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 $\lesssim 0.2M_{\odot }$, 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.

Accepted to MNRAS with minor revisions since v1. 12 pages, 9 figures, plus appendices

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