Gravitational instability in an Omega(0) = 1, four-component universe

The linear adiabatic modes in an Omega(0) = 1 universe containing, in addition to baryonic matter (BM) and radiation, two forms of dark matter, one cold and one hot (CDM and HDM), are evaluated; the latter is assumed to consist of light massive neutrinos. The evolution in time of the mass variance is thence derived as a function of the parameters of interest, and which mass goes nonlinear, and when, is estimated. For the scale-invariant primordial spectrum, n = 1, it is concluded that, in spite of the catalytic influence of condensation seeds of CDM and BM, in the linear regime HDM is capable of appreciable growth only from the galactic scale upward, while on the smaller scales growth remains insufficient due to free-streaming. The timely formation of the existing cosmic structure requires in particular that HDM does not exceed one-third or one-fourth of the total density (depending on whether all three or only one of the three existing neutrinos are massive). On the other hand, this upper limit is raised to one-half for n = 2 and entirely removed for n = 3.