MOND is an alternative paradigm of dynamics, seeking to replace Newtonian dynamics and general relativity. It aims to account for the ubiquitous mass discrepancies in the Universe, without invoking the dark matter that is required if one adheres to standard dynamics. MOND departs from standard dynamics at accelerations smaller than \\( a_0\\): a new constant with the dimensions of acceleration that MOND introduces into physics. Such accelerations characterize galactic systems and the Universe at large. The other central tenet of MOND is space-time scale-invariance of this low-acceleration limit. This symmetry, together with the pivotal role of acceleration, immediately leads to the prediction of many clear-cut laws of galactic dynamics (analogous to, and extending Kepler’s laws), such as asymptotic flatness of rotation curves, and a mass-asymptotic-speed relation \\(V^4_\infty\propto MG a_0\\). The latter had predicted the so called “baryonic Tully-Fisher relation”, confirmed later. Another obvious prediction of MOND, also clearly confirmed, is a tight correlation between the observed mass discrepancy in galactic systems, and the accelerations in them. In general, MOND predicts very well the observed dynamics of individual galaxies of all types (from dwarf to giant spirals, ellipticals, dwarf spheroidals, etc.), and of galaxy groups, based only on the distribution of visible matter (and no dark matter). Specifically, the general laws of galactic dynamics predicted by MOND’s basic tenets are well obeyed by the data, with \\( a_0\\) appearing in these laws in different, independent roles (as \\(\hbar\\) appears in disparate quantum phenomena). Significantly perhaps, it’s measured value coincides with acceleration parameters of cosmological relevance, namely, \\(\bar a_{0}\equiv 2\pi a_0\approx cH_0\approx c^2(\Lambda/3)^{1/2}\\) (\\(H_0\\) is the Hubble constant, and \\(\Lambda\\) the cosmological constant). This adds to several other mysterious coincidences that characterize the mass-discrepancy conundrum, and may provide an important clue to the origin of MOND. For galaxy clusters, MOND reduces greatly the observed mass discrepancy: from a factor of \\(\sim 10\\), required by standard dynamics, to a factor of about 2. But, this systematically remnant discrepancy is yet to be accounted for. It could be due to, e.g., the presence of some small fraction of the yet undetected, “missing baryons”, which are known to exist (unlike the bulk of the putative “dark matter”, which cannot be made of baryons). MOND, as a set of new laws, affords new tools for astronomical measurements–such as of masses and distances of far away objects–in ways not afforded by standard dynamics. Beyond the basic tenets, we need to construct full-fledged theories, generalizing Newtonian dynamics and general relativity, that satisfy the basic tenets, that are, preferably, derived from an action, and that can be applied to any system and situation. There exist several nonrelativistic theories of MOND as modified gravity incorporating its basic tenets. Recent years have seen the advent of several relativistic formulations of MOND. These account well for the observed gravitational lensing, but do not yet provide a satisfactory description of cosmology and structure formation. While these theories are useful in many ways, it may well be that none of them points to the correct MOND theory. We do not know if MOND is only relevant to gravitational phenomena, or should also affect in some way other phenomena, such as electromagnetism. ## Contents * 1 MOND introduced * 2 Rudiments of MOND phenomenology: MOND laws of galactic dynamics * 2.1 Systems embedded in an external field and the external-field effect * 3 The significance of the MOND acceleration constant * 4 MOND phenomenology in detail * 4.1 Disc galaxies * 4.2 Pressure-supported systems * 4.3 Elliptical galaxies * 4.4 Dwarf spheroidals and tidal dwarfs * 4.5 Galaxy groups * 4.6 Galaxy clusters * 4.6.1 The “Bullet” cluster * 5 MOND as an astronomical tool * 6 MOND theories * 6.1 Nonrelativistic theories * 6.1.1 Modified Poisson gravity * 6.1.2 Quasilinear MOND * 6.1.3 Generalizations * 6.1.4 Modified inertia theories * 6.2 Relativistic theories * 6.2.1 TeVeS * 6.2.2 MOND adaptations of Einstein-Aether theories * 6.2.3 Bimetric MOND * 6.2.4 Nonlocal single-metric theories * 6.2.5 Dipolar dark matter * 6.3 Microscopic and other theories * 7 Cosmology and structure formation * 8 Summary * 9 Footnotes * 10 References * 10.1 MOND Reviews * 10.2 Useful links # [edit] MOND introduced Newtonian analysis of galaxy dynamics leads ubiquitously to large mass discrepancies: The masses directly observed fall far short of the dynamical masses: those needed to account for the observed motions in galaxies and systems of galaxies. Adherence to standard dynamics has thus lead to the idea of “dark matter” (DM): galactic gravity is much stronger than meets the eye because galactic systems contain large quantities of yet undetected matter, in a yet unknown form. In a similar vein, observations in cosmology require, within standard dynamics, two dark components: DM, which might economically be assumed of the same kind as the galactic remedy, and another, even less constrained, component called “dark energy”. In contradistinction, MOND posits that the observed discrepancies are due to failures of standard dynamics in the realm of galactic systems and the cosmos at large; failures that lead to artificially large dynamical masses. The dynamical masses and their distributions as calculated with MOND should agree with those of the baryonic masses observed directly, without DM. MOND, conceived in mid 1981, was enunciated in January 1982 in a series of three papers, published, after some struggle, in 1983 (Milgrom, 1983a; Milgrom, 1983b; Milgrom, 1983c). The main observational fact on which it drew was the rough tendency of disc-galaxy rotation curves then available to become flat in their outer parts. MOND then elevated the asymptotic flatness of rotation curves to an axiomatic requirement for the paradigm. In addition, known constraints on the observed slopes of the Tully-Fisher relation where reckoned with. The crucial novelty of MOND was the imputation of the mass discrepancies in galactic systems to the low accelerations in them. Then, by generalization, it posited a sweeping departure from standard dynamics at low accelerations. Famaey & McGaugh, 2012 is an extensive recent review of MOND. The motivation for considering alternatives to standard dynamics plus dark entities is severalfold. Foremost is the fact that there is an alternative, such as MOND, that works well and is much more predictive in the realm of the galaxies. Second, it is known that the DM paradigm is beset by many problems when confronted with the data on galaxies (e.g., Famaey & McGaugh, 2012; Kroupa, 2012; Boylan-Kolchin, et al. 2012; Weinberg, et al., 2013). Also, the major potential obviator of alternatives: the direct detection of DM, has not materialized, despite many searches over many years. Three basic tenets capture the essence of the MOND paradigm: 1. Departure from standard dynamics occurs at low accelerations, i.e., below some acceleration constant, \\( a_0\\), that MOND introduces into physics. (This is analogous to relativity departing from Newtonian dynamics for speeds near the speed of light, or to quantum theory departing from classical physics for values of the action of order or smaller than \\(\hbar\\).) 2. At high accelerations (corresponding to taking the formal limit \\( a_0 {\rightarrow} 0\\) in a MOND based theory, or relation) standard dynamics is restored. 3. In the low-acceleration limit, for purely gravitational systems of relativistically weak fields (but not necessarily slow motions of particles), the MOND equations of motion are space-time scale invariant; namely, symmetric under stretching (or contracting) all times and all lengths measured in a system by the same factor, \\((t, { \bf{r}}) {\rightarrow} {\lambda}(t, { \bf{r}})\\) (Milgrom, 2009a).1\\(^,\\)2 The deep-MOND limit of a theory, or a relation, can be effected by taking \\( a_0 {\rightarrow}\infty\\) and \\(G {\rightarrow} 0\\), keeping \\( { {\mathcal{A}}_0}\equiv G a_0\\) finite. (Equivalently: inflate all lengths and times in a system by a factor \\( {\lambda}\\), and let \\( {\lambda} {\rightarrow}\infty\\). If the limit exists, it is automatically scale invariant, since further, finite scalings have no effect.) In such a limiting theory, neither \\( a_0\\) not \\(G\\) can appear; the only dimensioned constants that can appear are \\( { {\mathcal{A}}_0}\\) and masses (and \\(c\\) in relativistic theories).3 Thus, \\( { {\mathcal{A}}_0}\\) is the “scale invariant” gravitational constant that replaces \\(G\\) in the deep-MOND limit. It might have been more appropriate to introduce this limit and \\( { {\mathcal{A}}_0}\\) first, and then introduce \\(a_0\equiv { {\mathcal{A}}_0}/G\\) as delineating the boundary between the \\(G\\)-controlled standard dynamics and the \\({ {\mathcal{A}}_0}\\)-controlled deep-MOND limit. Had the world been governed by deep-MOND dynamics, we would not have known about \\(G\\) or \\( a_0\\), only the fact that there is a Newtonian range of phenomena brings them to light. For a collection of point, test masses \\(m_i\\), scale invariance means that if trajectories \\({\bf r}_i(t)\\) are a solution of the theory, so are \\(\lambda{\bf r}_i(t/\lambda)\\) [with velocities \\({\bf v}_i(t/\lambda)\\)] for any \\(\lambda>0\\) (with appropriately scaled initial conditions). For a continuum mass distribution: if the density-velocity fields \\(\rho({\bf r},t)\\), \\({\bf v}({\bf r},t)\\) are a solution, so is \\(\lambda^{-3}\rho({\bf r}/\lambda,t/\lambda)\\), \\({\bf v}({\bf r}/\lambda,t/\lambda)\\). This is illustrated in Figure 1. Figure 1: Two systems of masses that are related by space-time scaling by a factor \\(b\\). A snapshot of the first is shown at time \\(t\\), and of the second at time \\(bt\\), when it is an image of the first inflated by a factor \\(b\\) (these are not two configurations of the same system shown at two times; the first system might look very different at time \\(bt\\)). Note that the velocities are the same in the two configurations. With the additional knowledge that only the constant \\(\mathcal{A}_0\\) appears in the theory one can deduce that the theory is invariant to a larger, two-parameter family of scalings: If \\({\bf r}_i(t)\\) is a system history for masses \\(m_i\\), then \\(\alpha{\bf r}_i(t/\beta)\\) [with velocities \\((\alpha/\beta){\bf v}_i(t/\beta)\\)], is a system history for masses \\((\alpha/\beta)^4 m_i\\) for any \\(\alpha,\beta>0\\). For continuum systems, if \\(\rho({\bf r},t)\\), \\({\bf v}({\bf r},t)\\) is a solution, so is \\(\alpha\beta^{-4}\rho({\bf r}/\alpha,t/\beta)\\), \\((\alpha/\beta){\bf v}({\bf r}/\alpha,t/\beta)\\). In the relativistic context it may be useful to view the MOND length, \\(\ell_M\equiv c^2/ a_0\\) as more fundamental. However, the threshold for galactic phenomena is defined by an acceleration, \\( a_0\\), not by a length. The MOND mass \\(M_M=c^4/ { {\mathcal{A}}_0}\\) is also a useful reference in some contexts. MOND is generically nonlinear. This means that the effect felt by a test particle under the gravitational influence of a system of masses is not the simple sum of the effects produced by the constituents separately. (Linearity is rather unique to Newtonian gravity. The theory of general relativity is also nonlinear.) All our observational constraints on the mass discrepancies, and hence on MOND, come from systems whose dynamics is by far dominated by gravity. Neither these constraints, then, nor existing theoretical considerations, tell us whether MOND applies only to gravity, such as if it is underlaid by modification of gravity, or whether it should be applied as well to all other phenomena, such as electromagnetism -- as would be the case if it is underlaid by “modification of inertia”. This is an important issue to explore. # [edit] Rudiments of MOND phenomenology: MOND laws of galactic dynamics Until MOND is put on firmer theoretical grounds, and is underlaid by a first-principle theory, phenomenology remains its foremost raison d’être. Clearly, for very detailed predictions of MOND we need a theory, but it turns out that many robust predictions can be made based on the basic tenets alone (Milgrom, 2014).4 Some of these apply to deep-MOND phenomena and follow from scale invariance, and some follow from the existence of a transition to the MOND regime, all revolving around \\(a_0\\). For example, it is readily seen that asymptotically far from a central mass, \\(M\\), the effective gravitational field should become invariant to dilatations; i.e., the effective potential5 is logarithmic, and so the MOND acceleration, \\(g\\), is inversely proportional to the distance, \\(r\\), from \\(M\\). The fact that only \\(\mathcal{A}_0\\) and \\(M\\) can appear in the deep-MOND limit dictates, in itself, that in the spherically symmetric, asymptotic limit we must have \\(g\propto (M \mathcal{A}_0)^{1/2}/r\\), since this is the only expression with the dimensions of acceleration that can be formed from \\(M\\), \\(\mathcal{A}_0\\), and \\(r\\). The basic tenets imply this proportionality, but the exact normalization of \\(a0\\) (and hence \\(\mathcal{A}_0\\)) is still free. It is conventional to normalize \\( a_0\\) so that equality holds.6 Thus, MOND predicts for the asymptotic gravitational acceleration \\[g= \frac{(M\mathcal{A}_0)^{1/2}}{r}=\left(\frac{M}{M_M}\right)^{1/2}\frac{c^2}{r}=a_0\frac{r_M}{r}.\tag{1}\\] Here, \\(r_M=(MG/a_0)^{1/2}=(M/M_M)^{1/2}\ell_M\\) is the MOND radius of the mass, where we cross from standard dynamics to the MOND regime. (\\(r_M\\) is analogous to the Schwarzschild radius \\(r_s=2MG/c^2\\), which for a mass \\(M\\) marks the transition from the relativistic to the Newtonian dynamics.) This asymptotic behavior is valid only when we are far outside the mass distribution, and far outside \\(r_M\\), but not too far, so the mass may be considered “isolated”, and unaffected by the fields of other masses in the Universe. Since typical background accelerations in the present Universe are a few percents of \\(a_0\\), the above asymptotic expression is valid for \\(r\\) roughly between a few \\(r_M\\) and a few tens \\(r_M\\). The MOND acceleration vs. the distance from a point mass are compared with those for the Newtonian acceleration, \\(g_N=MG/r^2\\), in Figure 2. Figure 2: The MOND acceleration produced by an isolated mass \\(M\\), as a function of radius from the mass (in heavy lines), for a star of \\(1 M_{\odot}\\) (red), a globular cluster of \\(10^5 M_{\odot}\\) (blue), a galaxy of \\(3\times 10^{10} M_{\odot}\\) (green), and a galaxy cluster of \\(3\times 10^{13} {M_{\odot}}\\) (magenta). The Newtonian accelerations are shown as dashed lines. Departure of MOND from Newtonian dynamics occurs at different radii for different central masses (the respective MOND radii of these masses), but always at the same value of the acceleration, \\( a_0\\), below which we are in the MOND regime, and above which we are in the Newtonian regime. Succinctly formulated, some of the predicted MOND laws are (propounded in the original MOND papers, except as noted): 1. Speeds along an orbit around any isolated, bounded mass, \\(M\\), become independent of the size of the orbit for asymptotically large radii. For example, the velocity on a circular orbit becomes independent of the orbital radius, \\(r\\), for very large \\(r\\) we have \\(V(r) {\rightarrow} {V_{\infty}}(M)\\). This contrasts with Kepler’s 3rd law, which rests on Newtonian dynamics, according to which \\(V\propto \sqrt{M/r}\\). 2. \\( {V_{\infty}}(M)=(M { {\mathcal{A}}_0})^{1/4}=c(M/M_M)^{1/4}\\). 3. Define the discrepancy as the ratio , \\(\eta=g/g_N\\), of the observed acceleration, \\(g\\), to the Newtonian accelerations, \\(g_N\\) (calculated from the observed mass alone), at a given position. In a system where \\(g(r)\\) varies with radius, a discrepancy is predicted to appear at the radius where \\(g(r)\\) (or, equivalently, \\(g_N\\)) crosses \\( a_0\\). For example, in a disc galaxy with measured rotational speed \\(V(r)\\), the discrepancy is predicted to always start around the radius where \\(g(r)=V^2(r)/r= a_0\\). 4. In system where \\(g,~g_N< a_0\\) everywhere, a discrepancy is predicted everywhere, with \\(\eta\approx a_0/g\\). Observationally, low acceleration (small \\(MG/r^2\\)) is synonymous with low surface density (small \\(M/r^2\\)), or low surface brightness (luminosity per unit area). 5. Many galactic systems -- such as globular clusters, dwarf spheroidal, and elliptical galaxies, galaxy clusters -- may be described as Quasi-isothermal systems: systems where gravity is balanced by random motions of their constituents, whose velocity dispersion is roughly independent of radius. For such systems MOND predicts that they must have mean mass surface densities \\(\bar\Sigma\lesssim { {\Sigma}_M}\equiv a_0/ 2\pi G\\). 6. In an isolated, quasi-isothermal or deep-MOND system of mass \\(M\\), a characteristic velocity dispersion \\(\sigma\sim (M { {\mathcal{A}}_0})^{1/4}=c(M/M_M)^{1/4}\\) is predicted.7 If one interprets MOND consequences as being due to a DM halo, then MOND predicts the following for this fictitious halo: 7. The acceleration produced by such a fictitious halo can never much exceed \\( a_0\\) (Brada and Milgrom, 1999a). 8. The central surface density such “dark halos” is \\(\lesssim \Sigma_M\\), with \\(\Sigma_M\\) being an accumulation point, with near equality holding for many halos (Milgrom, 2009b). 9. The MOND central-surface-densities relation (CSDR): The `dynamical' central surface density of a disc galaxy, \\(\Sigma^0_D\equiv -(4\pi G)^{-1}\int_{-\infty}^{\infty}\nabla\cdot {\bf g}(z,r=0)dz\\) (i.e., the total dynamical column density along the galaxy's symmetry \\(z\\)-axis, baryonic plus phantom) is strongly correlated with the baryonic central surface density of the disc, \\(\Sigma^0_B\\). Specifically, in the presently known modified-gravity theories (see section on theories) the two attributes are functionally related: \\(\Sigma^0_D=\Sigma_M\mathcal{S}(\Sigma^0_B/\Sigma_M) \\). The exact form of \\(\mathcal{S}(x)\\) can be calculated and depends somewhat on the specific MOND theory. But its asymptotes are fixed by the basic tenets of MOND: \\(\mathcal{S}(x\gg 1)\approx x\\), and \\(\mathcal{S}(x\ll 1)\approx 2x^{1/2}\\) (Milgrom, 2016). 10. Scale invariance of the relativistically-weak-field limit, extends laws (1) and (2) to gravitational light bending (Milgrom, 2014b): The bending angle, \\(\theta\\), for light rays from a distant source, passing at a distance \\(b\gg r_M\\) from an isolated body of mass \\(M\\), is constant. \\(\theta\\) can depend only on \\((M\mathcal{A}_0)^{1/2}/c^2=V^2_{\infty}(M)/c^2\ll 1\\). If in a MOND theory, \\(\theta\\) is first order in this quantity, as is plausible, this behavior is the same as that of GR with a gravitational acceleration proportional to that in eq. (1). The proportionality constant depends on the theory. Some additional, more qualitative, predictions of MOND are: 11. The complete study of the dynamics of a spiral galaxy includes study of motions perpendicular to its galactic disc. MOND predicts that, all considered, analysis in the framework of Newtonian dynamics will require not only a spheroidal halo, but also a thin disc of putative “dark matter” in such a galaxy. 12. MOND endows self gravitating systems with an increased, but limited stability (Milgrom, 1989a; Brada & Milgrom, 1999b; Banik, Milgrom, & Zhao, 2018). 13. High-acceleration systems should show no discrepancies. Since these laws follow essentially from only the basic tenets, they should be shared in one way or another, by all MOND theories that embody these tenets. They are also independent as phenomenological laws8, and would require independent explanations in the framework of the DM paradigm. In fact, some of these laws, when interpreted in terms of DM, would describe properties of the “DM” alone [e.g., laws (1), (7), and (8)], of the baryons alone [e.g., law (5)], or relations between the two [e.g., laws (2) and (3)]. Law (2), the “mass-asymptotic-speed relation (MASR)”, is a most robust and clear-cut prediction of the basic tenets. In the phenomenological context, it is the prediction of a specific “baryonic Tully-Fisher relation” (BTFR). The original Tully-Fisher relation, in its different varieties, is a phenomenological correlation between the luminosity of a disc galaxy, in some photometric band, and some measure of its rotational speed (e.g., some measure of the 21 cm line width). Unlike this, the MOND MASR dictates the following: (i) Correlate the total (baryonic) mass of the galaxy, not its luminosity, which, at best, is a measure of the stellar mass alone. In particular, the MASR stresses the need to include the mass of the gas (Milgrom & Braun, 1988), since this can contribute substantially to the total mass. (ii) Use the asymptotic value of the rotational speed as a velocity measure (this requires measuring the rotation curve, not just some integrated line profile). The predicted MASR has been clearly confirmed, as shown, e.g., in Figure 3, and see, e.g., Sanders, 1996; Noordermeer & Verheijen, 2007; McGaugh, 2011; McGaugh, 2012; den Heijer et al., 2015; Papastergis et al., 2016. There are many studies in the literature, said to plot a BTFR. They are `baryonic' in that they use an estimate of the total baryonic mass. However, many use other measures of the rotational speed, not \\(V_{\infty}\\) (see, e.g., the meta-analysis in Bradford et al., 2016). I reserve `MASR' for the relation that uses the MOND prescription. This prediction, together with law (1), both encapsulated in eq. (1), were also tested, statistically, on a large sample of galaxies of all types, using weak gravitational lensing, as shown in Figure 4 from Milgrom (2013). See also weak-lensing testing of MOND by Brouwer et al. (2021), shown in Figure 5. This method measures distortions of background galaxy images by foreground galaxies of all kinds, to statistically map the gravitational fields of the latter. It tests MOND in a wide variety of galaxies, using relativistic test particles (light), and down to very low accelerations -- much lower than are accessible to rotation-curve analysis, as clearly demonstrated in Figure 5. Rotation-curve tests are, however, more accurate, and can be applied to individual galaxies, while the weak-lensing technique is statistical. Figure 3: Data for galaxy baryonic mass plotted against the measured asymptotic rotation speed, compared with the MOND prediction (line). Left: a large sample of disc galaxies of all types (circles for gas-rich, squares for star-dominated galaxies). Middle: the same test with only gas-rich galaxies included, for which the baryonic mass is insensitive to adopted stellar mass-to-light ratios (McGaugh, 2011). In both plots, the line is the MOND prediction using the value of \\( a_0\\) determined earlier from rotation-curve analysis of 11 galaxies (McGaugh, 2012). Right: distribution of \\(V_f^4/MG\\) for the latter sub-sample, compared with that expected from measurement errors alone; showing that the observed scatter is consistent with no intrinsic scatter in the observed relation. Figure 4: The MOND predictions (Milgrom, 2013) of the velocity-dispersion (\\(\sigma\\))-Luminosity (\\(L\\)) relations that are deduced from galaxy-galaxy weak lensing, shown for baryonic mass-to-light ratios \\(M/L=1,~1.5,~3,~6\\) solar values (these ratios are needed to translate the observed luminosities to masses, which appear in the predictions). The measurements (Brimioulle, et al. 2013) are for ‘blue’ (blue squares) and ‘red’ (red triangles) lenses, respectively. The predicted lines for \\(M/L\\) of 1.5 and 6 are practically identical to the best-fit relations found in Brimioulle, et al. 2013 for ‘blue’ (aka spiral or disc) and ‘red’ (aka elliptical) lenses, respectively. In Brimioulle, et al. 2013, the strength of the asymptotic logarithmic potential [predicted by MOND [eq. (1)], and verified separately in a preliminary step] is \\(2\sigma^2\\). MOND predicts the relation \\(\sigma=(\mathcal{A}_0/4)^{1/4}(M/L)^{1/4}L^{1/4}\\). Figure 5: The MOND test with the weak-lensing analysis of Brouwer et al. (2021). The accelerations produced at many radii in many galaxies -- shown as various data points, and measured by gravitational lensing -- are plotted against the accelerations that would have been produced in Newtonian dynamics (with no dark matter) by the observed (baryonic) matter. This is compared with the MOND prediction in eq. (3) below (from Milgrom, 1983a, with an interpolating function used by McGaugh et al., 2016). Also shown are the data points testing the same relation, from McGaugh et al., 2016, using rotation curves of the galaxies in the SPARC sample. We see that the lensing test reaches much lower accelerations than can be probed with rotation curves. We see, then, that MOND generally predicts a tight correlation between the observed \\(\eta\\) and the acceleration, \\(g\\) [laws (3)(4)(13)] as follows: (1) No discrepancy for \\(g,~g_N\gg a_0\\). (2) The discrepancy appears around, and develops below, \\(a_0\\). (3) Far below \\(a_0\\), we should have \\(\eta\approx a_0/g\\), or equivalently, \\(\eta\approx (a_0/{g_N})^{1/2}\\). This discrepancy-acceleration relation is encapsuled in the interpolated form given below in eq. (3) (from Milgrom, 1983a). This has been checked and confirmed several times, starting with Sanders, 1990 (Fig. 4 there), and McGaugh, 1999 (Fig. 7 there), for rotationally-supported, disc galaxies, and with Scarpa, 2003 (Figs. 6-8 there) and Scarpa, 2006, for pressure-supported systems. Then, with more and better data for disc galaxies, in McGaugh, 2004 (Figs. 4, 5 there), Tiret & Combes, 2009 (Fig. 3 there), Wu & Kroupa, 2015 (Fig. 1 there), McGaugh & al., 2016 (Fig. 3 there); and for early-type (elliptical) galaxies by Janz & al., 2016 and by Tian & Ko, 2017. An updated test of this crucial MOND prediction is shown in Famaey & McGaugh, 2012 for disc galaxies, a modified version of which (provided by Stacy McGaugh) is reproduced here in Figure 6. The important lesson from Figure 6 contains several sub-lessons that are worth appreciating: (1) Many of the points in the \\(g_N\ll a_0\\) region come from the asymptotic regions of galaxies. Their obeying \\(\eta\approx (a_0/g_N)^{1/2}\\) is then a recapitulation of law (2). (2) Many of the points in this same region come from the bulk regions of galaxies whose accelerations are \\(\ll a_0\\) everywhere. They too satisfy \\(\eta\approx (a_0/g_N)^{1/2}\\), which is a new lesson. (3) The asymptotic (red) line [\\(\eta=(a_0/g_N)^{1/2}\\)] is drawn for an \\(a_0\\) value that is derived from law (2), namely based on the very outskirts of disc galaxies. It can be read from the intersection of the red and blue lines. We learn from Figure 6 that this same value constitutes also the "boundary constant" that separates the Newtonian and the deep-MOND regimes. (4) We see that the transition between the two regimes occur within a \\(g_N\\) range roughly between \\(a_0/2\\) and \\(2a_0\\). From points (3) and (4) we learn that MOND does not involve a new large (or small) dimensionless constants. Law (4) was a surprising, major prediction of MOND (Milgrom, 1983b), before it was found observationally that, indeed, all low-acceleration galaxies show large mass discrepancies. Figure 6: The measured discrepancy: the ratio, \\(\eta=g/{g_N}\\), at many radii, in 73 disc galaxies (courtesy of Stacy McGaugh). In the upper panel, \\(\eta\\) is plotted against radius, where we see no correlation. In the lower panel, it is plotted against \\( {g_N}\\). As predicted by MOND (Milgrom, 1983a), the discrepancy is a tight function of \\( {g_N}\\) [the function \\( {\nu}( {g_N}/ a_0)\\) defined in eq. (3) below], and departs from 1 for accelerations smaller than \\(\sim 10^{-8}~{\rm cm~s^{-2}}(=10^{-10}~ {\rm m~s^{-2}})\\). The lines show the two asymptotic behaviors predicted by MOND: the Newtonian limit, in blue, \\(\eta=1\\), and the deep-MOND limit, in red, \\(\eta=(a_0/g_N)^{1/2}\\). Figure 7: The MOND central-surface-densities relation (CSDR) (Milgrom, 2016) plotted with the data of Lelli et al. (2016). The thicker, blue line (full and dashed) is the equality line (the Newtonian asymptote of the MOND prediction). The thinner, red line (full and dashed) is the predicted, deep-MOND asymptote. The thinnest, black line is the full MOND relation. For the data, the Toomre surface density, \\(\Sigma^0_T\\), is taken as a proxy for \\(\Sigma^0_D\\), and the proxy for \\(\Sigma^0_B\\) is the central, stellar surface density, \\(\Sigma^0_*\\). The dotted line is the best-fit to the data in Lelli et al. (2016), with some 3-parameter formula (not theoretically motivated). No fitting is involved in the MOND curves. The values of the MOND surface density, \\(\Sigma_M\\) is marked. Law (9) is compared with the relevant data of Lelli et al. (2016) in Figure 7 (see more details in Milgrom, (2016)). It is another prediction of an exact, functional relation. Unlike the MASR, it involves a local baryonic attribute, and a global dynamical attribute, and instead of being concerned with the outer parts of a galaxy, it pertains to the inner parts. Thus, for instance, in the language of dark matter paradigm: For disc galaxies that differ only in their central, baryonic surface densities (they may have the same total mass, for example), sitting even within overwhelmingly dominant halos, the halos must know to have their total column densities conform with the MOND CSDR, for the specific baryon surface density at the center. This is a tall order, indeed. Law (13) is quite unexpected in the DM paradigm. It pertains to (and holds well in) globular clusters, the inner parts of elliptical galaxies (which are high surface brightness systems) and of high-surface-brightness disc galaxies (see below, and Figure 6), and to compact dwarf galaxies (Scarpa, 2005), but, on the face of it, does not hold for the cores of galaxy clusters (see below). A recent review of those (and other) MOND predictions for disc galaxies, and of how they compare with the data, can also be found in McGaugh, 2020. ## [edit] Systems embedded in an external field and the external-field effect In many cases we deal with a relatively-small subsystem, embedded, or falling freely, in the field of a possibly larger and more massive mother system. For example, stars, gas clouds, globular clusters, or satellite galaxies falling in the field of a mother galaxy, or a galaxy in the field of a galaxy cluster, or of the background field produced by large-scale structure. The internal accelerations, \\(g_{in}\\), inside the subsystem - i.e. those relative to its center of mass -- can be smaller (as in many dwarf satellites) or larger (as in stars) than its free-fall acceleration, \\(g_{ex}\\). In light of the inherent nonlinearity of MOND, two questions then arise. The first is: `how do the motions internal to the subsystem affect its motion in the mother system? In existing MOND theories the answer is that the center-of-mass motion of the subsystem is not affected by the internal structure and dynamics in the limit of small and light subsystems (Bekenstein & Milgrom, 1984; Milgrom, 2010a). This shows that the `weak-equivalence principle', aka `universality of free fall', holds in these theories: All systems that are small compared with the scale over which the external field varies, fall in the same way in that external field, independent of their internal dynamics (including the magnitude of the internal accelerations). The second relevant question is `how are internal dynamics affected by the external field \\(g_{ex}\\)? In Newtonian dynamics and in general relativity, a constant external acceleration field does not affect the internal dynamics. This is encapsuled in the `strong equivalence principle', which implies that within a small `laboratory', such as the space vehicle, freely-falling in a gravitational field (constant within the extent of the `laboratory') no effect of either the field or the acceleration can be felt. In MOND, the situation is very different. Although the exact effect may depend on the MOND theory, the generic answer is that the internal dynamics are affected by the external field, through the so called MOND external-field effect (EFE) (Milgrom, 1983a; Bekenstein & Milgrom, 1984; Milgrom, 2014). For example, if \\(g_{ex}\gg a_0\\) (as is the case on Earth), the internal dynamics is Newtonian. If \\(g_{in}\ll g_{ex}\ll a_0\\), the internal dynamics is approximately Newtonian, but with a much larger effective gravitational constant \\(G_{eff}\sim Ga_0/g_{ex}\\). Even when \\(a_0\gg g_{in}\gg g_{ex}\\) there will be a small effect on the internal dynamics because \\(g_{ex}\\) dictates the boundary conditions. But in this case it is difficult to give a general rule. The detailed dependance on the specific MOND theory (e.g., Milgrom, 2014) is particularly important when \\(g_{ex}\sim a_0\\), where the departure of MOND from Newtonian dynamics is small, but possibly not negligible. This fact is relevant, e.g., for suggested tests in small systems in the Milky way, not far from the sun, where the galactic acceleration, \\(g_{ex}=(1.5-2)a_0\\). For example, the study of dynamics perpendicular to the galactic disc, or that using wide binaries (Hernandez, Jimenez & Allen, 2012; Pittordis & Sutherland, 2018; Banik & Zhao, 2018). These are expected to depart only a little from Newtonian behavior; but exactly how little depends strongly on the MOND formulation. Some noteworthy aspects of the EFE are: 1\. The EFE, hinging as it does on accelerations, is unique to MOND. It is not accounted for within the dark-matter paradigm, which is based on general relativity, which in turn satisfies the strong equivalent principle.(General relativity is the only full-fledged relativistic theory known to satisfy the this principle, apart from its unripe predecessor -- the Nordstrom theory). 2\. A robust observational verification of the EFE may point to the existence of an absolute inertial frame -- giving meaning to absolute, not only relative, accelerations. For some reason, this inertial frame makes itself felt clearly at low accelerations (at or below \\(a_0\\)). It was suggested in Milgrom, 2011b that this frame may be defined by the quantum vacuum. This fact makes the EFE one of several aspects of MOND that may point to deeper and far-reaching principles. (Another aspect is the a0-cosmology connection.) The relevance of the EFE in this connection is detailed in Milgrom, 2020. 3\. The EFE is responsible for the inefficacy of Earth, and inner-solar-system experimental tests of MOND. These are very-high-acceleration environments, which strongly suppresses any MOND departures from standard dynamics. For example, only gravitational-wave modes that are compatible with general relativity can penetrate and propagate in the inner solar system, even if other modes exist according to some MOND theory. 4\. The EFE acts not only when the external acceleration dominates over the internal ones. Even when the external acceleration is small it can subtly affect the internal dynamics (see examples below). 5\. The exact form of the effect is known for specific modified-gravity formulations of MOND, such as the modified-Poisson and the QUMOND theory (see below), but a wider scope of possibilities is open for other theories. Following is succinct list of landmarks in the study and application of the EFE: The effect was introduced as a generic MOND effect, and discussed in connection with various galactic systems in Milgrom, 1983a, where it was it was also identified as potentially of deep implications, and as offering a major test of MOND. The effect was derived exactly, for the case of a dominant external field, in the first full-fledged, modified-gravity formulation of MOND in Bekenstein & Milgrom, 1984, and further elaborated on, for this theory, in Milgrom, 1986. The exact form in QUMOND was derived in Milgrom, 2010a. Brada & Milgrom, 2000 showed that the EFE by the Magellanic clouds on the Milky Way might account for the Galaxy's warp in its outer parts. Brada & Milgrom, 2000a, using both analytic considerations and numerical calculations, described the EFE of a mother galaxy on dwarf satellites. In particular, the emphasis was on effects due to the variable external field as the satellite moves in its non-circular orbit, changing its distance and aspect relative to the mother galaxy. Tiret, et al. 2007 discussed the EFE on satellites of elliptical galaxies. A glimpse of EFE versions that can emerge in modified-inertia MOND theories was given in Milgrom, 2011b. The EFE was implemented as part of the study of the observed internal dynamics of dwarf satellites in McGaugh & Milgrom, 2013b, and in McGaugh, 2016. In particular, the former, which studied the dwarf satellites of the Andromeda galaxy, noted the appearance of expected differences in internal velocity dispersions between dwarfs that look the same (have the same luminosity and size), but are subject to different external fields due to their different distances from Andromeda. MOND predicts asymptotically-flat rotation curves for isolated disc galaxies. However, the EFE due to large-scale structure, or to near-bye massive bodies, such as galaxy clusters, is predicted to cause a decline in the outer parts of the rotation curve, depending on the strength of the external field relative to the centripetal acceleration in the galaxy (Milgrom, 1983a). Haghi, et al. 2016, and Hees, et al. 2016, showed that the MOND predictions of rotation curves of disc galaxies improve in the outer parts -- better accounting for a slight decline observed in some galaxies -- if one allows for the action of some EFE. (But they did not attempt to correlate this decline with the actual external fields present in each case.) Chae, et al. 2020 have elevated this observation to a full-fledged test. Much improving on earlier studies, they used a large sample of galaxies with observed and analyzed rotation curves. And, importantly, they estimated for each galaxy, the external field in which it is falling, due to surrounding structures. They found a statistically quite significant agreement between their estimated external field, and the observed decline in the rotation curve, of the magnitude predicted by the MOND EFE. # [edit] The significance of the MOND acceleration constant The central role of an acceleration constant, \\(a_0\\), in various facets of galaxy phenomenology, is now well established (e.g., Milgrom, 1983b; Milgrom, 1983c; Sanders, 1990; McGaugh, 2004; Scarpa, 2003; Tiret & Combes, 2009; Milgrom, 2009b; Famaey & McGaugh, 2012; Milgrom, 2014; Trippe, 2014; Walker & Loeb, 2014). It marks the boundary below which the mass discrepancies appear, and it also appears in various regularities. These aspects of galaxy phenomenology are here to stay, whether one views MOND as a modification of dynamics or not. They call for an explanation in any paradigm claiming to account for the mass discrepancies. But, the appearance of a critical acceleration constant does not follow in any known DM scenario. \\(a_0\\) can be determined from several of the MOND laws in which it appears, as well as from more detailed analyses, such as of full rotation curves of galaxies. All of these give consistently \\( a_0\approx (1.2\pm 0.2)\times 10^{-8}{\rm cm~s^{-2}}\\). It was noticed early on (Milgrom, 1983a; Milgrom, 1989; Milgrom, 1994) that this value is of the order of cosmologically relevant accelerations. \\[\bar a_0\equiv 2\pi a_0\approx cH_0\approx c^2(\Lambda/3)^{1/2}, \tag{2}\\] where \\(H_0\\) is the Hubble constant, and \\(\Lambda\\) the cosmological constant. In other words, the MOND length, \\(\ell_M\approx 7.5\times 10^{28}{\rm cm} \approx 2.5\times 10^4{\rm Mpc} \\), is of order of today’s Hubble distance, namely, \\(\ell_M\approx 2\pi \ell_H\\) (\\(\ell_H\equiv c/H_0\\)), or of the de Sitter radius associated with \\(\Lambda\\), namely, \\(\ell_M\approx 2\pi \ell_S\\). The MOND mass, \\(M_M\approx 10^{57}{\rm gr}\\), is then \\(M_M\approx 2\pi c^3/GH_0\approx 2\pi c^2/G(\Lambda/3)^{1/2}\\), of the order of the closure mass within today’s horizon, or the total energy within the Universe observable today. Thus, to the already mysterious coincidences concerning the dark sector (the roughly similar densities of baryons and dark matter, and the fact that, today, these also are of the same order as the “dark energy” density), MOND has pointed out another: The appearance of the cosmological acceleration parameters in local dynamics in systems very small on the cosmological scale. This “coincidence” may be an important hint for understanding the origin of MOND, and for constructing MOND theories. If indeed fundamental, it may point to the most far-reaching implication of MOND: The state of the Universe at large strongly enters local dynamics of small systems.9 Alternatively, such a coincidence could come about if the same fundamental parameter enters both cosmology, as a cosmological constant, and local dynamics, as \\( a_0\\). This connection may underlie the reason a break in the dynamical behavior occurs at some critical acceleration as entailed in MOND (and not, e.g., as one crosses a critical distance) (Milgrom, 1994): An acceleration, \\(a\\), of a body or a system defines a length, \\(\ell_a\equiv c^2/a\\), that plays different roles. For example, it defines the scale of the Rindler horizon associated with \\(a\\); it is the characteristic wavelength of the Unruh effect corresponding to \\(a\\); it defines the maximal size of a locally freely falling frame that can be erected around the body, etc. When \\(a\gg a_0\\) we have \\(\ell_a\ll \ell_M\sim\ell_H,~\ell_S\\), while \\(a\ll a_0\\) corresponds to \\(\ell_a\gg \ell_H,~\ell_S\\). Thus, if in some way, yet to be established (but see below, and Milgrom, 1999; Pikhitsa, 2010; Li & Chang, 2011; Kiselev & Timofeev, 2011; Klinkhamer & Kopp, 2011; van Putten, 2014), a body of acceleration \\(a\\) is probing distances \\(\sim \ell_a\\), then a body with \\(a\gg a_0\\) does not probe the nontrivial (curved) geometry of the Universe, while a body with \\(a\ll a_0\\) does. And this could establish \\(a_0\\) as a transition acceleration. This is analogous to the fact entailed in quantum physics that particles with momentum \\(P\\) define a length of order of their de Broglie wavelength, \\(h/P\\), as dictated by the uncertainty principle. So, for example, in a box of size \\(L\\) a transition momentum, \\(P_0=h/L\\), is defined; the state spectrum for \\(P\gg P_0\\) is oblivious to the presence of the box, but not so for \\(P\lesssim P_0\\). If \\(a\\) above is the gravitational acceleration produced by a mass \\(M\\) at a distance \\(R\\), we have \\(\ell_a/R\approx 2R/R_S\\) in the Newtonian regime, and \\(\ell_a/R\approx c^2/V^2_{\infty}\\) in the deep-MOND regime, where \\(R_S\\) and \\(V_{\infty}\\) are, respectively, the Schwarzschild radius and the asymptotic circular speed for \\(M\\). So, \\(\ell_a\\) is always larger than the system size \\(R\\), with near equality occurring for black holes or the Universe at large. If, indeed, \\(cH_0\\) (and not only the cosmological constant) is causally related to \\(a_0\\), then, since \\(H\\) varies with cosmic time, by its definition, so may \\(a_0\\). For example, if always \\(a_0\sim cH/2\pi\\), then \\(a_0\\) decreases as \\(H\\) does. Such variations could be identified directly from MOND analysis of objects at high redshift, which are seen at early cosmic times. For example, by measuring a redshift dependence of the proportionality coefficient in the mass-velocity relations. Such variations can also be discerned or constrained because they would have caused secular evolution of galactic systems (Milgrom, 1989; Milgrom, 2015) due to the adiabatic changes in \\(a_0\\), which enters the dynamics of these systems. For example, a system starting in the deep-MOND regime, will exhibit velocities that vary as \\(V^4\propto MGa_0\\), or \\(V\propto a_0^{1/4}\\), and, if adiabatic invariance implies \\(RV=constant\\), the lengths in the system would have varied as \\(R\propto a_0^{-1/4}\\). Then \\(V^2/Ra_0\propto a_0^{-1/4}\\); so it increases with decreasing \\(a_0\\), and may reach unity. As regions in the system reach \\(a\approx a_0\\) they would have stopped varying due to the \\(a_0\\) variations. As long as a system is wholly in the deep-MOND regime, it expands homologously with decreasing \\(a_0\\), with velocities decreasing everywhere by the same factor (as can be seen from the scale invariance of this limit). Recent analysis (Milgrom, 2017) found that the rotation curves of distant disc galaxies presented by Genzel et al. (2017) may already preclude large variations of \\(a_0\\) with cosmic time. Some “practical” consequences of this nearness in values, eq. (2), are: (i) If a system of mass \\(M\\), and size \\(R< \ell_H\\), produces gravitational accelerations \\(MG/R^2< a_0\\), then \\(MG/R< c^2/2\pi\\): Namely, no system smaller than today’s cosmological horizon requires for its description both a relativistic, strong-field (\\(MG/R\sim c^2\\)) and deep-MOND description. This means that to describe all phenomena except the Universe at large we need only a relativistically-weak-field theory. Since MONDian dynamics is probably a derived, effective concept, it is not clear that an effective MOND theory needs to have a consistent relativistic deep-MOND limit. (ii) Strong lensing (e.g., image splitting) of cosmological sources (such as quasars) by a much nearer lens cannot probe the MOND regime. (iii) Energy losses of high-energy particles by Cherenkov radiation of subluminal gravitational waves, which may occur in MOND theories, are unimportant for sub-Hubble travel (Milgrom, 2011a). (iv) For a gravitational wave of dimensionless amplitude \\(h\\) we can define a MOND-relevant acceleration attribute: \\(g_W=hc^2/\lambda\\), where \\(\lambda\\) is the wavelength. Consider a wave generated by a highly relativistic process, involving most of the system's mass (such as the final stages of a merger of two black holes of similar mass). Then, due to relation (2), \\(g_W\\) remains above \\(a_0\\) for distances from the source comparable with the Hubble distance (Milgrom, 2014b). # [edit] MOND phenomenology in detail ## [edit] Disc galaxies Rotation curves of disc galaxies afford the most accurate and clear-cut tests of MOND: They probe the accelerations in the plane of the discs to relatively large radii, rather low accelerations (down to about \\(0.1 a_0\\)), and using test particles whose (nearly circular) motions are, by and large, well known. Given the observed (baryonic) mass distribution in a galaxy, MOND predicts it rotation curve, which can then be compared with the measured curve \\(V(r)\\). All existing MOND theories predict very similar rotation curves for a given mass distribution. The main properties of the predicted curve follow, anyhow, from the basic tenets alone: the asymptotic flatness, the value of the asymptotic velocity, the transition radius from the Newtonian to the MOND regime, and the validity of the Newtonian prediction in the high-acceleration regime. In most analyses, the straightforward-to-use prediction of “modified inertia” theories is used (see below), which predict for rotation curves (Milgrom, 1994) a universal algebraic relation between the Newtonian acceleration, \\(g_N\\), gotten from the mass distribution, and the MOND acceleration, \\(g=V^2(r)/r\\) (Milgrom, 1983a), \\[g= g_N\nu(g_N/ a_0), ~~~~~~~~~~~~~~~~~~~~ g_N=g\mu(g/a_0)\tag{3}\\] (shown in two commonly used forms that are mutual inverses), where the interpolating function, \\({\nu}(y)\\), and its inverse-related \\(\mu(x)\\), is universal for a given theory and is derived from the action of the theory specialized to circular orbits. The basic MOND tenets require \\( {\nu}(y\gg 1)\approx 1\\), and \\( {\nu}(y\ll 1)\propto y^{-1/2}\\); \\(\mu(x\gg 1)\approx 1\\), and \\(\mu(x\ll 1)\propto x\\). The above-chosen normalization of \\(a_0\\) fixes \\( {\nu}(y\ll 1)\approx y^{-1/2}\\), and \\(\mu(x\ll 1)\approx x\\). At present, \\(\mu(x)\\), or \\(\nu(y)\\), is put in by hand to interpolate between the standard and the deep-MOND regimes (see the discussion of theories below). For spherical systems, eq. (3) is predicted also in all existing modified-gravity MOND theories. Its asymptotic form for an isolated system of mass \\(M\\), \\(g\approx ( {g_N} a_0)^{1/2}=(M { {\mathcal{A}}_0})^{1/2}r^{-1}\\), follows, as we saw in eq. (1), from the basic tenets for modified gravity theories, and for circular orbits in modified inertia theories. Low-surface-density disc galaxies, which are, by definition, low acceleration galaxies (with \\(g\ll a_0\\) everywhere in the galaxy) afford particularly acute tests of MOND: They were predicted [law (4)] to exhibit large mass discrepancies everywhere in the disc, long before their dynamics were measured (Milgrom, 1983b), 10 as indeed they have proven to do. They happen to contain much gas mass compared with the stellar mass, making the MOND prediction relatively free of the knowledge of the stellar mass-to-light ratios (needed in order to convert observed luminosities to masses). Since they are wholly in the deep-MOND regime MOND prediction of their rotation curve is free of the remaining latitude in the choice of the interpolating function, since we work in the region where \\(\mu(x)\approx x\\). Since they are predicted to, and do, show large mass discrepancies, i.e., the predicted departure from standard dynamics is very large, the comparison is more clear-cut. Many MOND rotation-curve analyses have been presented to date starting some years after the advent of MOND (Milgrom & Bekenstein, 1987; Kent, 1987; Milgrom, 1988; Begeman, Broeils & Sanders, 1991; Sanders, 1996; Sanders & Verheijen, 1998; de Blok & McGaugh, 1998; Bottema, et al., 2002; Begum & Chengalur, 2004; Gentile, et al., 2004; Gentile, et al., 2007a; Corbelli & Salucci, 2007; Barnes, et al., 2007; Sanders & Noordermeer, 2007; Milgrom & Sanders, 2007; Swaters, Sanders, & McGaugh 2010; Gentile, et al., 2011; Famaey & McGaugh, 2012; Randriamampandry & Carignan, 2014; Hees, et al., 2016; Haghi, et al., 2016; Li, et al., 2018; Sanders, 2019). The results for a few galaxies of different mean accelerations are shown in Figure 8. Figure 8: Observed rotation curves of five galaxies, (data points) compared with the MOND predictions (solid lines going through the data points). The three leftmost from Begeman, Broeils, & Sanders, 1991, the lowest-right from Swaters, Sanders, & McGaugh, 2010, and the upper right from Sanders, 2006. Other lines in the figures are the Newtonian curves for various baryonic components. Of particular note is the analysis of Sanders (2019). He analyzed a small sample of gas-dominated, low-surface-density, disc galaxies. Because stars contribute rather little to their baryonic mass, the uncertainty introduced by converting starlight to stellar mass hardly matters. Also, as explained above, as these galaxies are wholly in the deep-MOND regime, MOND predictions are practically independent of the choice of interpolating function. For such galaxies MOND makes parameter-free predictions of the full rotation curves. (There is still some uncertainty in the exact distance and inclination of the galaxies; but Sanders (2019) did not adjust these.) His results are shown in Figure 9. Figure 9: MOND predictions of rotation curves of gas-rich disc galaxies from Sanders, 2019. Each galaxy is represented in two panels. The upper panel shows the surface densities of gas (dashed) and stars (dotted) as functions of radius. The lower panel shows the observed rotation curve (points), the Newtonian rotation curve for the baryonic components (long dashed curve), and MOND rotation curve calculated from it. Because stars contibute very little to the baryonic mass (as is seen in the upper panel for each galaxy), the exact conversion of starlight to stellar mass (mass-to-light ratio) is practically immaterial. To boot, the mass discrepancies are very large everywhere in these galaxies (as seen in the lower panels: the observed velocities are much higher than the baryonic ones). So the MOND predictions depend only very little on the exact form of the interpolating function. Rotation-curve tests of MOND differ conceptually from rotation-curve fits within the dark-matter paradigm: In MOND, the observed baryon-mass distribution in a given galaxy leads to a unique prediction of the rotation curve, which can be compared with the observed curve. In the dark-matter paradigm, the relations between the baryons and the total mass distribution strongly depend on the unknowable formation and evolution history of the particular galaxy. It is thus not possible to predict the rotation curve (mostly dominated by dark matter) from the baryon distribution. At best one can fit a few-parameters dark halo to reproduce the observed rotation curve. Some observed rotation curves show features that are clearly traced back to features in the baryonic mass distribution. Some examples are evident in Figure 8. These features are predicted in MOND where the rotation curves are determined by baryons alone, but are not reproduced when a (featureless) dark halo dominates the rotation curve at the position of the feature. A summary of MOND analysis of many rotation curves is shown in Figure 6, which shows the mass discrepancy, namely, the ratio of acceleration measured from the rotation curve, \\(g=V^2/r\\), to the Newtonian value, \\( {g_N}\\), calculated from the observed baryon distribution, plotted as a function of \\( {g_N}\\). It shows collectively that as predicted: the discrepancy is a function of the acceleration and develops below \\( a_0\\), and at low accelerations the discrepancy is \\(\approx ( a_0/ {g_N})^{1/2}\approx a_0/g\\). As regards the MOND laws listed above that pertain to disc galaxies: Law (1) is clearly seen to hold in the many observed rotation curves that go to large enough radii (and is collectively subsumed in Figure 6). Law (7) (Brada & Milgrom, 1999a) was tested in Milgrom & Sanders, 2005 and Milgrom, 2009b. Law (8) was pointed out and tested in Milgrom, 2009b. ## [edit] Pressure-supported systems A pressure-supported system is a self gravitating system of masses in long-term equilibrium, in which gravity is balanced by roughly random motions of the constituents (unlike the discs of spiral galaxies where gravity is balanced by ordered, quasi-circular motions). These include globular clusters, elliptical galaxies (and bulges of spirals), dwarf spheroidal galaxies, galaxy groups and clusters, etc. Determining their dynamical masses from the measured line-of-sight velocities of their constituents is based on the same physical laws, but requires specialized tools (e.g., when we measure only one component of the velocities).11 One such tool is a general MOND virial relation. It applies to an isolated, self gravitating, deep-MOND-limit system of pointlike masses, \\(m_p\\), at positions \\( {\bf r}_p\\), subject to forces \\( {\bf F}_p\\), and reads (Milgrom, 1997; Milgrom, 2010a) \\[ \sum_p {\bf r}_p\cdot {\bf F}_p=-\frac{2}{3}\mathcal{A}_0^{1/2}[(\sum_p m_p)^{3/2}-\sum_p m_p^{3/2}].\tag{4}\\] This is now known to hold in all modified-gravity MOND theories, where it was shown to follow from only the basic tenets (Milgrom, 2014a). Interestingly, the virial (the left hand side) can thus be expressed only in terms of the masses. This contrasts materially with the Newtonian expression \\(\sum_p {\bf r}_p\cdot {\bf F}_p=-\sum_{p0\\) [under which \\( \phi({\bf r})\rightarrow\phi({\bf r}/\lambda)\\)], and to inversion about a sphere of any radius \\(a\\), centered at any point \\( {\bf r}_0\\), namely, to \\[{\bf r}\rightarrow{\bf R}={\bf r}_0+\frac{a^2}{|{\bf r}-{\bf r}_0|^2}({\bf r}-{\bf r}_0),\tag{10}\\] with \\( {\phi}( {\bf r}) {\rightarrow}\hat {\phi}( { {\bf R}})= {\phi}[ {\bf r}({\bf R})]\\), and \\( {\rho}( {\bf r}) {\rightarrow}\hat {\rho}( {\bf R})=J^{-1} {\rho}[ { {\bf r}}( { {\bf R}})]\\), where \\(J\\) is the Jacobian of the transformation (10). This ten-parameter conformal symmetry group of eq. (9) is known to be the same as the isometry (geometric symmetry) group of a 4-dimensional de Sitter space-time, with possible deep implications, perhaps pointing to another connection of MOND with cosmology (Milgrom, 2009a). This conformal symmetry also has a very useful application, helping to derive various analytic results in this highly nonlinear theory (Milgrom, 1997). ### [edit] Quasilinear MOND Another theory, Quasilinear MOND (QUMOND) (Milgrom, 2010a) (also derived from an action), involves two potentials -- \\(\phi\\), which dictates particle accelerations, and an auxiliary potential \\(\phi_N\\) -- whose field equations are \\[\Delta\phi_N= 4\pi G \rho,~~~~~~~~~\Delta\phi=\vec \nabla\cdot[\nu(|\vec\nabla\phi_N|/a_0)\vec\nabla\phi_N],\tag{11}\\] requiring solving only the linear Poisson equation twice. Here \\(\nu(y)\\) plays the same role as the interpolating function in eq. (3). The deep-MOND limit, which is (defining \\(\psi\equiv a_0\phi_N\\)) \\[\Delta\psi=4\pi { {\mathcal{A}}_0}\rho,~~~~~~~\Delta\phi=\vec\nabla(|\nabla\psi|^{-1/2}\nabla\psi), \tag{12}\\] is space-dilatation invariant,14 but, apparently, not conformally invariant. ### [edit] Generalizations The above two theories are special cases in a class of two-potential, modified-gravity theories (Milgrom, 2010a). These have a gravitational Lagrangian containing only first derivatives of the potentials \\( {\mathcal{L}}= \mathcal{L}_g+(1/2)\rho {\bf v}^2( {\bf r})\\), with \\( \mathcal{L}_g=- {\rho} \phi({\bf r})+ {\mathcal{L}}_f[( { {\vec\nabla}\phi})^2,( {\vec\nabla}\psi)^2, { {\vec\nabla}\phi}\cdot \vec\nabla\psi]\\) that embody the MOND tenets. They have a deep-MOND limit of the form \\[ \mathcal{L}_f\rightarrow\mathcal{A}_0^{-1}\sum_{a,b} s_{ab}[(\vec\nabla\phi)^2]^{a+3/2}[(\vec\nabla\psi)^2]^{a+b(2-p)/2}(\vec\nabla\phi\cdot\vec\nabla\psi)^{b(p-1)-2a}, \tag{13}\\] where \\(p\\) is fixed for a given theory. The 3rd tenet is satisfied for any set of \\(a,~b\\). The dimensions of \\( \phi\\) and \\(\psi\\) are, respectively, \\([\phi]=[\ell]^{2}[t]^{-2}\\) and, if \\(b\not= 0\\), \\([\psi]=[\ell]^{2-p}[t]^{2(p-1)}\\) (for \\(b=0\\), the dimensions of \\(\psi\\) are arbitrary); \\(s_{ab}\\) are dimensionless. For any \\(p\\), this reduces to the nonlinear Poisson theory for \\(a=b=0\\). QUMOND is gotten for \\(p=-1\\) with two terms with \\(a=-3/2,~b=1\\) and \\(a=-b=-3/2\\). For \\(p=0\\) and any combination of \\(a,~b\\), the deep-MOND limit is conformally invariant. Likewise for \\(b=0\\), in which case \\(p\\) does not enter.15 ### [edit] Modified inertia theories “Modified-inertia” formulations of MOND are those in which the matter action is modified, with or without modifying the gravitational action.16 While this approach to MOND is very promising, we do not yet have a full-fledged theory of this type. At the nonrelativistic level, simplistic theories have been considered (Milgrom, 1994; Milgrom, 2011b) in which only the kinetic Lagrangian of particles, \\(\int \frac{1}{2} mv^2~dt\\) is modified,17 while the gravitational potential is still determined from the Poisson equation. The particle equation of motion is then of the form \\[\textbf{A}[\\{{\bf r}(t)\\},a_0]=-\vec\nabla\phi, \tag{14}\\] instead of \\(\ddot {\bf r}=- \vec\nabla\phi\\); \\(\textbf{A}\\) is a functional of the whole trajectory \\(\\{ { {\bf r}}(t)\\}\\), with the dimensions of acceleration. For \\( a_0 {\rightarrow} 0\\), \\(\textbf{A} {\rightarrow} \ddot { {\bf r}}\\). Some interesting general deductions can be made for such theories (Milgrom, 1994). For example, if such an equation of motion is to follow from an action principle, enjoy Galilei invariance, and have the correct Newtonian and MOND limits, it has to be time nonlocal. Another important and robust prediction shared by all such theories is that for circular trajectories in an axisymmetric potential, eq. (14) has to take the form of eq.(3). The "interpolating function", \\( {\nu}(y)\\), is then a derived concept and relevant only for circular orbits; it does not appear in the action itself, as in the above modified-gravity theories. It is this relation (3) that has been used in most MOND rotation-curve analyses to date. The possibility that inertia is a derived attribute, and in particular, that it has to do with the influence of the Universe at large, is old. “Mach’s principle” is an example, whereby it is the totality of matter in the Universe that interacts with local bodies to endow them with inertia. Modified-inertia MOND resonates well with this idea, especially that local MOND, as applied to small systems such as galaxies, bears a clear (if not established) imprint of the Universe. It is, indeed, natural in such a picture (Milgrom, 1999) for inertia to have different scaling properties for accelerations smaller or larger than the de Sitter acceleration. For example, by the heuristic idea put forth in Milgrom, 1999, it is the quantum vacuum – which is shaped by the state of the Universe – that is the inertia-giving agent. The origin of \\( a_0\\) in cosmology also emerges, and is indeed \\(\sim c^2\Lambda^{1/2}\\). The vacuum then serves as an absolute inertial frame (acceleration with respect to the vacuum is detectable, e.g., through the Unruh effect). Here, it is cosmology that enters local dynamics to give rise to the MOND-cosmology coincidence. The “interpolating function” is not put in by hand, but emerges. It could be calculated only for the very special (and impractical) case of eternally constant, linear acceleration, \\(a\\). If we generalize Newton's 2nd law to \\(F=mA(a)\\), then one finds \\(A(a)=(a^2+c^4\Lambda/3)^{1/2}-(c^4\Lambda/3)^{1/2}\\). At high accelerations, \\(a\gg (c^4\Lambda/3)^{1/2}\\), it gives the Newtonian expression, \\(A=a\\), while at low accelerations, \\(a\ll (c^4\Lambda/3)^{1/2}\\), we have \\(A=a^2/(4c^4\Lambda/3)^{1/2}\\). This is exactly the required MOND behavior; furthermore, the observed relation \\(a_0\sim(c^4\Lambda/3)^{1/2}\\) is gotten. The "interpolating function" underlying this result is analogous to the "interpolating function" that enters the relation between the kinetic energy, \\(E_K\\), and the momentum, \\(P\\), of a particle of mass \\(M\\), in special relativity: \\(E_K=E-Mc^2=(P^2c^2+M^2c^4)^{1/2}-Mc^2\\), which in the limit \\(P\gg Mc\\) gives \\(E_k=Pc\\), and at low momenta, \\(P\ll Mc\\), gives \\(E_K=P^2/2M\\). ## [edit] Relativistic theories ### [edit] TeVeS The Tensor-Vector-Scalar (TeVeS) theory, the bellwether of relativistic MOND, was put forth by Bekenstein (Bekenstein, 2004), building on ideas by Sanders (Sanders, 1997). The theory has been discussed and reviewed extensively (e.g., by Skordis, 2009 and by Ferreira & Starkman, 2009). Its advent helped greatly put MOND on firmer ground. In TeVeS, gravity is carried by a metric \\( {g_{\alpha\beta}}\\), a vector field \\({\cal U}_ {\alpha}\\), and a scalar field \\( {\phi}\\), while matter degrees of freedom couple in the standard, general relativistic, way to the “physical” metric \\({\tilde g}_{\alpha\beta} =e^{-2 {\phi}}( {g_{\alpha\beta}} + {\cal U}_ {\alpha} {\cal U}_ {\beta}) - e^{2 {\phi}} {\cal U}_\alpha {\cal U}_ {\beta}\\). TeVeS reproduces MOND phenomenology for galactic systems in the nonrelativistic limit, with a certain combination of its constants playing the role of \\( a_0\\). In particular, when \\( a_0 {\rightarrow} 0\\), the nonrelativistic limit goes to Newtonian gravity. However, the relativistic theory itself does not tend exactly to general relativity at high accelerations (which would seem an apt desideratum). This remaining departure from general relativity even for very high accelerations has subjected TeVeS to challenging constraints from the solar system (e.g., Sagi, 2009), and from binary compact stars (Freire, et al., 2012). These constraints do not pertain, however, to MOND (low acceleration) aspects of TeVeS. As in general relativity, the potential that appears in the expression for lensing by nonrelativistic masses (such as galactic systems) is the same as that which governs the motion of massive particles. Cosmology, the CMB, and structure formation in TeVeS have been considered by Dodelson & Liguori, 2006, Skordis & Mota, 2006, Skordis, 2006, Skordis, 2008, and Zlosnik, et al., 2008. It was shown that there are elements in TeVeS that could mimic cosmological DM, although no fully satisfactory application of TeVeS to cosmology has been demonstrated. Gravitational waves in TeVeS-like theories have been considered in Bekenstein, 2004, Sagi, 2010, and Skordis & Zlosnik, 2019. In particular, Skordis & Zlosnik, 2019 advanced a new version of TeVeS in which (unlike the original TeVeS) the tensor gravitational waves and electromagnetic waves always travel at the same speed. Galileon k-mouflage MOND adaptations (Babichev, et al., 2011), making use of an extended Vainshtein mechanism, are found to help TeVeS avoid the above-mentioned high-acceleration constraints. ### [edit] MOND adaptations of Einstein-Aether theories Einstein-Aether theories (e.g., Jacobson & Mattingly, 2001) have been adapted to account for MOND phenomenology (Zlosnik, et al., 2007). Gravity is carried by a metric, \\(g_{ {\mu} {\nu}}\\), as well as a vector field, \\(A^ {\alpha}\\). To the standard Einstein-Hilbert Lagrangian for the metric one adds the terms \\[\mathcal{L}(A,g)=\frac{a_0^2}{16\pi G}\mathcal{F}(\mathcal{K})+ \mathcal{L}_L,\tag{15} \\] where \\[\mathcal{K}=a_0^{-2}\mathcal{K}^{\alpha\beta}_{\gamma\sigma}A^{\gamma}_{;\alpha}A^{\sigma}_{;\beta}.\\] \\[ {\mathcal{K}}^{ {\alpha} {\beta}}_{ {\gamma} {\sigma}}=c_1g^{ {\alpha} {\beta}}g_{ {\gamma} {\sigma}}+c_2 {\delta}^ {\alpha}_ {\gamma} {\delta}^ {\beta}_\sigma +c_3 {\delta}^ {\alpha}_ {\sigma} {\delta}^ {\beta}_ {\gamma}+c_4A^ {\alpha} A^ {\beta} g_{ {\gamma} {\sigma}},\tag{16}\\] and \\( {\mathcal{L}}_L\\) is a Lagrange multiplier term that forces the vector to be of unit length. The asymptotic behaviors of \\( {\mathcal{F}}\\), at small and large arguments, give the deep-MOND behavior and general relativity, respectively. This relativistic MOND formulation was rediscovered by Hossenfelder, 2017. ### [edit] Bimetric MOND Bimetric MOND gravity (BIMOND) (Milgrom, 2009; Milgrom, 2010b; Milgrom, 2010c) is a class of relativistic theories governed by the action \\[I=-\frac{1}{16\pi G}\int[\beta g^{1/2} R \+ \alpha{\hat g}^{1/2} \hat R -2(g\hat g)^{1/4}a_0^2\mathcal{M}] d^4x +I_M(g_{\mu\nu},\psi_i)+\hat I_M(\hat g_{\mu\nu},\hat\psi_i). \tag{17} \\] It involves two metrics, \\( g_{\mu \nu}\\) and \\( \hat g_{\mu \nu}\\), whose Ricci scalars are \\(R\\) and \\(\hat R\\) (\\(c=1\\) is assumed here). The interaction term between the two metrics, \\( {\mathcal{M}}\\), is a dimensionless, scalar function of the two metrics and their first derivatives. The novelty in BIMOND is in the choice of the interaction term. The difference of the two Levi-Civita connections \\[C^{\alpha}_{\beta\gamma}=\Gamma^{\alpha}_{\beta\gamma}-\hat\Gamma^{\alpha}_{\beta\gamma}, \tag{18} \\] is a tensor that acts like the relative gravitational accelerations of the two sectors. This is particularly germane in the context of MOND, where, with \\( a_0\\) at our disposal, we can construct from \\( a_0^{-1}C^{ {\alpha}}_{ {\beta} {\gamma}}\\) dimensionless scalars to serve as variables on which \\( {\mathcal{M}}\\) depends. In particular, the scalars constructed from the quadratic tensor \\[ \Upsilon_{\mu\nu}\equiv C^{\gamma}_{\mu\lambda}C^{\lambda}_{\nu\gamma} -C^{\gamma}_{\mu\nu}C^{\lambda}_{\lambda\gamma}, \tag{19} \\] such as \\( {\Upsilon}= g^{\mu \nu} \Upsilon_{\mu\nu},~~~\hat \Upsilon= {\hat g}^{\mu \nu}\Upsilon_{\mu\nu}\\), have particular appeal. \\(I_M\\) and \\(\hat I_M\\) are the matter actions for standard matter and for putative twin matter, whose existence is suggested (but not required) by the double metric nature of the theory. Matter degrees of freedom \\(\psi_i\\) couple only to the standard metric \\( {g_{ {\mu} {\nu}}}\\), while twin matter couples only to \\( {\hat g_{ {\mu} {\nu}}}\\). BIMOND cosmology is preliminarily discussed in Milgrom, 2009; Clifton & Zlosnik, 2010; and Milgrom, 2010b. Some aspects of structure formation are discussed in Milgrom, 2010b. The weak-field limit--which is scale invariant--and a preliminary account of gravitational waves are discussed in Milgrom, 2014b. BIMOND has several attractive features (shared by the modified Einstein-Aether theories): It tends to general relativity for \\( a_0\rightarrow 0\\); it has a simple nonrelativistic limit; it describes gravitational lensing correctly; and, it has a generic appearance of a cosmological-constant term that is of order \\( a_0^2/c^4\\), as observed. In this case, the MOND-cosmology coincidence occurs not because cosmology affects local dynamics, but because the same quantity (the MOND length, or \\( a_0\\)) enters both. ### [edit] Nonlocal single-metric theories Nonlocal metric MOND theories (Soussa & Woodard, 2003; Deffayet, et al. 2011; Deffayet, et al. 2014) are pure metric, but highly nonlocal in that they involve operators that are functions of the 4-Laplacian. They agree with general relativity in the weak-field regime appropriate to the solar system, but possess an ultra-weak field regime when the gravitational acceleration becomes comparable to \\( a_0\\). In this regime, the models reproduce the MOND force without dark matter and also give enough gravitational lensing to be consistent with existing data. It has been proposed that these theories might emerge from quantum corrections to the effective field equations. ### [edit] Dipolar dark matter Blanchet, 2007 pointed out that the analogy of the MOND formulation of eq. (8) with the equation for the electric potential in a nonlinear dielectric medium could underlie a MOND origin in a gravitationally polarizable medium. This idea has been developed into a relativistic formulation (Blanchet & Le Tiec, 2008; Blanchet & Le Tiec, 2009) that invokes an omnipresent medium (a field) of a novel type of matter, dubbed “dipolar dark matter”, capable of being gravitationally polarized by baryonic matter, much in the way that a dielectric medium is polarized by electric charges. The polarization then enhances the effective gravitational attraction of baryonic masses. By choosing an appropriate field potential – which plays the role of an interpolating function, and which involves a constant that plays the role of \\( a_0\\) – we can get eq. (8) in the nonrelativistic limit. The constant \\( a_0\\), if it is the only new one allowed, also appears in a cosmological constant term, which might account for the MOND-cosmology coincidence. Another parameter of the theory controls the role of the medium as dark matter that acts gravitationally beside its polarization effect. It can, thus, double as cosmological dark matter (Blanchet & Le Tiec, 2009; Blanchet, et al., 2013). Outgrowths of these ideas that involve bimetric extensions of general relativity were discussed in Bernard & Blanchet, 2014 and Blanchet & Heisenberg, 2015. ## [edit] Microscopic and other theories There are suggestions to obtain MOND phenomenology in various microscopic-physics scenarios, in addition to the above mentioned gravitationally bipolar medium. For example, an omnipresent superfluid (Berezhiani & Khoury, 2016; Berezhiani & Khoury, 2015; Cai & Wang, 2016), and the “dark fluid” approach (Zhao & Li, 2010). Others assume properly tailored baryon-DM interactions (Bruneton, et al., 2009), various versions of gravity as a thermodynamic or holographic effect (Ho & Minic, 2010; Ho, et al., 2012; Pikhitsa, 2010; Li & Chang, 2011; Kiselev & Timofeev, 2011; Klinkhamer & Kopp, 2011; Pazy & Argaman, 2012; Pazy, 2013; van Putten, 2014; Verlinde, 2017; Smolin, 2017), vacuum effects (Milgrom, 1999), and other constructions (Bettoni, et al., 2011; Bernal, et al., 2011; Hidalgo, et al., 2012; Trippe, 2013). Khoury, 2015 has suggested a certain extension of MOND. A brane-world origin of MOND was proposed in Milgrom, 2002a and extended in Milgrom, 2018: our `Universe' is described as a 4-dimensional brane (membrane) residing in a higher dimensional space. Gravity as it appears in on-brane physics represents distortions of the brane in the higher-dimensional space. In the nonrelativistic limit of MOND, the gravitational potential stands for the extra coordinate in a one-up dimensional embedding space. The meaning of the MOND acceleration constant, its connection with cosmology expressed in eq. (2), the transition from the Newtonian to the deep-MOND regime, and from the nonrelativistic to the relativistic regimes, all attain geometrical meanings. The question of why the very large quantum-vacuum energy density is not reflected in the small measured cosmological constant may also be addressed within this picture. Figure 14 shows some elements of this idea. Figure 14: Schematics of the brane picture for MOND: Our directly-observable Universe is a 3- or 4- dimensional `membrane' (a brane) of density \\(\sigma\\), embedded in a higher-dimensional flat space. The brane -- through its density -- and `masses' (\\(m\\)) that are confined to the barane (representing matter), couple to a spherical potential \\(\varepsilon\\) in the embedding space (encircled \\(\varepsilon\\) in the Figure). The brane has an on-average spherical shape of radius \\(\ell_0\\) (which represents the MOND length, \\(\ell_M\\)), for which the force due to brane pressure (or tension), \\(T\\), which per unit volume is \\(\beta T/\ell_0\\) is balanced by the force per unit volume \\(\sigma\varepsilon'(\ell_0)\equiv\sigma a_0\\) (encircled `a' in the Figure) (\\(\beta\\) is a geometrical factor of order unity). The speed of propagation of perturbations on the brane, \\(c\\) (playing the role of the speed of light), is given by \\(c^2=\alpha^{-1}T/\sigma\\) (\\(\alpha\\) of order unity). The nonrelaivistic gravitational potential \\(\phi\equiv a_0\zeta\\), where \\(\zeta\\) is the local departure from sphericity (encircled `z'). A shallow depression \\(\zeta\ll\ell_0\\), which is equivalent to \\(\phi\ll c^2\\), characterizes nonrelativistic gravity (encircled `b'), an approximation that is broken where \\(\zeta\not\ll\ell_0\\) (encircled `c'). \\(\theta\\) is the angle between the normal to the brane and the radial direction (encircled `d'). The limit \\(|\tan\theta|\ll 1\\) -- the same as \\(|\nabla\zeta|\ll 1\\), and the same as \\(|\nabla\phi|\ll a_0\\) -- corresponds to the deep-MOND limit (encircled `e'), while \\(|\tan\theta|\gg 1\\) is the Newtonian limit (encircled `f'). The \\(\varepsilon\\) force on a mass \\(m\\) is \\(m\varepsilon'\\), which for `nonrelativistic' depressions is \\(\approx m\varepsilon'(\ell_0)\approx m a_0\\) (encircled `g'). This breaks down when \\(\phi\not\ll c^2\\). It can be shown that a dynamics for the masses can result from this picture with MOND gravity, and \\(a_0\equiv\varepsilon(\ell_0)\\) playing the role of the MOND constant, which furthermore satisfies \\(a_0\approx c^2/\ell_0\\), accounting for the MOND-cosmology coincidence of eq. (2). A depression, \\(\zeta\\), that is both relativistic (\\(\zeta\not \ll 1\\)), and in the deep-MOND regime (\\(|\nabla\zeta|\ll 1\\)) is not possible, as it would have to have an extent \\(\gg \ell_0\\) (encircled `h').]. There are also other modifications of general relativity proposed, such as special cases of Hořava gravity (Romero, 2010; Sanders, 2011; Blanchet & Marsat, 2011), and theories based on Finslerian geometry (Chang & Li, 2008; Namouni, 2015). # [edit] Cosmology and structure formation For many years, we had not had a good account of cosmology and structure formation in the framework of the MOND paradigm. This want has been tightly connected with the wider want of a satisfactory “fundamental” theory of MOND -- a `FUNDAMOND' theory. In fact, these two desiderata are most probably one: In light of the various connections of MOND with cosmology–as evinced by the value of \\( a_0\\) and the various MOND symmetries–it is sensible to expect that the MOND theory and the understanding of cosmology within MOND will come as parts of the same move. There are some semi-quantitative observations that point to such a connection: There are several unexplained “coincidences” in the standard picture of cosmology: a. The required amount of DM is only about 5 times that of baryons, while these quantities are believed to have been determined at very different times in the history of the Universe, and by unrelated processes. b. The density of dark energy is only about thrice today’s matter density. c. MOND has revealed another striking “coincidence”: galaxy dynamics is strongly imprinted with an acceleration, \\( a_0\\) (or a length \\(\ell_M\\)), that coincides with cosmological parameters: For example, the MOND density \\(M_M/\ell_M^3\\) is about a fifth of the cosmological closure density today. Some of these “coincidences” would be natural in MOND. If cosmological DM does not exist, so its emergence in standard dynamics is only an artifact of using the wrong theory, and, in fact, baryons alone determine the dynamics (a hypothesis that has proven very successful in galaxies) then it is natural for the fictitious DM density to be of the order of the baryon density from which it is born. The coincidence between the MOND constants and analogous cosmological parameters (e.g. as derived from a cosmological constant) have also been shown to emerge naturally in various MOND contexts. The proximity of the “dark-energy” density and the matter density today might be attributable to antropic underpinnings (Milgrom, 1989) (with galaxy formation becoming possible as \\(cH_0\\) decreases and becomes of order or smaller than the constant \\( a_0\\)). MOND does offer elements that, in principle, can supplant roles of cosmological DM. For example, DM is needed in standard dynamics to hasten structure formation after matter-radiation decoupling. All studies show that the growth of structure is indeed faster in MOND than in standard dynamics without DM (sometimes too fast). Over the years, there have been studies of structure formation using various nonrelativistic schemes “in the spirit of MOND” can be found, e.g., in Sanders, 2001; Nusser, 2002; Llinares, et al., 2008; Angus, et al., 2013; Candlish, 2016. Aspects of cosmology and structure formation in existing relativistic formulations of MOND were discussed, e.g., in Milgrom, 2009; Clifton & Zlosnik, 2010; and Milgrom, 2010b in BIMOND, by Dodelson & Liguori, 2006; Skordis & Mota, 2006; Skordis, 2006; Skordis, 2008; and Zlosnik, et al., 2008, in the context of TeVeS, by Blanchet, et al., 2013 for dipolar dark matter, and by Deffayet, et al. 2014; and Kim, et al., 2016 for nonlocal metric MOND theories. However, these studies have been only preliminary, and were based on specific subclasses of the theories in question. Recently, Skordis and Zlosnik, 2021 have put forth a class of reativistic formulations of MOND (which they dubbed RelMOND, for relativistic MOND), which have been shown to reproduce in great detail the main observations in cosmology, including the cosmic-microwave-background power spectrum and the matter structure power spectrum. RelMOND is a subclass of the theories, mentioned above, of Skordis and Zlosnik, 2019, in which the speed of light and of tensor gravitational waves are always the same. # [edit] Summary In a way of summary, Figure 15 presents a chart encapsulating the state of the MOND program, with the still-elusive underlying `FUNDAMOND' at its bottom. Figure 15: The various aspects under study within the framework of MOND, as described in this article. # [edit] Footnotes 1. The pristine formulation of the deep-MOND limit, still used often today, posits a relation between the measured (MOND) acceleration \\(g\\) and the Newtonian value \\( {g_N}\\), of the form \\(g\approx (a_0 g_N)^{1/2}\\). This simple, deep-MOND relation, which satisfies scale invariance, captures many of the salient MOND predictions. However, it is neither exact, nor generally applicable. 2. It is not clear whether this symmetry has a deep origin, or just happens to be a symmetry of the deep-MOND-limit, on a par with the scaling symmetry \\((t, { {\bf r}}) {\rightarrow}( {\lambda} t, {\lambda}^{2/3} { {\bf r}})\\) enjoyed by Newtonian dynamics. Either way, the symmetry is powerfully predictive. 3. This assumes a certain normalization of the theory’s degrees of freedom, which can always be affected (Milgrom, 2014). With such a choice, a scaling transformation is equivalent to a change in the values of the constants that appear in the theory, according to their dimensions. Scale invariance then implies that the only constants that can appear are those whose value does not change if we simultaneously change the units of length and time by the same factor. This is true of masses, of \\(c\\), and of \\( { {\mathcal{A}}_0}\\) (whose dimensions are \\([ {\ell}]^4[t]^{-4}[m]^{-1}\\)), but not of \\(G\\) or \\( a_0\\). 4. Similarly, important predictions of general relativity were made prior to its advent, based on its basic tenets (Lorentz invariance and the equivalence principle), such as gravitational redshift, and the inevitability of light bending of a known order. The same was true of quantum theory. 5. Defined such that its gradient gives the acceleration of test particles. 6. Once a theory of dynamics is formulated, with \\(a_0\\) appearing, this appearance defines the normalization of \\(a_0\\). However at the phenomenological level, or at that of the basic tenets, we can normalize \\(a_0\\) so that it appears in a simple way in this or that relation. (This is similar to working sometimes with \\(h\\), sometimes with \\(\hbar\\), in quantum mechanics, or to normalizing \\(G\\) so that the Newtonian acceleration is \\(MG/r^2\\) with the cost of \\(4\pi G\\) appearing in the Poisson equation.) The acceleration constant appears in many phenomena that can suggest convenient normalizations. Arguably, the best determination of \\(a_0\\) is afforded by rotation-curve analysis of many galaxies (see below). But it is difficult to use this procedure for a practical way to define the normalization of \\(a_0\\). Our choice here seems the most convenient for further theoretical considerations. 7. Despite the similar appearance this is not the same as the law (2): the latter is exact and predicted to have no true scatter, while law (6) is only a correlation. Also, law (2) applies to the asymptotic, not to bulk velocities like law (6). 8. In the same sense that without the unifying framework of quantum dynamics, the different quantum phenomena–such as the black-body spectrum, the photoelectric effect, the hydrogen-atom spectrum, superconductivity, etc.–would appear unrelated phenomena that somehow involve the same constant \\(\hbar\\). 9. Mach’s principle, whereby inertia of a body is due to its interaction with far away matter in the Universe is an example of such an effect. 10. This is also true of low surface brightness spheroidal systems, such as dwarf spheroidals. 11. Methods that use external test particles, such as gravitational lensing are common to all systems. 12. This is the opposite of what is found in galaxies, where the discrepancy increases outwards. 13. If the baryonic mass is dominated by neutral gas, its derived mass and the measured flux in the 21 centimeter line give us the distance. If the stars’ contribution to the mass is dominant, we have to convert the MOND stellar mass to luminosity, which together with the measured flux gives the distance. 14. Generally, since the deep-MOND-limit equations of motion of a MOND theory are space-time scale invariant, those equations where time does not appear are invariant under spatial dilatations. 15. Many of these theories maybe unfit for various reasons. 16. The distinction is not always clear cut. 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Phys. 93, 126 (2015) Wikipedia: Modified Newtonian Dynamics R.H. Sanders, “The dark matter problem: a historical perspective”, Cambridge U. Press, (2010) Google Books R.H. Sanders, “Deconstructing Cosmology”, Cambridge U. Press, (2016) Google Books ## [edit] Useful links Papers with “MOND” in the abstract (from the Astrophysics Data System) Papers with “modified Newtonian dynamics” in the abstract (from the Astrophysics Data System) Popular articles discussing MOND Milgrom’s home page linking to “MOND: general resources” Stacy McGaugh’s “The MOND pages”