Maximal packing density of $n$-dimensional Euclidean space with equal balls

An estimate $\delta_n\le2^{-n(0,\!5237+o(1))}$ is obtained for the maximal packing density of $n$-dimensional Euclidean space with equal balls for $n\to\infty$. This result is based on an improvement in a corresponding upper estimate for the maximal packing density of the unit $(n-1)$-dimensional sphere with spherical caps of fixed angular radius.