Formation times, mass growth histories and concentrations of dark matter haloes

We develop a simple model for estimating the mass growth histories of dark matter haloes. The model is based on a fit to the formation time distribution, where formation is defined as the earliest time that the main branch of the merger tree contains a fraction f of the final mass M. Our analysis exploits the fact that the median formation time as a function of f is the same as the median of the main progenitor mass distribution as a function of time. When coupled with previous work showing that the concentration c of the final halo is related to the formation time tf associated with f˜ 0.04, our approach provides a simple algorithm for estimating how the distribution of halo concentrations may be expected to depend on mass, redshift and the expansion history of the background cosmology. We also show that one can predict log10c with a precision of about 0.13 and 0.09 dex if only its mass or both mass and tf are known. Moreover, conversely, one can predict log10tf from mass or c with a precision of 0.12 and 0.09 dex, approximately independent of f. Adding the mass to the c-based estimate does not result in further improvement. These latter results may be useful for studies which seek to compare the age of the stars in the central galaxy in a halo with the time the core was first assembled.