Annu. Rev. Astron. Astrophys. 1999. 37: 127-189
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3.1. Mass Reconstruction with Arclets

3.1.1. Recent Developments

The use of arc(let)s and multiple images has already been discussed in detail in Fort & Mellier (1994). In the meantime, with the refurbishment of the HST, spectacular images of arc(let)s and multiply imaged galaxies have permitted enormous progress in this field. It turns out that giant arcs are no longer the strongest constraints on cluster mass distribution, because similar and even better information can be obtained with spatially resolved HST images of arclets.

The usual mass reconstruction technique with arc(let)s present in the innermost regions (close to the critical lines where arcs are located) is based on the assumption that the cluster mass density is smoothly distributed and can be expressed analytically, possibly with addition of some substructures, and on the hypothesis that the observed arc(let)s correspond to rather generic lens configurations, such as folds, cups or lisp caustics. These assumptions have already provided some convincing results with the use of ground-based images, in particular with predictions of the position of additional images (counter images) associated with arc(let)s (Hammer & Rigaud 1989, Mellier et al 1993, Kneib et al 1993, 1995). In all cases, it was found that the core radius of the dark matter distribution is small (< 50h100-1 kpc) and that its geometry is compatible with the faint extended envelopes of light surrounding the giant cluster galaxies. The new investigations using the detailed morphology of the numerous arc(let)s visible in the HST images (see Figure 3) have provided more refined constraints on the dark matter distribution on the 100 kpc-scale (Hammer et al 1997, Gioia et al 1998, Kneib el al 1996, Tyson et al 1998). They confirm the trends inferred from previous ground-based data.

One critical issue concerning the approach described above is the possible sensitivity of the result to the analytical mass profile used for the modeling. Because none of the mass distribution has an unrealistic shape, the large-scale global property is expected to be well described. However, a direct comparison of the detailed mass distribution with theoretical expectations seen in simulations is difficult. Furthermore, the redshift distribution inferred from the lensing inversion (see section 5) can be strongly affected by the properties of the analytical model. AbdelSalam et al (1998a, b) have recently proposed a non-parametric mass reconstruction algorithm which helps to overcome the limitation of analytical modeling. The technique uses arc(let)s with known redshifts as strong constraints to recover a pixelised mass map of the lensing-cluster. The pixel-mass reconstruction uses the smoothed projected light distribution of the galaxy distribution which is then pixelised exactly like the projected mass map. A fit of a pixelised Mass-to-Light ratio (M/L) permits one to relate the projected light distribution to the projected mass distribution for each pixel. The results found for A370 and A2218 (for A2218, weak and strong lensing features are used) are similar to those obtained otherwise, but this approach appears to be a very interesting alternative that permits a complete lens modeling based only on arc(let)s properties. Dye & Taylor (1998) attempted to generalize this approach in order to compute the convergence and the shear, in the weak lensing regime.

3.1.2. The X-ray/Lensing Mass Discrepancy

The use of X-ray and optical images (ground-based or from the HST) of arc(let)s reveals that the X-ray peaks are located at the center of the most massive clumps of dark matter (Kneib et al 1995, Pierre et al 1996, AbdelSalam et al 1998a, Gioia et al 1998, Hammer et al 1997, Kneib el al 1996, Ota et al 1998, Kneib et al in preparation). On the other hand, the apparent contradiction between the mass estimated from X-ray data and the lensing mass (Mlensing approx 2-3 MX), initially raised by Miralda-Escudé & Babul (1995), is not totally clear. The puzzling results obtained on several clusters, sometimes on the same cluster but analyzed by different groups, have not yet provided conclusive statements about the mass density profile and the X-ray versus dark matter dynamics. Böhringer et al (1998) find an excellent agreement between X-ray and lensing masses in A2390 which confirms the view claimed by Pierre et al (1996);, Gioia et al (1998) show that the disagreement reaches a factor of 2 at least in MS0440+0204; Schindler et al (1997) find a factor of 2-3 discrepancy for the massive cluster RX 1347.4-1145, but Sahu et al (1998) claim that the disagreement is marginal and may not exist; Ota et al (1998), Wu & Fang (1997) agree that there are important discrepancies in A370, Cl0500-24 and Cl2244-02.

As yet, there are no definitive interpretations of these contradictory results. It could be that the modeling of the gravitational mass from the X-ray distribution is not as simple. By comparing the geometry of the X-ray isophotes of A2218 to the mass isodensity contours of the reconstruction, Kneib et al (1995) found significant discrepancies in the innermost parts. The numerous substructures visible in the X-ray image have orientations which do not follow the projected mass density. They interpret these features as shocks produced by the infalling X-ray gas, which implies that the current description of the dynamical stage of the inner X-ray gas is oversimplified (see Markevitch 1997, Girardi et al 1997 for similar views). Recent ASCA observations of three lensing-clusters corroborate the view that substructures are the major source of uncertainties (Ota et al 1998).

To study this possibility in more detail, Smail et al (1997a), Allen (1998) have performed a detailed comparison between the lensing mass and X-ray mass for a significant number of lensing clusters. Both works conclude that the substructures have a significant impact on the estimate of X-ray mass. More remarkably, the X-ray clusters where cooling flows are present do not show a significant discrepancy with X-ray mass, whereas other X-ray clusters do (Allen 1998). This confirms that the discrepancy is certainly due to wrong assumptions on the physical state of the gas. These two studies provide strong presumptions that we are now close to understanding the origin of the X-ray and lensing discrepancy.

An alternative has been suggested by Navarro et al (1997) who proposed that the analytical models currently used for modeling mass distributions may be inappropriate. Instead, they argue that the universal profile of the mass distribution produced in numerical simulations of hierarchical clustering may reconcile the lensing and X-ray masses. This is an attractive possibility because the universal profile is a natural outcome from the simulations that do not use external prescriptions. However, Bartelmann (1996) emphasized that the caustics produced by the universal profile predict that radial arcs should be thicker than observed in MS2137-23 (Fort et al 1992;, Mellier et al 1994;, Hammer et al 1997) and in A370 (Smail et al 1995b), unless the sources are very thin (approx 0.6 arcsecond for MS2137-23). This is not a strong argument against the universal profile because this is possible in view of the shapes of some faint galaxies observed with HST that some distant galaxies are indeed very thin. However, it is surprising that no radial arcs produced by "thick galaxies" have been detected so far. Even a selection bias would probably favor the observation of large sources rather than small, thin and hardly visible ones. Evans & Wilkinson (1998) have explored the range of slopes of cusp-like mass profiles which would produce radial arcs with thicknesses similar to those observed. As found by Bartelmann, they too found that the universal profile does not work well, but that a more singular mass profile could be satisfactory. They do not mention, however, whether these new profiles are compatible with the numerical simulations of Navarro et al (1997).

3.1.3. Probing the Clumpiness of Clusters

HST images have also revealed the clumpiness of the cluster mass distribution on small scales. Although most of the HST images of lensing-clusters show arc(let)s with a coherent polarization on scales of 100 kpc, numerous perturbations are visible on scales of about 10 kpc. The long-range pattern is disrupted around most of the bright galaxies and shows saddle-shaped configurations as expected for clumpy mass distributions. In some extreme cases, giant arcs appear as broken filaments, probably disrupted by the halos around the brightest galaxies. With so much detail, one can therefore make a full mass reconstruction which takes into account all these clumps and possibly constrains the mass of individual cluster galaxies. For giant arcs, this has already been stressed by Kassiola et al (1992a), Mellier et al (1993), Kneib et al (1993), Dressler et al (1994), Wallington et al (1995), Kneib et al (1996). They used the breaks (or the absence of breaks) in arcs to put upper limits to the masses of a few cluster galaxies which are superimposed on the arcs. The masses found for these cluster galaxies range between 1010 Modot and 2 × 1011 Modot, with typical mass-to-light ratios between 5 and 15.

With the details visible on the HST images of arclets in A2218, AC114 or A2390, the sample of halos which can be constrained by this method is much larger and can provide more significant results. The number of details also permits use of more sophisticated methods of investigation. The most recent procedure uses the galaxy-galaxy lensing analysis. This technique is described in Section 5, but because the clumpiness of dark matter in clusters is strongly related to the halos of cluster galaxies, I present the use of galaxy-galaxy lensing for cluster galaxies in this section.

The simplest strategy is to start with an analytical potential that reproduces the general features of the shear pattern of HST images, and in a second step, to include analytical halos around the brightest cluster members in the model. In practice, additional mass components are put in the model in order to interpret the arc(let)s which cannot be easily explained by the simple mass distribution. Some guesses are made in order to pair unexplained multiple images. The colors of the arc(let)s as well as their morphology help to make these associations. This approach was proposed by Natarajan & Kneib (1997), Natarajan et al (1998). The detailed study done in AC114 by Natarajan et al indicates that about 10% of the dark matter is associated with halos of cluster galaxies. These halos have truncation radii smaller than field galaxies (rt approx 15 kpc) with a general trend of S0-galaxies to be even more truncated than the other galaxies. If this result is confirmed, it would be a direct evidence that truncation by tidal stripping is very efficient in rich clusters of galaxies. This result is somewhat contradictory with the absence of a clear decrease of rotation curves of spiral galaxies in nearby clusters (Amram et al 1993) which is interpreted as a proof that massive halos of galaxies still exist in cluster galaxies. However, it could be explained if the spirals that have been analyzed only appear to be in the cluster center because of projection effects, but actually are not located in the very dense region of the clusters where stripping is efficient.

Geiger & Schneider (1998, 1999) used a maximum likelihood analysis that explores simultaneously the distortions induced by the cluster as a whole and by its individual galaxies. They applied this analysis to the HST data of Cl0939+4713 and reached conclusions similar to those of Natarajan et al (1998). Several issues limit the reliability of their analyses and of the other methods as well (Geiger & Schneider 1998). First, depending on the slope of the mass profile of the cluster, the contributions of the cluster mass density and of the cluster galaxies may be difficult to separate. Second, it is necessary to have a realistic model for the redshift distribution of the background and foreground galaxies. Finally, the mass sheet degeneracy (see Section 3.2) is also an additional source of uncertainties. Regarding these limitations, Geiger & Schneider discuss the capability of the galaxy-galaxy lensing in clusters to provide valuable constraints on the galactic halos from the data they have in hand. Indeed, some of the issues they raised can be solved, such as the redshift distribution of the lensed galaxies. It would be interesting to take a more detailed look at how the analysis could be improved with more and better data.

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