Title: The statistical challenge of constraining the low-mass IMF in Local Group dwarf galaxies,
Abstract: We use Monte Carlo simulations to explore the statistical challenges of
constraining the characteristic mass ($m_c$) and width ($\sigma$) of a
lognormal sub-solar initial mass function (IMF) in Local Group dwarf galaxies
using direct star counts. For a typical Milky Way (MW) satellite ($M_{V} =
-8$), jointly constraining $m_c$ and $\sigma$ to a precision of $\lesssim 20\%$
requires that observations be complete to $\lesssim 0.2 M_{\odot}$, if the IMF
is similar to the MW IMF. A similar statistical precision can be obtained if
observations are only complete down to $0.4M_{\odot}$, but this requires
measurement of nearly 100$\times$ more stars, and thus, a significantly more
massive satellite ($M_{V} \sim -12$). In the absence of sufficiently deep data
to constrain the low-mass turnover, it is common practice to fit a
single-sloped power law to the low-mass IMF, or to fit $m_c$ for a lognormal
while holding $\sigma$ fixed. We show that the former approximation leads to
best-fit power law slopes that vary with the mass range observed and can
largely explain existing claims of low-mass IMF variations in MW satellites,
even if satellite galaxies have the same IMF as the MW. In addition, fixing
$\sigma$ during fitting leads to substantially underestimated uncertainties in
the recovered value of $m_c$ (by a factor of $\sim 4$ for typical
observations). If the IMFs of nearby dwarf galaxies are lognormal and do vary,
observations must reach down to $\sim m_c$ in order to robustly detect these
variations. The high-sensitivity, near-infrared capabilities of JWST and WFIRST
have the potential to dramatically improve constraints on the low-mass IMF. We
present an efficient observational strategy for using these facilities to
measure the IMFs of Local Group dwarf galaxies.