= Primordial helium response to the neutron-proton mass difference
{title2=$\delta\log Y_p\simeq-\delta Q/[T_f(1+e^{-Q/T_f})]$}
At fixed <cosmological weak freeze-out> temperature $T_f$ and fixed neutron survival factor, <neutron-proton chemical equilibrium> gives $r_f=e^{-Q/T_f}$ and <neutron-limited helium synthesis> gives $Y_p\propto2r_f/(1+r_f)$. Differentiating yields the displayed sensitivity. A finite change $Q\mapsto(1+\alpha)Q$ gives $Y_p'/Y_p=(1+e^{Q/T_f})/(1+e^{(1+\alpha)Q/T_f})$. Increasing the positive mass difference reduces helium production. Changes to the weak conversion rates, neutron lifetime or nuclear history must be supplied separately for a full abundance prediction.
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