On the central stellar mass density and the inside-out growth of
  early-type galaxies
  [Abridged] In this paper we derive the central stellar mass density within a
fixed radius and the effective stellar mass density within the effective radius
for a complete sample of 34 ETGs morphologically selected at 0.9<z_{spec}<2 and
compare them with those derived for a sample of ~900 local ETGs in the same
mass range. We find that the central stellar mass density of high-z ETGs spans
just an order of magnitude and it is similar to the one of local ETGs as
actually found in previous studies.However, we find that the effective stellar
mass density of high-z ETGs spans three orders of magnitude, exactly as the
local ETGs and that it is similar to the effective stellar mass density of
local ETGs showing that it has not changed since z~1.5, in the last 9-10 Gyr.
Thus, the wide spread of the effective stellar mass density observed up to
z~1.5 must originate earlier, at z>2. Also, we show that the small scatter of
the central mass density of ETGs compared to the large scatter of the effective
mass density is simply a peculiar feature of the Sersic profile hence,
independent of redshift and of any assembly history experienced by galaxies.
Thus, it has no connection with the possible inside-out growth of ETGs.
Finally, we find a tight correlation between the central stellar mass density
and the total stellar mass of ETGs in the sense that the central mass density
increases with mass as M^{~0.6}. This implies that the fraction of the central
stellar mass of ETGs decreases with the mass of the galaxy. These correlations
are valid for the whole population of ETGs considered independently of their
redshift suggesting that they originate in the early-phases of their formation.
