3.2. Galaxy and Cluster Centers
When the first high-resolution simulations of cold dark matter halos became available , they had a central density profile approximately (r) r-1, which has come to be known as the central "cusp." It was soon pointed out [32, 33] that this central behavior was inconsistent with the HI observations of dwarf galaxies that were then becoming available, which suggested that the central density is roughly constant, and also that the first cluster lensing observations appeared to be inconsistent with a r- central cusp with = 1 . Many additional rotation curves of low surface brightness (LSB) galaxies were measured, and they also were claimed to imply that the central density of these galaxies is rather flat. It was subsequently realized that the HI observations of galaxies were affected by finite resolution ("beam smearing"), and that when this was taken into account the disagreement with simulations is alleviated . More recently, higher resolution H and CO rotation curves have been obtained for a few nearby dwarf and low surface brightness galaxies , and the highest resolution two-dimensional data imply a variety of central density profiles ranging from 0 to 1, with evidence for radial motion especially in the 0 cases .
Meanwhile, theorists have done simulations with improving resolution. On the basis of simulations with tens of thousands of particles per dark matter halo, Navarro, Frenk, & White (NFW)  showed that halos from galaxy to cluster scales have density profiles that are described fairly well by the fitting function NFW(r) s(r / rs)-1(1 + r / rs)-2. Subsequently, James Bullock  and Risa Wechsler  improved our understanding of halo evolution in their dissertation research with me, which included analyzing thousands of dark matter halos in a high-resolution dissipationless cosmological simulation by Anatoly Klypin and Andrey Kravtsov. Defining the (virial) concentration cvir Rvir / rs (where Rvir is the virial radius, Bullock et al.  showed that at fixed halo mass cvir varies with redshift z as (1 + z)-1, and developed an approximate mathematical model that explained the dependence on mass and redshift. (An alternative model was proposed in , but it appears to be inconsistent with a recent simulation of small-mass halos .) Wechsler et al.  determined many halo structural merger trees, and showed that the central scale radius rs is typically set during the early phase of a halo's evolution when its mass is growing rapidly, while cvir subsequently grows with Rvir during the later slow mass accretion phase. Higher resolution simulations with roughly a million particles per halo gave central density profiles (r) r- with as steep as 1.5 , although more recent very high resolution simulations are finding less steep central profiles with 1 or shallower as r 0 . The disagreement between the theoretical 1 - 1.5 vs. the observed 0 - 1 may just reflect the effects of baryonic matter in the centers of galaxies and the difference between circular velocity and rotation speed likely to arise in gaseous disks embedded within triaxial halos .
The mean density V/2 inside the radius rV/2 (where the rotation velocity reaches half the maximum observed value) appears to be somewhat smaller than the CDM prediction with 8 = 0.9, but more consistent with CDM with 8 = 0.7  or "quintessence" models with equation of state parameter w < - 1 , but as we have mentioned these possibilities appear to be inconsistent with other data (e.g. ).
In several clusters of galaxies, after removing the baryonic contribution the central dark matter profile appears to be rather shallow, with 0.35 for cluster MS2137-23 . The apparent disagreement with CDM worsens if adiabatic compression of the dark matter by the infalling baryons is considered . However, dynamical friction of the dense galaxies moving in the smooth background of the cluster dark matter counteracts the effect of adiabatic compression, and can lead to energy transfer from the galaxies to the dark matter which heats up the central cuspy dark matter and softens the cusp. N-body simulations  show that the dark matter distribution can become very shallow, with 0.3 for a cluster like MS2137, in agreement with observations. Taking the triaxial shapes of cluster centers  into account may also help to bring theory and observations into agreement .