  As a prospective resolution of the Hubble tension, early dark energy (EDE)
suffers from the coincidence problem, why EDE is active just at
matter-radiation equality (equivalently why the slope of EDE potential is
required to approximately equal to the Hubble parameter at that time). In this
paper we present a dark-matter-trapped EDE mechanism, by which the bound on the
slope of EDE potential can be relieved. We show how this mechanism can work,
and discuss the possibility that after inflation ended EDE can settle down at
the initial conditions required by observations.
