With the growing success of gravitational lensing analysis of clusters, it has become possible to compare and test theoretical predictions against observations. With the rapid progress in high-resolution cosmological simulations of dark matter we now have a unique opportunity to directly compare properties of cluster dark matter halos derived from lensing studies. Many important physical questions with regard to the internal structures of halos, their dynamical evolution and the granularity of dark matter can now be tackled: the assembly process (role of merging sub-clusters); lensing cross sections; efficiency of lensing and super-lenses; selection effects; mass profiles; density profiles; ellipticity; alignments; abundances, and the mass concentration. We briefly outline below the results of recent studies on this topic.
7.1. Internal structure of cluster halos
In cosmological simulations of structure formation it is found that the density profiles of dark matter halos are well fit over many decades in mass from cluster mass scales down to dwarf galaxy-scales by the Navarro-Frenk-White profile (see Appendix A.3 for details). By combining strong and weak lensing constraints, as discussed above it has become possible to probe the mass profile of the clusters on scales of 0.15 Mpc, thus providing a valuable test of the universal form proposed by NFW on large scales (e.g. Okabe et al. 2010, Umetsu et al. 2011). As for the inner density profile slopes there appears to be similarly a large degree of variation, some like Cl0024+1654 (Kneib et al. 2003; Tu et al. 2008; Limousin et al. 2008) adequately fit the NFW form, others like RXJ1347-11 are found to have slopes shallower than the NFW prediction (Newman et al. 2011, Umetsu et al. 2011), while others like MS2137-23 are found to have steeper slopes (Sand et al. 2004). One caveat with the NFW prediction is that the functional form is derived from dark matter only simulations, whereas in reality it is clear that baryons in the inner regions close to the cD/BCG play a significant role both in terms of the mass budget and modifications to density profile slopes in the very center.
Lensing clusters are preferentially more significantly concentrated than all clusters (see Figure 44 and Comerford & Natarajan 2007; but also: Broadhurst et al. 2008; Oguri et al. 2009; and Meneghetti et al. 2011) and they typically tend to be outliers on the concentration-mass relationship predicted for clusters in the ΛCDM model. The origin of this enhanced concentration parameter is likely due to: i) high incidence of projected line of sight structures for massive lensing clusters; ii) elongated shapes that enhance lensing efficiency, factors that might observationally bias lensing selection; iii) baryons that could play an important role in the inner regions.
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Figure 44. Observed cluster concentrations and virial masses derived from lensing (filled circles) and X-ray (open circles) measurements. For reference, the solid lines depict the best-fit power law to our complete sample and its 1-σ scatter. The lensing concentrations appear systematically higher than the X-ray concentrations, and a Kolmogorov-Smirnov test confirms that the lensing results likely belong to a different parent distribution. This figure is from Comerford & Natarajan 2007. |
7.2. Mass function of substructure in cluster halos
Combining observed strong and weak lensing and exploiting galaxy-galaxy lensing inside clusters, it has been possible to map the granularity of the dark matter distribution (Natarajan & Kneib 1997; Natarajan & Springel 2005; Natarajan, DeLucia & Springel 2007) in clusters and compare them to predictions from the Millennium cosmological simulation (Springel et al. 2005). This is done by attributing local anisotropies in the observed shear field to the presence of dark matter sub-halos (Natarajan et al. 2009). The mass function thus derived for several clusters agrees well with that predicted entirely independently from high-resolution cosmological simulations of structure formation in the standard ΛCDM paradigm over the mass range 1011 − 1013 M⊙. The comparison was made with clusters that form in the Millennium Simulation (Springel et al. 2005). This excellent agreement of the mass function derived from these 2 independent methods demonstrates that there is no substructure problem (which was claimed earlier) on cluster scales in ΛCDM. This is a significant result as a substructure crisis has been claimed on galaxy-scales. Since ΛCDM is a self-similar theory, if the substructure problem had been endemic to the model, it would have been replicated on cluster scales. This suggests that the substructure discrepancy on galaxy-scales arises from the galaxy formation process or from some hitherto undiscovered coupling between baryons and dark matter particles. Therefore, lensing clusters have provided unanticipated insights into the dark matter model. Moving on from the Millennium Simulation, state of the art at the present time is the Mare Nostrum simulation which is promising in terms of mass resolution and larger volume probed and offers a new test-bed for comparison with lensing data from cluster surveys like the CLASH Hubble survey (Meneghetti et al. 2011).
7.3. Dynamical evolution of cluster halos
Exploiting strong and weak gravitational lensing signals inferred from panoramic Hubble Space Telescope imaging data, high-resolution reconstructions of the mass distributions are now available for clusters ranging from z = 0.2 − 0.5. Applying galaxy-galaxy lensing techniques inside clusters the fate of dark matter sub-halos can now be tracked as a function of projected cluster-centric radius out to 1-5 Mpc, well beyond the virial radius in some cases. There is now clear detection of the statistical lensing signal of dark matter sub-halos associated with both early-type and late-type galaxies in clusters. In fact, it appears now that late-type galaxies in clusters (which dominate the numbers in the outskirts but are rare in the inner regions of the cluster) also possess individual dark matter halos (Treu et al. 2002; Limousin et al. 2005, 2007; Moran et al. 2006; Natarajan et al. 2009). In the case of the cluster Cl0024+1656 that has been studied to beyond the virial radius, the mass of a fiducial dark matter halo that hosts an early-type L* galaxy varies from M = 6.3 ± 2.7 × 1011 M⊙ within r < 0.6 Mpc, to 1.3 ± 0.8 × 1012 M⊙ within r < 2.9 Mpc, and increases further to M = 3.7 ± 1.4 × 1012 M⊙ in the outskirts. The mass of a typical dark matter sub-halo that hosts an L* galaxy increases with projected cluster-centric radius in line with expectations from the tidal stripping hypothesis. Early-type galaxies appear to be hosted on average in more massive dark matter sub-halos compared to late-type galaxies. Early-type galaxies also trace the overall mass distribution of the cluster whereas late-type galaxies are biased tracers. The findings in this cluster and others are interpreted as evidence for the active re-distribution of mass via tidal stripping in galaxy clusters. Upon comparison of the masses of dark matter sub-halos as a function of projected cluster-centric with the equivalent mass function derived from clusters in the Millennium Run very good agreement is found (see Figure 45 and Natarajan, De Lucia & Springel 2007). However, simulated sub-halos appear to be more efficiently stripped than lensing observations suggest (see Figure 46). This is likely an artifact of comparison with a dark matter only simulation. Future simulations that simultaneously follow the detailed evolution of the baryonic component during cluster assembly will be needed for a more detailed comparison. Lensing has proved to be a powerful probe of how clusters assemble and grow, and it appears that our findings ratify the ΛCDM paradigm, hierarchical growth of structure and the key role played by tidal stripping during cluster assembly.
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Figure 45. Comparison between substructure mass function retrieved from the galaxy-galaxy lensing analysis (red shaded histograms) and results from haloes selected from the Millennium Simulation. The black solid line in each panel represents the average sub-halo mass function of haloes selected at the redshift of the observed lensing cluster (see text for details). The grey shaded region represents, for each value of the sub-halo mass, the min-max number of substructures found in the simulated haloes (Figure from Natarajan, DeLucia & Springel 2007). |
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Figure 46. Variation of the mass of a dark matter sub-halo that hosts an early-type L* galaxy as a function of cluster centric radius. The results from the likelihood analysis are used to derive the sub-halo mass for the galaxy-galaxy lensing results and the counterparts are derived from the Millennium simulation with an embedded semi-analytic galaxy formation model. This enables selection of dark matter halos that host a single L* galaxy akin to our assumption in the lensing analysis. The solid circles are the data points from the galaxy-galaxy lensing analysis and the solid squares are from the Millennium simulation. The upper solid square in the core region marks the value of the sub-halo mass with correction by a factor of 2 as found in Natarajan, De Lucia & Springel (2007). The solid triangle is the galaxy-galaxy lensing data point for the sub-halo associated with a late-type L* galaxy. The radial trend derived from lensing is in very good agreement with simulations and demonstrate that tidal stripping is operational with higher efficiency in the central regions as expected. |
7.4. Constraints on the nature of dark matter
While it is clear that clusters are vast repositories of dark matter, the nature of dark matter remains elusive. A plethora of astronomical observations from the early Universe to the present time are consistent with the dark matter being a cold, collision-less fluid that does not couple to baryons. However, there is potential for dark matter self-interactions and lensing observations offer a unique window to probe this further (e.g. Miralda-Escude 2002). Limits on the dark matter interaction cross section can be placed from lensing observation of clusters, however, these are currently not particularly constraining or illuminating. Two distinct arguments have been used to obtain limits that strongly support the collision-less nature of dark matter. One involves the distribution of the sizes of tidally truncated sub-halos in clusters (Natarajan et al. 2002); and the second involves estimates from the separation between the dark matter and X-ray gas in the extreme merging system, the Bullet Cluster (Clowe et al. 2006; Bradač et al. 2006) wherein they find σ / m < 4 gm−1 cm2 assuming that the two colliding sub-clusters experienced a head-on collision in the plane of the sky. Similar results were also found from the so-called “Baby Bullet” cluster (Bradač et al. 2008). Exploring these merging clusters is certainly an avenue where lensing observations may provide constraints and insights on the nature of dark matter.