  A weak-rate library aimed at investigating the sensitivity of astrophysical
environments to variations of electron-capture rates on medium-heavy nuclei has
been developed. With this library, the sensitivity of the core-collapse and
early post-bounce phases of core-collapse supernovae to nuclear
electron-capture is examined by systematically and statistically varying
electron-capture rates of individual nuclei. The rates are adjusted by factors
consistent with uncertainties indicated by comparing theoretical rates to those
deduced from charge-exchange and $\beta$-decay measurements. To ensure a model
independent assessment, sensitivity studies across a comprehensive set of
progenitors and equations of state are performed. In our systematic study, we
find a +16/-4 % range in the mass of the inner-core at the time of shock
formation and a $\pm$20% range of peak {\nu}e-luminosity during the
deleptonization burst. These ranges are each five times as large as those seen
from a separate progenitor study in which we evaluated the sensitivity of these
parameters to 32 presupernova stellar models. It is also found that the
simulations are more sensitive to a reduction in the electron-capture rates
than an enhancement, and in particular to the reduction in the rates for
neutron-rich nuclei near the N = 50 closed neutron-shell. As measurements for
medium-heavy (A > 65) and neutron-rich nuclei are sparse, and because accurate
theoretical models which account for nuclear structure considerations on the
individual nucleus level are not readily available, rates for these nuclei may
be overestimated. If more accurate estimates confirm this, results from this
study indicate that significant changes to the core-collapse trajectory can be
expected. For this reason, experimental and theoretical efforts should focus in
this region of the nuclear chart.
