  We study the time evolution of small classical perturbations in a gauge
invariant way for a complex scalar field in the early zero curvature
Friedmann-Lema\^{\i}tre universe. We, thus, generalize the analysis which has
been done so far for a real scalar field. We give also a derivation of the
Jeans wavenumber in the Newtonian regime starting from the general relativistic
equations, avoiding the so-called Jeans swindle.
  During the inflationary phase, whose length depends on the value of the
bosonic charge, the behavior of the perturbations turns out to be the same as
for a real scalar field. In the oscillatory phase the time evolution of the
perturbations can be determined analytically as long as the bosonic charge of
the corresponding background solution is sufficiently large. This is not
possible for the real scalar field, since the corresponding bosonic charge
vanishes.
