One of the most remarkable results of N-body simulations is that virialized dark matter halos follow a universal density profile, regardless of their mass, the initial density fluctuation
Figure 3.8: As Fig. 3.6, but the clusters are shown with symbols indicating the result of the Dressler- Shectman statistics. Clusters shown as large symbols have at least 16 spectroscopic members, and we have calculated the Dressler-Shectman statistic for them. Of these, those that are shown in red have values ofP <0.05, i.e. they could have significant substructure. The small circles show clusters with less than 16 spectroscopic members, i.e. too few to compute a meaningful measure of substructure.
spectrum, and cosmological parameters (Navarro et al. 1997): ρ(x) = ρ0
x(1 +x)2 , (3.2)
where x is the distance from the center expressed in units of a scale radius, i.e. x = r/rs. Although the slope of the NFW profile is shallower at small radii than at large radii, it is not zero at the center. In this aspect, it differs from other models (such as the King profile), which call for a constant density at the center, i.e. acore. Profiles where the density increases towards the center, such as the NFW profile, are said to have acusp.
Although this does not necessarily imply that also the galaxy distribution should follow the same profile, several authors find that the NFW profile provides an adequate description for the galaxy number density (e.g. Carlberg et al. 1997; Lin et al. 2004). Others find that the galaxy distributions display a core, i.e. the number density flattens towards the center (Popesso et al. 2007a). Adami et al. (1998) note that this may be dependent on the magni- tude limit of the galaxy sample: they find that bright galaxies follow a cuspy profile, such as the NFW profile, whereas the faint galaxy population displays a core. Furthermore, in the cluster sample studies by Popesso et al. (2007a), the more massive clusters display a cusp, whereas the less massive ones have cores.
Fig. 3.9 the composite number density profile of our clusters, constructed from the galaxies brighter thanM0.1
3.5 Number density profile
Figure 3.9: Number density profile of a composite cluster, assembled by stacking all the clusters in our sample. Galaxies brighter than M0.1
r < −18.5 (but fainter than the respective BCG) are
considered, and are weighted by their cluster membership probability prad. For each radial bin, the
number of galaxies is normalized both by the bin area, and the total number of galaxies. The error bars indicate the 68% confidence intervals, derived from bootstrapping the clusters. The solid line is the best fit NFW profile.
redshift confirmation are weighted by the cluster membership probabilityprad, which is derived in radial bins from the cluster center (Sect. 3.1.2.3). Between 0.05 and 1R200, the galaxy distribution is well described by an NFW profile, with a concentration parameter ofcg = 3.6. consistent with previous works (e.g. cg = 3.7, Carlberg et al. 1997). But our study is the first to probe cluster regions very close to the BCG. Our data points clearly refute a core at the cluster center; rather, the center is cuspy, with possibly even a slight excess number density than predicted by the NFW profile that best describes the outer radii (note that due to the much larger number of galaxies, the fit is constrained almost entirely by the outer radial bins).
Popesso et al. (2007a) found that the core is present predominantly in lower mass halos. In the left panel of Fig. 3.10, we have computed the number density profile in four bins of cluster velocity dispersion. Also here, we do not find a central core. Rather, the lower mass clusters appear to be slightly more cuspy than the higher mass ones. A steeper central slope implies a higher concentration, as is expected from simulations for the dark matter profiles (Dolag et al. 2004). This is not the case for the galaxy number density: there is a slight trend of increasing concentration with cluster mass. Due to the larger error bars in the center, the best fit is constrained mainly by the outer bins, i.e. it is sensitive to radii beyond∼0.05R200. In the right panel of Fig. 3.10, we have repeated the same analysis, but splitting the cluster sample by total optical light. Again, there is no indication of a core. The trend of concentration
0.01 0.1 1 1 10 100 0.01 0.1 1 1 10 100
Figure 3.10: The number density profiles of composite clusters assembled from cluster subsamples split by mass, as traced by velocity dispersion (left) or total optical light (right). The subsamples have been chosen so that each contains a similar number of galaxies (∼12000). The blue symbols indicate the lowest mass sample (the number density has been shifted downward by a factor of 2.5 to avoid confusion); the green ones the lower intermediate masses; the black ones the higher intermediate masses (shifted upward by a factor of 2.5); the red ones the most massive ones (shifted upward by a factor of 6.3). The solid lines show the respective best-fit NFW profile.
with halo mass is less pronounced than when the clusters are split by total light (although one should keep in mind that R200 is computed fromσ).
Our results differ significantly from those of Popesso et al. (2007a), although both studies are based on SDSS data. Popesso et al. find a pronounced core, and even a drop in the number density in their central bin (at 0.05R200). The lower magnitude limit is similar in both works; our study usesMr0.1 <−18.5 incmodel magnitudes for our study, Popesso et al. use Mr <−18.5 inmodelmagnitudes. The difference is therefore not due to different distributions of bright and faint galaxies, as proposed by Adami et al. (1998). There are a number of other reasons which might explain this discrepancy:
• A significant difference in the two studies lies in the background subtraction. Although both employ a statistical subtraction method, our method takes into account spectroscopic redshifts as far as available, as well as galaxy color. Popesso et al. do not make use of the spectroscopic data, and only measure the background number density in bins of apparent magnitude, but not in color. Galaxies on the red sequence make up a large fraction of the cluster population, and taking into account this color criterion substantially boosts the contrast of clusters against the background galaxy population, so much that two-band imaging surveys have become a powerful method to identify clusters (Gladders & Yee 2000). At the cluster core, &80% of the galaxies are red (Sect. 4.4.1). At least for these galaxies, our method should provide more accurate cluster membership criteria.
• Although both studies have a similar magnitude limit in terms of absolute magnitudes, Popesso et al. probe galaxies to much fainter apparent magnitudes, as their cluster sample