On the basis of a new convergence study of high-resolution N-body simulations , my colleagues and I now agree that the Navarro , Frenk , & White ( 1996 ) density profile \rho _ { NFW } ( r ) \propto r ^ { -1 } ( r + r _ { s } ) ^ { -2 } is a good representation of typical dark matter halos of galactic mass .
Comparing simulations of the same halo with numbers of particles ranging from \sim 10 ^ { 3 } to \sim 10 ^ { 6 } , we have also shown that r _ { s } , the radius where the log-slope is -2 , can be determined accurately for halos with as few as \sim 10 ^ { 3 } particles .
Based on a study of thousands of halos at many redshifts in an Adaptive Refinement Tree ( ART ) simulation of a cosmological volume in a \Lambda CDM cosmology , we have found that the concentration c _ { vir } \equiv R _ { vir } / r _ { s } has a log-normal distribution , with 1 \sigma \Delta ( \log c _ { vir } ) = 0.18 at a given mass , corresponding to a scatter in maximum rotation velocities of \Delta V _ { max } / V _ { max } = 0.12 .
The average concentration declines with redshift at fixed mass as c _ { vir } ( z ) \propto ( 1 + z ) ^ { -1 } .
This may have important implications for galaxy rotation curves .
Finally , we have found that the velocity function determined from galaxy luminosity functions plus luminosity-velocity relations agrees with the predictions from our \Lambda CDM simulations .
But we also note that the very limited evolution with redshift of the velocity function predicted by \Lambda CDM conflicts with the data that is becoming available on the number density of bright galaxies unless there is significant evolution of the luminosity-velocity relation at z > 1 .