We discuss how the effective radius R _ { e } function ( ERF ) recently worked out by Bernardi et al .
( 2009 ) represents a new testbed to improve the current understanding of Semi-analytic Models of Galaxy formation .
In particular , we here show that a detailed hierarchical model of structure formation can broadly reproduce the correct peak in the size distribution of local early-type galaxies , although it significantly overpredicts the number of very compact and very large galaxies .
This in turn is reflected in the predicted size-mass relation , much flatter than the observed one , due to too large ( \gtrsim 3 kpc ) low-mass galaxies ( < 10 ^ { 11 } { M _ { \odot } } ) , and to a non-negligible fraction of compact ( \lesssim 0.5 - 1 kpc ) and massive galaxies ( \gtrsim 10 ^ { 11 } { M _ { \odot } } ) .
We also find that the latter discrepancy is smaller than previously claimed , and limited to only ultracompact ( R _ { e } \lesssim 0.5 kpc ) galaxies when considering elliptical-dominated samples .
We explore several causes behind these effects .
We conclude that the former problem might be linked to the initial conditions , given that large and low-mass galaxies are present at all epochs in the model .
The survival of compact and massive galaxies might instead be linked to their very old ages and peculiar merger histories .
Overall , knowledge of the galactic stellar mass and size distributions allows a better understanding of where and how to improve models .