A Parametric Galactic Model toward the Galactic Bulge Based on Gaia and Microlensing Data

We developed a parametric Galactic model toward the Galactic bulge by fitting to spatial distributions of the Gaia DR2 disk velocity, VVV proper motion, BRAVA radial velocity, OGLE-III red clump star count, and OGLE-IV star count and microlens rate, optimized for use in microlensing studies. We include the asymmetric drift of Galactic disk stars and the dependence of velocity dispersion on Galactic location in the kinematic model, which has been ignored in most previous models used for microlensing studies. We show that our model predicts a microlensing parameter distribution significantly different from those typically used in previous studies. We estimate various fundamental model parameters for our Galaxy through our modeling, including the initial mass function (IMF) in the inner Galaxy. Combined constraints from star counts and the microlensing event timescale distribution from the OGLE-IV survey, in addition to a prior on the bulge stellar mass, enable us to successfully measure IMF slopes using a broken power-law form over a broad mass range, $\alpha_{\rm bd}=0.22^{+0.20}_{-0.55}$ for $M<0.08\,M_{\odot}$, $\alpha_{\rm ms} = 1.16^{+0.08}_{-0.15}$ for $0.08\,M_{\odot}\leq\,M<M_{\rm br}$, and $\alpha_{\rm hm} = 2.32^{+0.14}_{-0.10}$ for $M\geq\,M_{\rm br}$, as well as a break mass at $M_{\rm br} = 0.90^{+0.05}_{-0.14} \, M_{\odot}$. This is significantly different from the Kroupa IMF for local stars, but similar to the Zoccali IMF measured from a bulge luminosity function. We also estimate the dark matter mass fraction in the bulge region of $28\pm7$\% which could be larger than a previous estimate. Because our model is purely parametric, it can be universally applied using the parameters provided in this paper.