Abstract
A detailed analysis of the 2006 November 15 data release Xray surface density Σmap and the strong and weak gravitational lensing convergence κmap for the Bullet Cluster 1E0657558 is performed and the results are compared with the predictions of a modified gravity (MOG) and dark matter. Our surface density Σmodel is computed using a King βmodel density, and a mass profile of the main cluster and an isothermal temperature profile are determined by the MOG. We find that the main cluster thermal profile is nearly isothermal. The MOG prediction of the isothermal temperature of the main cluster is T= 15.5 ± 3.9 keV, in good agreement with the experimental value T= 14.8^{+2.0}_{−1.7} keV. Excellent fits to the 2D convergence κmap data are obtained without nonbaryonic dark matter, accounting for the 8σ spatial offset between the Σmap and the κmap reported in Clowe et al. The MOG prediction for the κmap results in two baryonic components distributed across the Bullet Cluster 1E0657558 with averaged mass fraction of 83 per cent intracluster medium (ICM) gas and 17 per cent galaxies. Conversely, the Newtonian dark matter κmodel has on average 76 per cent dark matter (neglecting the indeterminant contribution due to the galaxies) and 24 per cent ICM gas for a baryon to dark matter mass fraction of 0.32, a statistically significant result when compared to the predicted Λcold dark matter cosmological baryon mass fraction of 0.176^{+0.019}_{−0.012}.
1 INTRODUCTION
1.1 The question of missing mass
Galaxy cluster masses have been known to require some form of energy density that makes its presence felt only by its gravitational effects since Zwicky (1933) analysed the velocity dispersion for the Coma cluster. Application of the Newtonian 1/r^{2} gravitational force law inevitably points to the question of missing mass, and may be explained by dark matter (Oort 1932). The amount of nonbaryonic dark matter required to maintain consistency with Newtonian physics increases as the mass scale increases so that the masstolight ratio of clusters of galaxies exceeds the masstolight ratio for individual galaxies by as much as a factor of ∼6, which exceeds the masstolight ratio for the luminous inner core of galaxies by as much as a factor of ∼10. In clusters of galaxies, the dark matter paradigm leads to a masstolight ratio as much as 300 M_{⊙}/L_{⊙}. In this scenario, nonbaryonic dark matter dominates over baryons outside the cores of galaxies (by ≈80–90 per cent).
Dark matter has dominated cosmology for the last five decades, although the search for dark matter has to this point come up empty. Regardless, one of the triumphs of cosmology has been the precise determination of the standard (powerlaw flat, ΛCDM) cosmological model parameters. The highly anticipated thirdyear results from the Wilkinson Microwave Anisotropy Probe team have determined the cosmological baryon mass fraction (to nonbaryonic dark matter) to be 0.176^{+0.019}_{−0.012} (Spergel et al. 2006). This ratio may be inverted, so that for every gram of baryonic matter, there are 5.68 grams of nonbaryonic dark matter – at least on cosmological scales. There seems to be no evidence of dark matter on the scale of the Solar system, and the cores of galaxies also seem to be devoid of dark matter.
Galaxy clusters and superclusters are the largest virialized (gravitationally bound) objects in the Universe and make ideal laboratories for gravitational physicists. The data come from three sources:
Xray imaging of the hot intracluster medium (ICM).
Hubble, Spitzer and Magellan telescope images of the galaxies comprising the clusters.
strong and weak gravitational lensing surveys which may be used to calculate the mass distribution projected on to the sky (within a particular theory of gravity).
The alternative to the dark matter paradigm is to modify the Newtonian 1/r^{2} gravitational force law so that the ordinary (visible) baryonic matter accounts for the observed gravitational effect. An analysis of the Bullet Cluster 1E0657558 surface density Σmap and convergence κmap data by Angus, Famaey & Zhao (2006) and Angus et al. (2007) based on Milgrom's Modified Newtonian Dynamics (MOND) model (Milgrom 1983; Sanders & McGaugh 2002) and Bekenstein's relativistic version of MOND (Bekenstein 2004) failed to fit the data without dark matter. More recently, further evidence that MOND needs dark matter in weak lensing of clusters has been obtained by Takahashi & Chiba (2007). Problems with fitting Xray temperature profiles with Milgrom's MOND model without dark matter were shown in Aguirre, Schaye & Quataert (2001), Sanders (2006), Pointecouteau & Silk (2005) and Brownstein & Moffat (2006b). Neutrino matter with an electron neutrino mass m_{ν}∼ 2 eV can fit the Bullet Cluster data (Angus et al. 2006, 2007; Sanders 2006). This mass is at the upper bound obtained from observations. The Karlsruhe Tritium Neutrino (KATRIN) experiment will be able to falsify 2 eV electron neutrinos at 95 per cent confidence level within months of taking data in 2009.
The theory of modified gravity – or MOG model – based on a covariant generalization of Einstein's theory with auxiliary (gravitational) fields in addition to the metric was proposed in Moffat (2005, 2006a) including metric–skew–tensor gravity (MSTG) theory and scalar–tensor–vector gravity (STVG) theory. Both versions of MOG, MSTG and STVG modify the Newtonian 1/r^{2} gravitational force law in the same way so that it is valid at small distances, say at terrestrial scales.
Brownstein & Moffat (2006a) applied MOG to the question of galaxy rotation curves, and presented the fits to a large sample of over 100 low surface brightness (LSB), high surface brightness (HSB) and dwarf galaxies. Each galaxy rotation curve was fit without dark matter using only the available photometric data (stellar matter and visible gas) and alternatively a twoparameter mass distribution model which made no assumption regarding the masstolight ratio. The results were compared to MOND and were nearly indistinguishably right out to the edge of the rotation curve data, where MOND predicts a forever flat rotation curve, but MOG predicts an eventual return to the familiar 1/r^{2} gravitational force law. The masstolight ratio varied between 2 and 5 M_{⊙}/L_{⊙} across the sample of 101 galaxies in contradiction to the dark matter paradigm which predicts a masstolight ratio typically as high as 50 M_{⊙}/L_{⊙}.
In a sequel, Brownstein & Moffat (2006b) applied MOG to the question of galaxy cluster masses, and presented the fits to a large sample of over 100 Xray galaxy clusters of temperatures ranging from 0.52 keV (six million kelvin) to 13.29 keV (150 million kelvin). For each of the 106 galaxy clusters, the MOG provided a parameterfree prediction for the ICM gas mass profile, which reasonably matched the Xray observations (King βmodel) for the same sample compiled by Reiprich (2001) and Reiprich & Böhringer (2002). The MOND predictions were presented for each galaxy cluster, but failed to fit the data. The Newtonian dark matter result outweighed the visible ICM gas mass profiles by an order of magnixstude.
In the Solar system, the Doppler data from the Pioneer 10 and 11 spacecraft suggest a deviation from the Newtonian 1/r^{2} gravitational force law beyond Saturn's orbit. Brownstein & Moffat (2006c) applied MOG to fit the available anomalous acceleration data (Nieto & Anderson 2005) for the Pioneer 10/11 spacecraft. The solution showed a remarkably low variance of residuals corresponding to a reduced χ^{2} per degree of freedom of 0.42, signalling a good fit. The magnitude of the satellite acceleration exceeds the MOND critical acceleration, negating the MOND solution (Sanders 2006). The dark matter paradigm is severely limited within the Solar system by stability issues of the Sun, and precision gravitational experiments including satellite, lunar laser ranging and measurements of the Gaussian gravitational constant and Kepler's law of planetary motion. Without an actual theory of dark matter, no attempt to fit the Pioneer anomaly with dark matter has been suggested. Remarkably, MOG provides a closely fit solution to the Pioneer 10/11 anomaly and is consistent with the accurate equivalence principle, all current satellite, laser ranging observations for the inner planets and the precession of perihelion for all of the planets.
A fit to the acoustical wave peaks observed in the cosmic microwave background (CMB) data using MOG has been achieved without dark matter. Moreover, a possible explanation for the accelerated expansion of the Universe has been obtained in MOG (Moffat 2007).
Presently, on both an empirical and theoretical level, MOG is the most successful alternative to dark matter. The successful application of MOG across scales ranging from clusters of galaxies (Mpc) to HSB, LSB and dwarf galaxies (kpc), to the Solar system (au) provides a clue to the question of missing mass. The apparent necessity of the dark matter paradigm may be an artefact of applying the Newtonian 1/r^{2} gravitational force law to scales where it is not valid, where a theory such as MOG takes over. The ‘excess gravity’ that MOG accounts for may have nothing to do with the hypothesized missing mass of dark matter. But how can we distinguish the two? In most observable systems, gravity creates a central potential, where the baryon density is naturally the highest. So in most situations, the matter which is creating the gravity potential occupies the same volume as the visible matter. Clowe et al. (2006b) describe this as a degeneracy between whether gravity comes from dark matter or from the observed baryonic mass of the hot ICM and visible galaxies where the excess gravity is due to MOG. This degeneracy may be split by examining a system that is out of steady state, where there is spatial separation between the hot ICM and visible galaxies. This is precisely the case in galaxy cluster mergers: the galaxies will experience a different gravitational potential created by the hot ICM than if they were concentrated at the centre of the ICM. Moffat (2006b) considered the possibility that MOG may provide the explanation of the recently reported ‘extra gravity’ without nonbaryonic dark matter which has so far been interpreted as direct evidence of dark matter. The research presented here addresses the fullsky data product for the Bullet Cluster 1E0657558, recently released to the public (Clowe et al. 2006c).
1.2 The latest results from the Bullet Cluster 1E0657558
The Chandra Peer Review has declared the Bullet Cluster 1E0657558 to be the most interesting cluster in the sky. This system, located at a redshift z= 0.296, has the highest Xray luminosity and temperature (T= 14.1 ± 0.2 keV, ∼1.65 × 10^{8} K), and demonstrates a spectacular merger in the plane of the sky exhibiting a supersonic shock front, with Mach number as high as 3.0 ± 0.4 (Markevitch 2006). The Bullet Cluster 1E0657558 has provided a rich data set in the Xray spectrum which has been modelled to high precision. From the extra long 5.2 × 10^{5} s Chandra space satellite Xray image, the surface mass density, Σ(x, y), was reconstructed providing a highresolution map of the ICM gas (Clowe et al. 2006b). The Σmap, shown in a false colour composite map (in red) in Fig. 1, is the result of a normalized geometric mass model based on a 16 × 16 arcmin field in the plane of the sky that covers the entire cluster and is composed of a square grid of 185 × 185 pixels (∼8000 data points).1 Our analysis of the Σmap provides first published results for the King βmodel density, as shown in Table 2, and mass profiles of the main cluster and the isothermal temperature profile as determined by MOG, as shown in Fig. 9.
Based on observations made with the NASA/ESA Hubble Space Telescope (HST), the Spitzer Space Telescope and the 6.5m Magellan telescopes, Clowe et al. (2006a), Bradač et al. (2006) and Clowe et al. (2006b) report on a combined strong and weak gravitational lensing survey used to reconstruct a highresolution, absolutely calibrated convergence κmap of the region of sky surrounding Bullet Cluster 1E0657558, without assumptions on the underlying gravitational potential. The κmap is shown in the false colour composite map (in blue) in Fig. 1. The gravitational lensing reconstruction of the convergence map is a remarkable result, considering it is based on a catalogue of strong and weak lensing events and relies on a thorough understanding of the distances involved – ranging from the redshift of the Bullet Cluster 1E0657558 (z= 0.296) which puts it at a distance of the order of one billion parsec away. Additionally, the typical angular diameter distances to the lensing event sources (z∼ 0.8 to ∼1.0) are several billion parsec distant! This is perhaps the greatest source of error in the κmap which limits its precision. Regardless, we are able to learn much about the convergence map and its peaks.
King model  Main cluster  Subcluster 
β  0.803 ± 0.013  … 
r_{c}  278.0 ± 6.8 kpc  … 
ρ_{0}  3.34 × 10^{5} M_{⊙} kpc^{−2}  … 
M_{gas}  3.87 × 10^{14} M_{⊙}  2.58 × 10^{13} M_{⊙} 
r_{out}  2620 kpc  … 
King model  Main cluster  Subcluster 
β  0.803 ± 0.013  … 
r_{c}  278.0 ± 6.8 kpc  … 
ρ_{0}  3.34 × 10^{5} M_{⊙} kpc^{−2}  … 
M_{gas}  3.87 × 10^{14} M_{⊙}  2.58 × 10^{13} M_{⊙} 
r_{out}  2620 kpc  … 
Both the Σmap and the κmap are 2D distributions based on lineofsight integration. We are fortunate, indeed, that the Bullet Cluster 1E0657558 is not only one of the hottest, most supersonic, most massive cluster mergers seen, but the plane of the merger is also aligned with our sky! As exhibited in Fig. 1, the latest results from the Bullet Cluster 1E0657558 show, beyond a shadow of doubt, that the Σmap, which is a direct measure of the hot ICM gas, is offset from the κmap, which is a direct measure of the curvature (convergence) of space–time. The fact that the κmap is centred on the galaxies, and not on the ICM gas mass, is certainly evidence of either ‘missing mass’, as in the case of the dark matter paradigm, or ‘extra gravity’, as in the case of MOG. Clowe et al. (2006b) state
“one would expect that this (the offset Σ and κpeaks) indicates that dark matter must be present regardless of the gravitational force law, but in some alternative gravity models, the multiple peaks can alter the lensing surface potential so that the strength of the peaks is no longer directly related to the matter density in them. As such, all of the alternative gravity models have to be tested individually against the observations.”
The data from the Bullet Cluster 1E0657558 provide a laboratory of the greatest scale, where the degeneracy between ‘missing mass’ and ‘extra gravity’ may be distinguished. We demonstrate that in MOG, the convergence κmap correctly accounts for all of the baryons in each of the main and subclusters, including all of the galaxies in the regions near the main central dominant (CD) and the subcluster's brightest central galaxy (BCG), without nonbaryonic dark matter.
1.3 How MOG may account for the Bullet Cluster 1E0657558 evidence
We will show how MOG may account for the Bullet Cluster 1E0657558 evidence, without dominant dark matter, by deriving the modifications to the gravitational lensing equations from MOG. Concurrently, we will provide comparisons to the equivalent Einstein–Newton results utilizing dark matter to explain the missing mass. The paper is divided as follows.
Section 2 is dedicated to the theory used to perform all of the derivations and numerical computations and is separated into three pieces: Section 2.1 presents the Poisson equations in MOG for a nonspherical distribution of matter and the corresponding derivation of the acceleration law and the dimensionless gravitational coupling, . Section 2.2 presents the King βmodel density profile, ρ. Section 2.3 presents the derivation of the weighted surface mass density, , from the convergence . The effect of the dimensionless gravitational coupling, , is to carry more weight away from the centre of the system. If the galaxies occur away from the centre of the ICM gas, as in the Bullet Cluster 1E0657558, their contribution to the κmap will be weighted by as much as a factor of 6 as shown in Fig. 7.
Section 3 is dedicated to the surface density Σmap from the Xray imaging observations of the Bullet Cluster 1E0657558 from the 2006 November 15 data release Σmap (Clowe et al. 2006c). Section 3.1 presents a visualization of the Σmap and our low χ^{2} bestfitting King βmodel (neglecting the subcluster). Section 3.2 presents a determination of for the Bullet Cluster 1E0657558 based on the (>100) galaxy cluster survey of Brownstein & Moffat (2006b). Section 3.3 presents the cylindrical mass profile, , about the main cluster Σmap peak. Section 3.4 presents the isothermal spherical mass profile from which we have derived a parameterfree (unique) prediction for the Xray temperature of the Bullet Cluster 1E0657558 of T= 15.5 ± 3.9 keV which agrees with the experimental value of T= 14.8^{+2.0}_{−1.7} keV, within the uncertainty. Section 3.5 presents the details of the separation of the Σmap into the main cluster and subcluster components.
Section 4 is dedicated to the convergence κmap from the weak and strong gravitational lensing survey of the Bullet Cluster 1E0657558 from the 2006 November 15 data release κmap (Clowe et al. 2006c). Section 4.1 presents a visualization of the κmap and some remarks on the evidence for dark matter or conversely, extra gravity. Section 4.2 presents the MOG solution which uses a projective approximation for the density profile to facilitate numerical integration, although the full nonspherically symmetric expressions are provided in Section 2.1. To a zerothorder approximation, we present the spherically symmetric solution, which does not fit the Bullet Cluster 1E0657558. In our next approximation, the density profile of the Bullet Cluster 1E0657558, ρ, is taken as the bestfitting King βmodel (spherically symmetric), but the dimensionless gravitational coupling, , also assumed to be spherically symmetric, has a different centre – in the direction closer to the subcluster. We determined that our bestfitting κmodel corresponds to a location of the MOG centre 140 kpc away from the main cluster Σmap towards the subcluster Σmap peak. Section 4.3 presents the MOG prediction of the galaxy surface mass density, computed by taking the difference between the κmap data and our κmodel of the Σmap (ICM gas) data. 4.4 presents the mass profile of dark matter computed by taking the difference between the κmap data and scaled Σmap (ICM gas) data. This corresponds to the amount of dark matter (not a falsifiable prediction) necessary to explain the Bullet Cluster 1E0657558 data using Einstein/Newton gravity theory.
2 THE THEORY
2.1 Modified gravity
Analysis of the recent Xray data from the Bullet Cluster 1E0657558 (Markevitch 2006), probed by our computation in MOG, provides direct evidence that the convergence κmap reconstructed from strong and weak gravitational lensing observations (Clowe et al. 2006a; Bradač et al. 2006; Clowe et al. 2006b) correctly accounts for all of the baryons in each of the main and subclusters including all of the galaxies in the regions near the main CD and the subcluster's BCG. The available baryonic mass, in addition to a secondrank skew symmetric tensor field (in MSTG), or massive vector field (in STVG), are the only properties of the system which contribute to the running gravitational coupling, G(r). It is precisely this effect which allows MOG to fulfil its requirement as a relativistic theory of gravitation to correctly describe astrophysical phenomena without the necessity of dark matter (Moffat 2005, 2006a,b). MOG contains a running gravitational coupling – in the infrared (IR) at astrophysical scales – which has successfully been applied to galaxy rotation curves (Brownstein & Moffat 2006a), Xray cluster masses (Brownstein & Moffat 2006b), and is within limits set by Solar system observations (Brownstein & Moffat 2006c).
The weak field, point particle spherically symmetric acceleration law in MOG is obtained from the action principle for the relativistic equations of motion of a test particle in Moffat (2005, 2006a). The weak field point particle gravitational potential for a static spherically symmetric system consists of two parts:
where and denote the Newtonian and Yukawa potentials, respectively. M denotes the total constant mass of the system and μ denotes the effective mass of the vector particle in STVG. The Poisson equations for Φ_{N}(r) and Φ_{Y}(r) are given by and respectively. For sufficiently weak fields, we can assume that the Poisson equations (4) and (5) are uncoupled and determine the potentials Φ_{N}(r) and Φ_{Y}(r) for nonspherically symmetric systems, which can be solved analytically and numerically. The Green's function for the Yukawa Poisson equation is given by The full solutions to the potentials are given by and For a delta function source density we obtain the point particle solutions of equations (2) and (3).The modified acceleration law is obtained from
Let us set where M_{0} is a constant and G_{N} denotes Newton's gravitational constant. From equations (7), (8) and (10), we obtain We can write this equation in the form: where For a static spherically symmetric point particle system, we obtain using equation (9) the effective modified acceleration law: whereHere, M is the total baryonic mass of the system and we have set μ= 1/r_{0} and r_{0} is a distance range parameter. We observe that G(r) → G_{N} as r→ 0.
For the spherically symmetric static solution in MOG, the modified acceleration law equation (16) is determined by the baryon density and the parameters G_{N}, M_{0} and r_{0}. However, the parameters M_{0} and r_{0} are scaling parameters that vary with distance, r, according to the field equations for the scalar fields ω(r) ∝M_{0}(r) and μ(r) = 1/r_{0}(r), obtained from the action principle (Moffat 2006a). In principle, solutions of the effective field equations for the variations of ω(r) and μ(r) can be derived given the potentials V(ω) and V(μ) in equations (24) and (26) of Moffat (2006a). However, at present the variations of ω(r) and μ(r) are empirically determined from the galaxy rotation curve and Xray cluster mass data (Brownstein & Moffat 2006a,b).
In Fig. 2, we display the values of M_{0} and r_{0} for spherically symmetric systems obtained from the published fits to the galaxy rotation curves for dwarf galaxies, elliptical galaxies and spiral galaxies, and the fits to (normal and dwarf) Xray cluster masses. A complete continuous relation between M_{0} and r_{0} at all mass scales needs to be determined from the MOG that fits the empirical mass scales and distance scales shown in Fig. 2. This will be a subject of future investigations.
The modifications to gravity leading to equation (17) would be negated by the vanishing of either M_{0}→ 0 or the r_{0}→∞ IR limit. These scale parameters are not to be taken as universal constants, but are sourcedependent and scale according to the system. This is precisely the opposite case in MOND, where the Milgrom acceleration, a_{0Milgrom}= 1.2 × 10^{−8} cm s^{−2} (Milgrom 1983), is a phenomenologically derived universal constant – arising from a classical modification to the Newtonian potential. Additionally, MOND has an arbitrary function, μ(x), which should be counted among the degrees of freedom of that theory. Conversely, MOG does not arise from a classical modification but from the equations of motion of a relativistic modification to general relativity.
The cases we have examined until now have been modelled assuming spherical symmetry. These include 101 galaxy rotation curves (Brownstein & Moffat 2006a), 106 Xray cluster masses (Brownstein & Moffat 2006b) and the gravitational solution to the Pioneer 10/11 anomaly in the Solar system (Brownstein & Moffat 2006c). In applications of MOG, we vary the gravitational coupling G, the vector field φ_{μ} coupling to matter and the effective mass μ of the vector field according to a renormalization group (RG) flow description of quantum gravity theory formulated in terms of an effective classical action (Moffat 2005, 2006a). Large IR renormalization effects may cause the effective coupling constants to run with momentum. A cutoff leads to spatially varying values of G, M_{0} and r_{0} and these values increase in size according to the mass scale and distance scale of a physical system (Reuter & Weyer 2006).
The spatially varying dimensionless gravitational coupling is given by
whereG_{N}= 6.6742 × 10^{−11} m^{3} kg s^{−2} is the ordinary (terrestrial) Newtonian gravitational constant measured experimentally,2
M(r) is the total (ordinary) mass enclosed in a sphere or radius, r. This may include all of the visible (Xray) ICM gas and all of the galactic (baryonic) matter, but none of the nonbaryonic dark matter.
M_{0} is the MOG mass scale (usually measured in units of [M_{⊙}]),
r_{0} is the MOG range parameter (usually measured in units of [kpc]).
The dimensionless gravitational coupling, , of equation (18), approaches an asymptotic value as r→∞:
where M is the total baryonic mass of the system. Conversely, taken in the limit of r≪r_{0}, G(r) → G_{N} or down to the Planck length.In order to apply the MOG model of equation (18) to the Bullet Cluster 1E0657558, we must first generalize the spherically symmetric case. Our approach follows a sequence of approximations:

Treat the subcluster as a perturbation, and neglect it as a zerothorder approximation.

Treat the subcluster as a perturbation, and shift the origin of the gravitational coupling, , towards the subcluster (towards the centreofmass of the system).

Use the concentric cylinder mass M(R) as an approximation for M(r), and shift that towards the subcluster (towards the MOG centre).

Treat the subcluster as a perturbation, and utilize the isothermal sphere model to approximate M(R) and shift that towards the subcluster [towards the centreofmass of the system – where the origin of is located].
2.2 The King βmodel for the Σmap
Starting with the zerothorder approximation, we neglect the subcluster, and perform a best fit to determine a spherically symmetric King βmodel density of the main cluster. We assume the main cluster gas (neglecting the subcluster) is in nearly hydrostatic equilibrium with the gravitational potential of the galaxy cluster. Within a few core radii, the distribution of gas within a galaxy cluster may be fit by a King ‘βmodel’. The observed surface brightness of the Xray cluster can be fit to a radial distribution profile (Chandrasekhar 1960; King 1966):
resulting in bestfitting parameters, β and r_{c}. A deprojection of the βmodel of equation (20) assuming a nearly isothermal gas sphere then results in a physical gas density distribution (Cavaliere & FuscoFemiano 1976): where ρ(r) is the ICM mass density profile and ρ_{0} denotes the central density. The mass profile associated with this density is given by where M(r) is the total mass contained within a sphere of radius r. Galaxy clusters are observed to have finite spatial extent. This allows an approximate determination of the total mass of the galaxy cluster by first solving equation (21) for the position, r_{out}, at which the density, ρ(r_{out}), drops to ≈10^{−28} g cm^{−3}, or 250 times the mean cosmological density of baryons: Then, the total mass of the ICM gas may be taken as M_{gas}≈M(r_{out}): To make contact with the experimental data, we must calculate the surface mass density by integrating ρ(r) of equation (21) along the lineofsight: where Substituting equation (21) into equation (25), we obtain This integral becomes tractable by making a substitution of variables: so that where we have made use of the hypergeometric function, F([a, b], [c], z). Substituting equation (28) into equation (29) gives We next define which we substitute into equation (30), yielding In the limit z_{out}≫r_{c}, the hypergeometric functions simplify to Γ functions, and equations (31) and (32) result in the simple, approximate solutions: and which we may, in principle, fit to the Σmap data to determine the King βmodel parameters, β, r_{c} and ρ_{0}.2.3 Deriving the weighted surface mass density from the convergence κmap
The goal of the strong and weak lensing survey of Clowe et al. (2006a), Bradač et al. (2006) and Clowe et al. (2006b) was to obtain a convergence κmap by measuring the distortion of images of background galaxies (sources) caused by the deflection of light as it passes the Bullet Cluster 1E0657558 (lens). The distortions in image ellipticity are only measurable statistically with large numbers of sources. The data were first corrected for smearing by the point spread function in the image, resulting in a noisy, but direct, measurement of the reduced shear g=γ/(1 −κ). The shear, γ, is the anisotropic stretching of the galaxy image, and the convergence, κ, is the shapeindependent change in the size of the image. By recovering the κmap from the measured reduced shear field, a measure of the local curvature is obtained. In Einstein's general relativity, the local curvature is related to the distribution of mass/energy, as it is in MOG. In Newtonian gravity theory, the relationship between the κmap and the surface mass density becomes very simple, allowing one to refer to κ as the scaled surface mass density [see e.g. chapter 4 of Peacock (2003) for a derivation]:
where is the surface mass density and is the Newtonian critical surface mass density (with vanishing shear), D_{s} is the angular distance to a source (background) galaxy, D_{l} is the angular distance to the lens (Bullet Cluster 1E0657558) and D_{ls} is the angular distance from the Bullet Cluster 1E0657558 to a source galaxy. The result of equation (37) is equivalent to Since there is a multitude of source galaxies, these distances become ‘effective’, as is the numeric value presented in equation (37), quoted from Clowe, Gonzalez & Markevitch (2004) without estimate of the uncertainty. In fact, due to the multitude of sources in the lensing survey, both D_{s} and D_{ls} are distributions in (x, y). However, it is a common practice to move them outside the integral, as a necessary approximation.We may obtain a similar result in MOG, as was shown in Moffat (2006b), by promoting the Newtonian gravitational constant, G_{N} to the running gravitational coupling, G(r), but approximating G(r) as sufficiently slow varying to allow it to be removed from the integral. We have in general,
and we assume that where we applied equation (18). However, equations (39) and (40) are only valid in the thin lens approximation. For the Bullet Cluster 1E0657558, the thin lens approximation is inappropriate, and we must use the correct relationship between κ and Σ: where is the weighted surface mass density [weighted by the dimensionless gravitational coupling of equation 18] and Σ_{c} is the usual Newtonian critical surface mass density equation (37).For the remainder of this paper, we will use equations (18), (37), (41) and (42) to reconcile the experimental observations of the gravitational lensing κmap with the Xray imaging Σmap. We can already see from these equations that how in MOG the convergence, κ, is now related to the weighted surface mass density, , so
“…κ is no longer a measurement of the surface density, but is a nonlocal function whose overall level is still tied to the amount of mass. For complicated system geometries, such as a merging cluster, the multiple peaks can deflect, suppress or enhance some of the peaks (Clowe et al. 2006b).”
3 THE SURFACE DENSITY MAP FROM XRAY IMAGE OBSERVATIONS
3.1 The Σmap
With an advance of the 2006 November 15 data release (Clowe et al. 2006c), we began a precision analysis to model the gross features of the surface density Σmap data in order to gain insight into the 3D matter distribution, ρ(r), and to separate the components into a model representing the main cluster and the subcluster – the remainder after subtraction.
The Σmap is shown in false colour in Fig. 3. There are two distinct peaks in the surface density Σmap – the primary peak centred at the main cluster and the secondary peak centred at the subcluster. The main cluster gas is the brightly glowing (yellow) region to the lefthand side of the subcluster gas, which is the nearly equally bright shockwave region (arrowhead shape to the righthand side). The κmap observed peaks, the CD galaxy of the main cluster, the brightest cluster galaxy (BCG) of the subcluster and the MOG predicted gravitational centre are shown in Fig. 3 for comparison. J2000 and map (x, y) coordinates are listed in Table 1.
Observation  J2000 coordinates  Σmap (x, y)  κmap (x, y)  
RA  Dec.  
Main cluster Σmap peak  (89, 89)  (340, 321)  
Subcluster Σmap peak  (135, 98)  (365, 326)  
Main cluster κmap peak  (70, 80)  (329, 317)  
Subcluster κmap peak  (149, 102)  (374, 327)  
Main cluster CD  (71, 88)  (330, 320)  
Subcluster BCG  (154, 98)  (375, 326)  
MOG Centre  (105, 92)  (348, 322) 
Observation  J2000 coordinates  Σmap (x, y)  κmap (x, y)  
RA  Dec.  
Main cluster Σmap peak  (89, 89)  (340, 321)  
Subcluster Σmap peak  (135, 98)  (365, 326)  
Main cluster κmap peak  (70, 80)  (329, 317)  
Subcluster κmap peak  (149, 102)  (374, 327)  
Main cluster CD  (71, 88)  (330, 320)  
Subcluster BCG  (154, 98)  (375, 326)  
MOG Centre  (105, 92)  (348, 322) 
We may now proceed to calculate the bestfitting parameters, β, r_{c} and ρ_{0}, of the King βmodel of equations (33) and (34) by applying a nonlinear leastsquares fitting routine (including estimated errors) to the entire Σmap, or alternatively by performing the fit to a subset of the Σmap on a straight line connecting the main cluster Σmap peak (R≡ 0) to the main CD, and then extrapolating the fit to the entire map. This reduces the complexity of the calculation to a simple algorithm, but is not guaranteed to yield a global best fit. However, our approximate best fit will prove to agree with the Σmap everywhere, except at the subcluster (which is neglected for the best fit).
The scaled surface density Σmap data are shown in solid red in Fig. 4. The unmodelled peak (at R∼ 300 kpc) is due to the subcluster. The best fit to the King βmodel of equation (34) is shown in Fig. 4 in shortdashed blue, and corresponds to
where the value of the Σmap at the main cluster peak is constrained to the observed value,We may now solve equation (33) for the central density of the main cluster,
The values of the parameters β, r_{c} and ρ_{0} completely determine the King βmodel for the density, ρ(r), of equation (21) of the main cluster Xray gas. We provide a comparison of the full Σmap data (Fig. 5, in red) with the result of the surface density Σmap derived from the bestfitting King βmodel to the main cluster (Fig. 5, in blue).
Substituting equations (43), (44) and (46) into equation (23), we obtain the main cluster outer radial extent,
the distance at which the density, ρ(r_{out}), drops to ≈ 10^{−28} g cm^{−3}, or 250 times the mean cosmological density of baryons. The total mass of the main cluster may be calculated by substituting equations (43), (44) and (46) into equation (24):3.2 The gravitational coupling for the main cluster
As discussed in Section 2.1, in order to apply the MOG model of equation (18) to the Bullet Cluster 1E0657558, we must first generalize the spherically symmetric case by treating the subcluster as a perturbation. In the zerothorder approximation, we begin by neglecting the subcluster. According to the large (>100) galaxy cluster survey of Brownstein & Moffat (2006b), the MOG mass scale, M_{0}, is determined by a power law, depending only on the computed total cluster mass, M_{gas}:
Substituting the result of equation (48) for the main cluster into equation (49), we obtain and substituting the result of equation (48) for M and equation (50) for the main cluster of Bullet Cluster 1E0657558 into equation (19), we obtain From Brownstein & Moffat (2006b), the MOG range parameter, r_{0}, depends only on the computed outer radial extent, r_{out}: from which we obtain for the main cluster.The bestfitting King βmodel parameters for the main cluster are listed in Table 2, and the MOG parameters for the main cluster are listed in Table 3. A plot of the dimensionless gravitational coupling of equation (18) using the MOG parameter results of equations (50) and (53) for the main cluster (neglecting the subcluster) is plotted in Fig. 7 using a linear scale for the raxis (lefthand side), and on a logarithmic scale for the raxis (righthand side).
MOG  Main cluster  Subcluster  Best fit to κmap 
M _{0}  1.02 × 10^{16} M_{⊙}  3.56 × 10^{15} M_{⊙}  3.07 × 10^{15} M_{⊙} 
r _{0}  139.2 kpc  …  208.8 kpc 
6.14  …  3.82  
T  15.5 ± 3.9 keV  …  … 
MOG  Main cluster  Subcluster  Best fit to κmap 
M _{0}  1.02 × 10^{16} M_{⊙}  3.56 × 10^{15} M_{⊙}  3.07 × 10^{15} M_{⊙} 
r _{0}  139.2 kpc  …  208.8 kpc 
6.14  …  3.82  
T  15.5 ± 3.9 keV  …  … 
3.3 The cylindrical mass profile
Σ(x, y) is an integrated density along the lineofsight, z, as shown in equation (36). By summing the Σmap pixelbypixel, starting from the centre of the main cluster Σmap peak, one is performing an integration of the surface density, yielding the mass,
enclosed by concentric cylinders of radius, . We performed such a sum over the Σmap data, and compared the result to the bestfitting King βmodel derived in Section 3.1, with the parameters listed in Table 2. The results are shown in Fig. 8. The fact that the data and model are in good agreement provides evidence that the King βmodel is valid in all directions from the main cluster Σmap peak, and not just on the straight line connecting the peak to the main CD, where the fit was performed. The model deviates slightly from the data, underpredicting M(R) for R > 350 kpc, which may be explained by the subcluster (which is included in the data, but not the model). The integrated mass profile arising from the dark matter analysis of the κmap of equation (83) is shown for comparison on the same figure.The ratio of dark matter to ICM gas for the main cluster is >3, which is significantly less than the average cosmological ratio of 5.68 discussed in Section 1.1, whereas one should expect the highest masstolight ratios from large, hot, luminous galaxy clusters, and the Bullet Cluster 1E0657558 is certainly one of the largest and hottest.
3.4 The isothermal spherical mass profile
For a spherical system in hydrostatic equilibrium, the structure equation can be derived from the collisionless Boltzmann equation
where Φ(r) is the gravitational potential for a point source, σ_{r} and σ_{θ,φ} are massweighted velocity dispersions in the radial (r) and tangential (θ, φ) directions, respectively. For an isotropic system,The pressure profile, P(r), can be related to these quantities by
Combining equations (55), (56) and (57), the result for the isotropic sphere is For a gas sphere with temperature profile, T(r), the velocity dispersion becomes where k is Boltzmann's constant, μ_{A}≈ 0.609 is the mean atomic weight and m_{p} is the proton mass. We may now substitute equations (57) and (59) into equation (58) to obtain Performing the differentiation on the lefthand side of equation (58), we may solve for the gravitational acceleration: For the isothermal isotropic gas sphere, the temperature derivative on the righthand side of equation (61) vanishes and the remaining derivative can be evaluated using the βmodel of equation (21):The dynamical mass in Newton's theory of gravitation can be obtained as a function of radial position by replacing the gravitational acceleration with Newton's law:
so that equation (61) can be rewritten as and the isothermal βmodel result of equation (62) can be rewritten as Similarly, the dynamical mass in MOG can be obtained as a function of radial position by substituting the MOG gravitational acceleration law (Moffat 2005, 2006a; Brownstein & Moffat 2006a,b) so that our result for the isothermal βmodel becomes We can express this result as a scaled version of equation (64) or the isothermal case of equation (65): where we have substituted equation (18) for . Equation (68) may be solved explicitly for M_{MOG}(r) by squaring both sides and determining the positive root of the quadratic equation.The scaling of the Newtonian dynamical mass by according to equation (68) answered the question of missing mass for the galaxy clusters of the >100 galaxy cluster survey of Brownstein & Moffat (2006b). The unperturbed Bullet Cluster 1E0657558 is no exception! In Fig. 9, we plotted the MOG and the Newtonian dynamical masses, M_{MOG}(r) and M_{N}(r), respectively, and compared it to the spherically integrated bestfitting King βmodel for the main cluster gas mass of equations (21) and (22) using the parameters listed in Table 2. The MOG temperature prediction due to the best fit is listed in Table 4 and is compared to experimental values.
Year  Source – theory or experiment  T(keV)  Per cent error 
2007  MOG Prediction  15.5 ± 3.9  
2002  Accepted experimental value  14.8^{+1.7}_{−1.2}  4.5 
1999  ASCA+ROSAT fit  14.5^{+2.0}_{1.7}  6.5 
1998  ASCA fit  17.4 ± 2.5  12.3 
Year  Source – theory or experiment  T(keV)  Per cent error 
2007  MOG Prediction  15.5 ± 3.9  
2002  Accepted experimental value  14.8^{+1.7}_{−1.2}  4.5 
1999  ASCA+ROSAT fit  14.5^{+2.0}_{1.7}  6.5 
1998  ASCA fit  17.4 ± 2.5  12.3 
As shown in Fig. 9, across the range of the raxis, and throughout the radial extent of the Bullet Cluster 1E0657558, the 1σ correlation between the gas mass, M(r), and the MOG dynamical mass, M_{MOG}(r), provides excellent agreement between theory and experiment. The same cannot be said of any theory of dark matter in which the Xray surface density map is negligible in relation to the DM density. So, dark matter makes no prediction for the isothermal temperature, which has been measured to reasonable precision for many clusters, but simply ‘accounts for missing mass’. Since there is no mysterious missing mass in MOG, the prediction should be taken seriously as direct confirmation of the theory, or at least as hard evidence for MOG.
3.5 The subcluster subtraction
Provided the surface density Σmap derived from the bestfitting King βmodel of equation (34) is sufficiently close to the Σmap data (consider Figs 4 and 5), the difference between the data and the βmodel is the surface density Σmap due to the subcluster. Our subcluster subtraction, shown in Figs 10 and 11, is based on a highprecision (χ^{2} < 0.2) bestfitting King βmodel to the main cluster, which agrees with the main cluster surface mass Σmap (data) within 1 per cent everywhere and to a mean uncertainty of 0.8 per cent. The subcluster subtraction is accurate down to ρ= 10^{−28} g cm^{−3}∼ 563.2 M_{⊙} pc^{−3} baryonic background density. After subtraction, the subcluster Σmap peak takes a value of 1.30 × 10^{8} M_{⊙} kpc^{−2}, whereas the full Σmap has a value of 2.32 × 10^{8} M_{⊙} kpc^{−2} at the subcluster Σmap peak. Thus, the subcluster (at its most dense position) provides only ≈ 56 per cent of the Xray ICM, the rest is due to the extended distribution of the main cluster. The subcluster subtraction surface density Σmap shown in Fig. 10 uses the same colourscale as the full Σmap shown in Fig. 3, for comparison. Fig. 11 is a stereogram of the subclustersubtracted surface density Σmap and the subcluster superposed on to the surface density Σmap of the bestfitting King βmodel to the main cluster. There is an odd Xray bulge in Figs 10 and 11 which may be an artefact of the subtraction, or perhaps evidence of an as yet unidentified component.
Since the outer radial extent of the subcluster gas is less than 400 kpc, the Σmap completely contains all of the subcluster gas mass. By summing the subclustersubtracted Σmap pixelbypixel over the entire Σmap peak, one is performing an integration of the surface density, yielding the total subcluster mass. We performed such a sum over the subclustersubtracted Σmap data, obtaining
for the mass of the subcluster gas, which is less than 6.7 per cent of the mass of main cluster gas (the bestfitting King βmodel parameters are listed in Table 2.). This justifies our initial assumption that the subcluster may be treated as a perturbation in order to fit the main cluster to the King βmodel. Our subsequent analysis of the thermal profile confirms that the main cluster Xray temperature is nearly isothermal, lending further support to the validity of the King βmodel.We may now calculate the MOG mass scale, M_{0}, of the subcluster by substituting the subcluster gas mass, M_{gas}, of equation (69) into the powerlaw relation of equation (49), yielding
as shown in Fig. 6. The computed values of M_{gas} and M_{0} for the main and subcluster are listed in Table 3.4 THE CONVERGENCE MAP FROM LENSING ANALYSIS
4.1 The κmap
As tempting as it is to see the convergence κmap of Fig. 12– a false colour image of the strong and weak gravitational lensing reconstruction (Clowe et al. 2006a,b; Bradač et al. 2006) – as a photograph of the ‘curvature’ around the Bullet Cluster 1E0657558, it is actually a reconstruction of all of the bending of light over the entire distance from the lensing event source towards the HST. The source of the κmap is , along the lineofsite, as in equation (35), as in equation (39), or as in equation (41). For the Bullet Cluster 1E0657558, we are looking along a lineofsight which is at least as long as indicated by a redshift of z= 0.296 (Gpc scale). The sources of the lensing events are in a large neighbourhood of redshifts, an estimated z= 0.85 ± 0.15. This fantastic scale (several Gpc) is naturally far in excess of the distance scales involved in the Xray imaging surface density Σmap. It is an accumulated effect, but only over the range of the Xray source – as much as 2.2 Mpc. A comparison of these two scales indicates that the distance scales within the Σmap are 10^{−3} below the Gpc's scale of the κmap.
Preliminary comments on the 2006 November 15 data release (Clowe et al. 2006c):
The conclusion of Clowe et al. (2006a) and Bradač et al. (2006), that the κmap shows direct evidence for the existence of dark matter, may be premature. Until dark matter has been detected in the lab, it remains an open question whether a MOG theory, such as MOG, can account for the κmap without nonbaryonic dark matter. MOG, due to the varying gravitational coupling, equation (18), gives the Newtonian 1/r^{2} gravitational force law a considerable boost –‘extra gravity’ as much as for the Bullet Cluster 1E0657558.
It may be feasible that the mysterious plateau in the northeast corner of the κmap is from some distribution of mass unrelated to the Bullet Cluster 1E0657558. This ‘background curvature’ contribution to the κmap is one part of the uncertainty in the reconstruction. The second, dominant source of uncertainty must certainly be the estimate of the angular diameter distances between the source of the lensing event and the Bullet Cluster 1E0657558, as shown in equation (37).
We have completed a large (>100) galaxy cluster survey in Brownstein & Moffat (2006b) that provides a statistically significant answer to the question of how much dark matter is expected. The relative abundance across the scales of the Xray clusters would imply that the κmap should peak at ∼1.0 as opposed to the 2006 November 15 data release (Clowe et al. 2006c), which peaks at a value of κ≈ 0.38 for the main cluster κmap peak.
The dark matter paradigm cannot statistically account for such an observation without resorting to further ad hoc explanation, so the question of ‘missing matter’ may be irrelevant. The problem of ‘extra gravity’ due to MOG may be a step in the right direction, with our solution as the subject of the next section.
4.2 The MOG solution
The MOG κmodel we have developed in equations (41) and (42) can now be applied to the Bullet Cluster 1E0657558. In order to model the κmap, shown in Figs 12 and 14, in MOG, we must integrate the product of the dimensionless gravitational coupling of equation (18) with the mass density, ρ(r), over the volume of the Bullet Cluster 1E0657558. As discussed in Section 3.3, we may integrate the surface density Σmap data according to equation (54) to obtain the integrated cluster gas mass about concentric cylinders centred about R= 0. However, what we require in the calculation of the of equation (18) is the integrated mass about concentric spheres,
The calculation of M(r) using equation (71) is nontrivial, and we proceed by making a suitable approximation to the density, ρ(r). As discussed in Section 2.1, we may apply a sequence of approximations to develop a MOG solution.Zerothorder approximation We have obtained the spherically integrated bestfitting King βmodel for the main cluster gas mass of equations (21) and (22) using the parameters listed in Table 2. By neglecting the subcluster, we have generated a baseline solution similar to every other spherically symmetric galaxy cluster. The result of substituting the spherically symmetric King βmodel mass profile of equation (22) into equation (18) is shown in Fig. 7 for the dimensionless gravitational coupling, . We obtain a zerothorder, spherically symmetric approximation to the κmap by substituting ρ(r) and into equations (41) and (42) and integrating over the volume.
MOG centre Treat the subcluster as a perturbation, and shift the origin of towards the subcluster (towards the centreofmass of the system). In this approximation, we continue to use the spherically integrated bestfitting King βmodel for the main cluster gas mass of equations (21) and (22) using the parameters listed in Table 2, but we allow the subcluster to perturb (shift) the origin of towards the true centreofmass of the system. The integrals of equations (41) and (42) become nontrivial as the integrand is no longer spherically symmetric. We were able to obtain a full numerical integration, but the computation proved to be too time consuming (∼70 000 numerical integrations to cover the 185 × 185 pixel^{2}) to treat by means of a nonlinear leastsquares fitting routine.
Projective approximation Approximate M(r) with the concentric cylinder mass, M(R) of equation (54), calculated directly from the Σmap, where the pixelbypixel sum proceeds from the MOG centre. This is a poor approximation for small R (few pixels), but becomes very good for large R (many pixels), as can be seen by comparing Figs 8 and 9. The cylindrical mass, M(R), can be computed directly from the Σmap data using equation (54) without the need of a King βmodel.
Isothermal βmodel approximation Treat the subcluster as a perturbation, and utilize the isothermal βmodel of equation (68) to approximate M(r) and shift that towards the subcluster (towards the MOG centre). This is a fully analytic expression, allowing ease of integration and an iterative fitting routine.
We will base our analysis on our best fit determined by the isothermal βmodel approximation of equation (68), with located at the MOG centre, a distance away from the main cluster Σmap peak towards the subcluster Σmap peak. Since the dimensionless gravitational coupling, , of equation (18) depends on M(r), and the isothermal βmodel of equation (68) depends on , we must solve the system simultaneously. Let us rewrite equation (18) as
where We may solve equation (72) for and equate this to the isothermal βmodel of equation (68): and we may solve the quadratic equation for where M_{N} is the isothermal βmodel Newtonian dynamic mass of equation (65). The result of equation (76) is a fully analytic function, as opposed to a hypergeometric integral, and may be easily computed across the full κmap. There was a notable parameter degeneracy in choosing the MOG centre, the best fit corresponded to a distance of 140 kpc away from the main cluster Σmap peak towards the subcluster Σmap peak. This is reasonable since the Σmap and κmap data are 2D ‘surface projections’, due to the lineofsight integral. The full simulation was run iteratively over a range of positions for the MOG centre, while covarying the MOG parameters, M_{0} and r_{0}, the MOG mass scale and MOG range parameter, respectively. This yielded a bestfitting MOG model for the dimensionless gravitational coupling, of equation (18), where r is the distance from the MOG centre, as listed in Table 3. Our best fit to the κmap corresponds to the MOG κmodel of equations (41) and (42). The bestfitting location of the MOG centre is provided on the Σmap in Fig. 3, and the coordinates are listed in Table 1.We show a 3D visualization of the convergence κmap data in Fig. 14 (top left). The zerothorder approximation result, rescaled, is shown in the same figure (top right). The twin humps at the peaks of our prediction will be a generic prediction for any spherically symmetric galaxy cluster (nonmergers). The bestfitting MOG κmodel is shown in Fig. 14 (bottom left) along the line connecting the MOG centre to the main cluster κmap peak. We show a 3D visualization of the full convergence κmap model in the same figure (bottom right).
4.3 Including the galaxies
In considering the MOG κmodel resulting from the zerothorder approximation, as shown in Fig. 14 (top right), it is tempting to try to explain the entire convergence κmap by the Xray gas mass, just by shifting the solid black line to the left by ∼200 kpc, but then there would be no way to explain the subcluster κmap peak. We next proceed to account for the effect of the subcluster on the dimensionless gravitational coupling, , of MOG, as shown in the bestfitting κmodel of Fig. 14 (bottom). Remarkably, as the MOG centre is separated from the main cluster Σmap peak, say due to the gravitational effect of the subcluster, the centroid naturally shifts towards the κmap peak, and the predicted height of the κmap drops. Let us take the difference between the κmap data and our bestfitting κmodel, to see if there is any ‘missing mass’. We hypothesize that the difference can be explained by including the galaxies,
where is the weighted surface mass density of equation (42), and the bestfitting κmodel of is derived from equations (41) and (42). Therefore, the galaxies contribute a ‘measurable’ surface mass density, where corresponds to the bestfitting model of equation (18) listed in Table 3. The result of the galaxy subtraction of equation (78) is shown in Fig. 13. Now we may interpret Fig. 14 (bottom right) as the total convergence κmap where the black surface is the contribution from the weighted surface density of the ICM gas, , and the red surface is the remainder of the κmap due to the contribution of the weighted surface density of the galaxies, . We may calculate the total mass of the galaxies, We were able to perform the integration within a 100 kpc radius aperture about the main cluster CD and subcluster BCG, separately, the results of which are listed in Table 5, where they are compared with the upper limits on galaxy masses set by HST observations. If the hypothesis that the predicted M_{galax} is below the bound set by HST observations is true, then it follows that requires no addition of nonbaryonic dark matter. The results of our best fit for M_{gas}, M_{galax} and M_{bary} of equation (80) are listed in Table 5.Component  Main cluster  Subcluster  Central ICM  Total 
M_{gas}  7.0 × 10^{12} M_{⊙}  5.8 × 10^{12} M_{⊙}  6.3 × 10^{12} M_{⊙}  2.2 × 10^{14} M_{⊙} 
M_{galax}  1.8 × 10^{12} M_{⊙}  3.1 × 10^{12} M_{⊙}  2.4 × 10^{10} M_{⊙}  3.8 × 10^{13} M_{⊙} 
M_{bary}  8.8 × 10^{12} M_{⊙}  9.0 × 10^{12} M_{⊙}  4.9 × 10^{12} M_{⊙}  2.6 × 10^{14} M_{⊙} 
M_{DM}  2.1 × 10^{13} M_{⊙}  1.7 × 10^{13} M_{⊙}  1.4 × 10^{13} M_{⊙}  6.8 × 10^{14} M_{⊙} 
M_{galax}/M_{gas}  26 per cent  53 per cent  0.4 per cent  17 per cent 
M_{gas}/M_{DM}  33 per cent  34 per cent  45 per cent  32 per cent 
Component  Main cluster  Subcluster  Central ICM  Total 
M_{gas}  7.0 × 10^{12} M_{⊙}  5.8 × 10^{12} M_{⊙}  6.3 × 10^{12} M_{⊙}  2.2 × 10^{14} M_{⊙} 
M_{galax}  1.8 × 10^{12} M_{⊙}  3.1 × 10^{12} M_{⊙}  2.4 × 10^{10} M_{⊙}  3.8 × 10^{13} M_{⊙} 
M_{bary}  8.8 × 10^{12} M_{⊙}  9.0 × 10^{12} M_{⊙}  4.9 × 10^{12} M_{⊙}  2.6 × 10^{14} M_{⊙} 
M_{DM}  2.1 × 10^{13} M_{⊙}  1.7 × 10^{13} M_{⊙}  1.4 × 10^{13} M_{⊙}  6.8 × 10^{14} M_{⊙} 
M_{galax}/M_{gas}  26 per cent  53 per cent  0.4 per cent  17 per cent 
M_{gas}/M_{DM}  33 per cent  34 per cent  45 per cent  32 per cent 
4.4 Dark matter
From the alternative point of view, dark matter is hypothesized to account for all of the ‘missing mass’ which results in applying Newton/Einstein gravity. This means, for the 2006 November 15 data release (Clowe et al. 2006a,b,c; Bradač et al. 2006), that the ‘detected’ dark matter must contribute a surface mass density,
with an associated total mass, The integral of equation (82) becomes trivial on substitution of equation (81): where we have neglected M_{galax} in equation (83) because as is usually argued, the contribution from the galaxies in the dark matter paradigm is ≤ 1– 4 per cent of M_{total}. The calculation of M_{DM} in equation (83) was performed by a pixelbypixel sum over the convergence κmap data and surface density Σmap data, within a 100 kpc radius aperture around the main and subcluster κmap peaks, respectively. The result of our computation is included in Table 5.We emphasize, here, that for each of the main cluster, subcluster and total Σmap, our results of Table 5 indicate that
implying that we have successfully put Bullet Cluster 1E0657558 on a lean diet! This seems to us to be a proper use of Occam's razor. The mass ratios, M_{galax}/M_{gas}, for the main and subcluster and central ICM are shown at the bottom of Table 5. The result of M_{galax}/M_{gas}≈ 0.4 per cent in the central ICM is due to the excellent fit in MOG across the hundreds of kpc separating the main and subcluster. The dark matter result of M_{gas}/M_{DM}≈ 45 per cent in the central ICM implies that the evolutionary scenario does not lead to a spatial dissociation between the dark matter and the ICM gas, which indicates that the merger is ongoing. In contrast, the MOG result shows a true dissociation between the galaxies and the ICM gas as required by the evolutionary scenario. The baryontodark matter fraction over the full Σmap is 32 per cent, which is significantly higher than the ΛCDM cosmological baryon mass fraction of 17^{+1.9}_{−1.2} per cent (Spergel et al. 2006). The distribution of mass predicted by MOG versus the dark matter paradigm is shown in Fig. 15.5 CONCLUSIONS
The MOG theory provides a fit to the κmap of the 2006 November 15 data release (Clowe et al. 2006c). The model, derived purely from the Xray imaging Σmap observations combined with the galaxy Σmap predicted by MOG, accounts for the κmap peak amplitudes and their spatial dissociation without the introduction of nonbaryonic dark matter.
The question of the internal degrees of freedom in MOG has to be further investigated. It would be desirable to derive a theoretical prediction from the MOG field equations that fit the empirically determined mass and distance scales in Fig. 6.
It could be argued that any modified theory designed to solve the dark matter conundrum, such as MOND or MOG, has less freedom than dark matter. So, the important question to resolve is precisely how much freedom the MOG solution has. On one hand we said, definitely, that there was no freedom in choosing the pair of M_{0} and r_{0} for the main cluster since it was well described by an isothermal sphere, to an excellent approximation. We further argued that the subcluster was (per mass) a small perturbation to the ICM. But if MOG has more freedom than MOND, but less freedom than dark matter, then what is the additional degree of freedom that enters the Bullet Cluster observations? The question is resolved in that there is a physical degree of freedom due to a lack of spherical symmetry in the Bullet Cluster, and whence the galaxies sped outward, beyond the ICM gas clouds which lagged behind – effectively allowing the galaxies to climb out of the spherical minimum of the Newtonian core where MOG effects are small (inside the MOG range r_{0}) upwards along the divergence of the stressenergy tensor (Newtonian potential, if you prefer a simple choice) towards the farIR region of large gravitational coupling, G_{∞}.
In fact, the Bullet Cluster data results describe, to a remarkable precision, a simple King βModel. Our analysis, with the result to the best fit shown in Table 2, uniquely determines the mass profile ρ(r) of equation (21) used throughout our computations. We permitted only a single further degree of freedom to account for the fits of Figs 14 and the predictions of Figs 13 and 15; this was the location of the MOG centre, where the gravitational coupling, , is a minimum at the Newtonian core. Remarkably, the data did not permit a vanishing MOG centre, with respect to the peak of the ICM gas ρ(0). We have shown the location of the MOG centre as determined by a numerical simulation of convergence map according to equations (41) and (42) in each of Figs 3, 10, 12 and 13 and provided the coordinates in Table 1.
The surface density Σmap derived from Xray imaging observations is separable into the main cluster and the subclustersubtracted surface density Σmap through a low χ^{2}fitting King βmodel. Following the (>100) galaxy cluster survey of Brownstein & Moffat (2006b), we have derived a parameterfree (unique) prediction for the Xray temperature of the Bullet Cluster 1E0657558 which has already been experimentally confirmed. In equations (41) and (42), we have derived a weighted surface mass density, , from the convergence κmap which produced a bestfitting model (bottom of Fig. 14 and Table 3). We have computed the dark matter and the MOG predicted galaxies and baryons (Fig. 15), and noted the tremendous predictive power of MOG as a means of utilizing strong and weak gravitational lensing to do galactic photometry – a powerful tool simply not provided by any candidate dark matter (Fig. 13). The predictions for galaxy photometry will be the subject of future investigations in MOG, and the availability of weak and strong gravitational lensing surveys will prove invaluable in the future.
Although dark matter allows us to continue to use Einstein (weak field) and Newtonian gravity theory, these theories may be misleading at astrophysical scales. By searching for dark matter, we may have arrived at a means to answer one of the most fundamental questions remaining in astrophysics and cosmology: how much matter (energy) is there in the Universe and how is it distributed? For the Bullet Cluster 1E0657558, dark matter dominance is a ready answer, but in MOG we must answer the question with only the visible contribution of galaxies, ICM gas and gravity.
This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). We thank Douglas Clowe and Scott Randall for providing early releases of the gravitational lensing convergence data and Xray surface mass density data, respectively, and for stimulating and helpful discussions. Research at Perimeter Institute for Theoretical Physics is supported in part by the Government of Canada through NSERC and by the Province of Ontario through the Ministry of Research and Innovation (MRI).