= Radiation-era adiabatic potentials with free-streaming neutrinos
{title2=$\phi=(1+2f_\nu/5)\psi$}
For regular <adiabatic initial conditions> on <superhorizon scales> during <radiation domination>, the unweighted <neutrino Boltzmann hierarchy> gives $\Theta_1=\psi k\eta/2$ and $\Theta_2=\psi(k\eta)^2/6$. With $f_\nu=\bar\rho_\nu/(\bar\rho_\nu+\bar\rho_\gamma)$ and neutrinos the only source of <scalar anisotropic stress>, the trace-free <Einstein field equations> gives $\phi=(1+2f_\nu/5)\psi$. For the sign convention $\mathcal R=-\phi-\psi/2$,
$$
\psi=-\frac{10}{15+4f_\nu}\mathcal R,\qquad\phi=-\frac{10+4f_\nu}{15+4f_\nu}\mathcal R.
$$
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