Gaia DR3 in 6D: The search for fast hypervelocity stars and constraints on the Galactic Centre environment

The third data release (DR3) of the European Space Agency satellite Gaia provides coordinates, parallaxes, proper motions, and radial velocities for a sample of $\sim 34$ million stars. We use the combined 6-dimensional phase space information to search for hypervelocity stars (HVSs), unbound stars accelerated by dynamical processes happening in the Galactic Centre. By looking at the kinematics of Gaia DR3 stars in Galactocentric coordinates and by integrating their orbits in the Galactic potential, we do not identify any HVS candidates with a velocity higher than $700$ km s$^{-1}$ and robustly observed kinematics. Assuming a scenario wherein the interaction between a stellar binary and the massive black hole Sgr A$^*$ is responsible for HVS ejections from the Galactic Centre, we derive degenerate limits on the ejection rate of HVSs and the slope of the initial mass function of the primary star among binaries in the Galactic Centre. Our results indicate that the HVS ejection rate is $\lesssim 8\times10^{-5}$ yr$^{-1}$ assuming a Salpeter mass function, and this upper limit becomes progressively smaller for an increasingly top-heavy mass distribution. A fiducial HVS ejection rate of $10^{-4}$ yr$^{-1}$ prefers a mass function slope $\lesssim -2.35$, disfavouring previously claimed top-heavy initial mass functions among stars in the Galactic Centre.