A new technique for the determination of the initial mass function in unresolved stellar populations

We present a new technique for the determination of the low-mass slope (${\alpha }_1$; $M_{\ast }<0.5M_{\odot }$) of the present day stellar mass function (PDMF) using the pixel space fitting of integrated light spectra. It can be used to constrain the initial mass function (IMF) of stellar systems with relaxation timescales exceeding the Hubble time and testing the IMF universality hypothesis. We provide two versions of the technique: (1) a fully unconstrained determination of the age, metallicity, and ${\alpha }_1$ and (2) a constrained fitting by imposing the externally determined mass-to-light ratio of the stellar population. We have tested our approach by Monte-Carlo simulations using mock spectra and conclude that: (a) age, metallicity and ${\alpha }_1$ can be precisely determined by applying the unconstrained version of the code to high signal-to-noise datasets (S/N=100, R=7000 yield $\Delta \alpha_1 \approx 0.1$);(b)the$M/L$ constraint significantly improves the precision and reduces the degeneracies, however its systematic errors will cause biased ${\alpha }_1$ estimates; © standard Lick indices cannot constrain the PDMF because they miss most of the mass function sensitive spectral features; (d) the ${\alpha }_1$ determination remains unaffected by the high-mass IMF shape (${\alpha }_3$; $M_{\ast }\ge 1M_{\odot }$) variation for stellar systems older than 8 Gyr, while the intermediate-mass IMF slope (${\alpha }_2$; $0.5 \le M_* < 1 M_{\odot}$)mayintroducebiasesintothebest-fitting$\alpha_1$valuesifitisdifferentfromthecanonicalvalue$\alpha_2 = 2.3$. We analysed observed intermediate resolution spectra of ultracompact dwarf galaxies with our technique and demonstrated its applicability to real data.