosmic microwave background temperature and polarization spectra. The kernel acts on $\dot h$, so a constant superhorizon tensor mode is unaffected at leading order.
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