  The second and third-named authors (arXiv:1705.04115) established a Central
Limit Theorem for the error term in the Sato-Tate law for families of modular
forms. This method was adapted to families of elliptic curves in by the first
and second-named authors (arXiv:1705.09229). In this context, a Central Limit
Theorem was established only under a strong hypothesis going beyond the Riemann
Hypothesis. In the present paper, we consider a smoothed version of the
Sato-Tate conjecture, which allows us to overcome several limitations. In
particular, for the smoothed version, we are able to establish a Central Limit
Theorem for much smaller families of modular forms, and we succeed in proving a
theorem of this type for families of elliptic curves under the Riemann
Hypothesis for $L$-functions associated to Hecke eigenforms for the full
modular group.
