Extensive analyses of statistical strong gravitational lensing are performed based on the final Cosmic Lens All‐Sky Survey (CLASS) well‐defined statistical sample of flat‐spectrum radio sources and current estimates of galaxy luminosity functions per morphological type. The analyses are carried out under the assumption that galactic lenses are well‐approximated by singular isothermal ellipsoids and early‐type galaxies evolved passively since redshift z∼ 1. Two goals of the analyses are: (i) to constrain cosmological parameters independently of other techniques (e.g. Type Ia supernovae magnitude‐redshift relation, cosmic microwave background anisotropies, galaxy matter power spectra); and (ii) to constrain the characteristic line‐of‐sight velocity dispersion and the mean projected mass ellipticity for the early‐type galaxy population. Depending on how the late‐type galaxy population is treated (i.e. whether its characteristic velocity dispersion is constrained or not), we find for a flat universe with a classical cosmological constant that the matter fraction of the present critical density Ωm= 0.31+0.27−0.14 (68 per cent) for the unconstrained case or 0.40+0.28−0.16 (68 per cent) for the constrained case, with an additional systematic uncertainty of ≈0.11 arising from the present uncertainty in the distribution of CLASS sources in redshift and flux density. For a flat universe with a constant equation of state for dark energy w=px(pressure)/ρx(energy density) and the prior constraint w≥−1, we find that −1 ≤w < −0.55+0.18−0.11 (68 per cent) for the unconstrained case or −1 ≤w < −0.41+0.28−0.16 (68 per cent) for the constrained case, where w=−1 corresponds to a classical cosmological constant. The determined value of the early‐type characteristic velocity dispersion [σ(e)*] depends on the faint‐end slope of the early‐type luminosity function [α(e)] and the intrinsic shape distribution of galaxies; for equal frequencies of oblates and prolates, we find that σ(e)*(0.3 ≲z≲ 1) = 198+22−18 km s−1 (68 per cent) for a ‘steep’α(e)=−1 or σ(e)*(0.3 ≲z≲ 1) = 181+18−15 km s−1 (68 per cent) for a ‘shallow’α(e)=−0.54. Finally, from the relative frequencies of doubly imaged sources and quadruply imaged sources, we find that a mean projected mass ellipticity of early‐type galaxies with a 68 per cent lower limit of 0.28 assuming equal frequencies of oblates and prolates.

The strong equivalence principle (SEP) distinguishes general relativity (GR) from other viable theories of gravity. The SEP demands that the internal dynamics of a self-gravitating system under freefall in an external gravitational field should not depend on the external field strength. We test the SEP by investigating the external field effect (EFE) in Milgromian dynamics (MOND), proposed as an alternative to dark matter in interpreting galactic kinematics. We report a detection of this EFE using galaxies from the Spitzer Photometry and Accurate Rotation Curves (SPARC) sample together with estimates of the large-scale external gravitational field from an all-sky galaxy catalog. Our detection is threefold: (1) the EFE is individually detected at 8σ to 11σ in “golden” galaxies subjected to exceptionally strong external fields, while it is not detected in exceptionally isolated galaxies, (2) the EFE is statistically detected at more than 4σ from a blind test of 153 SPARC rotating galaxies, giving a mean value of the external field consistent with an independent estimate from the galaxies’ environments, and (3) we detect a systematic downward trend in the weak gravity part of the radial acceleration relation at the right acceleration predicted by the EFE of the MOND modified gravity. Tidal effects from neighboring galaxies in the Λ cold dark matter (CDM) context are not strong enough to explain these phenomena. They are not predicted by existing ΛCDM models of galaxy formation and evolution, adding a new small-scale challenge to the ΛCDM paradigm. Our results point to a breakdown of the SEP, supporting modified gravity theories beyond GR.

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.