It is known that in high dimensions, the measure of the spherical cap is small, due to the measure concentration for the sphere. In particular, we have the following inequality in $n$ dimension: $$ 1-\mu(A_t)\leq 2e^{-t^2n/2} $$ where $A$ is any set with measure at least 1/2 and $A_t$ is its $t$-neighborhood. For the case of spherical caps, $A$ can be a hemisphere. Thus we have a small upper bound on the measure of spherical caps.
My question is, is there a lower bound for the measure of spherical caps? In particular I would like to know how small $t$ should be, such that $1-\mu(A_t)$ is at least a constant fraction of the measure of the whole sphere? Thanks.