  There appears to exist a conflict between the standard structure formation
theory and the existence of the faint dwarf spheroidal galaxies in the Local
Group. Theoretical considerations suggest that a cold dark matter universe has
been a very hostile place for the formation of small galaxies. In particular,
gas would not have been able to cool and condense to form stars inside dark
matter halos with a velocity dispersion <10km/s since before the cosmological
reionization epoch. In other words, one should not expect to see any major star
formation activities in dwarf galaxies with a velocity dispersion <10km/s
during the past 12Gyr, according to the current structure formation theory. The
real universe, on the other hand, shows that all dwarf spheroidal galaxies in
the Local Group have velocity dispersions <10km/s and extended and recent star
formation activities are quite common in them. This apparent conflict between
cold dark matter models and local observations can be resolved, if one allows
the cold dark matter particles to decay relatively recently. The resolution
comes about in that the dwarf spheroidal galaxies with a velocity dispersion of
\~10km/s seen today were born, in this picture, in a small fraction of halos
with a velocity dispersion of ~20km/s, where gas can be retained and is able to
cool and contract to form stars. The presently observed lower velocity
dispersion ~10km/s of these dwarf spheroidal galaxies is a consequence of the
decay of one half of the dark matter in and subsequent expansion of the halos
since redshift z~2.
