ANTECEDENTES HISTÓRICOS DEL APLAZAMIENTO

After achieving a good match to the convergence map, we needed to compute the corresponding mass distribution, which in MOND is not linearly related to the

potential applied.

The average mass density ρ(< r) or the total mass (e.g., baryons and neutri- nos) of the system enclosed inside any radius rcentered onany positionwill thus be estimated from the divergence theorem. If you recall Eq 1.8

4πGM(r) = Z ∇ ·[µ_∇Φ]dV = Z µ_∇Φ_·dA, (2.6) which gives Mbary(r) +Mν(r) = Z _{∂Φ(r, θ, ψ)} ∂r µdA 4πG, (2.7)

where the surface area element dA =rsin(θ)dθdψ and the interpolating function

µ(x), where x = |∇_a0Φ|. The case µ _→ 1 corresponds to General Relativity. We use the simple interpolating function (Eq 1.7) and also the standard (Eq 1.6) for comparison with other works.

Applying Eq 2.7 to our potential model that matches the convergence map allows us to predict the matter volume density in the clusters within certain annuloids, e.g., the values given in Table 2.2. Integrating over the line of sight, we note that the convergence contours are slightly different from that of projected matter contours in non-linear gravities (cf. dashed blue contours of Fig 2.3). However, this non-linear effect appears much milder than suggested in Angus et al. (2006), mainly because gravity is strong (greater thanao) everywhere across the bullet cluster.

In order to match the observed X-ray gas mass, which is a minor contributor to the lensing map, we use the asymmetry in the calculated surface density to subtract off all the dark matter centred on the galaxies (CM1 and CM2). The key here is to notice the symmetry of the contours around the dashed line joining the centres of the two galaxy clusters (cf. Fig 2.3 upper panel). If we fold the map over the axis of symmetry subtracting the lower part from the upper part we are left with the majority of the gas since it lies significantly above the line. Then we performed a straight forward numerical integration over the areas given in Table 2.2.

The inset panel of Fig 2.3 demonstrates that this technique works well in separating the surface density of gas from the collisionless matter. The values for the gas mass for our three gravities are given in Table 2.2.

close to the axis of symmetry and thus much gas is cancelled out by other gas. For GR only we can directly compare the gas corresponding to the potential and that calculated by our subtraction method. For the main cluster, we find that integration of the surface density gives 2.3_×1013_M

within the 180kpc aperture

which is 15% more gas than estimated by symmetry. For the sub cluster we find 5.7_×1012_M

from integration within the 100kpc aperture of the gas center, 73%

more than from symmetry. As such, in MOND we can expect the gas masses to increase by similar amounts and this helps to explain the low gas masses found in the sub cluster (XR2).

The reason ourκ-map is skewed towards the gas peaks is a feature of the cored isothermal potentials. Table 2.2 shows we pack too much gas into the central 100kpc of the main cluster compared to that observed only for it to balance by 180kpc. Using a potential that correctly matches the gas density would most likely not skew the map.

A more sophistocated approach to the problem would be to go from density to potential (here we’ve done the opposite) and input all the observed matter and work backwards to find the DM distribution required to match the convergence map.

Table 2.2 also compares the B06 and C06 projected mass within a 250 kpc circular aperture centred on CM1 and CM2 with our total mass within these aper- tures for three gravities (GR, simple µ and standard µ). Clearly, these amounts of mass exceed the observed baryons in gas and galaxies over the same apertures, by a factor of 3 even in MOND. While very dense clumps of cold gas or MACHOs are still easily allowed by BBN limits to reside in galaxy clusters without many collisions, we will focus on the possibilities of fermionic particles being the dark matter in the lensing peaks.

Following Tremaine & Gunn (1979), we use the densest regions of the dark matter to set limits on the mass of active/sterile neutrinos. A cluster core made of neutrinos of mass 2eV would have a maximum density (see Eq 2.30) satisfying

ρmax

ν = 0.04Mpc−3 where we adopted the temperature of 14 keV for the main

cluster. Comparison with the regions of the highest volume density of matter shown in Fig 2.3 upper panel suggests that the relatively diffuse phase space density in the bullet cluster is still consistent with active 2eV neutrinos making up the dark matter component. This prediction is applicable to sterile neutrinos as

well. To be conservative, we avoid extrapolating our predictions to scales smaller than 45 kpc, where the weak lensing data from C06 is limited by smoothing. In B06, the convergence near in the central regions is greatly increased using strong lensing which probes this more efficiently.

Note that while a better fit to the gas mass and the lensing map could be produced by using several ellipsoidal potential components as opposed to the rigid four spheres with fixed centers here, the present model suffices as a demonstration.