How do galaxies populate dark matter haloes

For any assumed standard stellar initial mass function, the Sloan Digital Sky Survey (SDSS) gives a precise determination of the abundance of galaxies as a function of their stellar mass over the full stellar mass range 108 M(circle dot) < M(*) < 1012 M(circle dot). Within the concordance Lambda cold dark matter (Lambda CDM) cosmology, the Millennium Simulations give precise halo abundances as a function of mass and redshift for all haloes within which galaxies can form. Under the plausible hypothesis that the stellar mass of a galaxy is an increasing function of the maximum mass ever attained by its halo, these results combine to give halo mass as a function of stellar mass. The result agrees quite well with observational estimates of mean halo mass as a function of stellar mass from stacking analyses of the gravitational lensing signal and the satellite dynamics of SDSS galaxies. For M(*) similar to 5.5 x 1010 M(circle dot), the stellar mass usually assumed for the Milky Way (MW), the implied halo mass is similar to 2 x 1012 M(circle dot), consistent with most recent direct estimates and inferences from the MW/M31 timing argument. The fraction of the baryons associated with each halo which are present as stars in its central galaxy reaches a maximum of 20 per cent at masses somewhat below that of the MW and falls rapidly at both higher and lower masses. These conversion efficiencies are lower than in almost all recent high-resolution simulations of galaxy formation, showing that these are not yet viable models for the formation of typical members of the galaxy population. When inserted in the Millennium-II Simulation, our derived relation between stellar mass and halo mass predicts a stellar mass autocorrelation function in excellent agreement with that measured directly in the SDSS. The implied Tully-Fisher relation also appears consistent with observation, suggesting that galaxy luminosity functions and Tully-Fisher relations can be reproduced simultaneously in a Lambda CDM cosmology.