8.3 Strong and weak lensing by galaxy clusters

Gravitational lensing is a complementary technique to the hydrostatic equilibrium of the X-ray emitting gas (Section 6.6.4) to probe the mass distribution of galaxy clusters. Since clusters are the most recently formed structures, they could be slightly out of equilibrium, which makes gravitational lensing extremely interesting as this technique is fully independent from the relaxed or unrelaxed nature of the lens. A famous example of such a clearly unrelaxed object is the cluster 1E0657-56, known as the Bullet Cluster (Figure 42View Image). It is actually a pair of clusters which collided at high speed (> 3100 km ∕s) at z = 0.3. In the collision, the dissipational hot X-ray emitting gas which dominates the baryonic matter was separated from the negligible and collisionless galaxies and any presumed collisionless dark matter. Using background galaxies to map the shear field, the convergence map of the cluster was provided by [102Jump To The Next Citation Point], a convergence very conspicuously centered where the collisionless dark matter should be 59. It would appear difficult to reconstruct such a configuration merely by modifying gravity, but the non-linearity of MOND does not guarantee that the convergence from a two-center baryonic distribution would indeed be centered on the two centers. Indeed, while the linear relation between the matter density and the gravitational potential implies that the convergence parameter is a direct measurement of the projected surface density in the weak-field limit of GR, this is not the case anymore in MOND due to the non-linearity of the modified Poisson equation. Actually, it has been shown that, in MOND, it is possible to have a non-zero convergence along a line of sight where there is zero projected matter [15]. What is more, the gravitational environment might play an important role on the internal gravitational field too [113, 259], and the additional degrees of freedom of the various relativistic theories might play a non-negligible role, especially in non-static situations [112]. Neglecting possible effects of the gravitational environment and non-trivial features of the additional fields of the relativistic theories out of equilibrium, i.e., simply assuming that the physical metric is given by Ψ = − Φ in Eq. 73View Equation, and that Φ obeys Eq. 17View Equation, a MOND model of the bullet cluster was produced [17Jump To The Next Citation Point], in which a parametrized potential was fitted to the convergence map to then determine the underlying mass distribution from Eq. 17View Equation. The result is displayed in Figure 43View Image, and exactly the same conclusion was reached by going from the baryonic density to the convergence map [147]. The main conclusions are that (i) the amount of residual missing mass needed to account for the convergence map of the bullet cluster is the same as in all other clusters (Section 6.6.4 and [449]), but that (ii) if it is made of dark baryons, they must be in a collisionless form, since the residual missing mass is centered on the collisionless galaxies and not on the dissipational hot gas. The dense molecular gas clouds proposed by Milgrom [310] (see discussion in Section 6.6.4) satisfy this criterion, and would mostly behave like individual stars. Like in most clusters with T > 4 keV, ordinary neutrinos with a 2 eV mass would be broadly sufficient to account for the missing mass deduced from weak lensing (and, obviously, heavier exotic hot dark matter particles such as 10 eV sterile neutrinos would do the job too).

For TeVeS (Section 7.4) and GEA (Section 7.7), the growth of the spatial part of the vector perturbation in the course of cosmological evolution can successfully seed the growth of baryonic structures, just as dark matter does, and it is possible to reconstruct the gravitational field of the bullet cluster without any extra matter but with a substantial contribution from the vector field. However, why the dynamical evolution of the vector field perturbations would lead to precisely such a configuration remains unclear. Similarly, the massive scalar field of Section 7.6 or the monopolar part of the dipolar DM of Section 7.9 could, in principle, provide the off-centered missing mass too, but again, why they would appear distributed as they do remains unclear, especially in the case of dipolar DM, which is supposed to cluster only very weakly, and, in principle, not to appear as densely clustered. Whether the twin matter of BIMOND (Section 7.8) could help providing the right convergence map also remains to be seen, while for non-local models (Section 7.10), there is a strong dependence on the past light-cone, meaning that recently-disturbed systems, such as the Bullet, may be far from the static MOND limit (but in that case, it would not be clear why all the other clusters from Section 6.6.4 exhibit the same amount of residual missing mass). So, while the bullet cluster clearly does not represent the MOND-killer that it was supposed to be, explaining its convergence map remains an outstanding challenge for all MOND theories. However, the bullet cluster also represents an outstanding challenge to ΛCDM (see Section 4.2), due to its high collision speed [249Jump To The Next Citation Point]. In that respect, MOND is much more promising [16Jump To The Next Citation Point].

On the other hand, a comprehensive weak lensing mass reconstruction of the rich galaxy cluster Cl0024+17 at z = 0.4 [211Jump To The Next Citation Point] has been argued to have revealed the first dark matter structure that is offset from both the gas and galaxies in a cluster. This structure is ringlike, located between r ∼ 60 ′′ and r ∼ 85′′. It was, again, argued to be the result of a collision of two massive clusters 1 – 2 Gyr in the past, but this time along the line-of-sight. It has also been argued [211Jump To The Next Citation Point] that this offset was hard to explain in MOND. Assuming that this ringlike structure is real and not caused by instrumental bias or spurious effects in the weak lensing analysis (due, e.g., to the unification of strong and weak-lensing or to the use of spherical/circular priors), and that cluster stars and galaxies do not make up a high fraction of the mass in the ring (which would be too faint to observe anyway), it has been shown that, for certain interpolating functions with a sharp transition, this is actually natural in MOND [325]. A peak in the phantom dark matter distribution generically appears close to the transition radius of MOND rt = (GM ∕a0 )1∕2, especially when most of the mass of the system is well-contained inside this radius (which is the case for the cluster Cl0024+17). This means that the ring in Cl0024+17 could be the first manifestation of this pure MOND phenomenon, and thus be a resounding success for MOND in galaxy clusters. However, the sharpness of this phantom dark matter peak strongly depends on the choice of the μ-function, and for some popular ones (such as the “simple” μ-function) the ring cannot be adequately reproduced by this pure MOND phenomenon. In this case, a collisional scenario would be needed in MOND too, in order to explain the feature as a peak of cluster dark matter. Indeed, we already know that there is a mass discrepancy in MOND clusters, and we know that this dark matter must be in collisionless form (e.g., neutrinos or dense clumps of cold gas). So the results of the simulation with purely collisionless dark particles [211] would surely be very similar in MOND gravity. Again, it was shown that the density of missing mass was compatible with 2 eV ordinary neutrinos, like in most clusters with T > 4 keV [139]. Finally, let us note that strong lensing was also recently used as a robust probe of the matter distribution on scales of 100 kpc in galaxy clusters, especially in the cluster Abell 2390 [149]. A residual missing mass was again found, compatible with the densities provided by fermionic hot dark matter candidates only for masses of ∼ 10 eV and heavier. All in all, the problem posed by gravitational lensing from galaxy clusters is thus very similar to the one posed by the temperature profiles of their X-ray emitting gas (Section 6.6.4), and remains one of the two main current problems of MOND, together with its problem at reproducing the CMB anisotropies (see Section 9.2).

Finally, let us note in passing that another (non-lensing) test of relativistic MOND theories in galaxy clusters has been performed by analysing the gravitational redshifts of galaxies in 7800 galaxy clusters [489], which were originally found to be difficult to reconcile with MOND: however, this original analysis assumed a distribution of residual missing mass in MOND by simply scaling down the Newtonian dynamical mass represented by a NFW halo by a factor 0.8, and the analysis confused the interpolating functions μ(x) and &tidle;μ(s) (see Section 6.2). A subsequent analysis [41] showed that these gravitational redshifts were in accordance with relativistic MOND when the correct residual mass and acceptable μ-functions were used.

View Image

Figure 42: The bullet cluster 1E0657-56. The hot gas stripped from both subclusters after the collision is colored red-yellow. The green and white curves are the isocontours of the lensing convergence parameter κ (Eq. 113View Equation). The two peaks of κ do not coincide with those of the gas, which makes up most of the baryonic mass, but are skewed in the direction of the galaxies. The white bar corresponds to 200 kpc. Image courtesy of Clowe, reproduced by permission from [102Jump To The Next Citation Point], copyright by AAS.
View Image

Figure 43: A MOND model of the bullet cluster [17Jump To The Next Citation Point]. The fitted κ-map (solid black lines) is overplotted on the convergence map of [102] (dotted red lines). The four centers of the parametrized potential used are the red stars. Also overplotted (blue dashed line) are two contours of surface density. Note slight distortions compared to the contours of κ. The green shaded region corresponds to the clustering of 2 eV neutrinos. Inset: The surface density of the gas in the model of the bullet cluster. Image reproduced by permission from [17], copyright by AAS.

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