We present an approach to the design of distribution functions that depend on the phase-space coordinates through the action integrals. The approach makes it easy to construct a dynamical model of a given stellar component. We illustrate the approach
by deriving distribution functions that self-consistently generate several popular stellar systems, including the Hernquist, Jaffe, and Navarro, Frenk and White models. We focus on non-rotating spherical systems, but extension to flattened and rotating systems is trivial. Our distribution functions are easily added to each other and to previously published distribution functions for discs to create self-consistent multi-component galaxies. The models this approach makes possible should prove valuable both for the interpretation of observational data and for exploring the non-equilibrium dynamics of galaxies via N-body simulation.
Early-type galaxies (ETGs) are observed to be more compact, on average, at $z gtrsim 2$ than at $zsimeq 0$, at fixed stellar mass. Recent observational works suggest that such size evolution could reflect the similar evolution of the host dark matter
halo density as a function of the time of galaxy quenching. We explore this hypothesis by studying the distribution of halo central velocity dispersion ($sigma_0$) and half-mass radius ($r_{rm h}$) as functions of halo mass $M$ and redshift $z$, in a cosmological $Lambda$-CDM $N$-body simulation. In the range $0lesssim zlesssim 2.5$, we find $sigma_0propto M^{0.31-0.37}$ and $r_{rm h}propto M^{0.28-0.32}$, close to the values expected for homologous virialized systems. At fixed $M$ in the range $10^{11} M_odot lesssim Mlesssim 5.5 times 10^{14} M_odot$ we find $sigma_0propto(1+z)^{0.35}$ and $r_{rm h}propto(1+z)^{-0.7}$. We show that such evolution of the halo scaling laws is driven by individual haloes growing in mass following the evolutionary tracks $sigma_0propto M^{0.2}$ and $r_{rm h}propto M^{0.6}$, consistent with simple dissipationless merging models in which the encounter orbital energy is accounted for. We compare the $N$-body data with ETGs observed at $0lesssim zlesssim3$ by populating the haloes with a stellar component under simple but justified assumptions: the resulting galaxies evolve consistently with the observed ETGs up to $z simeq 2$, but the model has difficulty reproducing the fast evolution observed at $zgtrsim 2$. We conclude that a substantial fraction of the size evolution of ETGs can be ascribed to a systematic dependence on redshift of the dark matter haloes structural properties.
