  Presenting population models of high-mass X-ray binaries (HMXBs) formed after
bursts of star formation, we investigate the effect of electron-capture
supernovae (ECS) of massive ONeMg white dwarfs and the hypothesis that ECS
events are associated with typically low supernova kicks imparted to the
nascent neutron stars. We identify an interesting ECS bump in the time
evolution of HMXB numbers; this bump is caused by significantly increased
production of wind-fed HMXBs 20-60 Myr post starburst. The amplitude and age
extent of the ECS bump depend on the strength of ECS kicks and the mass range
of ECS progenitors. We also find that ECS-HMXBs form through a specific
evolutionary channel that is expected to lead to binaries with Be donors in
wide orbits. These characteristics, along with their sensitivity to ECS
properties, provide us with an intriguing opportunity to probe ECS physics and
progenitors through studies of starbursts of different ages. Specifically, the
case of the Small Magellanic Cloud, with a significant observed population of
Be HMXBs and starburst activity 30-60 Myr ago, arises as a promising laboratory
for understanding the role of electron-capture supernovae in neutron star
formation.
