Gas depletion in primordial globular clusters due to accretion on to stellar-mass black holes

We consider the effect of compact stellar remnants on the interstellar medium of a massive star cluster following the initial burst of star formation. We argue that accretion on to stellar-mass black holes is an effective mechanism for rapid gas depletion in clusters of all masses, as long as they contain progenitor stars more massive than greater than or similar to 50 M-circle dot. This scenario appears especially attractive for the progenitor systems of present-day massive globular clusters which likely had masses above M greater than or similar to 10(7) M-circle dot. In such clusters, alternative mechanisms such as supernovae and stellar winds cannot provide a plausible explanation for the sudden removal of the primordial gas reservoir that is required to explain their complex chemical enrichment history. In order to consider different regimes in the rate of gas accretion on to stellar-mass black holes, we consider both the Bondi-Hoyle approximation as well as Eddington-limited accretion. For either model, our results show that the cluster gas can be significantly depleted within only a few tens of Myr. In addition, this process will affect the distribution of black hole masses and, by extension, may accelerate the dynamical decoupling of the black hole population and, ultimately, their dynamical ejection. Moreover, the time-scales for gas depletion are sufficiently short that the accreting black holes could significantly affect the chemistry of subsequent star formation episodes. The gas depletion times and final mass in black holes are not only sensitive to the assumed model for the accretion rate, but also to the initial mass of the most massive black hole which, in turn, is determined by the upper mass cut-off of the stellar initial mass function. Given that the mass function of 'dark' remnants is a crucial parameter for their dynamical ejection, our results imply that their accretion history can have an important bearing on the observed present-day cluster mass-to-light ratio. In particular, we show that the expected increase of the upper mass cut-off with decreasing metallicity could contribute to the observed anticorrelation between the mass-to-light ratio and the metallicity of globular clusters.