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Already in the early 1990s, high resolution simulations of individual galaxy halos in CDM were finding rho(r) ~ r- with alpha ~ 1. This behavior implies that the rotation velocity at the centers of galaxies should increase as r1/2, but the data, especially that on dark-matter-dominated dwarf galaxies, instead showed a linear increase with radius, corresponding to roughly constant density in the centers of galaxies. This disagreement of theory with data led to concern that CDM might be in serious trouble [65, 66].

Subsequently, NFW [58] found that halos in all variants of CDM are well fit by the rhoNFW(r) given above, while Moore's group proposed an alternative (r) propto r-3/2(r + rM)-3/2 based on a small number of very-high-resolution simulations of individual halos [67, 68, 69]. Klypin and collaborators (including me) initially claimed that typical CDM halos have shallow inner profiles with alpha approx 0.2 [72], but we subsequently realized that the convergence tests that we had performed on these simulations were inadequate. We now have simulated a small number of galaxy-size halos with very high resolution [59], and find that they range between rhoNFW and rhoM. Actually, these two analytic density profiles are almost indistinguishable unless galaxies are probed at scales smaller than about 1 kpc, which is difficult but sometimes possible.

Figure 2

Figure 2. High resolution Halpha rotation curves (filled circles, solid lines) and HI rotation curves for the same galaxies (open circles, dotted lines) from Ref. 64. The horizontal bar shows the FWHM beam size of the HI observations. From Swaters, Madore, and Trewhella [75].

Meanwhile, the observational situation is improving. The rotation curves of dark matter dominated low surface brightness (LSB) galaxies were measured with radio telescopes during the 1990s, and the rotation velocity was typically found to rise linearly at their centers [70, 71, 72]. But a group led by van den Bosch [73] showed that in many cases the large beam size of the radio telescopes did not adequately resolve the inner parts of the rotation curves, and they concluded that after correcting for beam smearing the data are on the whole consistent with expectations from CDM. Similar conclusions were reached for dwarf galaxies [74]. Swaters and collaborators showed that optical (Halpha) rotation curves of some of the LSB galaxies rose significantly faster than the radio (HI) data on these same galaxies [75] (see Fig. 2), and these rotation curves (except for F568-3) appear to be more consistent with NFW [76]. At a conference in March 2000 at the Institute for Theoretical Physics in Santa Barbara, Swaters also showed a Halpha rotation curve for the nearby dwarf galaxy DDO154, which had long been considered to be a problem for CDM [65, 66]; but the new, higher-resolution data appeared consistent with an inner density profile alpha approx 1. (5)

Very recently, a large set of high-resolution optical rotation curves has been analyzed for LSB galaxies, including many new observations [77]. The first conclusion that I reach in looking at the density profiles presented is that the NFW profile often appears to be a good fit down to about 1 kpc. However, some of these galaxies appear to have shallower density profiles at smaller radii. Of the 48 cases presented (representing 47 galaxies, since two different data sets are shown for F568-3), in a quarter of the cases the data do not probe inside 1 kpc, and in many of the remaining cases the resolution is not really adequate for definite conclusions, or the interpretation is complicated by the fact that the galaxies are nearly edge-on. Of the dozen cases where the inner profile is adequately probed, about half appear to be roughly consistent with the cuspy NFW profile (with fit alpha gtapprox 0.5), while half are shallower. This is not necessarily inconsistent with CDM, since observational biases such as seeing and slight misalignment of the slit lead to shallower profiles [78]. Perhaps it is significant that the cases where the innermost data points have the smallest errors are cuspier.

I think that this data set may be consistent with an inner density profile alpha ~ 1 but probably not steeper, so it is definitely inconsistent with the claims of the Moore group that alpha gtapprox 1.5. But very recent work by Navarro and collaborators [79] has shown that Moore's simulations did not have adequate resolution to support their claimed steep central cusp; the highest-resolution simulations appear to be consistent with NFW, or even shallower with alpha approx 0.75. Further simulations and observations, including measurement of CO rotation curves [80], may help to clarify the nature of the dark matter.

It is something of a scandal that, after all these years of simulating dark matter halos, we still do not have a quantative - or even a qualitative - theory explaining their radial density profiles. In her dissertation research [63], Risa Wechsler found that the central density profile and the value of rs are typically established during the early, rapidly merging phase of halo evolution, and that, during the usually slower mass accretion afterward, rs changes little. The mass added on the halo periphery increases Rvir, and thus the concentration cvir ident Rvir/rs. Now we want to understand this analytically. Earlier attempts to model the result of sequences of mergers (e.g., [81, 82]) led to density profiles that depend strongly on the power spectrum of initial fluctuations, in conflict with simulations (e.g. [83]). Perhaps it will be possible to improve on the simple analytic model of mass loss due to tidal stripping during satellite inspiral that we presented in [101]. Avishai Dekel and his students have recently shown that including the tidal puffing up of the inspiralling satellite before tidal stripping can perhaps account for the origin of the cusp seen in dissipationless simulations, independent of the power spectrum. They argue that the profile must be steeper than alpha = 1 as long as enough satellites make it into the halo inner regions, simply because for flatter profiles the tidal force causes dilation rather than stripping. The proper modeling of the puffing and stripping in the merger process of CDM halos may also provide a theoretical framework for understanding the observed flat cores as a result of gas processes; work on this by Ari Maller and Dekel is in progress. Reionization and feedback into the baryonic component of small satellites would make their cores puff up before merging. This could cause them to be torn apart before they penetrate into the halo centers, and thus allow alpha < 1 cores.

Another possible explanation for flatter central density profiles involving the baryonic component in galaxies has recently been proposed [84], in which the baryons form a bar that transfers angular momentum into the inner parts of the halo. It is not clear, however, that this effect could be very important in dark matter dominated dwarf and LSB galaxies that have small or nonexistent bulge components.

It would be interesting to see whether CDM can give a consistent account of the distribution of matter near the centers of big galaxies, but this is not easy to test. One might think that big bright galaxies like the Milky Way could help to test the predicted CDM profile, but the centers of such galaxies are dominated by ordinary matter (stars) rather than dark matter. (6)

5 Swaters (private communication) and Hoffman have subsequently confirmed this with better data, which they are preparing for publication. Back.

6 Navarro and Steinmetz had claimed that the Milky Way is inconsistent with the NFW profile [85], but they have now shown that LambdaCDM simulations with a proper fluctuation spectrum are actually consistent with the data [86]. Back.

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