  Theoretical treatments of the BCS-BEC crossover need to provide as accurate
as possible descriptions of the two regimes where the diluteness condition
applies, either in terms of the constituent fermions (BCS limit) or of the
composite bosons which form as bound-fermion pairs (BEC limit). This has to
occur via a single fermionic theory that bridges across these two limiting
representations. In this paper, we set up successive improvements of the
fermionic theory, that result into composite bosons described at the level of
either the Bogoliubov or the Popov approximations for point-like bosons. This
work bears on the recent experimental advances on the BCS-BEC crossover with
trapped Fermi atoms, which show the need for accurate theoretical descriptions
of BEC side of the crossover.
