Generalized Transport Equation for the Autocovariance Function of the Density Field and Mass Invariant in Star-forming Clouds

In this Letter, we study the evolution of the autocovariance function of density-field fluctuations in star-forming clouds and thus of the correlation length lc(ρ) of these fluctuations, which can be identified as the average size of the most correlated structures within the cloud. Generalizing the transport equation derived by Chandrasekhar for static, homogeneous turbulence, we show that the mass contained within these structures is an invariant, i.e., that the average mass contained in the most correlated structures remains constant during the evolution of the cloud, whatever dominates the global dynamics (gravity or turbulence). We show that the growing impact of gravity on the turbulent flow yields an increase of the variance of the density fluctuations and thus a drastic decrease of the correlation length. Theoretical relations are successfully compared to numerical simulations. This picture brings a robust support to star formation paradigms where the mass concentration in turbulent star-forming clouds evolves from initially large, weakly correlated filamentary structures to smaller, denser, more correlated ones, and eventually to small, tightly correlated, prestellar cores. We stress that the present results rely on a pure statistical approach of density fluctuations and do not involve any specific condition for the formation of prestellar cores. Interestingly enough, we show that, under average conditions typical of Milky-Way molecular clouds, this invariant average mass is about a solar mass, providing an appealing explanation for the apparent universality of the IMF in such environments.