  In this paper we investigate the overcooling problem and propose some
possible solutions. We show that the overcooling problem is generic to the
hierarchical picture of structure formation, as long as the cooling is actually
possible in small halos at high redshift. Solutions to this problem are likely
to be associated with some feedback mechanism, and probably imply the existence
of a warm IGM containing most of the cosmological baryons. We concentrate on
some possible solutions, mainly photoionization and bulk-heating of the IGM. We
show photoionization can act as a significant feedback mechanism but is not
strong enough to solve the entire overcooling problem. We therefore assume that
the IGM is maintained hot by some energy injection provided by supernova and
galaxy formation is then limited by this feedback mechanism. Such a self
regulated scheme allows us to compute the thermal history of the IGM...
(abridged)
