# The MOND paradigm of modified dynamics

Mordehai Milgrom (2014), Scholarpedia, 9(6):31410. | doi:10.4249/scholarpedia.31410 | revision #197233 [link to/cite this article] |

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. MOND has predicted many clear-cut laws of galactic dynamics (analogous to, and extending Kepler’s laws), most of which involve \(a_0\)in different roles. In this way, MOND has unearthed a number of unsuspected laws of galactic dynamics, predicting them a priori, and leading to their subsequent tests and verification with data of ever increasing quality. One of these phenomenological laws is the baryonic Tully-Fisher relation, which is underlaid by the MOND mass-asymptotic-speed relation (MASR). This is a relation between the asymptotic rotational speed around a galaxy, \( V_{\infty}\) (predicted by MOND to be constant), and the total (baryonic) mass, \(M\), of the galaxy: \(V^4_{\infty}=MGa_0\). Another prediction of MOND is a tight correlation between the observed mass discrepancy in galactic systems, and the accelerations in them. This predicted mass-discrepancy-acceleration relation (MDAR), aka radial acceleration relation (RAR), has also been confirmed by many subsequent analyses.

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.

# 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.

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.

# 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 potential^{5} 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.

Succinctly formulated, some of the predicted MOND laws are (propounded in the original MOND papers, except as noted):

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 \(V\propto \sqrt{M/r}\).

\( {V_{\infty}}(M)=(M { {\mathcal{A}}_0})^{1/4}=c(M/M_M)^{1/4}\).

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 \(g(r)=V^2(r)/r= a_0\).

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).

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 \(\bar\Sigma\lesssim { {\Sigma}_M}\equiv a_0/ 2\pi G\).

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}The acceleration produced by such a fictitious halo can never much exceed \( a_0\) (Brada and Milgrom, 1999a).

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).

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).

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:

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.

MOND endows self gravitating systems with an increased, but limited stability (Milgrom, 1989a; Brada & Milgrom, 1999b; Banik, Milgrom, & Zhao, 2018).

High-acceleration systems should show no discrepancies.

For a mass \(M\) of a rotating galaxy component, MOND defines a reference value of the specific angular momentum, \(j_M\equiv M^{3/4}(G^3/a_0)^{1/4}\), which plays a role in correlation involving the angular momentum.

If one interprets MOND consequences as being due to a DM halo, then MOND predicts the following for this fictitious halo:

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 laws^{8}, 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). This method measures distortions of background galaxy images by foreground galaxies of all kinds, to statistically map the gravitational fields of the latter.

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 is a central prediction of MOND called the mass-discrepancy-acceleration relation (MDAR, also known as the radial-acceleration relation --RAR). It is encapsuled in the interpolated form given below in eq. (3) (from Milgrom, 1983a). This has been testeded 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 example of such a test is reproduced here in Figure 5. The important lesson from Figure 5 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 5 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.

A test of the MOND MDAR/RAR using gravitational weak lensing by Brouwer et al. (2021), is shown in Figure 6. This method has the great advantage that it tests MOND in a wide variety of galaxies (the lenses producing the effect); unlike other methods that use slow-moving test particles to map the gravitational field, this uses relativistic test particles (light); and importantly, it tests MOND down to very low accelerations -- much lower than are accessible to rotation-curve analysis, as clearly demonstrated in Figure 6. Rotation-curve tests are, however, more accurate, and can be applied to individual galaxies, while the weak-lensing technique is statistical (measuring average properties of many lenses).

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.

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 5), and to compact dwarf galaxies (Scarpa, 2005), but, on the face of it, does not hold for the cores of galaxy clusters (see below).

Law (14), Dictates and underlies correlations between properties of rotating galaxies that involve their angular momentum Milgrom, (2021), in particular the mass-specific-angular-momentum correlation.

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.

## 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.

# 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).

# MOND phenomenology in detail

## 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. This is the MOND MDAR/RAR discussed above.

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.

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.

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 5, 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 5).

Law (7) (Brada & Milgrom, 1999a) was tested in Milgrom & Sanders, 2005 and Milgrom, 2009b. Law (8) was pointed out and tested in Milgrom, 2009b.

## 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 \(\sum_p {\bf r}_p\cdot {\bf F}_p=-\sum_{p<q}Gm_pm_q| {\bf r}_p- {\bf r}_q|^{-1}\), which depends also on the exact structure of the system (i.e., on where the masses are located).

Among the various corollaries of this relation, two are particularly noteworthy: 1. The deep-MOND limit (attractive) force between two masses \(m_1,~m_2\), a distance \(\ell\) apart, is \[F(m_1,m_2,\ell)= \frac{2}{3}\frac{\mathcal{A}_0^{1/2}}{\ell}[(m_1+m_2)^{3/2}-m_1^{3/2}-m_2^{3/2}],\tag{5}\] replacing the Newtonian expression \(F(m_1,m_2,\ell)=Gm_1m_2/\ell^2.\) 2. A general relation between the root-mean-square velocity, \( \sigma\), in a system as defined above, and the masses, of the form \[ \sigma^2=\frac{2}{3}(M\mathcal{A}_0)^{1/2}[1-\sum_p(m_p/M)^{3/2}], \tag{6}\] where \(M=\sum_p m_p\). For a system made of many small masses, with \(\sum_p (m_p/M)^{3/2}\ll 1\) (such as a galaxy of stars), we have universally \[\sigma^2=\frac{2}{3}(M\mathcal{A}_0)^{1/2}.\tag{7}\] These last two are vital tools for predicting global velocity dispersions from the baryonic masses in pressure-supported systems (see below). Remarkably, only the masses appear in these relations.

Such results are facets of the general MOND law (6), which predicts a mass-velocity-dispersion correlation for pressure-supported systems (for example, they underlie the so called Faber-Jackson relation between luminosity and velocity dispersion of elliptical galaxies; see, e.g., Fig. 7 of Famaey & McGaugh, 2012).

## Elliptical galaxies

Testing MOND in individual elliptical galaxies is far less advanced than in spiral galaxies. This is because a. it is difficult to find test particles that probe the field of an elliptical far from the center, where MOND effects are appreciable. b. even nearer the center, the probes we do find, such as stars, planetary nebulae, and globular clusters, move on unknown orbits, a factor that adds to the uncertainties. As explained above in connection with the coincidence eq. (2), strong gravitational lensing, which is free of the second limitation, probes only the high-acceleration region, where MOND effects are small. All these had led to ambiguous and conflicting claims on whether MOND does explain completely the mass discrepancies in the near vicinity of ellipticals, or still leaves a small discrepancy (Milgrom & Sanders, 2003, Chen & Zhao, 2006; Ferreras, Sakellariadou, & Yusaf, 2008; Chiu, Tian, & Ko, 2008; Sanders & Land, 2008; Weijmans, et al., 2008; Chiu, et al., 2011; Ferreras, et al., 2012; Schuberth, et al., 2012). More recent, extensive studies using stellar dynamics (Tortora, et al., 2014; Durazo, et al., 2018), strong gravitational lensing (Sanders, 2014), and planetary nebulae as probes (Tian & Ko, 2016) find MOND to be consistent with the observed dynamics of ellipticals with reasonable mass-to-light ratios .

It has also become possible to test MOND in ellipticals in ways that are free of the above limitations; i.e., probing the dynamics to very large radii (thus, to very small accelerations), with probes of well understood motions. The first analysis applied to two rare, individual cases of isolated ellipticals (Milgrom, 2012a) that are enshrouded by spherical clouds of hot, x-ray emitting gas. Assuming, plausibly, that these are in hydrostatic equilibrium enables measurements of the acceleration field around these ellipticals (Humphrey, et al., 2011; Humphrey, et al., 2012), to radii as large as \(100\) and \(200 {\rm kpc}\), with acceleration span from more than \(10 a_0\) near the center, down to about \(0.1 a_0\). For both ellipticals, good agreement with the MOND predictions [eq. (3) for spherical systems] have been found. One of the two cases is shown in Figure 10.

The MASR for a sample of `early type' galaxies, including ellipticals, has been plotted by den Heijer et al., 2015, and it agrees with the prediction of MOND.

Another method to study dynamics is the above-mentioned statistical analysis of weak-gravitational-lensing (Figure 4), which applies to all galaxy types, including many elliptical galaxies. It showed that the MOND predictions work as well for the asymptotic fields of such galaxies.

## Dwarf spheroidals and tidal dwarfs

Large galaxies, such as the Milky Way and our neighbor, the Andromeda galaxy, have many dwarf-spheroidal satellites. These are are very tenuous, which implies that they have low internal accelerations, which, in turn, makes them particularly amenable to testing MOND, which predicts large mass discrepancies in them. (As we saw, this is much the case for dwarf spiral galaxies, as it is for pressure-supported dwarfs.) The measured velocity dispersions of dwarf spheroidal satellites of the Milky Way and the Andromeda galaxy point to very large mass discrepancies, typically between 10 and 100, in general accordance with MOND. Indeed, quantitative studies showed that, by and large, MOND accounts well for the observed kinematics with only the stars contributing to gravity, and no DM.

A detailed MOND analysis of the radius-dependent kinematics of eight Milky-Way satellites is described in Angus, 2008, Serra, et al., 2010, and Alexander, et al., 2017. In all but two, MOND requires dynamical-mass-to-light ratios of order 1 solar unit (compared with about 10-400 in Newtonian analysis), consistent with stars only. In two cases, the required value is still high for stars alone, but this could be attributed to these systems not being in the presumed equilibrium, for example due to their being subject to tidal effects or still showing the effects of their history.

Particularly noteworthy is the MOND prediction of the velocity dispersion of the “Feeble Giant”, a very-low-surface-brightness, dwarf, Milky-Way satellite, Crater II (McGaugh, 2016). The external-field effect plays an important role in this dwarf. The MOND prediction differs greatly from the expectation from dark-matter, and was later confirmed rather decidedly by Caldwell, 2017.

Observations of the farther dwarf satellites of Andromeda does not permit a radius-dependent analysis, only a global one. The practice is then to use the measured stellar luminosity of a dwarf (and if necessary its size and distance from Andromeda) and use MOND to predict its internal root-mean-square velocity dispersion. There are now around 30 such dwarfs with measured velocity dispersions (Tollerud, et al., 2012, Tollerud, et al., 2013, Collins, et al., 2013) for which the test has been conducted. They all exhibit large mass discrepancies within standard dynamics – typically, of a factor of ten to several tens if the stellar mass-to-light ratios are about 2 solar units.

McGaugh & Milgrom, 2013a compared the MOND predicted velocity dispersions for 17 Andromeda dwarfs with the then existing measurements, and also presented the predictions for 10 more dwarfs with no measured dispersions. These have been measured later. McGaugh & Milgrom, 2013b discussed these new measurements and added predictions for two newly discovered Andromeda dwarfs, and Pawlowski & McGaugh, 2014 made predictions for yet another; the velocity dispersions for these last three dwarfs were then measured and compared with the MOND predictions (Martin, et al., 2014). A comparison of the predictions with the measured dispersions is shown in Figure 11. Considering the remaining systematics, MOND eliminates the mass discrepancies, and accounts quite well for the kinematics of the Andromeda dwarfs with the gravity of the stars alone, and no DM.A very acute test of MOND vs. DM is afforded by the possibly related class of tidal-dwarf galaxies: typically small galaxies that form from the debris (in the form of tidal tails) ejected from larger, gassy galaxies when they collide. The now prevailing cold-dark-matter paradigm predicts robustly that hardly any of the DM in the parent, colliding galaxies should find its way into the tidal dwarfs. They should thus be practically devoid of DM, and thus show no mass discrepancy to speak of. MOND predicts a discrepancy wherever the accelerations are small irrespective of the formation history.

Well established tidal dwarfs with measured rotation curves were described by Bournaud, et al., 2007. They were found to exhibit mass discrepancies (by a factor of about 3 within the studied region). They indeed have measured accelerations a few times lower than \( a_0\), and their analysis (Gentile, et al., 2007, Milgrom, 2007) showed good agreement with MOND. However, the analyses of Lelli, et al., 2015, and even more so, of Flores, et al., 2016, show that these systems are not suitable for proper dynamical analysis, because, among other reasons, they are not in virial equilibrium.

It has been suggested that a large fraction of the dwarf spheroidal satellites of the Milky way and Andromeda (and of other galaxies) formed as tidal dwarfs due to past interactions of these galaxies (with each other or with other galaxies) (Kroupa, 2012; Dabringhausen & Kroupa, 2013; Pawlowski & Kroupa, 2013; Zhao, et al., 2013). This is based largely on the fact that a large fraction of these satellites reside in large, relatively thin, coherently rotating discs around the Milky Way (Pawlowski & Kroupa, 2013) and the neighboring Andromeda galaxy (Ibata, et al., 2013, Conn, et al., 2013). This would be naturally explained by a tidal origin in MOND, for example due to a past encounter between the Milky Way and Andromeda (Bilek & al., 2018; Banik, O'Ryan, & Zhao, 2018), but not by a primordial origin, which would imply a more random distribution. The cold-dark-matter paradigm, however, conflicts with this finding, since a. The dwarfs show very large internal mass discrepancies, in line with the predictions of MOND -- since they are all very-low-acceleration systems -- while cold dark matter predicts that tidal dwarfs should not show palpable mass discrepancies. b. The cold-dark-matter paradigm predicts orders of magnitude more primordial dwarf satellites then are observed; eliminating many of latter out as non-primordial, only aggravates this so called “missing satellite” problem.

## Galaxy groups

Intermediate-richness galaxy groups are a crucial testing ground for MOND. They form a distinct class of galactic systems, with their own evolution paths and underlying physical processes, and so, in the context of the dark-matter paradigm would require separate accounting (see, e.g., Oehm & Kroupa, 2018). And, they probe regions of parameter space that cannot be probed in other systems -- such as individual galaxies, or rich groups and galaxy clusters. For example, they have baryonic masses similar to massive galaxies, and similar velocity dispersions, but much larger sizes -- similar to galaxy-cluster cores (or even to galaxy clusters), but much lower dispersions. Importantly in the context of MOND, they have the lowest internal accelerations reachable inside galactic systems, and thus test MOND to very low accelerations.

Binary galaxies constitute the “simplest” galaxy groups. However, their MOND (and Newtonian) analysis (starting with Milgrom, 1983c) is beset by thorny problems of identifying bound pairs, and of disentangling projection effects. So, by and large, they have not afforded very strong tests of MOND. But, see Scarpa, et al., 2022 for a more significant confirmation of MOND predictions concerning galaxy pairs.

One particularly interesting galaxy pair is our own; i.e., that made of the Milky Way and Andromeda. It’s detailed discussion in the context of MOND can be found in Zhao, et al., 2013. Aspects of interactions in galaxy pairs were treated succinctly in the context of MOND by Tiret & Combes, 2008.

Newtonian analysis of small galaxy groups, whose individual dynamics have large uncertainties, show very large mass discrepancies. For example, Tully, et al., 2002 find Newtonian mass-to-light ratios of up to 1200 solar units. They also report a systematic difference between dwarf-only groups and ones containing massive galaxies, with the former showing larger M/L-values (by a factor of 5).

With MOND [using eqs. (6)(7)] these discrepancies disappear (Milgrom, 2002), and furthermore, the Newtonian disparity between the two classes is explained by the dwarfs-only groups having systematically smaller observed intrinsic accelerations (they have similar sizes but rather smaller velocity dispersions). (See also earlier analysis of groups in Milgrom, 1983c).

Milgrom, 2018a and Milgrom, 2019 analyzed the MOND dynamics of galaxy groups in large group catalogs, using the MOND prediction (6)(7). In particular, Milgrom, 2019 analyzed a sample of 56 groups with large numbers of observed, individual-galaxy velocities, culled from the catalog of Makarov & Karachentsev, 2011. It was found that the observed (MOND) baryonic masses needed to account for their observed velocity dispersions match the observed luminosities with mass-to-light ratios of order 1 solar unit (compared with the very large Newtonian values of several tens), i.e., with no need for dark matter. These results are shown in Figure 12. The Figure also shows the positions on the luminosity-velocity plot of dwarf-spheroidal satellites of Andromeda (those that are not subject to an external-field effect) to demonstrate that the same predicted MOND relation is obeyed by very disparate kinds of systems, as the dwarfs are two-three orders smaller in size than the groups, and have seven-eight orders lower masses.

Figure 13 shows the results of the analysis of Milgrom, 2019 as the mass-discrepancy-acceleration relation for the groups, which is tantamount to plotting them on Figure 5 (albeit in a somewhat different presentation). The values for groups place them in Figure 5 near the continuation of the red line (the deep-MOND prediction) down to values of \(g_N\) as low as \(10^{-14}~ {\rm m~s^{-2}}\) and of \(\eta\approx 100\). Importantly, this shows that, at least as far as intermediate-richness groups are concerned, there is now breakdown in MOND down to these very low accelerations of \(g_N\approx 10^{-4}a_0\). Also importantly, some of the groups analyzed in Milgrom, 2019 have harmonic radii nearing a Megaparsec, which are comparable with those of galaxy clusters, yet MOND's predictions hold even in these groups. This argues against a breakdown of MOND at large radii.

## Galaxy clusters

It has been evident from early on (The and White, 1988; Gerbal, et al., 1992; Sanders, 1994; Sanders, 1999; Sanders, 2003; Aguirre, et al., 2001; Pointecouteau and Silk, 2005; Takahashi and Chiba, 2007; Angus, et al., 2008a; Milgrom, 2008; Ettori, et al., 2019) that MOND does not fully explain away the mass discrepancy in galaxy clusters. One then has to attribute the remaining discrepancy to yet undetected standard-model matter. It is likely that this matter is baryonic in some yet unidentified form, such as cold, dense clouds (Milgrom, 2008). Another possibility that has been considered are massive neutrinos (Sanders, 2003; Angus, et al., 2008). Modified versions of MOND were also proposed to account for this remaining discrepancy (Zhao & Famaey, 2012).

The following rough picture emerges regarding the mass discrepancy in clusters, based on observations of galaxy motions, x-ray emitting gas, weak and strong lensing, and Sunyaev-Zeldovich signals: In Newtonian dynamics, the discrepancy decreases with distance from the cluster’s center. In the cluster’s core it is of the order of 10-50 and decreases with radius to a value of \(\sim5-10\), at \((1-2){\rm Mpc}\).^{12}

The measured accelerations at intermediate radii (of about 1 Megaparsec) in the clusters are roughly \( (0.2-0.5)a_0\). So, MOND reduces the discrepancy in clusters at these radii to only a factor of \(\sim 2-3\) (Sanders, 1999; Ettori, et al., 2019). At larger radii (about 2 Megaparsecs) MOND leaves even smaller disscrepancies, as low as only 15 percent typically (Ettori, et al., 2019). In other words, MOND still requires yet-undetected matter in clusters whose amount is only a fraction of the already-observed baryonic matter, which is largely in the form of hot gas. If this undetected matter is in some form of baryons, it adds only little to the baryonic budget in the Universe: A rough estimate is that less than 5% of the amount of baryons implied by Big-Bang nucleosynthesis would suffice to account for this extra baryons that MOND requires in clusters. Since a large fraction of the baryons in the Universe are still missing, and their whereabouts are anyway unknown (e.g., Bhattacharjee, 2012), a small fraction of them could account for the remaining MOND discrepancy in clusters.

The typical observed accelerations in the cores of clusters are of the order of, or a few times larger than, \( a_0\). Thus, MOND implies only a small correction there. So most of the discrepancy observed in the core must be due to the extra, undetected matter. This matter is thus found to be more centrally concentrated than the observed baryons, which are largely in the form of hot gas. In fact, its distribution resembles that of the galaxies in a cluster (which make up only a small fraction –about 20 percent – of the observed baryonic mass).

If the extra matter required in MOND is made of compact macroscopic objects–as is most likely–then like the galaxies, these must have sloshed across the cluster, through the hot gas, many times. So, when two clusters collide, we expect this extra matter to follow the galaxies in going through the collision zone, and not to be greatly affected even in head-on collisions, while the gas components of the two clusters coalesce at the center.

### The “Bullet” cluster

The “Bullet” system of two galaxy clusters in collision (Clowe, et al., 2006) has been brought up as “proof of the existence of DM”, and as marking the demise of modified-dynamics alternatives to DM. This claim is based on the fact that the mass discrepancies in this system (deduced from weak lensing analysis of background galaxies) appear in places that do not coincide with the bulk of observed baryons. But why should they? It is not the case that modified-dynamics theories predict that the discrepancies (or the fictitious “phantom matter”) would appear to reside where the baryons are. On the contrary; in MOND, for example, the “phantom matter” should appear where accelerations are small, and this is usually found away from the baryons. So clearly the bullet observations are anything but a refutation of modified dynamics in general (as an alternative to DM). But, specifically, MOND itself indeed does not explain away the discrepancies in the bullet cluster as it does not explain them away in individual clusters. In fact, as already explained, the observations of the bullet cluster match what is expected of two colliding clusters having the matter distribution described above deduced from the observations of single clusters. The observations of the “Bullet” do tell us that the collisional mean free path of the yet undetected matter has to be larger than the core size (both on the gas and on itself) so it goes through the collision zone almost unaffected (like the galaxies in the two clusters). But, as explained above, even this is expected from the observations of individual clusters, at least for the case of macroscopic constituents. At any rate, these observations of the “Bullet” do not prove that a new type of matter that fits the bill of nonbaryonic DM, such as “particle DM”, has been discovered.

# MOND as an astronomical tool

We use physical laws as tools to measure attributes of far away astronomical systems. For example, the Doppler effect is used to measure velocities of, and inside, such systems, and the black-body law is used to estimate the sizes of unresolved objects (e.g., stars) from their spectral temperature and luminosity. Since MOND is a new system of laws that differ from those of standard dynamics, it affords measurements of astronomical quantities in ways that are not possible in standard dynamics. Here are some examples:

MOND relates velocities with masses in low-acceleration astronomical systems, as seen above in laws (2)(6) and eqs. (6)-(7), enabling us to measure masses from the readily observed velocities, without having to know sizes (hence distances), as is required in standard dynamics. Moreover, since MOND dynamics is accounted for with no DM, it can be used to determine directly stellar masses (in star dominated galaxies)
or gas masses (in gas dominated systems), without having to reckon with the putative, unknown of the DM content. This can help us determine distances directly.^{13}
Conversely, if the distance can be measured by other means we get information on mass-to-light ratios, which can be used to constrain stellar evolution, galaxy-formation models, etc., as used, e.g., in Tortora, et al., 2014 and in Chen & Ko, 2015.

MOND offers an even more incisive tool for measuring directly the distance to an object whose velocity profile is measured – e.g., a galaxy whose rotation curve is known – and the acceleration in which crosses from over \( a_0\) to under \( a_0\). In principle, MOND can account for the rotation curve of such a galaxy only for one distance: The measured asymptotic speed gives the total mass, whose distribution can be determined without knowledge of the distance. Then the transition radius, where \(V^2(r)/r= a_0\) can be determined both in angular and absolute terms.

A program to determine the Hubble constant using MOND has not been attempted. It is clear, however, off hand, that while MOND works well for a value of \(H_0\) around \(70~{\rm km~s^{-1}Mpc^{-1}}\), it would have failed with a value of order 50 or 100 \( {\rm km~s^{-1}Mpc^{-1}}\) (all other observations of galaxies being the same).

# MOND theories

We are still a far cry from understanding the theoretical basis of MOND (see review in Milgrom, 2015a). We have today several nonrelativistic, and several relativistic, full-fledged MOND theories to be listed below. They are all derived from an action; they all involve \( a_0\) (in addition to \(G\) and \(c\)), and have a nonrelativistic version that satisfies the basic premises of MOND; and, they all involve an interpolating function that has to be put in by hand, to artificially interpolate between the MOND and the high-acceleration regimes. Also, all the theories that do not invoke new matter components constitute modified-gravity theories. By this we mean a theory that can be written such that the standard matter actions remain intact, and only the gravitational action is modified: the Poisson action in the nonrelativistic case, and the Einstein-Hilbert action in the relativistic case.

In contrast, a theory that also modifies the matter actions, in particular the free matter actions, we term a “modified inertia” theory, since inertia of the various matter degrees of freedom is encapsulated in their free actions. The Einstein-Hilbert action for the metric, or the Poisson action for the nonrelativistic gravitational potential, may be viewed as the free actions of gravity. It is thus plausible that they too are modified if the matter free actions are. Still, we may consider modifications of only the matter actions.

It may be said that we have so far explored only a small subset of possible MOND theories. Past experience, with relativity and quantum mechanics, showed that a new regime of physical phenomena, separated from the old one by a newly introduced constant (\(c\) or \(\hbar\)), is not artificially connected to it by some interpolating function that appears in the foundations of the theory (e.g., in the action). Furthermore, different relativistic and quantum phenomena involve different “interpolating functions” (such as the Lorentz factor, or the particle dispersion relation in relativity, and the black-body function, or the specific heat of solids as a function of temperature, in quantum theory). These functions are derived within an all-encompassing theory. Thus, the appearance of an artificial interpolating function in present MOND theories indicates that none is the basic MOND theory we are after -- a `FUNDAMOND' theory. Another indication that MOND, as we now know it, is an emergent theory, comes from the above-mentioned coincidence, eq. (2). Thus, the existing MOND theories are, at best, only effective, approximate theories, of limited validity.

They are nonetheless very useful: They have provided proofs of various concepts, first that MOND theories can be written that are derived from an action, satisfy all the standard conservation laws, give the correct center-of-mass motion of a composite body, etc. Then, the advent of relativistic theories showed that covariant theories can be written with, e.g., correct lensing. Also, these theories can be used for various calculations, since they are complete theories, in the hope that they are close enough approximations for the problem at hand. This hope is founded on the fact that they all satisfy the basic tenets, and so all make the same salient predictions (Milgrom, 2014). Further confidence in them is gained by noting that they also make very similar, if not always quite the same, predictions of more detailed aspects such as full rotation curves of spiral discs (Brada & Milgrom, 1995; Angus, et al., 2012; Milgrom, 2012b), or of gravitational lensing. Still, we do not know what their exact validity domain is, and how far to trust their predictions beyond those that are anchored in the basic tenets.

Many ideas have already been suggested that depart from the above scheme of constructing MOND theories. Some are more advanced, some less, but none has yet lead to a full-fledged theory. There are, e.g., quite a few ideas to obtain MOND phenomenology from “microscopic” (quantum) approaches.

A promising direction is to construct MOND theories where the matter actions are modified.

## Nonrelativistic theories

### Modified Poisson gravity

In this theory (Bekenstein & Milgrom, 1984), the Poisson action is modified, and the Poisson equation for the gravitational potential --which dictates particle accelerations according to \({\bf a}=-\vec\nabla\phi\) -- is replaced by a nonlinear version \[\vec\nabla\cdot[\mu(|\vec\nabla\phi|/a_0)\vec\nabla\phi]= 4\pi G\rho. \tag{8}\] It reproduces a behavior similar, but not quite the same, as eq. (3), with the same interpolating function \(\mu(x)\).

Very interestingly, its deep-MOND limit, \[\vec\nabla(|\vec\nabla\phi|\vec\nabla\phi)= 4\pi \mathcal{A}_0\rho, \tag{9}\] is invariant under space conformal transformations (Milgrom, 1997): Namely, beside its obvious invariance to translations and rotations, eq. (9) is invariant to dilatations, \( {\bf r}\rightarrow\lambda{\bf r}\), \( \rho({\bf r})\rightarrow\lambda^{-3} \rho({\bf r}/\lambda)\) for any constant \( \lambda>0\) [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).

### 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.

### 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}

### 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 \(E_K=P^2/2M\).

## Relativistic theories

### 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.

### 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.

### 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.

### 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.

### 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.

## 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.

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).

# 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.

# 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.

# Footnotes

- 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.
- 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.
- 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 \( a_0\).
- 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.
- Defined such that its gradient gives the acceleration of test particles.
- 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.
- 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).
- 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 \(\hbar\).
- 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.
- This is also true of low surface brightness spheroidal systems, such as dwarf spheroidals.
- Methods that use external test particles, such as gravitational lensing are common to all systems.
- This is the opposite of what is found in galaxies, where the discrepancy increases outwards.
- 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.
- 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.
- Many of these theories maybe unfit for various reasons.
- The distinction is not always clear cut. For example the Brans-Dicke theory and many of its descendants can be described equivalently by a modification of the gravitational Einstein-Hilbert action, or by modifying the matter action so that matter couples to a modified metric, not the Einstein one.
- If we modify the relativistic particle action \(-mc^2\int d\tau\), then in the nonrelativistic limit, the way matter sources gravity is also modified, in principle, not only its kinetic Lagrangian.

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## MOND Reviews

D. Merritt: A Philosophical Approach to MOND: Assessing the Milgromian Research Program in Cosmology, Cambridge University Press (2020) (Google books)

I. Banik and HS. Zhao: From galactic bars to the Hubble tension − weighing up the astrophysical evidence for Milgromian gravity , Symmetry (2021)

S. McGaugh: Predictions and Outcomes for the Dynamics of Rotating Galaxies , Galaxies, vol. 8, issue 2, p. 35 (2020)

M. Milgrom: MOND vs. dark matter in light of historical parallels, Studies in History and Philosophy of Modern Physics (2020)

B. Famaey and S.S. McGaugh: Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions, Living Reviews in Relativity, 15, 10 (2012)

R.H. Sanders: A historical perspective on modified Newtonian dynamics, Canadian J. 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

## 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”