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We compare state-of-the-art semi-analytic models of galaxy formation as wellas advanced sub-halo abundance matching models with a large sample ofearly-type galaxies from SDSS at z < 0.3. We focus our attention on thedependence of median sizes of central galaxies on host halo mass. The data donot show any difference in the structural properties of early-type galaxieswith environment, at fixed stellar mass. All hierarchical models considered inthis work instead tend to predict a moderate to strong environmentaldependence, with the median size increasing by a factor of about 1.5-3 whenmoving from low to high mass host haloes. At face value the discrepancy withthe data is highly significant, especially at the cluster scale, for haloesabove log Mhalo > 14. The convolution with (correlated) observational errorsreduces some of the tension. Despite the observational uncertainties, the datatend to disfavour hierarchical models characterized by a relevant contributionof disc instabilities to the formation of spheroids, strong gas dissipation in(major) mergers, short dynamical friction timescales, and very short quenchingtimescales in infalling satellites. We also discuss a variety of additionalrelated issues, such as the slope and scatter in the local size-stellar massrelation, the fraction of gas in local early-type galaxies, and the generalpredictions on satellite galaxies.

Environmental dependence of bulge-dominated galaxy sizes in hierarchical models of galaxy formation. Comparison with the local Universe