Spectroscopic characterization of the known O-star population in Cygnus OB2 Evidence of multiple star-forming bursts

Context. Cygnus OB2 provides a unique insight into the high-mass stellar content in one of the largest groups of young massive stars in our Galaxy. Although several studies of its massive population have been carried out over the last decades, an extensive spectroscopic study of the whole known O-star population in the association is still lacking. Aims. We aim to carry out a spectroscopic characterization of all the currently known O stars in Cygnus OB2, determining the distribution of rotational velocities and accurate stellar parameters to obtain an improved view of the evolutionary status of the region. Methods. Based on existing and new optical spectroscopy, we performed a detailed quantitative spectroscopic analysis of all the known O-type stars identified in the association. For this purpose, we used the user-friendly iacob-broad and iacob-gbat automatized tools, FASTWIND stellar models, and astrometry provided by the Gaia second data release. Results. We created the most complete spectroscopic census of O stars carried out so far in Cygnus OB2 using already existing and new spectroscopy. We present the spectra for 78 O-type stars, from which we identify new binary systems, obtain the distribution of rotational velocities, and determine the main stellar parameters for all the stars in the region that have not been detected as double-line spectroscopic binaries. We also derive radii, luminosities, and masses for those stars with reliable Gaia astrometry, in addition to creating the Hertzsprung-Russell Diagram to interpret the evolutionary status of the association. Finally, we inspect the dynamical state of the population and identify runaway candidates. Conclusions. Our spectroscopic analysis of the O-star population in Cygnus OB2 has led to the discovery of two new binary systems and the determination of the main stellar parameters, including rotational velocities, luminosities, masses, and radii for all identified stars. This work has shown the improvement reached when using accurate spectroscopic parameters and astrometry for the interpretation of the evolutionary status of a population, revealing, in the case of Cygnus OB2, at least two star-forming bursts at similar to 3 and similar to 5 Myr. We find an apparent deficit of very fast rotators in the distribution of rotational velocities. The inspection of the dynamical distribution of the sample has allowed us to identify nine O stars with peculiar proper motions and discuss a possible dynamical ejection scenario or past supernova explosions in the region.