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.

4 ApJ pages, submitted to ApJ Letters