BGM FASt: Besanc{c}on Galaxy Model for Big Data. Simultaneous inference of the IMF, SFH and density in the Solar Neighbourhood

We develop a new theoretical framework to generate Besan\c{c}on Galaxy Model fast approximate simulations (BGM FASt) to address fundamental questions of the Galactic structure and evolution performing multi-parameter inference. As a first application of our strategy we simultaneously infer the IMF, the star formation history and the stellar mass density in the Solar Neighbourhood. The BGM FASt strategy is based on a reweighing scheme, that uses a specific pre-sampled simulation, and on the assumption that the distribution function of the generated stars in the Galaxy can be described by an analytical expression. To validate BGM FASt we execute a set of tests. Finally, we use BGM FASt with an approximate Bayesian computation algorithm to obtain the posterior PDF of the inferred parameters, by comparing synthetic versus Tycho-2 colour-magnitude diagrams. Results: The validation shows a very good agreement between BGM FASt and the standard BGM, with BGM FASt being $\approx 10^4$ times faster. By analysing Tycho-2 data we obtain a thin disc star formation history decreasing in time and a present rate of $1.2 \pm 0.2 M_\odot/yr$. The resulting total stellar mass density in the Solar Neighbourhood is $0.051_{-0.005}^{+0.002} M_\odot/pc^3$ and the local dark matter density is $0.012 \pm 0.001 M_\odot/pc^3$. For the composite IMF we obtain a slope of $\alpha_2={2.1}_{-0.3}^{+0.1}$ in the mass range between $0.5 M_\odot$ and $1.53M_\odot$. The results of the slope at the high mass range are trustable up to $4M_\odot$ and highly depend on the choice of the extinction map (obtaining $\alpha_3={2.9}_{-0.2}^{+0.2}$ and $\alpha_3={3.7}_{-0.2}^{+0.2}$ respectively, for two different extinction maps). Systematic uncertainties are not included. Conclusions: The good performance of BGM FASt demonstrates that it is a very valuable tool to perform multi-parameter inference using Gaia data releases.