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

It has been argued in the past that bulges of galaxies could not 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 incorrect. Observations give estimates of the coarsely grained phase-space densities of galaxies, f'=ρ_s/σ_Rσ_φσ_z, where ρ_s is the stellar density and σ_R, σ_φ and σ_z are the radial, tangential and vertical rms velocities of stars, respectively. Using high-resolution N-body simulations, we study the evolution of f' in stellar discs of Galaxy-size models. During the secular evolution, the discs, which are embedded in live cold dark matter haloes, form a bar and then a thick, dynamically hot, central mass concentration. During the course of evolution f' declines at all radii. 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 becomes 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 that of the inner disc turns out to be an argument in favour of secular formation of bulges, not against it.