Do the majority of stars form as gravitationally unbound?

Some of the youngest stars (age $\lesssim 10$ Myr) are clustered, while many others are observed scattered throughout star forming regions or in complete isolation. It has been intensively debated whether the scattered or isolated stars originate in star clusters, or if they form truly isolated, which could help constrain the possibilities how massive stars are formed. We adopt the assumption that all stars form in gravitationally bound star clusters embedded in molecular cloud cores ($\Gamma$-$1\;$ model), which expel their natal gas, and compare the fraction of stars found in clusters with observational data. The star clusters are modelled by the code nbody6, which includes stellar and circumbinary evolution, gas expulsion, and the external gravitational field of their host galaxy. We find that small changes in the assumptions in the current theoretical model estimating the fraction, $\Gamma$, of stars forming in embedded clusters have a large influence on the results, and we present a counterexample as an illustration. This calls into question theoretical arguments about $\Gamma$ in embedded clusters, and it suggests that there is no firm theoretical ground for low $\Gamma$ in galaxies with lower star formation rates (SFRs). Instead, the assumption that all stars form in embedded clusters is in agreement with observational data for the youngest stars (age $\lesssim 10$ Myr). In the $\Gamma$-$1\;$ scenario, the observed fraction of the youngest stars in clusters increases with the SFR only weakly; the increase is caused by the presence of more massive clusters in galaxies with higher SFRs, which release fewer stars to the field in proportion to their mass. The $\Gamma$-$1\;$ model yields a higher fraction of stars in clusters for older stars (age between $10$ and $300$ Myr) than what is observed. This discrepancy can be caused by interactions with molecular clouds.
Comment: 16 pages, 9 figures, accepted to A&A