This is a very well written paper exploring the prospects for using the evolution of the phase space density of the entire stellar population of all galaxies to constrain the possible progenitors of galaxies in the local universe. The analysis primarily relies on the "mixing inequality" derived by Tremaine et al. (1986) that says the cumulative stellar mass in a given volume of phase space (ordered from highest to lowest phase space densities) can never increase through collisionless evolution.
The stellar phase space distribution for a single galaxy is modeled by separating the galaxy into bulge and disk components. The cumulative stellar mass as a function of phase space volume, M(V), for the bulge and disk components are simply added together to get M(V) for the whole galaxy under the assumption that these volumes are mostly nonoverlapping. Likewise the author obtains the M(V) for the entire stellar population within all galaxies by summing the M(V) distributions for each galaxy.
The distribution of galaxy shapes and stellar masses is derived from the public Millennium Galaxy Catalogue, with the final result for the observed M(V) given in fig. 4 of the paper. Plotted in fig. 4 are 2 curves: M(V) for 46x10^10 [tex]M_{\odot}[/tex] spheroids, and M(V) for 15x10^8 [tex]M_{\odot}[/tex] galaxies that are presumed to represent progenitors of the 10^10 M_sun galaxies. If they really were representative of the progenitors then the 10^8 [tex]M_{\odot}[/tex] curve would have larger M(V) values for all V than the 10^10 [tex]M_{\odot}[/tex] curve, according to the "mixing inequality". This is not true for intermediate volumes leading the author to conclude that "no combination of collisionless merger processes bringing small systems like these together could have produced the final large spheroids".
Because the violation of the mixing inequality only involves a small fraction of the stellar mass, it is not taken as a significant problem for models of hierarchical galaxy formation. However, before considering such conclusions, I was not able to understand how observational selection effects might have impacted the observed M(V) distributions. That is the 10^8 [tex]M_{\odot}[/tex] population that is taken as possible progenitors is presumably not completely sampled (or the bulgedisk decomposition is not accurate) as the magnitude limit of the survey is reached. And, what is the expected distribution of progenitor masses inferred from simulations for the 10^10 [tex]M_{\odot}[/tex] spheroid population?
[1008.4252] The Stellar Phase Density of the Local Universe and its Implications for Galaxy Evolution
Authors:  Michael R. Merrifield (University of Nottingham) 
Abstract:  This paper introduces the idea that the general mixing inequality obeyed by evolving stellar phase densities may place useful constraints on the possible history of the overall galaxy population. We construct simple models for the full stellar phase space distributions of galaxies' disk and spheroidal components, and reproduce the wellknown result that the maximum phase density of an elliptical galaxy is too high to be produced collisionlessly from a disk system, although we also show that the inclusion of a bulge component in the disk removes this evolutionary impediment. In order to draw more general conclusions about the evolution of the galaxy population, we use the Millennium Galaxy Catalogue to construct a model of the entire phase density distribution of stars in a representative sample of the local Universe. In such a composite population, we show that the mixing inequality rules out some evolutionary paths that are not prohibited by consideration of the maximum phase density alone, and thus show that the massive ellipticals in this population could not have formed purely from collisionless mergers of a low mass galaxy population like that found in the local Universe. Although the violation of the mixing inequality is in this case quite minor, and hence avoidable with a modest amount of noncollisionless star formation in the merger process, it does confirm the potential of this approach. The future measurement of stellar phase densities at higher redshift will allow this potential to be fully exploited, offering a new way to look at the possible pathways for galaxy evolution, and to learn about the environment of star formation through the way that this phase space becomes populated over time. 
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