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Evidently there is a consensus about the density profile of dark matter at large distances from the centers of galaxies, but it is not known whether their density profiles are "shallow" or "cuspy". What does this mean? Is this a statement about whether the density is getting larger towards the center, or the rate at which it is getting larger (or smaller)?

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The issue here is how concentrated the dark matter density profile is near the center of galaxies, and how sharply the density falls off with distance. Cuspy profiles are peaked near the center and fall-off rapidly, and shallow profiles do not exhibit a strong peak and taper off more slowly. Here I am intentionally being vague, see the references below for more details.

http://en.wikipedia.org/wiki/Cuspy_halo_problem

http://arxiv.org/abs/0910.3538

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