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We study galaxy evolution from   z = 1  to 0 as a function of velocity dispersion σ for galaxies with  σ≳ 95 km s −1   based on the measured and Monte Carlo realized local velocity dispersion functions (VDFs) of galaxies and the revised statistical properties of 30 strongly lensed sources from the Cosmic Lens All‐Sky Survey, the PMN–NVSS Extragalactic Lens Survey and the  Hubble Space Telescope  Snapshot survey. We assume that the total (luminous plus dark) mass profile of a galaxy is isothermal in the optical region for  0 ≤ z ≤ 1  as suggested by mass modelling of lensing galaxies. This study is the first to investigate the evolution of the VDF shape as well as the overall number density. It is also the first to study the evolution of the total and the late‐type VDFs in addition to the early‐type VDF. For the evolutionary behaviours of the VDFs, we find that: (1) the number density of massive (mostly early‐type) galaxies with  σ≳ 200 km s −1   evolves differentially in the way that the number density evolution is greater at a higher velocity dispersion; (2) the number density of intermediate‐ and low‐mass early‐type galaxies  (95 km s −1 ≲σ≲ 200 km s −1 )  is nearly constant and (3) the late‐type VDF transformed from the Monte Carlo realized circular velocity function is consistent with no evolution in its shape or integrated number density consistent with galaxy survey results. These evolutionary behaviours of the VDFs are strikingly similar to those of the dark halo mass function (DMF) from  N ‐body simulations and the stellar mass function (SMF) predicted by recent semi‐analytic models of galaxy formation under the current Λ cold dark matter hierarchical structure formation paradigm. Interestingly, the VDF evolutions appear to be qualitatively different from ‘stellar‐mass‐downsizing’ evolutions obtained by many galaxy surveys. The co‐evolution of the DMF, the VDF and the SMF is investigated in quantitative detail based on up‐to‐date theoretical and observational results in a following paper. We consider several possible systematic errors for the lensing analysis and find that they are not likely to alter the conclusions.

monthly notices of the royal astronomical society

Galaxy evolution from strong‐lensing statistics: the differential evolution of the velocity dispersion function in concord with the Λ cold dark matter paradigm