  An Early Enrichment Population (EEP) has been theorized to produce the
observed metallicity in the intracluster medium (ICM) of galaxy clusters. This
population likely existed at high redshifts (z$\sim$10), relics of which we
posit exist today as dwarf galaxies. Previous work argues that the initial mass
function (IMF) of the EEP must be flatter than those found at lower redshifts,
but with considerable uncertainties. In this work, we present a more
quantitative model for the EEP and demonstrate how observational constraints
can be applied to the IMF using supernova Type Ia (SNIa) rates, delay time
distribution (DTD), and the luminosity function (LF) of galaxy clusters. We
determine best-fit values for the slope and mass break of the IMF by comparing
IMFs from literature with observed DTDs and the low-luminosity component
($M(R)< -12$) of the Coma LF. We derive two best-fit IMFs: (1) $\alpha_{lo} =
-0.13 \pm 0.24$ for $0.07 < M/M_\odot < 1.75$ and $\alpha_{hi} = 0.53 \pm 0.01$
for $1.75 < M/M_\odot < 150$, and (2) $\alpha_{lo} = 1.06 \pm 0.11$ for $0.07 <
M/M_\odot < 6$ and $\alpha_{hi} = 0.53 \pm 0.01$ for $6 < M/M_\odot < 150$. We
also compare these with sl-5 from Loewenstein (2013) with $\alpha=0.5$ for
$0.07 < M/M_\odot < 8$ and $\alpha=0.3$ for $8 < M/M_\odot < 150$. This EEP
model, along with stars formed at later times, can produce the observed
metallicity, is consistent with other observations, and predicts a significant
rise in the SNIa rate at increasing redshift.
