  We study an acceleration phenomenon arising in monostable
integro-differential equations with a weak Allee effect. Previous works have
shown its occurrence and have given correct upper bounds on the rate of
expansion in some particular cases, but precise lower bounds were still
missing. In this paper, we provide a sharp lower bound for this acceleration
rate, valid for a large class of dispersion operators. Our results manage to
cover fractional Laplace operators and standard convolutions in a unified way,
which is new in the literature. A first very important result of the paper is a
general flattening estimate of independent interest: this phenomenon appears
regularly in acceleration situations, but getting quantitative estimates is
most of the time open. This estimate at hand, we construct a very subtle
sub-solution that captures the expected dynamics of the accelerating solution
(rates of expansion and flattening) and identifies several various regimes that
appear in the dynamics depending on the parameters of the problem.
