 |
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. |
 |
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 , where the enclosed
density is 500 times the cosmic mean [284]. The detected baryon fraction ,
where is the detected baryonic mass, 0.17 is the universal baryon fraction [229], and
is the dynamical mass (baryonic + dark mass) enclosed by . 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 CDM. |
 |
Figure 3:
The Baryonic Tully–Fisher (mass–rotation velocity) relation for galaxies with
well-measured outer velocities . The baryonic mass is the combination of observed stars and gas:
. 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 [272]. The light blue points have [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, . The dashed line has slope 3 as expected in CDM
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. |
 |
Figure 4:
Histogram of the accelerations in (bottom axis) and natural
units [ where is the Planck mass] for galaxies with well measured . The data
are peaked around a characteristic value of ( in natural units). |
 |
Figure 5:
Residuals ( ) from the baryonic Tully–Fisher relation as a function of a galaxy’s
characteristic baryonic surface density ( [271], 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 ) 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 or . This is puzzling in the
conventional context, where should lead to a strong systematic residual [109]. |
 |
Figure 6:
The fractional contribution to the total velocity at the radius where the
contribution of the baryons peaks for both baryons ( , top) and dark matter ( ,
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 and depends on choice of stellar mass estimator,
but the fine-tuning between them must persist for any choice of . |
 |
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 ( ).
The dotted line represents a constant value of the acceleration parameter . 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. |
 |
Figure 8:
Size and surface density. The characteristic surface density of baryons as defined in
Figure 5 is plotted against their dynamical scale length 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 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
. 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
[299, 77]. |
 |
Figure 9:
The dynamical acceleration in units of plotted against the
characteristic baryonic surface density [275]. Points as per Figure 5. The dotted line shows the
relation 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
( ). 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. |
 |
Figure 10:
The mass discrepancy in spiral galaxies. The mass discrepancy is defined [270] as the
ratio where is the observed velocity and 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 , only when . 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 , while the bottom panel plots it against the
acceleration predicted by Newton from the observed baryonic surface density .
Note that the correlation appears a little better with because the data are stretched out
over a wider range in than in . 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. |
 |
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. |
 |
Figure 12:
The spiral galaxy NGC 6946 as it appears in the optical (color composite from the
bands, left; image obtained by SSM with Rachel Kuzio de Naray using the Kitt Peak
2.1 m telescope), near-infrared ( 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. |
 |
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 -band ( m)
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 H 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. |
 |
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]. |
 |
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]. |
 |
Figure 16:
Centripetal acceleration as a function of radius and surface density (color coding as
per Figure 14). The critical acceleration 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 , as anticipated by Milgrom [294]. |
 |
Figure 17:
Discretisation scheme of the BM modified Poisson equation (Eq. 17) and of the phantom
dark matter derivation in QUMOND. The node corresponds to on the upper panel.
The gradient components in (for Eq. 25) and (for Eq. 35) are estimated at the
and points. Image courtesy of Tiret, reproduced by permission from [457], copyright by ESO. |
 |
Figure 18:
(a) Baryonic density of a model galaxy made of a small Plummer bulge with a mass of
and Plummer radius of 185 pc, and of a Miyamoto–Nagai disk of , 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]) |
 |
Figure 19:
Various -functions. Dotted green line: the “Bekenstein” function of Eq. 46.
Dashed red line: the “simple” function of Eq. 46 and Eq. 49. Dot-dashed black line: the
“standard” function of Eq. 49. Solid blue line: the -function corresponding
to the -function defined in Eq. 52 and Eq. 53. The latter function closely retains the virtues of
the simple function in galaxies ( ), but approaches 1 much more quickly and
connects with the standard function as . |
 |
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. |
 |
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
function of Eq. 52 and Eq. 53, the only free parameter being the stellar mass-to-light
ratio. In the -band, the best fit value is for NGC 6946 and for
NGC 1560. In practice, the best fit mass-to-light ratio can co-vary with the distance to the galaxy
and ; here is held fixed ( ) 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. |
 |
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 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. |
 |
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 . 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. |
 |
Figure 24:
Examples of MOND fits (blue lines, using Eq. 53 with ) to two massive
galaxies [402]. With baryonic masses in excess of , 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). |
 |
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 ( ). It is the lowest point in Figure 3. |
 |
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 to
. All galaxies have high resolution interferometric 21 cm data for the gas and
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. 53 with
. |
 |
Figure 27:
MOND rotation curve fits for LSB galaxies [120] updated with high resolution H
data [242, 241] and using Eq. 53 with . LSB galaxies are important tests of MOND because
their low surface densities ( ) 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). |
 |
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 -band, left) and near-infrared (2.2 m -band, right)
are shown as a function of 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 and -band data. Some have
neither, instead having photometry in some other bandpass (e.g., or or ). |
 |
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 corresponds to the Centaurus spiral arm. |
 |
Figure 30:
The scaled growth-rate of the 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]. |
 |
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. |
 |
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. |
 |
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. |
 |
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]. |
 |
Figure 35:
Rotation curves of the three TDGs in the NGC 5291 system. In red: CDM
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”
-function assumed). Image reproduced by permission from [165], copyright by ESO. |
 |
Figure 36:
MOND phantom dark matter scaling relations in ellipticals. The circles display central
density , and central phase space density of the phantom dark halos predicted by MOND
for different masses of baryonic Hernquist profiles (with scale-radius related to the effective
radius by ). 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. |
 |
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 at large radii [399]. The stellar mass-to-light ratio is
. 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]. |
 |
Figure 38:
The characteristic acceleration, in units of , 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
and . 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). |
 |
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 ( ), 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 CDM (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. |
 |
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 Lyman 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. |
 |
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 in the lens plane are displayed in black, and their corresponding caustics in the
source plane in red. Image reproduced by permission from [419]. |
 |
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 (Eq. 113). The two peaks of 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. |
 |
Figure 43:
A MOND model of the bullet cluster [17]. The fitted -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 . 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. |
 |
Figure 44:
In solid blue, the Zhao–Famaey [508] -function (Eq. 79) of TeVeS (Section 7.3 and
7.4), compared to the original Bekenstein one (dashed green) with a discontinuity at [33]. The
ZF function provides a more natural transition from static systems (the positive side) to cosmology
(the negative side). |
 |
Figure 45:
The acoustic power spectrum of the cosmic microwave background as observed by
WMAP [229] together with the a priori predictions of CDM (red line) and no-CDM (blue line)
as they existed in 1999 [265] prior to observation of the acoustic peaks. CDM 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
CDM 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 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]. |
 |
Figure 46:
Estimates of the baryon density [where ] 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 [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 CDM; light
blue for no-CDM). No measurement of any isotope suggested a value greater than
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 CDM requires
. More recent measurements of 2H and 4He have migrated towards the CDM
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 CDM CMB value. In contrast, the amplitude of the second
peak of the microwave background is consistent with no-CDM and [269].
Consequently, from the perspective of MOND, the CMB, lithium, deuterium, and helium all give a
consistent baryon density given the uncertainties. |
 |
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: CDM fit. Solid line: HDM fit with a sterile
neutrino of mass 11 eV. Image courtesy of Angus, reproduced by permission from [9]. |
 |
Figure 48:
The acceleration parameter of extragalactic systems, spanning ten
decades in baryonic mass . 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 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 CDM (see Section 4.3). |