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6. Conclusions

a) Standard interpretation of rotation curves.

There is a general consensus about the history of galaxy formation and the establishment of the rotation curve of spirals. This standard history could be summarized as follows:

At Inflation, quantum mechanical fluctuations were generated and then survived until the epoch of Recombination. The Universe then became CDM dominated with a baryonic component as a minor constituent. By gravitational collapse the primordial fluctuation that evolved gave rise to small DM halos. Adjacent halos merged to produce larger halos and this merging process has continued until the present. A complete hierarchy of CDM halos has been produced, those produced later being larger and the size being limited by the finite time of the Universe.

Once a new halo is formed by merging, violent relaxation effects destroy part of the previous DM substructure, but not completely, leaving some CDM subhalos still identifiable. After that, baryons cool and concentrate at the CDM centre at any size of the hierarchy. Baryon concentrates then form galaxies and shine. Larger CDM halos, produced later, also produce larger visible galaxies.

Hierarchical CDM models predict universal halo density distribution profiles, the so called NFW profiles, irrespective of their size and position in the hierarchy, following as R-1 in the inner region and as R-3 in the outer one. The CDM halo density profiles decide the rotation curve of the visible galaxies which are small bright indicators of large and massive CDM halos.

Subhalos within a halo are possible and would correspond to the existence of small ancient visible satellite galaxies orbiting around a large galaxy or to normal galaxies in a large cluster. Some CDM subhalos were destroyed in the merging process but their visible baryonic aggregates, because of their high density, were able to survive. Visible galaxies also merge. The merging of two or more former disk galaxies produces a larger elliptical.

Some aspects in this short account of the long history are better known than others. If we restrict ourselves to the rotation of spiral galaxies, there are some problems that remain unclear or are insufficiently explained. We prefer now to select problems rather than to emphasize successes. Among the outstanding problems let us highlight the following:

- Theorists themselves are unhappy with the rotation curves obtained; in particular the Tully-Fisher relation is unsatisfactorily explained.

- It is not clear if the visible galaxies should now be at the centres of their "own" halos or if they lie off-centre in common halos shared with other galaxies. The degree of CDM substructure predicted by different authors varies. In the first pioneering steps it was suggested that dwarf satellites would be in the halo of the larger primary galaxy, having no smaller halos of their own. Observations, however, and new theoretical model developments, indicate that dwarf satellites not only have their own halos, but also that the dark/visible matter ratios are much larger.

- The rotation curves observed can be fitted to the so called Universal Rotation Curves, but their density profiles do not coincide at all with the theoretical density profiles. The universal rotation curves have a core, i.e. a region in which the density is more or less constant or slightly decreasing as R, and in the periphery they decrease as R-3/2.

- The so called "halo-disk conspiracy", i.e. why the disk and halo dominated regions have a featureless flat transition, is not completely answered. The fact that some galaxies have rising or declining curves does not explain the conspiracy problem in those galaxies that do possess a flat curve. The adiabatic compression of the inner halo due to the disk formation establishes a halo-disk relation that alleviates the conspiracy, but much work is still needed to model this interaction.

- Both universal rotation curves and universal halo density distributions shed some light on the absence of correlation between the orbital velocities of satellite galaxies and rotation velocity of the primary galaxies (or equivalently their luminosities, as Tully-Fisher relates both quantities). But both velocities should be determined by the halo and should partially correlate.

The so called Bosma relation, even if some authors have their doubts about its validity, indicates a relation between the circular velocities of both the dark halo and the gas. This relation is not only a general one, but in some galaxies, small scale changes are present in the rotation curve and in the gas distribution. This seems paradoxical, particularly if we consider that the dark matter is more or less spherically distributed and the gas lies in a disk.

Clearly, other authors would discuss other points that are still obscure and others would have preferred to focus on the agreements with observational facts, which are certainly encouraging and suggest that the basic scenario has been firmly established. There are, however, many other possible histories: we have seen that some authors consider dark matter to be baryonic and even that it is in the disk. These theories have considerable merit, especially because they must be developed against the general flow of ideas, and because they explain some observational facts in a simpler way.

Let us just outline another alternative history, rather different from the above mentioned standard one, if the word "standard" can be properly assigned to any of the present scenarios.

b) The magnetic interpretation of the rotation curve.

At inflation, quantum mechanical fluctuations were generated not only in the energy density distribution but also created a non uniform distribution of magnetic fields. Magnetic flux tubes, probably interconnected with other tubes forming a network, conserved their shapes during the radiation dominated era, but the strength decreased as the expansion proceeded. Finite conductivity effects destroyed the small scale magnetic field structures, but those that were large enough, larger than about 50 Mpc (comoving) survived, producing filaments in the energy density distribution - probably also sheets - which after Recombination became dark matter filaments (with baryons as a minor constituent), more than 100 Mpc long.

The scenario provided by hierarchical CDM models, assumed in the "standard" history previously summarized, is fully assumed here too, with the only exception that mergers and non-linear evolution took place inside the large density filaments and not in the voids in between. The heating produced by shocks in the merging events also affected the magnetic fields, which became disordered and amplified. Individual halos belonging to a visible galaxy at its centre were formed in the first generations of halos, but subsequent mergers produced common halos shared by satellites or groups of galaxies. Pairs, satellites, poor clusters and rich clusters developed their superhalos with no dark matter substructure. Only exceptional truly isolated visible counterparts would retain their own halo against merging or were the result of merging of the visible galaxies in a previous common halo.

The fact that hierarchical mergers only took place within the filaments and not in the large voids, assumed here to be of primordial origin, could alleviate a problem encountered in CDM hierarchical models: that of overproduction of halos. Furthermore, these models predict too little structure at scales larger than about 40 Mpc, a problem that was detected in the pioneering simulations.

The disk was formed out of magnetized gas and maintained a magnetic pressure equilibrium with the region outside the visible galaxy, frozen-in in the low-density uncondensed gas lying in the common halo. Equilibrium was possible because of the frequent outbursts of disk material and magnetic fields due to violent star formation events, as observed for instance in M82 and in other galaxies like ours. This magnetic field acquired a toroidal distribution, due to the differential rotation, which was able to generate a centripetal force which produced a higher rotation in the periphery of disks.

Magnetic fields responsible for the high rotation velocity also rendered the disk thicker, facilitating the fountain effect and escape. Magnetic fields would act inwards in the radial direction and outwards in the vertical direction. The escape from the disk and even from the galaxy would be a more important effect in dwarf irregulars, which indeed present larger outbursts of material; they are gas rich, and therefore have greater ability to amplify magnetic fields. Under this interpretation, irregulars are not DM rich galaxies, but magnetic field rich galaxies, because they are gas rich galaxies.

One question naturally arises in this scenario. What is the DM content of the other galaxies, particularly in ellipticals? Suppose a rich cluster which has a giant cD elliptical at the centre. It is evident that cD galaxies are then also at the centre of the cluster superhalo and therefore, they are beated in a region with large amounts of dark matter. Therefore, cD galaxies should have large amounts of DM, as seems to be the case in M87. It is less clear what is the situation with normal ellipticals. Therefore, we merely propose that giant ellipticals at the centre of large halos would have large quantities of DM; spirals, lying far from the giant common cluster halo centre would not posses dark matter halos. Even if they were embedded in DM, this would be more homogeneously distributed around the spiral, and hence it would not have such a decisive influence on the rotation curve. The DM halo potential could produce warps as a tidal effect.

In the "standard" history, we discussed some current problems and disagreements between theory and observations. In this alternative scenario, we comment on their advantages. In this paper we have reviewed a model that numerically accounts for the basic facts. From a qualitative point of view, without any precise developments, let us also consider:

- Galaxies with more gas would produce, and be subject to, higher magnetic fields, and precisely these galaxies were considered to have rotation curves with a greater discrepancy from the curve expected from the gravitation produced by disk and bulge.

- The Bosma relation, establishing a connection between gas and DM (which in this picture should be expressed as a relation between gas and magnetic fields) would be obvious.

- There would be no conspiracy problems, as the magnetic fields and gravitation forces ratio would be progressive and continuously increasing for increasing radii.

- The problems arising from the lack of correlations in binary galaxies would naturally disappear. The velocity observed at the higher radius where the signal is detected would be the result of internal magnetic fields, clearly unrelated to any orbital velocity, whether or not the pair lies in a common halo.

- The agreement with theoretical hierarchical CDM models is better, as these models have "no responsibility" to directly explain rotation curves, unless they include magnetic fields. For instance, the theoretical prediction that irregulars orbiting a bright galaxy (like ours) would be embedded in its halo and would not have their own halo, would not contradict the standard interpretation of observations that irregulars are particularly dark matter dominated.

c) Other alternatives.

Theories assuming galactic dark matter to be undetected gas must eventually answer two basic questions. How have these galaxies formed? and, if $ \Omega_{M}^{}$ $ \sim$ 0.3 and $ \Omega_{B}^{}$ $ \sim$ 0.03, where is the non-baryonic dark matter? (could the answer to the second question be the existence of common cluster halos?).

Theories proposing a modification of Newton's Second Law should clarify whether we should also reject General Relativity. Modifying Newtonian Mechanics is a tolerable sacrifice, but physics would probably require more solid proof before abandoning General Relativity.

Summarizing this summary, we are beginning to understand galaxy formation, the nature and distribution of dark matter in galaxies and rotation in what could be called the standard scenario. But there are other interesting alternatives that should not be disregarded without an intense debate. MOND is one of them. Gaseous dark matter is another. The magnetic alternative is not frontally opposed to CDM hierarchical scenarios, but is, rather, complementary. Only secondary phenomena are in clear contradiction. It is unrealistic to attempt to deal with rotation curves while ignoring magnetic fields. This could constitute a particular flaw in the standard model for rotation curves.

If after this review, the topic of galactic dark matter is less clear, we will have accomplished our mission.


We are very grateful to Kor Begeman, Adrick Broeils, Jordi Cepa, Carlos Frenk, Mareki Honma, Julio Navarro, Michael Merrifield, Yoshiaki Sofue, Rob Swaters and Juan Carlos Vega Beltrán, for their permission to include some of their figures and for their valuable help. We are specially grateful to Constantino Ferro-Fontán, for his encouragement and stimulus to undertake this review. Our thanks also to Jorge Jiménez-Vicente and Ana Guijarro. Support for this work was provided by the Ministerio de Educación y Cultura (PB96-1428) and by Junta de Andalucía (FQM-108).

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