The statistical challenge of constraining the low-mass IMF in local group dwarf galaxies

We use Monte Carlo simulations to explore the statistical challenges of constraining the characteristic mass (m_c) and width (σ) 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 mc and s to a precision of < 20 per cent requires that observations be complete to < 0.2M⊙, 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⊙, but this requires measurement of nearly 100× more stars, and thus, a significantly more massive satellite (MV ~ -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 mc for a lognormal while holding s fixed. We show that the former approximation leads to best-fitting power-law slopes that vary with the mass range observed and can largely explain existing claims of low-mass IMF variations inMWsatellites, even if satellite galaxies have the same IMF as the MW. In addition, fixing s during fitting leads to substantially underestimated uncertainties in the recovered value of m_c (by a factor of ~4 for typical observations). If the IMFs of nearby dwarf galaxies are lognormal and do vary, observations must reach down to ~mc in order to robustly detect these variations. The high-sensitivity, near-infrared capabilities of the James Webb Space Telescope and Wide-Field Infrared Survey Telescope 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.