### [1008.4252] The Stellar Phase Density of the Local Universe

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**August 26 2010**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 non-overlapping. 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 4-6x10^10 [tex]M_{\odot}[/tex] spheroids, and M(V) for 1-5x10^8 [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 bulge-disk 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?

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 non-overlapping. 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 4-6x10^10 [tex]M_{\odot}[/tex] spheroids, and M(V) for 1-5x10^8 [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 bulge-disk 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?