Time evolution of the perturbations for a complex scalar field in Friedmann-Lemaitre universe

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.