Is there a conflict with phase-space density in the secular bulge formation scenario? Abstract: Several observational pieces of evidence support the secular bulge formation scenario, at least for late type galaxies. However, the fact that the observational measure of phase-space density, f obs, is larger for the bulge than for the inner disk of Milky Way (and some other spirals), poses an apparent difficulty for the collisionless secular bulge formation scenario. We have investigated the evolution of f obs in a high-resolution disk+halo N-body simulation. A bulge-like structure forms in the disk. As expected from the Liouville's theorem, f obs never increases. However, inside the bulge-like structure, f obs ends being larger than in the inner disk, in agreement with observations from the Milky Way. We have also estimated f obs for the bulge and the inner disk of other galaxies spanning several Hubble types. We conclude that the collisionless secular scenario seems to work in the correct direction regarding phase-space density evolution. Acknowledgments Thanks to DGEP-UNAM & CONACYT for the support given for this work References Bissantz N., & Gerhard O., 2002, MNRAS, 330, 591 Carollo M., Ferguson H. & Wyse R. 1999, The formation of galactic bulges, Cambridge University Press Combes F., Debbasch F., Friedli D., & Pfenniger D., 1990, A&A, 233, 82 Lewis J., & Freeman K.C. 1989, AJ, 97, 139 Pfenniger D, & Norman C.,1990, ApJ, 363, 391 Raha N., Sellwood J. A. James R. A. Kahn F. D. 1991, Nature, 352, 411 Shapiro K., Gerssen J., & van der Marel P.R., 2003, AJ, 126, 2707 Valenzuela O. & Klypin A., 2003, MNRAS, 345, 406 Wyse R., 1998, MNRAS, 293, 429 1) Bulge formation and clues from observations The three scenarios for bulge formation are: 1) monolitical collapse, 2) violent mergers and 3) disk secular evolution. The latter scenario is very claimed at least for late type galxies. Several observational pieces of evidence support the secular scenario for late type galaxies (see for review Carrollo 03: photometric, chemical and kinematical continuities betwen bulge and inner disk, Sérsic surface brigthness profiles with n < 4, correlation betwen r e vs. h d, etc. Evidence in favor of the secular scenario were found recently even for some early-type bulge. N-body simulations are a powerful method for studying many body systems such as galaxies. Early works demostrated tha a a collisionless disk system evolves forming structures like bars, spiral arms and concentrations in the center. Three secular mechanisms were proposed for bulge formation: scattering of the starts at vertical resonances (Combes et al.91), dissolution of bars due to central concentration (Pfeniger & Norman 90), and collisionless buckling instability ( Raha et al.91). 2) A posible conflict for the secular scenario? The observed space phase density function, f obs, is defined as the stellar number density n s divided by the product of the three observed dispertion velocity. The Liouville theorem states that phase-space density is constant for a collisionless system: Asuming isotropy for the bulge we can set up a limit for f bulge The phase space density function of the disk can be aproximated by: NameTypenf b (r e /2)/f d (h d ) NGC 1068Sb2.1>8.6 NGC 2460Sa1.8>2.4 NGC 2775Sbab1.5>2.0 NGC 4030Sbc1.7>1.0 Milky WaySbc~1>1.5 Simulation During the disk secular evolution a bar is formed and the disk thickens (fig.3). An increasing with time central concentration of mass in excess of the inward extrapolation of the outer exponential disk (fig.5), and a significant thickening and dynamical heating of this mass concentration (fig.4), are observed. These are the criteria by which a bulge is defined observationally. We use kinematical bulge/disk observations by Shapiro et al. 03 of four spirals. Their surface brigthness profiles (fig.7) were used to estimate bulge and disk densities. 3) The numerical simulations With the aim to explore the problem posed by Wyse 98, we analyse results from a state-of-the-art high-resolution N-body simulation of a galaxy-like disk embedded in a live CDM halo (Valenzuela & Klypin 03). Now, as Wyse 98 stated “One should not find a higher phase- space density in stellar progeny, formed by a collisionless procees, then its stellar progenitor”. Therefore, if the system is collisionless, we should have that the ratio Wyse noted that for the Galaxy f b (100pc) is ~ 5 times larger than f d (2kpc), claiming for a serious difficulty for the collisionless secular scenario, unless disipation is included. Based on the Bissantz & Gerhard 02 bulge model, the bulge kinematics compilation by Tremaine et al. 02, the Lewis & Freeman 89 kinematical data for the disk, and assumig an exponential disk stellar surface brigthness density profile (h d =3kpc,  ʘ =41.3 M ʘ /pc 3 ) and h z =330pc we calculate the bulge and disk observational phase-space density profile for the Galaxy: Fig.4 Evolution of the radial, tan- gential, and vertical velocity-dispersion radial profiles (top panels and left bottom) and the evolution of the vertical scale length (rigth bottom panel). The disk is heated significantly in the center. 6) Conclusions By means of high resolution N-bdy simulations of a disk embedded in a live CDM halo we have found that a bulge-like structure arises due to collisionless secular evolution. The Liouville theorem is obeyed as expected, however, the measure of the phase-space density of the inner disk becomes lower than the one of the central structure. Therefore, we conclude that secular evolution yields an ``observational'' phase-space density radial profile in agreement with that it is observationally inferred for the Galaxy and probably for other galaxies. Fig.5 -Evolution of the disk surface (top) and co-planar volumetric (bottom) azhimutally averaged density for the simulation. As seen in fig.6 for the initial thin exponential disk in vertical and radial equilibrium, f obs increases along the disk. Due to the secular dynamical evolution, f obs decreases with time (mainly in the first Gyr) in a roughly constant way along the overall disk. However, in the central region, where namely the bulge-like structure arises, the radial profile of f obs develops an increasing with time depression, with a minimum approximately at the radius where the bulge-like structure ends, R inn ~ 2.5 kpc). In the innermost region, f obs remains almost the same with time. Fig.1 –Observational phase-space density profile for the Galaxy (bulge + disk). Is the conflict posed by Wyse 98 confirmed by N-body simulations? Fig.2 Selected rings of the disk at the final time step (4.4 Gyr). The disk and halo have 1.2E6 and 8E6 particles, respectively. Fig.3 Vertical evo- lution of the disk (x-z plane). The time evolution goes from initial disk at 0 Gyrs, 1 Gyr, 2Gyrs and 4.4 Gyrs (clockwise). 5) Observational phase-space density in external galaxies Our estimates of the bulge Sérsic index n and of the f b (r e /2)/f d (h d ) ratio are presented in the following table. Fig.7 Bulge(Sérsic)/disk(exponential) decomposition of four spirals with velocity dispersion profiles as measured by Shapiro et al ) Evolution of phase-space density We measure in the simulation the evolution of the phase-space density radial profile in similar way as observers do. The density and dispersion velocity profiles (figs.4,5) were used to calculate the evolution of f obs shown in fig.6 Fig. 6 Phase-space density profiles at diferent time steps using  r    z (a),  r  (b). Panel (c) same as (a) but for a lower resolution simulation. The results are the same qualitatively. Alejandro Carrillo 1, Vladimir Avila-Reese 1 & Octavio Valenzuela 2 1 Instituto de Astronomía UNAM, México 2 University of Washington, Seattle, USA titi tftf titi tftf