Secular evolution of galactic discs: constraints on phase-space density

It was argued in the past that bulges of galaxies cannot be formed through collisionless secular evolution because that would violate constraints on the phase-space density: the phase-space density in bulges is several times larger than in the inner parts of discs. We show that these arguments against secular evolution are not correct. Observations give estimates of the coarsely grained phase-space densities of galaxies, f'=rho_s/(sigma_R sigma_phi sigma_z), where rho_s is stellar density and sigma_R, sigma_phi, sigma_z are the radial, tangential, and vertical rms velocities of stars. Using high-resolution N-body simulations, we study the evolution of f' in discs of Galaxy-size models. During the secular evolution, the discs, which are embedded in live CDM haloes, form a bar and then a thick, dynamically hot, central mass concentration. In the course of evolution f' declines at all radii, not just in the central region. However, the decline is different in different parts of the disc. In the inner disc, f'(R) develops a valley with a minimum around the end of the central mass concentration. The final result is that the values of f' in the central regions are significantly larger than those in the inner disc. The minimum, which gets deeper with time, seems to be due to a large phase mixing produced by the outer bar. We find that the shape and the amplitude of f'(R) for different simulations agree qualitatively with the observed f'(R) in our Galaxy. Curiously enough, the fact that the coarsely grained phase-space density of the bulge is significantly larger than the one of the inner disc turns out to be an argument in favor of secular formation of bulges, not against it.