  The mass aggregation and merger histories of present-day distinct haloes
selected from the cosmological Millennium Simulations I and II are mapped into
stellar mass aggregation and galaxy merger histories of central galaxies by
using empirical stellar-to-halo and stellar-to-gas mass relations. The growth
of bulges driven by the galaxy mergers/interactions is calculated using
dynamical prescriptions. The predicted bulge demographics at redshift z~0 is
consistent with observations (Zavala+2012). Here we present the evolution of
the morphological mix (traced by the bulge-to-total mass ratio, B/T) as a
function of mass up to z=3. This mix remains qualitatively the same up to z~1:
B/T<0.1 galaxies dominate at low masses, 0.1<B/T<0.45 at intermediate masses,
and B/T>0.45 at large masses. At z>1, the fractions of disc-dominated and
bulgeless galaxies increase strongly, and by z~2 the era of pure disc galaxies
is reached. Bulge-dominated galaxies acquire such a morphology, and most of
their mass, following a downsizing trend. Since our results are consistent with
most of the recent observational studies of the morphological mix at different
redshifts, a LCDM-based scenario of merger-driven bulge assembly does not seem
to face critical issues. However, if the stellar-to-halo mass relation changes
too little with redshift, then some tensions with observations appear.
