On the onset of the Jeans instability in a two-component fluid

Conditions for the establishment of small density perturbations in a self-gravitating two component fluid mixture are studied using a dynamical system approach. It is shown that besides the existence of exponentially growing and decaying modes, which are present for values of the perturbation wave-number $k$ smaller than a critical value $k_{{}_M}$, two other, pure oscillatory, modes exist at all scales. For $k<k_{{}_M}$, the growing mode always affects both components of the fluid and not only one of them. Due to the existence of a resonance between the baryonic and the dark perturbations, it is shown that the onset of structure formation in the post recombination epoch is substantially enhanced in a narrow scale band around another critical value $k_c$. For dark matter particles having a mass $\sim 30$ eV, the corresponding critical mass scale for the establishment of density perturbations at the time of recombination is of the same order of magnitude as the galactic one.