 Radiation-era adiabatic potentials with free-streaming neutrinos

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 $\varphi =(1+2f_{\nu }/5)\psi$. For the sign convention $\mathcal{R}=-\varphi -\psi /2$,