Current cosmological CDM simulations predict that dark matter haloes are cuspy (e.g. White, this volume), although there is still no agreement reached about the value of the inner slope. On the other hand, observations have shown that, at least in a number of cases where the data are of sufficiently good quality, haloes have cores (e.g. Bosma, or de Blok, this volume). If haloes were indeed formed with a cusp and now have a core, then some mechanism during galaxy formation and/or evolution should be responsible for this change. Several have been so far proposed, of which one involves a bar. Indeed, Hernquist & Weinberg (1992) and Weinberg & Katz (2002), based on analytical calculations and on N-body simulations of a cusped halo containing a rigid bar and no disc, proposed that the bar could flatten the cusp by giving angular momentum to the halo. This, however, was not generally accepted, since the fully self-consistent simulations of Sellwood (2003; S03) and Valenzuela & Klypin (2003; VK) showed the contrary, i.e. that during the evolution the halo radial profile showed a small, yet clear, steepening. These authors attribute the flattening found by the previous simulations to the fact that they do not include a live disc, and thus neglect the slow gradual change of the resonant positions due to the slowdown of the bar, and the disc responsiveness. This explanation did not, in turn, satisfy HBWK, whose fully self-consistent simulations showed a clear flattening of the halo cusp (HBWK) and thus argued that self-consistency, or the lack of it, could not be the factor determining why some of the previous simulations showed a flattening and others a steepening. HBWK instead argue that the cusp steepening found by S03 and VK is spurious, and due to the fact that their simulations are noisy and thus do not describe the resonances sufficiently accurately. Numerical noise could indeed, if sufficiently strong, prevent the halo resonances from absorbing the correct amount of angular momentum and thus lead to a wrong evolution. Other numerical shortcomings, however, e.g. leading to a wrong coupling between the planar and vertical resonances, could also influence the results, producing a spurious flattening or steepening.
Faced with the difficult task of reviewing this unsatisfactory state of affairs, I turned to my own simulations (AM02, A02, A03) to see whether they could provide any clues. They are neither grid based, nor use SCF, and thus could provide an independent view. They were not specifically designed to tackle this problem, so the initial halo has a core (albeit sometimes small). They can, nevertheless, lead to a number of insightful conclusions, which I will discuss briefly below. In particular, my study of the resonances allows me to single out and to follow the resonant stars. I could thus make sure that they cover the phase space adequately. Furthermore, in A02 I showed that there is a large trapping of the particles at resonances, which shows that they are not unduly knocked off their trajectories by noise during the evolution. Thus, my simulations fulfill all the necessary conditions set by HBWK to perform the test at hand.
In order to check whether the density profile in the inner parts flattens or steepens, I simply measured the amount of mass in concentric spherical shells 1. Since the existence of a bar is necessary for the mechanism to work, I confined myself to times after the bar had grown. I find that, in the vast majority of the cases I checked, there is a steepening of the halo radial density profile, albeit not large. In a couple of cases, however, I noted a very small flattening. Since this is even smaller in amplitude than the small steepening found in the remaining cases, I was ready to discard it as insignificant, until I noticed that it occurred in the more centrally concentrated cases. This prompted me to investigate the problem further.
The analytical predictions are clear and follow directly from what was discussed in the previous sections : Halo material at (near-) resonance should absorb angular momentum and move to larger cylindrical radii 2. For the case of the ILR, such material is located in the inner parts of the halo and this should lead to a flattening of the cusp. I have already shown in the previous sections that my simulations confirm the theoretical predictions about the existence of the resonances and the angular momentum exchanged. Showing the increase of the cylindrical radius of particles on orbits at, or around, the ILR is no an easy task, since they are trapped around periodic orbits of the x1 tree 3, for which it is not always easy to calculate the frequencies and the time average of the radius. Moreover, there are relatively few particles at ILR in the models I simulated (Fig. 1). On the other hand, it is very easy to see the increase in cylindrical radius in the case of CR, which does not suffer from the above drawbacks.
This, however, is not the complete picture, since there are three effects working against the previous one, and thus leading to a steepening of the halo :
As stressed by AM02, VK and O'Neill & Dubinski (2003), the radial density profile of the disc becomes considerably more centrally concentrated with time. In so doing, the disc material pulls the halo material also inwards, thus causing it to contract. This leads to an increase of the halo radial density profile in the inner parts.
As the halo gains angular momentum it will get flattened towards the disc equatorial plane and this will result in a decreasing of the spherical radius of the individual particles. As seen in the previous section, this happens mainly in the inner parts of the halo and will, therefore, lead to a steepening of the cusp.
During the evolution the bar becomes stronger and this influences the shape of the periodic orbits of the x1 tree and of the regular orbits trapped around them. Namely, their axial ratio a / b increases and they approach nearer to the center, so that sometimes the trapped orbits can actually cover the central area. This means that, although their average cylindrical radius may increase, the central-most area may have an increased density.
There are thus competing effects : On the one hand the resonant particles can move to larger cylindrical radii and thus tend to flatten the cusp. On the other hand, there are other effects, also linked to the angular momentum redistribution, which tend to diminish the spherical radii of the particles. It is the outcome of the competition between these effects that will decide whether the inner density profile will become steeper or shallower during the evolution. It is thus not necessarily surprising that HBWK find a flattening of the cusp, while S03 and VK find a steepening. The final result will depend on the distribution function of the halo, of which hardly anything is known, as well as that of the disc. In as far as the N-body simulations are concerned, it could also, unfortunately, depend on numerical effects, which could artificially modify the effect of the resonances.
The application of this mechanism to real galaxies faces, to my mind, two further serious problems. One is that haloes have substructure. Although this will not inhibit resonance driven evolution (Weinberg 2001), it could still perturb the effect of the resonances. The second, perhaps even more serious, problem is the presence of gas. All simulations that have so far studied this mechanism are purely stellar and have no gas. A gaseous component could increase further the central concentration of the disc material (e.g. Athanassoula 1992, Heller & Shlosman 1994) and thus make its inwards pull on the halo material yet stronger. It would then become yet more difficult for the halo (near-) resonant stars to overcome the inwards pull and achieve a flattening of the cusp.
1 Although very straightforward, this method is not the most appropriate for the problem at hand, because, as discussed in the previous section, the halo isodensities are triaxial and not spherical. Results based on other methods of calculating the density will be discussed elsewhere. Back.
2 It is important in this problem to distinguish between cylindrical and spherical radius. Back.
3 The x1 tree is the 3D orbital analog of the 2D x1 family and provides the backbone of the 3D bars (Skokos, Patsis & Athanassoula 2002). Back.