List of Footnotes

1 Up to now, all the dark-matter–particle candidates still elude both direct and indirect non-gravitational detection.
2 However, a way to effectively reproduce an apparent universal force law from an exotic dark component could be to enforce an intimate connection between the distribution of baryons, the dark component, and the gravitational field through, e.g., a fifth force effect. This possibility will be extensively discussed in Section 7, notably Section 7.9
3 The first four sections provide the observational evidence for the MOND phenomenology through the different appearances of a0 in galactic dynamics, but they are actually independent of any specific theory, while the reader more specifically interested in MOND per se could go directly to Section 5.
4 Arguably, a non-static, expanding or contracting Universe was an a priori prediction of general relativity in its original form, lacking the cosmological constant.
5 However, the WIMP miracle seems to fade away with modern particle physics constraints [23Jump To The Next Citation Point].
6 The simplest WIMPs are their own antiparticle.
7 In addition, the time-averaged value of the deceleration parameter q over the present age of the Universe is quite consistently ⟨q⟩ = 0 [473], another currently unexplained coincidence.
8 c = G = ¯h = 1.
9 We have that Λ ∼ a20∕c2 if expressed in inverse time-squared or Λ ∼ a20∕c4 if expressed in inverse length-squared (more precisely, the natural scale associated with the cosmological constant is ∘ - a0 ≈ (c2∕2π) (Λ ∕3)). Another way of expressing this coincidence is to say that predictions of GR from visible matter alone always break down for physics involving a length-scale constant on the order of the Hubble radius l ∼ Λ−1∕2 ∼ c2∕a0. This scale l could perhaps play a similar role to the Planck scale lP [233Jump To The Next Citation Point, 43], at the other end of the ladder (as we have l ≈ 1060lP ). However, this is not the length at which the modification would be seen, exactly as quantum mechanics does not depart from classical physics at a given length.
10 As we shall see (Section 5 and 6), MOND was constructed to predict a relation a = V4∕(GM ) 0 f for a point mass M (however, note that the slope of 4 is a pure consequence of the acceleration base; it is not possible to get an arbitrary slope from such an idea). Since spiral galaxies are not point masses, but rather flattened mass distributions that rotate faster than the equivalent spherical mass distribution [52Jump To The Next Citation Point], the empirical acceleration a is close to but not identical to a 0 in MOND. The geometric correction is about 20% so that a = 0.8a 0 [272Jump To The Next Citation Point].
11 The factor 10 arises from the commonly adopted definition of the virial radius of the dark matter halo at an overdensity of 200 times the critical density of the Universe [332Jump To The Next Citation Point].
12 Note that [180] claimed to measure a slope of 3 for the BTFR, but they relied on unresolved line-widths from single dish 21 cm observations to estimate rotation velocity rather than measuring Vf from resolved rotation curves. Line-widths give a systematically different estimate of the slope of the BTFR than Vf, even for the same galaxies [277, 340, 475], and they cannot be related at all to the circular velocity of the potential at the virial radius, nor to the prediction of MOND (Section 5 and 6).
13 The difference in phase space between gas and dark matter also prevents the accretion of tidal gas onto any dark matter sub-halos that may be present. It does not suffice for a tidal tail to intersect the location of a sub-halo in coordinate space, they must also dock in velocity space. The gas is moving at the characteristic velocity of the entire system (typically ∼ 200 km s−1), which, by definition, exceeds the escape speed of typical sub-halos (usually < 100 km s−1). Therefore, the odds of capture are effectively zero unless the tail and sub-halo happen to be on very nearly the same orbit initially, which is itself very unlikely because of the initial difference in their phase space distribution.
14 Note that this correlation with acceleration was looked at notably because it was pointed to by Milgrom’s law (see Section 5).
15 The Pioneer anomaly has an amplitude on the order of ∼ 10− 9 m s−2 but appears at a location in the solar system where the total gravitational acceleration is ∼ 10−6 m s−2. Thus, the discrepancy in Figure 11View Image is (V∕Vb)2 ≈ 1.001.
16 Note that such wiggles are often associated with spiral arm features (the existence of which in LSB galaxies being itself challenging in the presence of a massive dark matter halo, see Section 4.2), and hence associated with non-circular motions. It is conceivable that such observed wiggles are partly due to these, but the effect of local density contrasts due to spiral arms on the tangential velocity should be damped by the global effect of the spherical–dark-matter halo, which is apparently not the case.
17 Note that many of these relations were scrutinized during the last 30 years because they were pointed to by Milgrom’s law. Thus, this law has already achieved the important role of a theoretical idea, i.e., to point and direct observations and their arrangement
18 Of course, there is also a natural length scale associated with this acceleration constant, l = c2∕a0, but this length scale will enter the modification nonlinearly, and is thus not the length at which the modification would be seen in galaxies, as it is rather on the order of the Hubble radius
19 Note that the denominator 2πG comes from integrating the phantom dark matter density along a vertical line as per [313Jump To The Next Citation Point], which leads to a slightly smaller characteristic surface density for phantom dark matter than the Σ† defining Freeman limit in the 6th law.
20 Note that the main motivation for modifying dynamics is not to get rid of DM, but to explain why the observed gravitational field in galaxies is apparently mimicking a universal force law generated by the baryons alone. The simplest explanation is, of course, not that DM arranges itself by chance to mimic this force law, but rather that the force law itself is modified. Note that at a fundamental level, relativistic theories of modified gravity often will have to include new fields to reproduce this force law, so that dark matter is effectively replaced by “dark fields” in these theories, or even by dark matter exhibiting a new interaction with baryons (one could speak of “dark matter” if the stress-energy tensor of the new fields is numerically comparable to the density of baryons): this makes the confrontation between modified gravity and dark matter less clear than often believed. The actual confrontation is rather that between all sorts of theories embedding the phenomenology of Milgrom’s law vs. theories of DM made of simple self-uninteracting billiard balls assembling themselves in galactic halos under the sole influence of unmodified gravity, theories, which currently appear unable to explain the observed phenomenology of Milgrom’s law.
21 Generally covariant theories approaching these classical theories in the weak-field limit will then also be classified under this same MOND acronym, even if they really are Modified Einsteinian Dynamics (see Section 7)
22 The Newtonian mass density also satisfies the continuity equation ∂ρ∕∂t+ ∇.(ρv) = 0.
23 In general relativity, the first two terms ∫ 2 3 ρ(v ∕2− ΦN )d xdt are lumped together into the matter action (also containing the rest mass contribution in GR), and the last term is generalized by the Einstein–Hilbert action.
24 Let us note in passing that it would not be the first time that the kinetic action would be modified as special relativity does just this too, changing for a single particle mv2∕2 → − mc2γ−1 in Skin (where ∘ --------- γ(v) = 1∕ 1− (v∕c)2), leading for a moving body to a redefinition of the effective mass as meff = mγ(v). With this analogy in mind, a rather simplified view of the Lorentz-breaking modification of inertia needed in order to reproduce MOND would be that meff ≃ m μ(a), where a is the amplitude of the acceleration with respect to an absolute preferred inertial frame.
25 Such non-local theories, which also have to be nonlinear (like any MOND theory) are not easy to construct, and there is presently no real fully-fledged theory, which has been developed in this vein, although hints in this direction are summarized in Section 7.10.
26 Following the dielectric analogy (Section 5.1), this is akin to Maxwell’s first equation, Gauss’ law, in terms of free charge density ρf, i.e., ∇.[μ𝜖0E] = ρf, where E is the electric field and μ 𝜖0E = D is the electric displacement field. See [56Jump To The Next Citation Point] for a thorough discussion of the analogy.
27 This is similar to the Palatini formalism of GR, where the present auxiliary acceleration field is replaced by a connection
28 Confusing these two interpolating functions μ(x) and &tidle;μ(s) can lead to serious mistakes [489Jump To The Next Citation Point], as illustrated by [41Jump To The Next Citation Point].
29 In principle, α can be slightly larger, but if α ≫ 1, then in the range of gravities of interest for galaxy dynamics (between 0.1a0 and a few times a0) the scalar field contribution s is too small to account for the MOND effect, or said in another way, the corresponding Milgrom μ-function would deviate significantly from μ(x) = x (i.e., μ(x) > x, so that there would be less modification to the Newtonian prediction).
30 In principle, one could make − 1 A = α as small as desired in the α-family, by not limiting α to the range between 0 and 1, but passing solar system constraints would require α > 20, which would cancel the MOND effect in the range of interest for galaxy dynamics.
31 Note that, among the freedom of choice of that function, one could additionally even imagine that the μ-function is not a scalar function but a “tensor” μij such that the modification becomes anisotropic and the modified Poisson equation becomes something like ∂i[μijgj] = 4πGρ.
32 It is interesting to note that different MOND theories offer (very) different answers to the generic question “acceleration with respect to what?”. For instance, in the MOND-from-vacuum idea (see [304Jump To The Next Citation Point] and Section 7.10), the total acceleration is measured with respect to the quantum vacuum, which is well defined. In BIMOND (Section 7.8) it is the relative acceleration between the two metrics, which is also well defined through the difference of Christoffel symbols.
33 Thus, a Cavendish experiment in a freely falling satellite in Earth orbit would return a Newtonian result in MOND.
34 For instance, using the “simple” function μ(x) = x∕(1+ x) in Eq. 59View Equation would lead to ∘ -------------------------------------- g = [(gN a0 + gNge − 2gea0 − g2e)+ (2gea0 + ge2− gNa0 − gN ge) +4gN (a0 + ge)3]∕[2(a0 + ge)].
35 See also [203] and constraints excluding such functions also from Lunar Laser Ranging [138], neglecting the external field effect from the sun on the Earth-Moon system, since it is three orders of magnitude below the internal gravity of the system.
36 This is why, although the “simple” α = 1 function is known to represent very well the gravitational field of spiral galaxies [141Jump To The Next Citation Point, 166Jump To The Next Citation Point, 402Jump To The Next Citation Point, 508Jump To The Next Citation Point], we use, in Section 6.5.1, the γ = δ = 1 function of Eq. 52View Equation and Eq. 53View Equation in order to fit spiral galaxy rotation curves.
37 For the γ-family of Eq. 52View Equation and δ-family of Eq. 53View Equation, it means that even slightly sharper transitions than γ = 1 and δ = 1 might still be needed.
38 Note that the rotation curves of Figure 20View Image become flat only at larger radii than shown here; see [145Jump To The Next Citation Point].
39 If one assumes that a lot of dark baryons are present in the form of molecular gas, one can add another free parameter in the form of a factor multiplying the gas mass[460]. The, good MOND fits can still be obtained but with a lower value of a0.
40 However, the mass-to-light ratio is not really a constant in galaxies. Thus, Figure 22View Image gives an example of a rotation curve fit (to the Local Group galaxy M33), where the variation of the mass-to-light ratio according to the color-gradient has been included, even improving the MOND fit.
41 Separable models have also been investigated in [97].
42 The conventional baryon fraction of clusters increases monotonically with radius [426], only obtaining the cosmic value of 0.17 at or beyond the virial radius. Therefore, one might infer the presence of dark baryons in cluster cores in ΛCDM as well as in MOND.
43 If the action has the units of ¯h, the factor in front of the gravitational action is then c3∕16πG. And if one wishes to include a cosmological constant Λ, the integral then rather reads ∫ d4x√ − g(R − 2Λ).
44 With this signature, the proper-time is defined by 2 μ ν dτ = − gμνdx dx.
45 Note that, at 1PN, this weak-field metric can also be written as 2 2 g00 = − e2Φ∕c ,gij = e2Ψ∕c δij. Note also that Taylor expanding Eq. 70View Equation yields ∫ Spp = m (v2∕2− ΦN − c2)dt, so that the sum of the classical kinetic and internal actions for a point particle (see Eq. 13View Equation) are now lumped together into the matter action.
46 The derived lensing and dynamical masses are typically very close to each other but the data are not yet precise enough to ascertain that they are exactly identical.
47 The frame associated to the Einstein metric is called the “Einstein frame” as opposed to the “matter frame” or “Jordan frame”, associated to the physical metric.
48 k-essence fields have also recently been reintroduced as possible dark energy fluids, that could also drive inflation [20, 21, 92]. This name comes from the fact that their dynamics are dominated by their kinetic term f(X ) (in the case of RAQUAL, there is no potential at all), contrary to other dark energy models such as quintessence, in which the scalar field potential plays the crucial role.
49 Expressed in terms of U μ and its congugate momenta Pμ = ∂L∕∂U˙μ.
50 It is also important to remember that some interpolation functions (Section 6.2) are already excluded by solar system tests, and thus, it is useless to exclude these over and over again.
51 A characteristic matter density ρ0 thus becomes an additional order parameter, in the spirit of the velocity scale s0 of [36], see Eq. 3.1 of [82Jump To The Next Citation Point].
52 In the case of TeVeS and GEA theories, the dark fields do not really count as dark matter because their energy density is subdominant to the baryonic one.
53 This is to be contrasted with the time-like nature of TeVeS and GEA vector fields in the static weak-field limit.
54 And the current Jμ is conserved, i.e., ∇μJμ = 0
55 It can be shown that only the projection perpendicular to the four-velocity enters the field equations deduced from the action of Eq. 100View Equation. Thus, the dipole moment is always fully space-like.
56 This equality Ψ = − Φ in the weak-field metric is put in by hand in all TeVeS-like theories (Sections 7.2 to 7.6) through a disformal relation between the Einstein and physical metrics, and is a generic prediction of GEA (Section 7.7), BIMOND (Section 7.8) and DDM (Section 7.9) theories.
57 However, by this we do not mean that the MOND lensing can be computed from the projected surface density on the lens-plane as in GR, because the convergence parameter (Eq. 113View Equation below) is not a measure of the projected surface density anymore. This is also sometimes referred to as the “thin-lens approximation” in GR, and is not valid in MOND: two lenses with the same projected surface density can have different convergence parameters, because lensing also depends strongly on the distribution of the source mass along the line-of-sight in MOND.
58 Note that, in order for the problem to be well constrained, a regularization method was used in order to penalize solutions deviating from the fundamental plane as well as face-on solutions and solutions with an anomalous flux ratio or M/L ratio (see Eq. 21 of [419].)
59 Note, however, that this is not always the case in colliding clusters: Abell 520 actually provides a counter-example to the bullet cluster in which the mass peaks indicated by weak lensing do not behave as collisionless matter should [210].
60 Even if BAO features are present at high redshift in MOND, it is not clear that low redshift structures will correlate with the ISW in the CMB as they should in conventional cosmology because of the late time non-linearity of MOND.
61 Perhaps the most famous modern example of confirmation bias is in measurements of the Hubble constant [466], where over many years de Vaucouleurs persistently found H ≈ 100 km s−1 Mpc −1 0 while Sandage persistently found H ≈ 50 km s− 1 Mpc −1 0. Then, as now, there was a conflation of data with theory: the lower value of H 0 was more widely accepted because it was required for cosmology to be consistent with the ages of the oldest stars.
62 At ℓ ≈ 800, the third peak is only marginally resolved by WMAP. This scale is comparable to a single (frequency-dependent) beam size, and, as such, is extraordinarily sensitive to corrections for the instrumental point spread function [404].
63 Determining agreement between independent observations requires that we believe not just the result (e.g., the value of H0 from direct distance measurements) but also its uncertainty. The latter has always been challenging in astronomy, and the history of cosmology is replete with examples of results that were simply wrong. While we may have entered the era of precision cosmology, we have yet to reach an era when data are so accurate that we can hope to challenge cosmology with falsification if, for example, PLANCK data require H0 < 60 km s−1 Mpc−1, while galaxy distances require H0 > 70 km s−1 Mpc −1.
64 The third possibility actually means either non-local effects in non-local theories (Section 7.10), or the effect of additional fields in local modified gravity theories. The important difference with CDM is that these fields are not simply representative of collisionless massive particles, that their behavior is determined by the baryons in static configurations, and that they can be subdominant to the baryonic density. In theories where their energy density dominates that of baryons, these new fields then really act as dark matter in the early universe, which is also a possibility (see Section 7.6 and 7.9)
65 In TeVeS, the perturbations of the scalar field also play an important role in generating enhanced growth [146].
66 Ferreira’s talk, Alternative Gravities and Dark Matter Workshop, Edinburgh, April 2006.
67 The ISW effect can be cast as the integral of ′ ′ − (Φ + Ψ) + 2Φ, thus involving both a gravitational slip part and a growth rate part.
68 Note that the presence of non-baryonic matter in the form of massive neutrinos also helps damping the baryonic acoustic oscillations [127Jump To The Next Citation Point] at z = 0, as can be seen in Figure 4 of [430Jump To The Next Citation Point].
69 This means that, on surfaces of constant temperature, the densities of the various components (e.g. baryons, neutrinos, additional dark fields) are uniform, and that these components share a common velocity field.
70 [279Jump To The Next Citation Point] utilized this fact to predict that conventional analyses of LSB disks would infer abnormally high mass-to-light ratios for their stellar populations – a prediction that was subsequently confirmed [159, 371].
71 The observed shock velocity of ∼ 4700 km∕s is thought to be enhanced by hydrodynamical effects. The collision velocity is improbable after a substantial (∼ 1700 km∕s) correction for this [258].