Contribution of a decoupled real scalar to Neff
= Contribution of a decoupled real scalar to Neff
{title2=$\Delta N_{\rm eff}$}
If a real relativistic scalar decouples when the other coupled particles have entropy degrees of freedom $g_{*s}(T_d)$, then
$$
\Delta N_{\rm eff}=\frac47
\left(\frac{43}{4g_{*s}(T_d)}\right)^{4/3}.
$$
Earlier decoupling and the subsequent annihilation of more coupled species make the scalar colder and suppress its contribution.