Joint inference of the Milky Way star formation history and IMF from Gaia all-sky $G < 13$ data

Despite the fundamental importance of the Milky Way's star formation history (SFH) and initial mass function (IMF), their consistent derivation remains elusive. We aim to simultaneously infer the IMF and the SFH of the Galactic disc comparing Gaia data with the mock catalog resulting from the Besan\c{c}on population synthesis model (BGM). Our goal is also to estimate the impact of the systematics present in current stellar evolutionary models (SEMs) on this inference. We use a new implementation of the BGM Fast Approximate Simulations (BGM FASt) framework to fit the seven million star Gaia DR3 all-sky $G<13$ color-magnitude diagram (CMD) to the most updated dynamically self-consistent BGM. Our derived SFH supports an abrupt decrease of the star formation approximately 1-1.5 Gyr ago followed by a significant enhancement with a wide plateau in the range 2-6 Gyr ago. A remarkable hiatus appears around 5-7 Gyr ago with a $\sim$1 Gyr shift depending on the set of stellar models. A complex evolution at ages older than 8 Gyr deserves further investigation. Precise but discrepant values using different SEMs are found for the power-law indices of the IMF. In our fiducial execution with PARSEC SEM, the slope takes a value of $\alpha_2 = 1.45^{+0.19}_{-0.12}$ for the range [0.5-1.53]$M_\odot$, while for masses larger than 1.53 $M_\odot$ we obtain $\alpha_3 = 1.98^{+0.13}_{-0.05}$. Using STAREVOL SEM, the inferred values are $\alpha_2 = 2.48^{+0.09}_{-0.11}$ and $\alpha_3 = 1.64^{+0.15}_{-0.02}$. We find the solution with PARSEC to have a significantly higher likelihood than that obtained with STAREVOL. The BGM FASt framework is now ready to address executions fitting all-sky Gaia data up to 14-17 apparent limiting magnitude. This will naturally allow us to derive both a reliable SFH for the early epochs of the Galactic disc evolution and a precise slope for the IMF at low masses.