Galaxy cluster masses without non-baryonic dark matter

We apply the modified acceleration law obtained from Einstein gravity coupled to a massive skew symmetric field,   F _μνλ  , to the problem of explaining X-ray galaxy cluster masses without exotic dark matter. Utilizing X-ray observations to fit the gas mass profile and temperature profile of the hot intracluster medium (ICM) with King 'β-models', we show that the dynamical masses of the galaxy clusters resulting from our modified acceleration law fit the cluster gas masses for our sample of 106 clusters without the need of introducing a non-baryonic dark matter component. We are further able to show for our sample of 106 clusters that the distribution of gas in the ICM as a function of radial distance is well fitted by the dynamical mass distribution arising from our modified acceleration law without any additional dark matter component. In a previous work, we applied this theory to galaxy rotation curves and demonstrated good fits to our sample of 101 low surface brightness, high surface brightness and dwarf galaxies including 58 galaxies that were fitted photometrically with the single-parameter mass-to-light ratio ( M / L )_stars. The results obtained there were qualitatively similar to those obtained using Milgrom's phenomenological Modified Newtonian Dynamics (MOND) model, although the determined galaxy masses were quantitatively different, and MOND does not show a return to Keplerian behaviour at extragalactic distances. The results obtained here are compared to those obtained using Milgrom's phenomenological MOND model which does not fit the X-ray galaxy cluster masses unless an auxiliary dark matter component is included.     

The fundamental plane of elliptical galaxies with modified Newtonian dynamics

The modified Newtonian dynamics (MOND), suggested by Milgrom as an alternative to dark matter, implies that isothermal spheres with a fixed anisotropy parameter should exhibit a near-perfect relation between the mass and velocity dispersion of the form M ∝ σ  ⁴. This is consistent with the observed Faber–Jackson relation for elliptical galaxies: a luminosity–velocity dispersion relation with large scatter. However, the observable global properties of elliptical galaxies comprise a three-parameter family; they lie on a 'fundamental plane' in a logarithmic space consisting of central velocity dispersion, effective radius ( r _e) and luminosity. The scatter perpendicular to this plane is significantly less than that about the Faber–Jackson relation. I show here that, in order to match the observed properties of elliptical galaxies with MOND, models must deviate from being strictly isothermal and isotropic; such objects can be approximated by high-order polytropic spheres with a radial orbit anisotropy in the outer regions. MOND imposes boundary conditions on the inner Newtonian regions which restrict these models to a dynamical fundamental plane of the form where the exponents may differ from the Newtonian expectations ( α =2, γ =1). Scatter about this plane is relatively insensitive to the necessary deviations from homology.     

Neutrinos as cluster dark matter

The dynamical mass of clusters of galaxies, calculated in terms of MOdified Newtonian Dynamics (MOND), is a factor of 2 or 3 times smaller than the Newtonian dynamical mass but remains significantly larger than the observed baryonic mass in the form of hot gas and stars in galaxies. Here I consider further the suggestion that the undetected matter might be in the form of cosmological neutrinos with mass of the order of 2 eV. If the neutrinos and baryons have comparable velocity dispersions and if the two components maintain their cosmological density ratio, then the electron density in the cores of clusters should be proportional to T ^3/2, as appears to be true in non-cooling flow clusters. This is equivalent to the 'entropy floor' proposed to explain the steepness of the observed luminosity–temperature relation, but here preheating of the medium is not required. Two-fluid (neutrino–baryon) hydrostatic models of clusters, in the context of MOND, reproduce the observed luminosity–temperature relation of clusters. If the β law is imposed on the gas density distribution, then the self-consistent models predict the general form of the observed temperature profile in both cooling and non-cooling flow clusters.     

X-ray group and cluster mass profiles in MOND: unexplained mass on the group scale

Although very successful in explaining the observed conspiracy between the baryonic distribution and the gravitational field in spiral galaxies without resorting to dark matter (DM), the modified Newtonian dynamics (MOND) paradigm still requires DM in X-ray bright systems. Here, to get a handle on the distribution and importance of this DM, and thus on its possible form, we deconstruct the mass profiles of 26 X-ray emitting systems in MOND, with temperatures ranging from 0.5 to 9 keV. Initially, we compute the MOND dynamical mass as a function of radius, then subtract the known gas mass along with a component of galaxies which include the cD galaxy with   M / L _K = 1  . Next, we test the compatibility of the required DM with ordinary massive neutrinos at the experimental limit of detection  ( m _ν= 2 eV)  , with density given by the Tremaine–Gunn limit. Even by considering that the neutrino density stays constant and maximal within the central 100 or 150 kpc (which is the absolute upper limit of a possible neutrino contribution there), we show that these neutrinos can never account for the required DM within this region. The natural corollary of this finding is that, whereas clusters  ( T ≳ 3 keV)  might have most of their mass accounted for if ordinary neutrinos have a 2 eV mass, groups  ( T ≲ 2 keV)  cannot be explained by a 2 eV neutrino contribution. This means that, for instance, cluster baryonic dark matter (CBDM, Milgrom) or even sterile neutrinos would present a more satisfactory solution to the problem of missing mass in MOND X-ray emitting systems.     

Marriage à-la-MOND: Baryonic dark matter in galaxy clusters and the cooling flow puzzle

I start with a brief introduction to MOND phenomenology and its possible roots in cosmology—a notion that may turn out to be the most far reaching aspect of MOND. Next I discuss the implications of MOND for the dark matter (DM) doctrine: MOND’s successes imply that baryons determine everything. For DM this would mean that the puny tail of leftover baryons in galaxies wags the hefty DM dog. This has to occur in many intricate ways, and despite the haphazard construction history of galaxies—a very tall order. I then concentrate on galaxy clusters in light of MOND, which still requires some yet undetected cluster dark matter, presumably in some baryonic form (CBDM). This CBDM might contribute to the heating of the X-ray emitting gas and thus alleviate the cooling flow puzzle. MOND, qua theory of dynamics, does not directly enter the microphysics of the gas; however, it does force a new outlook on the role of DM in shaping the cluster gas dynamics: MOND tells us that the cluster DM is not cold dark matter, is not so abundant, and is not expected in galaxies; it is thus not subject to constraints on baryonic DM in galaxies. The mass in CBDM required in a whole cluster is, typically, similar to that in hot gas, but is rather more centrally concentrated, totally dominating the core. The CBDM contribution to the baryon budget in the universe is thus small. Its properties, deduced for isolated clusters, are consistent with the observations of the “bullet cluster”. Its kinetic energy reservoir is much larger than that of the hot gas in the core, and would suffice to keep the gas hot for many cooling times. Heating can be effected in various ways depending on the exact nature of the CBDM, from very massive black holes to cool, compact gas clouds.     

Note on a polytropic β-model to fit the X-ray surface brightness of clusters of galaxies

I suggest that the β -model used to fit the X-ray surface brightness profiles of extended sources, like groups and clusters of galaxies, has to be corrected when the counts are collected in a wide energy band comparable to the mean temperature of the source, and a significant gradient in the gas temperature is observed. I present a revised version of the β -model for the X-ray brightness that applies to an intracluster gas with temperature and density related by a polytropic equation and extends the standard version that is strictly valid for an isothermal gas. Given a temperature gradient observed through an energy window of 1–10 keV typical for the new generation of X-ray observatories, the β parameter can change systematically by up to 20 per cent from the value obtained under isothermal assumption, i.e. by an amount larger than any statistical uncertainty obtained from the present data. Within the virial regions of typical clusters of galaxies, these systematic corrections affect the total gravitating mass estimate by 5–10 per cent, the gas mass by 10–30 per cent and the gas fraction value up to 50 per cent, when compared with the measurements obtained under the isothermal assumption.     

MOND plus classical neutrinos are not enough for cluster lensing

Clusters of galaxies offer a robust test bed for probing the nature of dark matter that is insensitive to the assumption of the gravity theories. Both Modified Newtonian Dynamics (MOND) and General Relativity (GR) would require similar amounts of non-baryonic matter in clusters as MOND boosts the gravity only mildly on cluster scales. Gravitational lensing allows us to estimate the enclosed mass in clusters on small (∼20–50 kpc) and large (∼several 100 kpc) scales independent of the assumptions of equilibrium. Here, we show for the first time that a combination of strong and weak gravitational lensing effects can set interesting limits on the phase-space density of dark matter in the centres of clusters. The phase-space densities derived from lensing observations are inconsistent with neutrino masses ranging from 2–7 eV, and hence do not support the 2 eV-range particles required by MOND. To survive, the most plausible modification for MOND may be an additional degree of dynamical freedom in a covariant incarnation.     

Vertical dynamics of disc galaxies in modified Newtonian dynamics

We investigate the possibility of discriminating between modified Newtonian dynamics (MOND) and Newtonian gravity with dark matter, by studying the vertical dynamics of disc galaxies. We consider models with the same circular velocity in the equatorial plane (purely baryonic discs in MOND and the same discs in Newtonian gravity embedded in spherical dark matter haloes), and we construct their intrinsic and projected kinematical fields by solving the Jeans equations under the assumption of a two-integral distribution function. We find that the vertical velocity dispersion of deep MOND discs can be much larger than in the equivalent spherical Newtonian models. However, in the more realistic case of high surface density discs, this effect is significantly reduced, casting doubt on the possibility of discriminating between MOND and Newtonian gravity with dark matter by using current observations.     

The Bullet Cluster 1E0657-558 evidence shows modified gravity in the absence of dark matter

A detailed analysis of the 2006 November 15 data release X-ray surface density Σ-map and the strong and weak gravitational lensing convergence κ-map for the Bullet Cluster 1E0657-558 is performed and the results are compared with the predictions of a modified gravity (MOG) and dark matter. Our surface density Σ-model is computed using a King β-model 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 κ-map data are obtained without non-baryonic dark matter, accounting for the 8σ spatial offset between the Σ-map and the κ-map reported in Clowe et al. The MOG prediction for the κ-map results in two baryonic components distributed across the Bullet Cluster 1E0657-558 with averaged mass fraction of 83 per cent intracluster medium (ICM) gas and 17 per cent galaxies. Conversely, the Newtonian dark matter κ-model 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 Λ-cold dark matter cosmological baryon mass fraction of 0.176^+0.019_−0.012.     

Can cold dark matter paradigm explain the central-surface-densities relation?

Recently, a very strong correlation between the central surface density of stars and dynamical mass in 135 disk galaxies has been obtained. It has been shown that this central-surface-densities relation agrees very well with Modified Newtonian Dynamics(MOND). In this article, we show that if we assume the baryons have an isothermal distribution and dark matter exists, then it is possible to derive by means of the Jeans equation an analytic central-surface-densities relation connecting dark matter and baryons that agrees with the observed relation. We find that the observed central-surface-densities relation can also be accommodated in the context of dark matter provided the latter is described by an isothermal profile. Therefore, the observed relation is consistent with not only MOND.     

Milgromian dynamics and dwarf galaxies in galactic voids

We use kinematic data of 103 dwarf galaxies, obtained from the Sloan Digital Sky Survey catalog, to test the Milgromian dynamics (MOND) inside a galactic void. From this data, we compute the line-of-sight velocity dispersions of the dwarf galaxies in the frameworks of MOND and Newtonian dynamics without invoking any dark matter. The prediction for the line-of-sight velocity dispersions from MOND of 53 selected dwarf galaxies is compared with their measured values. For appropriate mass-to-light ratios in the range 1 to 5 for each individual dwarf galaxy, our results for the line-of-sight velocity dispersions predicted by MOND are more compatible with observations than those predicted by Newtonian dynamics.     

Supersonic motions of galaxies in clusters

We study motions of galaxies in galaxy clusters formed in the concordance Λ cold dark matter cosmology. We use high-resolution cosmological simulations that follow the dynamics of dark matter and gas and include various physical processes critical for galaxy formation: gas cooling, heating and star formation. Analysing the motions of galaxies and the properties of intracluster gas in a sample of eight simulated clusters at z = 0, we study the velocity dispersion profiles of the dark matter, gas and galaxies. We measure the mean velocity of galaxy motions and gas sound speed as a function of radius and calculate the average Mach number of galaxy motions. The simulations show that galaxies, on average, move supersonically with the average Mach number of ≈1.4, approximately independent of the cluster-centric radius. The supersonic motions of galaxies may potentially provide an important source of heating for the intracluster gas by driving weak shocks and via dynamical friction, although these heating processes appear to be inefficient in our simulations. We also find that galaxies move slightly faster than the dark matter particles. The magnitude of the velocity bias,   b _v ≈ 1.1  , is, however, smaller than the bias estimated for subhaloes in dissipationless simulations. Interestingly, we find velocity bias in the tangential component of the velocity dispersion, but not in the radial component. Finally, we find significant random bulk motions of gas. The typical gas velocities are of order ≈20–30 per cent of the gas sound speed. These random motions provide about 10 per cent of the total pressure support in our simulated clusters. The non-thermal pressure support, if neglected, will bias measurements of the total mass in the hydrostatic analyses of the X-ray cluster observations.     

Globular clusters in modified Newtonian dynamics: velocity dispersion profiles from self-consistent models

We test the modified Newtonian dynamics (MOND) theory with the velocity dispersion profiles of Galactic globular clusters populating the outermost region of the Milky Way halo, where the Galactic acceleration is lower than the characteristic MOND acceleration a ₀. For this purpose, we constructed self-consistent, spherical models of stellar systems in MOND, which are the analogues of the Newtonian King models. The models are spatially limited, reproduce well the surface brightness profiles of globular clusters and have velocity dispersion profiles that differ remarkably in shape from the corresponding Newtonian models. We present dynamical models of six globular clusters, which can be used to efficiently test MOND with the available observing facilities. A comparison with recent spectroscopic data obtained for NGC 2419 suggests that the kinematics of this cluster might be hard to explain in MOND.     

The relation between stellar mass and weak lensing signal around galaxies: implications for modified Newtonian dynamics

We study the amplitude of the weak gravitational lensing signal as a function of stellar mass around a sample of relatively isolated galaxies. This selection of lenses simplifies the interpretation of the observations, which consist of data from the Red-Sequence Cluster Survey and the Sloan Digital Sky Survey. We find that the amplitude of the lensing signal as a function of stellar mass is well described by a power law with a best-fitting slope  α= 0.74 ± 0.08  . This result is inconsistent with modified Newtonian dynamics (MOND), which predicts  α= 0.5  (we find  α > 0.5  with 99.7 per cent confidence). As a related test, we determine the MOND mass-to-light ratio as a function of luminosity. Our results require dark matter for the most luminous galaxies ( L ≳ 10¹¹ L_⊙). We rule out an extended halo of gas or active neutrinos as a way of reconciling our findings with MOND. Although we focus on a single alternative gravity model, we note that our results provide an important test for any alternative theory of gravity.     

Velocity dispersions of dwarf spheroidal galaxies: dark matter versus MOND

We present predictions for the line-of-sight velocity dispersion profiles of dwarf spheroidal galaxies and compare them to observations in the case of the Fornax dwarf. The predictions are made in the framework of standard dynamical theory of spherical systems with different velocity distributions. The stars are assumed to be distributed according to Sérsic laws with parameters fitted to observations. We compare predictions obtained assuming the presence of dark matter haloes (with density profiles adopted from N -body simulations) with those resulting from Modified Newtonian Dynamics (MOND). If the anisotropy of velocity distribution is treated as a free parameter, observational data for Fornax are reproduced equally well by models with dark matter and with MOND. If stellar mass-to-light ratio of 1 M_⊙/L_⊙ is assumed, the required mass of the dark halo is     , two orders of magnitude larger than the mass in stars. The derived MOND acceleration scale is     . In both cases a certain amount of tangential anisotropy in the velocity distribution is needed to reproduce the shape of the velocity dispersion profile in Fornax.     

The shape of 'dark matter' haloes of disc galaxies according to MOND

Analyses of halo shapes for disc galaxies are said to give conflicting results. I point out that the modified dynamics (MOND) predicts for disc galaxies a distribution of fictitious dark matter that comprises two components: a pure disc and a rounder halo. The former dominates the true disc in regions of small accelerations, where it controls the z -dynamics in the disc (disc flaring etc.); it has a finite total mass. It also dominates the round component near the centre where the geometry is nearly planar. The second component controls motions far from the plane, has a total enclosed mass that diverges linearly with radius, and determines the rotation curve at large radii. Its ellipticity may be appreciable at small radii but vanishes asymptotically. This prediction of MOND differs from what one expects from galaxy formation scenarios with dark matter. Analyses to date, which, as they do, assume one component – usually with a constant ellipticity – perforce give conflicting results for the best value of ellipticity, depending on whether they probe the disc or the sphere, small radii or large ones.     

Cosmological constraints from the X-ray gas mass fraction in relaxed lensing clusters observed with Chandra

We present precise measurements of the X-ray gas mass fraction for a sample of luminous, relatively relaxed clusters of galaxies observed with the Chandra observatory, for which independent confirmation of the mass results is available from gravitational lensing studies. Parametrizing the total (luminous plus dark matter) mass profiles using the model of Navarro, Frenk & White, we show that the X-ray gas mass fractions in the clusters asymptote towards an approximately constant value at a radius r ₂₅₀₀, where the mean interior density is 2500 times the critical density of the Universe at the redshifts of the clusters. Combining the Chandra results on the X-ray gas mass fraction and its apparent redshift dependence with recent measurements of the mean baryonic matter density in the Universe and the Hubble constant determined from the Hubble Key Project, we obtain a tight constraint on the mean total matter density of the Universe,     , and measure a positive cosmological constant,     . Our results are in good agreement with recent, independent findings based on analyses of anisotropies in the cosmic microwave background radiation, the properties of distant supernovae, and the large-scale distribution of galaxies.     

Gaseous drag on a gravitational perturber in Modified Newtonian Dynamics and the structure of the wake

We calculate the structure of a wake generated by, and the dynamical friction force on, a gravitational perturber travelling through a gaseous medium of uniform density and constant background acceleration   g _ext  , in the context of Modified Newtonian Dynamics (MOND). The wake is described as a linear superposition of two terms. The dominant part displays the same structure as the wake generated in the Newtonian gravity scaled up by a factor  μ⁻¹( g _ext/ a ₀)  , where a ₀ is the constant MOND acceleration and μ the interpolating function. The structure of the second term depends greatly on the angle between   g _ext  and the velocity of the perturber. We evaluate the dynamical drag force numerically and compare our MOND results with the Newtonian case. We mention the relevance of our calculations to orbit evolution of globular clusters and satellites in a gaseous protogalaxy. Potential differences in the X-ray emission of gravitational galactic wakes in MOND and in Newtonian gravity with a dark halo are highlighted.     

A comparison of different cluster mass estimates: consistency or discrepancy?

Rich and massive clusters of galaxies at intermediate redshift are capable of magnifying and distorting the images of background galaxies. A comparison of different mass estimators among these clusters can provide useful information about the distribution and composition of cluster matter and its dynamical evolution. Using the hitherto largest sample of lensing clusters drawn from the literature, we compare the gravitating masses of clusters derived from the strong/weak gravitational lensing phenomena, from the X-ray measurements based on the assumption of hydrostatic equilibrium, and from the conventional isothermal sphere model for the dark matter profile characterized by the velocity dispersion and core radius of galaxy distributions in clusters. While there is excellent agreement between the weak lensing, X-ray and isothermal sphere model-determined cluster masses, these methods are likely to underestimate the gravitating masses enclosed within the central cores of clusters by a factor of 2–4 as compared with the strong lensing results. Such a mass discrepancy has probably arisen from the inappropriate applications of the weak lensing technique and the hydrostatic equilibrium hypothesis to the central regions of clusters, as well as from assuming an unreasonably large core radius for both luminous and dark matter profiles. Nevertheless, it is pointed out that these cluster mass estimators may be safely applied on scales greater than the core sizes. Namely, the overall clusters of galaxies at intermediate redshift can still be regarded as the dynamically relaxed systems, in which the velocity dispersion of galaxies and the temperature of X-ray emitting gas are good indicators of the underlying gravitational potentials of clusters.     

Mond,dark matter and the cosmological constant

A possible connection between MOND (Modification of Newtonian Dynamics) proposed as an alternative hypothesis to dark matter in galaxies and clusters and a residual cosmological constant term dominating cosmological dynamics in a = 1 universe is explored.