A simplified model of the formation of structures in dark matter and a background of very long gravitational waves

The collapse of a rotating spheroid is approximated by a system of ordinary differential equations describing its dynamics. The gravitational potential is approximated by that of a uniform Maclaurin spheroid. Development of gravitational instability and collapse in the dark matter medium do not lead to any shock formation or radiation, but are characterized by non-collisional relaxation, which is accompanied by mass and angular momentum losses. The phenomenological account of these processes is performed in this model and the formation of the equilibrium configuration dynamics of collapse is investigated for several parameters, characterizing the configuration. A very long gravitational wave emission during the collapse is estimated, and a possible connection with the observed gravitational lenses is discussed.