List of Figures

View Image Figure 1:
Summary of the empirical roots of the missing mass problem (below line) and the generic possibilities for its solution (above line). Illustrated lines of evidence include the approximate flatness of the rotation curves of spiral galaxies, gravitational lensing in a cluster of galaxies, and the growth of large-scale structure from an initially very-nearly–homogeneous early Universe. Other historically-important lines of evidence include the Oort discrepancy, the need to stabilize galactic disks, motions of galaxies within clusters of galaxies and the hydrodynamics of hot, X-ray emitting gas therein, and the apparent excess of gravitating mass density over the mass density of baryons permitted by Big-Bang nucleosynthesis. From these many distinct problems grow several possible solutions. Generically, the observed discrepancies either imply the existence of dark matter, or the necessity to modify dynamical laws. Dark matter could, in principle, be any combination of non-luminous baryons and/or some non-baryonic form of mass-like neutrinos (hot dark matter) or some new particle, whose mass makes it dynamically cold or perhaps warm. Alternatively, the observed discrepancies might point to the need to modify the equation of gravity that is employed to infer the existence of dark matter, or perhaps some other fundamental dynamical assumption like the equivalence of inertial mass and gravitational charge. Many specific ideas of each of these types have been considered over the years. Note that none of these ideas are mutually exclusive, and that some form or the other of dark matter could happily cohabit with a modification of the gravitational law, or could even be itself the cause of an effective modification of the gravitational law. Question marks on some tree branches represent the fruit of ideas yet to be had. Perhaps these might also address the dark energy problem, with the most satisfactory result being a theory that would simultaneously explain the acceleration scale in the dark matter problem as well as the accelerating expansion of the Universe, and explain the coincidence of scales between these two problems, a coincidence exhibited in Section 4.1.
View Image Figure 2:
The fraction of the expected baryons that are detected as a function of potential-well depth (bottom axis) and mass (top). Measurements are referenced to the radius R500, where the enclosed density is 500 times the cosmic mean [284]. The detected baryon fraction fd = Mb ∕ (0.17M500 ), where Mb is the detected baryonic mass, 0.17 is the universal baryon fraction [229], and M500 is the dynamical mass (baryonic + dark mass) enclosed by R500. Each point is a bin representing many objects. Gray triangles represent galaxy clusters, which come close to containing the cosmic fraction. The detected baryon fraction declines systematically for smaller systems. Dark-blue circles represent star-dominated spiral galaxies. Light-blue circles represent gas-dominated disk galaxies. Orange squares represent Local Group dwarf satellites for which the baryon content can be less than 1% of the cosmic value. Where these missing baryons reside is one of the challenges currently faced by ΛCDM.
View Image Figure 3:
The Baryonic Tully–Fisher (mass–rotation velocity) relation for galaxies with well-measured outer velocities Vf. The baryonic mass is the combination of observed stars and gas: Mb = M ∗ + Mg. Galaxies have been selected that have well observed, extended rotation curves from 21 cm interferrometric observations providing a good measure of the outer, flat rotation velocity. The dark blue points are galaxies with M ∗ > Mg [272]. The light blue points have M ∗ < Mg [276] and are generally less precise in velocity, but more accurate in terms of the harmlessness on the result of possible systematics on the stellar mass-to-light ratio. For a detailed discussion of the stellar mass-to-light ratios used here, see [272, 276]. The dotted line has slope 4 corresponding to a constant acceleration parameter, 1.2 × 10−10 m s−2. The dashed line has slope 3 as expected in ΛCDM with the normalization expected if all of the baryons associated with dark matter halos are detected. The difference between these two lines is the origin of the variation in the detected baryon fraction in Figure 2.
View Image Figure 4:
Histogram of the accelerations a = V 4∕(GM ) f b in m s−2 (bottom axis) and natural units [4 c ∕(GmP ) where mP is the Planck mass] for galaxies with well measured Vf. The data are peaked around a characteristic value of ∼ 10 −10 m s−2 (∼ 2 × 10− 62 in natural units).
View Image Figure 5:
Residuals (δlog Vf) from the baryonic Tully–Fisher relation as a function of a galaxy’s characteristic baryonic surface density (Σb = 0.75Mb ∕R2p [271], Rp being the radius at which the contribution of baryons to the rotation curve peaks). Color differentiates between star (dark blue) and gas (light blue) dominated galaxies as in Figure 3, but not all galaxies there have sufficient data (especially of Rp) to plot here. Stellar masses have been estimated with stellar population synthesis models [42]. More accurate data, with uncertainty on rotation velocity less than 5%, are shown as larger points; less accurate data are shown as smaller points. The rotation velocity of galaxies shows no dependence on the distribution of baryons as measured by Σb or Rp. This is puzzling in the conventional context, where 2 V = GM ∕r should lead to a strong systematic residual [109].
View Image Figure 6:
The fractional contribution to the total velocity Vp at the radius RP where the contribution of the baryons peaks for both baryons (Vb∕Vp, top) and dark matter (VDM ∕Vp, bottom). Points as per Figure 5. As the baryonic surface density increases, the contribution of the baryons to the total gravitating mass increases. The dark matter contribution declines in compensation, maintaining a see-saw balance that manages to leave no residual in the BTFR (Figure 5). The absolute amplitude of Vb and VDM depends on choice of stellar mass estimator, but the fine-tuning between them must persist for any choice of M ∗∕L.
View Image Figure 7:
The Faber–Jackson relation for spheroidal galaxies, including both elliptical galaxies (red squares, [85, 232]) and Local Group dwarf satellites [285] (orange squares are satellites of the Milky Way; pink squares are satellites of M31). In analogy with the Tully–Fisher relation for spiral galaxies, spheroidal galaxies follow a relation between stellar mass and line of sight velocity dispersion (σ). The dotted line represents a constant value of the acceleration parameter 4 σ ∕ (GM ∗). Note, however, that this relation is different from the BTFR because it applies to the bulk velocity dispersion while the BTFR applies to the asymptotic circular velocity. In the context of Milgrom’s law (Section 5) the Faber–Jackson relation is predicted only when relying on assumptions such as isothermality, isotropy, and the slope of the baryonic density distribution (see 3rd law of motion in Section 5.2). In addition, not all pressure-supported systems are in the weak-acceleration regime. So, in the context of Milgrom’s law, deviations from the weak-field regime, from isothermality and from isotropy, as well as variations in the baryonic density distribution slope, would thus explain the scatter in this relation.
View Image Figure 8:
Size and surface density. The characteristic surface density of baryons as defined in Figure 5 is plotted against their dynamical scale length Rp in the left panel. The dark-blue points are star-dominated galaxies and the light-blue ones gas-dominated. High characteristic surface densities at low R p in the left panel are typical of bulge-dominated galaxies. The stellar disk component of most spiral galaxies is well approximated by the exponential disk with Σ (R) = Σ0e −R∕Rd. This disk-only central surface density and the exponential scale length of the stellar disk are plotted in the right panel. Galaxies exist over a wide range in both size and surface density. There is a maximum surface density threshold (sometimes referred to as Freeman’s limit) above which disks become very rare [264]. This is presumably a stability effect, as purely Newtonian disks are unstable [343, 415]. Stable disks only appear below a critical surface density Σ † ≈ a0∕G [299, 77].
View Image Figure 9:
The dynamical acceleration ap = Vp2∕Rp in units of a0 plotted against the characteristic baryonic surface density [275]. Points as per Figure 5. The dotted line shows the relation ap = GΣb that would be obtained if the visible baryons sufficed to explain the observed velocities in Newtonian dynamics. Though the data do not follow this line, they do show a correlation (1∕2 ap ∝ Σ b). This clearly indicates a dynamical role for the baryons, in contradiction to the simplest interpretation [109] of Figure 5 that dark matter completely dominates the dynamics.
View Image Figure 10:
The mass discrepancy in spiral galaxies. The mass discrepancy is defined [270] as the ratio V 2∕V 2b where V is the observed velocity and Vb is the velocity attributable to visible baryonic matter. The ratio of squared velocities is equivalent to the ratio of total-to-baryonic enclosed mass for spherical systems. No dark matter is required when V = V b, only when V > V b. Many hundreds of individual resolved measurements along the rotation curves of nearly one hundred spiral galaxies are plotted. The top panel plots the mass discrepancy as a function of radius. No particular linear scale is favored. Some galaxies exhibit mass discrepancies at small radii while others do not appear to need dark matter until quite large radii. The middle panel plots the mass discrepancy as a function of centripetal acceleration 2 a = V ∕r, while the bottom panel plots it against the acceleration 2 gN = V b ∕r predicted by Newton from the observed baryonic surface density Σb. Note that the correlation appears a little better with gN because the data are stretched out over a wider range in gN than in a. Note also that systematics on the stellar mass-to-light ratios can make this relation slightly more blurred than shown here, but the relation is nevertheless always present irrespective of the assumptions on stellar mass-to-light ratios [270]. Thus, there is a clear organization: the amplitude of the mass discrepancy increases systematically with decreasing acceleration and baryonic surface density.
View Image Figure 11:
The mass-discrepancy–acceleration relation from Figure 10 extended to solar-system scales (each planet is labelled). This illustrates the large gulf in scale between galaxies and the Solar system where high precision tests are possible. The need for dark matter only appears at very low accelerations.
View Image Figure 12:
The spiral galaxy NGC 6946 as it appears in the optical (color composite from the BV R bands, left; image obtained by SSM with Rachel Kuzio de Naray using the Kitt Peak 2.1 m telescope), near-infrared (J HK bands, middle [209]), and in atomic gas (21 cm radiaiton, right [481]). The images are shown at the same physical scale, illustrating how the atomic gas typically extends to greater radii than the stars. Images like these are used to construct mass models representing the observed distribution of baryonic mass.
View Image Figure 13:
Surface density profiles (top) and rotation curves (bottom) of two galaxies: the HSB spiral NGC 6946 (Figure 12, left) and the LSB galaxy NGC 1560 (right). The surface density of stars (blue circles) is estimated by azimuthal averaging in ellipses fit to the K-band (2.2μm) light distribution. Similarly, the gas surface density (green circles) is estimated by applying the same procedure to the 21 cm image. Note the different scale between LSB and HSB galaxies. Also note features like the central bulge of NGC 6946, which corresponds to a sharp increase in stellar surface density at small radius. In the lower panels, the observed rotation curves (data points) are shown together with the baryonic mass models (lines) constructed from the observed distribution of baryons. Velocity data for NGC 6946 include both HI data that define the outer, flat portion of the rotation curve [66] and Hα data from two independent observations [54, 114] that define the shape of the inner rotation curve. Velocity data for NGC 1560 come from two independent interferometric HI observations [28, 163]. Baryonic mass models are constructed from the surface density profiles by numerical solution of the Poisson equation using GIPSY [472]. The dashed blue line is the stellar disk, the red dot-dashed line is the central bulge, and the green dotted line is the gas. The solid black line is the sum of all baryonic components. This provides a decent match to the rotation curve at small radii in the HSB galaxy, but fails to explain the flat portion of the rotation curve at large radii. This discrepancy, and its systematic ubiquity in spiral galaxies, ranks as one of the primary motivations for dark matter. Note that the mass discrepancy is large at all radii in the LSB galaxy.
View Image Figure 14:
The mass discrepancy (as in Figure 10) as a function of radius in observed spiral galaxies. The curves for individual galaxies (lines) are color-coded by their characteristic baryonic surface density (as in Figure 5). In order to be completely empirical and fully independent of any assumption such as maximum disk, stellar masses have been estimated with population synthesis models [42]. The amplitude of the mass discrepancy is initially small in high–surface-density galaxies, and grows only slowly at large radii. As the baryonic surface densities of galaxies decline, the mass discrepancy becomes more severe and appears at smaller radii. This trend confirms one of the predictions of Milgrom’s law [294].
View Image Figure 15:
The shapes of observed rotation curves depend on baryonic surface density (color coding as per Figure 14). High–surface-density galaxies have rotation curves that rise steeply then become flat, or even fall somewhat to the asymptotic flat velocity. Low–surface-density galaxies have rotation curves that rise slowly to the asymptotic flat velocity. This trend confirms one of the predictions of Milgrom’s law [294].
View Image Figure 16:
Centripetal acceleration as a function of radius and surface density (color coding as per Figure 14). The critical acceleration a0 is denoted by the dotted line. Milgrom’s formula predicts that acceleration should decline with baryonic surface density, as observed. Moreover, high–surface-density galaxies transition from the Newtonian regime at small radii to the weak-field regime at large radii, whereas low–surface-density galaxies fall entirely in the regime of low acceleration a < a0, as anticipated by Milgrom [294].
View Image Figure 17:
Discretisation scheme of the BM modified Poisson equation (Eq. 17View Equation) and of the phantom dark matter derivation in QUMOND. The node (i,j,k) corresponds to A on the upper panel. The gradient components in μ(x) (for Eq. 25View Equation) and ν(y) (for Eq. 35View Equation) are estimated at the Li and Mi points. Image courtesy of Tiret, reproduced by permission from [457], copyright by ESO.
View Image Figure 18:
(a) Baryonic density of a model galaxy made of a small Plummer bulge with a mass of 2 × 108M ⊙ and Plummer radius of 185 pc, and of a Miyamoto–Nagai disk of 1.1 × 1010M ⊙, a scale-length of 750 pc and a scale-height of 300 pc. (b) The derived phantom dark matter density distribution: it is composed of a spheroidal component similar to a dark matter halo, and of a thin disk-like component (Figure made by Fabian Lüghausen [253])
View Image Figure 19:
Various μ-functions. Dotted green line: the α = 0 “Bekenstein” function of Eq. 46View Equation. Dashed red line: the α = n = 1 “simple” function of Eq. 46View Equation and Eq. 49View Equation. Dot-dashed black line: the n = 2 “standard” function of Eq. 49View Equation. Solid blue line: the γ = δ = 1 μ-function corresponding to the ν-function defined in Eq. 52View Equation and Eq. 53View Equation. The latter function closely retains the virtues of the n = 1 simple function in galaxies ( x < ∼ 10 ), but approaches 1 much more quickly and connects with the n = 2 standard function as x ≫ 10.
View Image Figure 20:
Comparison of theoretical rotation curves for the inner parts (before the rotation curve flattens) of an HSB exponential disk [145], computed with three different formulations of MOND. Green: Milgrom’s formula; Blue: Bekenstein–Milgrom MOND (AQUAL); Red: TeVeS-like multi-field theory. Image reproduced by permission from [145], copyright by APS.
View Image Figure 21:
Examples of detailed MOND rotation curve fits of the HSB and LSB galaxies of Figure 13 (NGC 6946 on the left and NGC 1560 on the right). The black line represents the Newtonian contribution of stars and gas as determined by numerical solution of the Newtonian Poisson equation for the observed light distribution, as per Figure 13. The blue line is the MOND fit with the γ = δ = 1 function of Eq. 52View Equation and Eq. 53View Equation, the only free parameter being the stellar mass-to-light ratio. In the K-band, the best fit value is 0.37M ⊙∕L ⊙ for NGC 6946 and 0.18M ⊙∕L ⊙ for NGC 1560. In practice, the best fit mass-to-light ratio can co-vary with the distance to the galaxy and a 0; here a 0 is held fixed (1.2 × 10−10 m s−2) and the distance has been held fixed to the best observed value (5.9 Mpc for NGC 6946 [220] and 3.45 Mpc for NGC 1560 [219]). Milgrom’s formula provides an effective mapping between the rotation curve predicted by the observed baryons and the observed rotation, including the bumps and wiggles.
View Image Figure 22:
The rotation curve [124] and MOND fit [384] of the Local Group spiral M33 assuming a constant stellar mass-to-light ratio (top panel). While the overall shape is a good match, there is a slight mismatch at ∼ 3 kpc and above 7 kpc. The observed color gradient implies a slight variation in the mass-to-light ratio, in the sense that the stars at small radii are slightly redder and heavier than those at large radii. Applying stellar population models [42] to the observed color gradient produces a slight adjustment of the Newtonian mass model. The dotted line in the lower panel reiterates the constant M ∕L model from the top panel, while the solid line has been corrected for the observed color gradient. This slight adjustment to the baryonic mass distribution considerably improves the fit.
View Image Figure 23:
Residuals of MOND fits to the rotation curves of 78 nearby galaxies (all data to which authors have access) including about two thousand individual resolved measurements. Data for 21 galaxies are either new or improved in terms of spatial resolution and velocity accuracy over those in [401]. More accurate points are illustrated with larger symbols. The histogram of residuals is plotted on the right panel, and is well fitted by a Gaussian of width Δv ∕v ∼ 0.04. The bulk of the more accurate data are in good accord with MOND. There are a few deviant points, mostly at small radii where non-circular motions are ubiquitous and observational resolution (beam smearing) can be a challenge. These are but a few trees outlying from a very clear forest.
View Image Figure 24:
Examples of MOND fits (blue lines, using Eq. 53View Equation with δ = 1) to two massive galaxies [402]. With baryonic masses in excess of 1011M ⊙, these are among the most massive, rapidly rotating disk galaxies known. Stars dominate the mass, and Newtonian dynamics suffices to explain the innermost regions because of the high acceleration, but the mass discrepancy becomes apparent as the Keplerian decline (black lines) falls well below the data at the enormous radii spanned by these giant disks (the diameter of UGC 2487 spans half a million lightyears).
View Image Figure 25:
Examples of MOND fits (blue lines) to two dwarf galaxies [324]. The data for DDO 210 come from [29], and those for UGC 11583 (also known as KK98 250) are from [30] augmented with high resolution data from [281, 242]. The high gas content of these galaxies make them strong tests of MOND, as the one fit-parameter – the mass-to-light ratio of the stars – has only a minor impact on the fit. What is more, as they are deep in the MOND regime, the exact form of the interpolating function (Section 6.2) also has little impact on the fits, making them the cleanest tests of MOND, with essentially no wiggle room. Note that, with a mass of only a few million solar masses (comparable in mass to the largest globular clusters), the Local Group dwarf DDO 210 is the smallest galaxy known to show clear rotation (Vf ∼ 15 km ∕s). It is the lowest point in Figure 3.
View Image Figure 26:
MOND rotation curve fits for representative galaxies from the THINGS survey [121, 166, 481]. Galaxies are chosen to illustrate a broad range of mass, from Mb ∼ 3 × 108M ⊙ to ∼ 3 × 1011M ⊙. All galaxies have high resolution interferometric 21 cm data for the gas and 3.6μ photometry for mapping the stars. The Newtonian baryonic mass model is shown as a black line and the MOND fit as a blue line (as in Figure 21). The fits use the interpolating function of Eq. 53View Equation with δ = 1.
View Image Figure 27:
MOND rotation curve fits for LSB galaxies [120] updated with high resolution Hα data [242, 241] and using Eq. 53View Equation with δ = 1. LSB galaxies are important tests of MOND because their low surface densities (Σ ≪ a0∕G) place them well into the MOND regime everywhere, and the exact form of the interpolating function is rather unimportant. Their baryonic mass models fall well short of explaining the observed rotation at any but the smallest radii in Newtonian dynamics, and MOND nevertheless provides the necessary additional force everywhere (lines as per Figure 21).
View Image Figure 28:
A comparison of the mass-to-light ratios obtained from MOND rotation curve fits (points) with the independent expectations of stellar population synthesis models (lines) [42]. The mass-to-light ratio in the optical (blue B-band, left) and near-infrared (2.2 μm K-band, right) are shown as a function of B − V color (the ratio of blue to green light). The one free parameter of MOND rotation curve fits reproduces the normalization, slope, and scatter expected from what we know about stars. Not all galaxies illustrated here have both B and K-band data. Some have neither, instead having photometry in some other bandpass (e.g., V or R or I).
View Image Figure 29:
The mass distribution of the Milky Way disk (left) inferred from fitting in MOND the observed bumps and wiggles in the rotation curve of the galaxy (right) [274]. The Newtonian contributions of the stellar and gas disk are shown as dashed and dotted lines as per Figure 13. The resulting model is consistent with independent star count data [155] and compares favorably to constraints on the rotation curve at radii beyond those included in the fit [494]. The prominent feature at R ≈ 6 kpc corresponds to the Centaurus spiral arm.
View Image Figure 30:
The scaled growth-rate of the m = 2 instability in Newtonian disks with a dark halo (dotted line) and MONDian disks (solid line) as a function of disk mass. In the MOND case, as the disk mass decreases, the surface density decreases and the disk sinks deeper into the MOND regime. However, at very low masses the growth-rate saturates. In the equivalent Newtonian case, the rotation curve is maintained at the MOND level by supplementing the force with a round stabilizing dark halo, which causes the growth-rate to crash [77, 401]. An ad-hoc dark disk could help maintain the growth rate in the dark matter context. Image reproduced by permission from [401].
View Image Figure 31:
(a) The galaxy ESO 509-98. (b) The galaxy NGC 1543. These are two examples of galaxies that exhibit clear ring and pseudo-ring structures. Image courtesy of Tiret, reproduced by permission from [458], copyright by ESO.
View Image Figure 32:
Simulations of ESO 509-98 and NGC 1543 in MOND, to be compared with Figure 31. Rings and pseudo-ring structures are well reproduced with modified gravity. Image courtesy of Tiret, reproduced by permission from [458], copyright by ESO.
View Image Figure 33:
Simulation of the Antennae with MOND (right, [459]) compared to the observations (left, [190]). In the observations, the gas is represented in blue and the stars in green. In the simulation the gas is in blue and the stars are in yellow/red. Image courtesy of Tiret, reproduced by permission from [459], copyright by ASP.
View Image Figure 34:
The NGC 5291 system [72]. VLA atomic hydrogen 21-cm map (blue) superimposed on an optical image (white). The UV emission observed by GALEX (red) traces dense star-forming concentrations. The most massive of these objects are rotating with the projected spin axis as indicated by dashed arrows. The three most massive ones are denoted as NGC5291N, NGC5291S, and NGC5291W. Image courtesy of Bournaud, reproduced by permission from [72].
View Image Figure 35:
Rotation curves of the three TDGs in the NGC 5291 system. In red: ΛCDM prediction (with no additional cold molecular gas), with the associated uncertainties. In black: MOND prediction with the associated uncertainties (prediction with zero free parameter, “simple” μ-function assumed). Image reproduced by permission from [165], copyright by ESO.
View Image Figure 36:
MOND phantom dark matter scaling relations in ellipticals. The circles display central density ρ0, and central phase space density f of the phantom dark halos predicted by MOND for different masses of baryonic Hernquist profiles (with scale-radius r H related to the effective radius by Reff = 1.815rH). The dotted lines are the scaling relations of [171], and the dashed lines those of [454], which exhibit a very large observational scatter in good agreement with the MOND prediction [363]. Image reproduced by permission from [363], copyright by ESO.
View Image Figure 37:
The surface brightness (a) and velocity dispersion (b) profiles of the elliptical galaxy NGC 7507 [375] fitted by MOND (lines [399]). Elliptical galaxies can be approximated in MOND as high-order polytropes with some radial orbit anisotropy [388]. This particular case has a polytropic index of 14 with anisotropy of the Osipkov–Merritt form with an anisotropy radius of 5 kpc and maximum anisotropy β = 0.75 at large radii [399]. The stellar mass-to-light ratio is ϒB = 3.03M ⊙∕L ⊙ ∗. This simple model captures the gross properties of both the surface brightness and velocity dispersion profiles. The galaxy is well-fitted by MOND, contrary to the claim of [375].
View Image Figure 38:
The characteristic acceleration, in units of a 0, in the smallest galaxies known: the dwarf satellites of the Milky Way (orange squares) and M31 (pink squares) [285]. The classical dwarfs, with thousands of velocity measurements of individual stars [477], are largely consistent with MOND. The more recently discovered “ultrafaint” dwarfs, tiny systems with only a handful of stars [427], typically are not, in the sense that their measured velocity dispersions and accelerations are too high. This could be due to systematic uncertainties in the data [230], as we must distinguish between −1 σ ≈ 2 km s and − 1 σ ≈ 5 km s. Nevertheless, there may be a good physical reason for the non-compliance of the ultrafaint galaxies in the context of MOND. The deviation of these objects only occurs in systems where the stars are close to filling their MONDian tidal radii: the left panel shows the half light radius relative to the tidal radius. Such systems may not be in equilibrium. Brada & Milgrom [78] note that systems will no longer respond adiabatically to the influence of their host galaxy when a star in a satellite galaxy can complete only a few orbits for every orbit the satellite makes about its host. The deviant dwarfs are in this regime (right panel).
View Image Figure 39:
The baryonic mass–X-ray temperature relation for rich clusters (gray triangles [359, 389]) and groups of galaxies (green triangles [12]). The solid line indicates the prediction of MOND: the data are reasonably consistent with the slope (M ∝ T 2), but not with the normalization. This is the residual missing baryon problem in MOND: there should be roughly twice as much mass (on average) as observed. Also shown is the scaling relation expected in ΛCDM (dashed line [137]). This is in better (if not perfect) agreement with the normalization of the data for rich clusters, but not the slope. The difference is sometimes attributed to preheating of the gas [496], which might also occur in MOND.
View Image Figure 40:
The baryon budget in the low redshift universe adopted from [421]. The census of baryons includes the detected Warm-Hot Intergalactic Medium (WHIM), the Lymanα forest, stars in galaxies, detected cold gas in galaxies (atomic HI and molecular H2), other gas associated with galaxies (the Circumgalactic Medium, CGM), and the Intracluster Medium (ICM) of groups and clusters of galaxies. The sum of known baryons falls short of the density of baryons expected from BBN: ∼ 30% are missing. These missing baryons presumably exist in some as yet undetected (i.e., dark) form. If a fraction of these dark baryons reside in clusters (an amount roughly comparable to that in the ICM) it would suffice to explain the residual mass discrepancy problem MOND suffers in galaxy clusters.
View Image Figure 41:
(a) The four images of the quasar Q2237+030 (known as the Einstein cross), gravitationally lensed by an isolated bulge-disk galaxy known as Huchra’s lens [197]. © ESA’s faint object camera on HST. (b) The empty squares denote the four observed positions of the images, and the filled square denotes the MOND-fit unique position of the source [419]. The critical curves for which M −1 = 0 in the lens plane are displayed in black, and their corresponding caustics in the source plane in red. Image reproduced by permission from [419].
View Image Figure 42:
The bullet cluster 1E0657-56. The hot gas stripped from both subclusters after the collision is colored red-yellow. The green and white curves are the isocontours of the lensing convergence parameter κ (Eq. 113View Equation). The two peaks of κ do not coincide with those of the gas, which makes up most of the baryonic mass, but are skewed in the direction of the galaxies. The white bar corresponds to 200 kpc. Image courtesy of Clowe, reproduced by permission from [102], copyright by AAS.
View Image Figure 43:
A MOND model of the bullet cluster [17]. The fitted κ-map (solid black lines) is overplotted on the convergence map of [102] (dotted red lines). The four centers of the parametrized potential used are the red stars. Also overplotted (blue dashed line) are two contours of surface density. Note slight distortions compared to the contours of κ. The green shaded region corresponds to the clustering of 2 eV neutrinos. Inset: The surface density of the gas in the model of the bullet cluster. Image reproduced by permission from [17], copyright by AAS.
View Image Figure 44:
In solid blue, the Zhao–Famaey [508] &tidle;μ(s)-function (Eq. 79View Equation) of TeVeS (Section 7.3 and 7.4), compared to the original Bekenstein one (dashed green) with a discontinuity at s = 0 [33]. The ZF function provides a more natural transition from static systems (the positive side) to cosmology (the negative side).
View Image Figure 45:
The acoustic power spectrum of the cosmic microwave background as observed by WMAP [229] together with the a priori predictions of ΛCDM (red line) and no-CDM (blue line) as they existed in 1999 [265] prior to observation of the acoustic peaks. ΛCDM correctly predicted the position of the first peak (the geometry is very nearly flat) but over-predicted the amplitude of both the second and third peak. The most favorable a priori case is shown; other plausible ΛCDM parameters [468] predicted an even larger second peak. The most important parameter adjustment necessary to obtain an a posteriori fit is an increase in the baryon density Ωb above what had previously been expected from BBN. In contrast, the no-CDM model ansatz made as a proxy for MOND successfully predicted the correct amplitude ratio of the first to second peak with no parameter adjustment [268, 269]. The no-CDM model was subsequently shown to under-predict the amplitude of the third peak [442].
View Image Figure 46:
Estimates of the baryon density Ω h2 b [where h = H ∕(100 km s−1 Mpc −1) 0] over time (updated [273] from [269]). BBN was already a well-established field prior to 1995; earlier contributions are summarized by compilations (green ovals [480, 107]) that gave the long-lived standard value Ωbh2 = 0.0125 [480]. More recent estimates from individual isotopes are shown as triangles (2H), squares (4He), diamonds (3He), and stars (7Li). Estimates of the baryon density based on analyses of the cosmic microwave background are shown by circles (dark blue for ΛCDM; light blue for no-CDM). No measurement of any isotope suggested a value greater than 2 Ωbh = 0.02 prior to observation of the acoustic peaks in the microwave background (dotted lines), which might be seen as a possible illustration of confirmation bias. Fitting the acoustic peaks in ΛCDM requires Ω h2 > 0.02 b. More recent measurements of 2H and 4He have migrated towards the ΛCDM CMB value, while 7Li remains persistently problematic [111]. It has been suggested that turbulent mixing might result in the depletion of primordial lithium necessary to reconcile lithium with the CMB (upward pointing arrow [287]), while others [405] argue that this would merely reconcile some discrepant stars with the bulk of the data defining the Spite plateau, which persists in giving a 7Li abundance discrepant from the ΛCDM CMB value. In contrast, the amplitude of the second peak of the microwave background is consistent with no-CDM and Ωbh2 = 0.014 ± 0.005 [269]. Consequently, from the perspective of MOND, the CMB, lithium, deuterium, and helium all give a consistent baryon density given the uncertainties.
View Image Figure 47:
CMB data as measured by the WMAP satellite year five data release (filled circles) and the ACBAR 2008 data release (triangles). Dashed line: ΛCDM fit. Solid line: HDM fit with a sterile neutrino of mass 11 eV. Image courtesy of Angus, reproduced by permission from [9].
View Image Figure 48:
The acceleration parameter ∼ V4f ∕ (GMb ) of extragalactic systems, spanning ten decades in baryonic mass M b. X-ray emitting galaxy groups and clusters are visibly offset from smaller systems, but by a remarkably modest amount over such a long baseline. The characteristic acceleration scale √ -- a0 ∼ Λ is in the data, irrespective of the interpretation. And it actually plays various other independent roles in observed galaxy phenomenology. This is natural in MOND (see Section 5.2), but not in ΛCDM (see Section 4.3).