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The MOND paradigm of modified dynamics
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Mordehai Milgrom (2014), Scholarpedia, 9(6):31410.
[3]doi:10.4249/scholarpedia.31410 revision #201915 [[4]link to/cite this article]
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Curator: [7]Mordehai Milgrom
Contributors:
0.20 -
[8]Olivier Minazzoli
0.20 -
[9]Nick Orbeck
[10]Benoit Famaey
[11]Stacy McGaugh
* [12]Prof. Mordehai Milgrom, Weizmann Institute of Science, Rehovot, Israel
MOND is an alternative paradigm of [13]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 [14]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 [15]Tully-Fisher relation, which
is underlain 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), undrlies MOND's predictions of rotation curves of
disc galaxies, and the dynamics in other galactic systems. It 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 (with some additional, plausible, non-MOND-specific
requirements) 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 [16]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.
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, that satisfy some additional, plausible requirements (such as
yielding unique solutions), 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, and have begun to tackle cosmology and structure
formation.
We do not know if MOND is only relevant to gravitational phenomena, or should
also affect in some way other phenomena, such as [17]electromagnetism.
Contents
* [18]1 MOND introduced
* [19]2 Rudiments of MOND phenomenology: MOND laws of galactic dynamics
+ [20]2.1 Systems embedded in an external field and the external-field
effect
* [21]3 The significance of the MOND acceleration constant
* [22]4 MOND phenomenology in detail
+ [23]4.1 Disc galaxies
+ [24]4.2 Pressure-supported systems
+ [25]4.3 Elliptical galaxies
+ [26]4.4 Dwarf spheroidals and tidal dwarfs
+ [27]4.5 Galaxy groups
+ [28]4.6 Galaxy clusters
o [29]4.6.1 The "Bullet" cluster
* [30]5 MOND as an astronomical tool
* [31]6 MOND theories
+ [32]6.1 Nonrelativistic theories
o [33]6.1.1 Modified Poisson gravity (AQUAL)
o [34]6.1.2 Quasilinear MOND (QUMOND)
o [35]6.1.3 Generalizations
o [36]6.1.4 Modified inertia theories
+ [37]6.2 Relativistic theories
o [38]6.2.1 Tensor-Vector-Scalar theories
o [39]6.2.2 MOND adaptations of Einstein-Aether theories
o [40]6.2.3 Bimetric MOND
o [41]6.2.4 Noncovariant and \(f(Q)\) MOND theories
o [42]6.2.5 Nonlocal single-metric theories
o [43]6.2.6 Dipolar dark matter
+ [44]6.3 Microscopic and other theories
* [45]7 Cosmology and structure formation
* [46]8 Summary
* [47]9 Footnotes
* [48]10 References
+ [49]10.1 MOND Reviews
+ [50]10.2 Useful links
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 "[51]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 ([52]Milgrom, 1983a; [53]Milgrom,
1983b; [54]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. Some extensive reviews,
with emphasis on different aspects of MOND, can be found in [55]Famaey & McGaugh,
2012; [56]Milgrom, 2020a; [57]McGaugh, 2020; [58]Banik & Zhao, 2022.
The [59]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., [60]Famaey & McGaugh, 2012;
[61]Kroupa, 2012; [62]Boylan-Kolchin, et al., 2012; [63]Weinberg, et al., 2013).
Also, the major potential outcome that would obviate all alternatives -- the
direct detection of DM -- has not materialized, despite many searches over many
years.
At the basic, minimalistic level, three underlying 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 - when all system attributes with the
dimensions of acceleration are much larger than \( a_0\) - standard dynamics is
restored. Such a limit can be effected by taking the formal limit \( a_0
{\rightarrow} 0\) in a MOND based theory, or in a relation predicted by such a
theory. This is analogous to going to the classical limit in a quantum relation
by taking \( \hbar {\rightarrow} 0\) (or to \( c {\rightarrow} \infty\) in
relativity). 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 multiplying all times and all lengths measured in a system by the same
factor, \((t, { \bf{r}}) {\rightarrow} {\lambda}(t, { \bf{r}})\) ([64]Milgrom,
2009a).[65]^1\(^,\)[66]^2 This deep-MOND limit of an equation that underlies a
theory, or of a relation predicted by such a theory, 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 relation by a factor \( {\lambda}\), and let \( {\lambda}
{\rightarrow}\infty\). If the limit exists, and is nontrivial, 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).[67]^3 The above tenets encapsulate the novelty
of MOND and are MOND specific (as they all refer to \( a_0\)). But, beyond these
axioms, a MOND theory is required to satisfy some additional plausible,
non-MOND-specific, requirements (see below).
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 [68]1.
Figure 1: Two systems of masses that are related by space-time scaling by a
factor \(b\). A snapshot of the first is shown at time \(t\), and of the second
at time \(bt\), when it is an image of the first inflated by a factor \(b\)
(these are not two configurations of the same system shown at two times; the
first system might look very different at time \(bt\)). Note that the velocities
are the same in the two configurations.
With the additional knowledge that only the constant \(\mathcal{A}_0\) appears in
the deep-MOND limit, one can deduce that this limit 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 underlain 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 underlain 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 underlain 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
complemented with some additional plausible requirements of a more general nature
([69]Milgrom, 2014).[70]^4 As a result, any MOND theory that embodies the basic
tenets (plus the extra requirement) will automatically make these predictions.
Such predictions are referred to as 'primary', or 'first-tier' MOND predictions,
in contradistinction from 'secondary', or 'lower-tier' predictions that depend on
the specific theory. This distinction is discussed at length, e.g., in
([71]Milgrom, 2023c). One additional requirement is that the theory yields unique
solution: we want, e.g., that the velocity on a circular orbit in an axisymmetric
system is unique for a given orbital radius. To make clear-cut predictions we
also require from a theory that it does not involve dimensionless constants,
\(q\), that differ much from unity. (Such is also the case in quantum theory and
relativity.) Otherwise, predictions could be characterized by two very different
acceleration constants, \(a_0\) and \(qa_0\). This implies, e.g., that the
Newtonian-to-deep-MOND transition occurs not only around \( a_0\), but also
within an acceleration range of the same order. We also require from a theory
that the dynamics (say of test particles) asymptotically far from a body of mass
\(M\) depend only on \(M\), but not on internal attributes of the body, such as
the mass distribution in it. This requirement is automatically satisfied by
modified-inertia MOND theories ([72]Milgrom, 2023c), where the attributes of the
body enter only through its Newtonian field. This requirement is also satisfied
by all MOND, modified-gravity theories propounded to date.
Some of the predictions that follow 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[73]^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.[74]^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 [75]2.
Figure 2: The MOND acceleration produced by an isolated point mass \(M\), as a
function of radius from the mass (in heavy lines). These are given for a mass
typical of a star of \(1 M_{\odot}\) (red), of a globular cluster of \(10^5
M_{\odot}\) (blue), of a galaxy of \(3\times 10^{10} M_{\odot}\) (green), and of
a galaxy cluster of \(3\times 10^{13} {M_{\odot}}\) (magenta). The Newtonian
accelerations are shown as dashed lines. Departure of MOND from Newtonian
dynamics occurs at different radii for different central masses (the respective
MOND radii of these masses), but always at the same value of the acceleration, \(
a_0\), below which we are in the MOND regime, and above which we are in the
Newtonian regime.
Succinctly formulated, some of the predicted MOND laws are (propounded in the
original MOND papers, except as noted):
1. Speeds along an orbit around any isolated, bounded mass, \(M\), become
independent of the size of the orbit for asymptotically large radii. For
example, the velocity on a circular orbit becomes independent of the orbital
radius, \(r\), for very large \(r\) we have \(V(r) {\rightarrow}
{V_{\infty}}(M)\). This contrasts with Kepler's 3rd law, which rests on
Newtonian dynamics, according to which \(V\propto \sqrt{M/r}\).
2. \( {V_{\infty}}(M)=(M { {\mathcal{A}}_0})^{1/4}=c(M/M_M)^{1/4}\).
3. Define the discrepancy as the ratio , \(\eta=g/g_N\), of the observed
acceleration, \(g\), to the Newtonian accelerations, \(g_N\) (calculated from
the observed mass alone), at a given position. In a system where \(g(r)\)
varies with radius, a discrepancy is predicted to appear at the radius where
\(g(r)\) (or, equivalently, \(g_N\)) crosses \( a_0\). For example, in a disc
galaxy with measured rotational speed \(V(r)\), the discrepancy is predicted
to always start around the radius where \(g(r)=V^2(r)/r= a_0\).
4. In system where \(g,~g_N< a_0\) everywhere, a discrepancy is predicted
everywhere, with \(\eta\approx a_0/g\). Observationally, low acceleration
(small \(MG/r^2\)) is synonymous with low surface density (small \(M/r^2\)),
or low surface brightness (luminosity per unit area).
5. Many galactic systems -- such as globular clusters, dwarf spheroidal, and
elliptical galaxies, galaxy clusters -- may be described as Quasi-isothermal
systems: systems where gravity is balanced by random motions of their
constituents, whose velocity dispersion is roughly independent of radius. For
such systems MOND predicts that they must have mean mass surface densities
\(\bar\Sigma\lesssim { {\Sigma}_M}\equiv a_0/ 2\pi G\).
6. In an isolated, quasi-isothermal or deep-MOND system of mass \(M\), a
characteristic velocity dispersion \(\sigma\sim (M {
{\mathcal{A}}_0})^{1/4}=c(M/M_M)^{1/4}\) is predicted.[76]^7
If one interprets MOND consequences as being due to a DM halo, then MOND
predicts the following for this fictitious halo:
7. The acceleration produced by such a fictitious halo can never much exceed \(
a_0\) ([77]Brada and Milgrom, 1999a).
8. The central surface density such "dark halos" is \(\lesssim \Sigma_M\), with
\(\Sigma_M\) being an accumulation point, with near equality holding for many
halos ([78]Milgrom, 2009b).
9. The MOND central-surface-densities relation (CSDR): The `dynamical' central
surface density of a disc galaxy, \(\Sigma^0_D\equiv -(4\pi
G)^{-1}\int_{-\infty}^{\infty}\nabla\cdot {\bf g}(z,r=0)dz\) (i.e., the total
dynamical column density along the galaxy's symmetry \(z\)-axis, baryonic
plus phantom) is strongly correlated with the baryonic central surface
density of the disc, \(\Sigma^0_B\). Specifically, in the presently known
modified-gravity theories (see section on theories) the two attributes are
functionally related: \(\Sigma^0_D=\Sigma_M\mathcal{S}(\Sigma^0_B/\Sigma_M)
\). The exact form of \(\mathcal{S}(x)\) can be calculated and depends
somewhat on the specific MOND theory. But its asymptotes are fixed by the
basic tenets of MOND: \(\mathcal{S}(x\gg 1)\approx x\), and
\(\mathcal{S}(x\ll 1)\approx 2x^{1/2}\) ([79]Milgrom, 2016; [80]Milgrom,
2024).
10. Scale invariance of the relativistically-weak-field limit, extends laws
([81]1) and ([82]2) to gravitational light bending ([83]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. ([84]1). The proportionality constant depends on the theory.
Some additional, more qualitative, predictions of MOND are:
11. The complete study of the dynamics of a spiral galaxy includes study of
motions perpendicular to its galactic disc. MOND predicts that, all
considered, analysis in the framework of Newtonian dynamics will require not
only a spheroidal halo, but also a thin disc of putative "dark matter" in
such a galaxy.
12. MOND endows self gravitating systems with an increased, but limited
[85]stability ([86]Milgrom, 1989a; [87]Brada & Milgrom, 1999b; [88]Banik,
Milgrom, & Zhao, 2018).
13. High-acceleration systems should show no discrepancies.
14. 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.
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[89]^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 ([90]1), ([91]7), and ([92]8)], of the baryons alone
[e.g., law ([93]5)], or relations between the two [e.g., laws ([94]2) and
([95]3)].
Law ([96]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 ([97]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 [98]3, and see, e.g., [99]Sanders,
1996; [100]Noordermeer & Verheijen, 2007; [101]McGaugh, 2011; [102]McGaugh, 2012;
[103]den Heijer et al., 2015; [104]Papastergis et al., 2016; [105]Lelli et al.,
2016a; [106]Di Teodoro et al., 2021; [107]Di Teodoro et al., 2022. (The laso two
references confirm this relation for the most massive disc galaxies in the nearby
universe.) 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 [108]Bradford et al., 2016). I reserve `MASR' for the
relation that uses the MOND prescription.
This prediction, together with law ([109]1), both encapsulated in eq. ([110]1),
were also tested, statistically, on a large sample of galaxies of all types,
using weak gravitational lensing, as shown in Figure [111]4 from [112]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.
Figure 3: Test of the mass-asmptotic-speed relation predicted by MOND: Data for
galaxy baryonic mass plotted against the measured asymptotic rotation speed,
compared with the MOND prediction (line). Left: a large sample of disc galaxies
of all types (circles for gas-rich, squares for star-dominated galaxies). Middle:
the same test with only gas-rich galaxies included, for which the baryonic mass
is insensitive to adopted stellar mass-to-light ratios ([113]McGaugh, 2011). In
both plots, the line is the MOND prediction using the value of \( a_0\)
determined earlier from rotation-curve analysis of 11 galaxies ([114]McGaugh,
2012). Right: distribution of \(V_f^4/MG\) for the latter sub-sample, compared
with that expected from measurement errors alone; showing that the observed
scatter is consistent with no intrinsic scatter in the observed relation.
Figure 4: The MOND predictions ([115]Milgrom, 2013) of the velocity-dispersion
(\(\sigma\))-Luminosity (\(L\)) relations that are deduced from galaxy-galaxy
weak lensing, shown for baryonic mass-to-light ratios \(M/L=1,~1.5,~3,~6\) solar
values (these ratios are needed to translate the observed luminosities to masses,
which appear in the predictions). The measurements ([116]Brimioulle, et al. 2013)
are for `blue' (blue squares) and `red' (red triangles) lenses, respectively. The
predicted lines for \(M/L\) of 1.5 and 6 are practically identical to the
best-fit relations found in [117]Brimioulle, et al. 2013 for `blue' (aka spiral
or disc) and `red' (aka elliptical) lenses, respectively. In [118]Brimioulle, et
al. 2013, the strength of the asymptotic logarithmic potential [predicted by MOND
[eq. ([119]1)], and verified separately in a preliminary step] is \(2\sigma^2\).
MOND predicts the relation \(\sigma=(\mathcal{A}_0/4)^{1/4}(M/L)^{1/4}L^{1/4}\).
MOND generally predicts a tight correlation between the observed \(\eta\) and the
acceleration, \(g\) [laws ([120]3)([121]4)([122]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. ([123]3) (from
[124]Milgrom, 1983a). This has been testeded and confirmed several times,
starting with [125]Sanders, 1990 (Fig. 4 there), and [126]McGaugh, 1999 (Fig. 7
there), for rotationally-supported, disc galaxies, and with [127]Scarpa, 2003
(Figs. 6-8 there) and [128]Scarpa, 2006, for pressure-supported systems. Then,
with more and better data for disc galaxies, in [129]McGaugh, 2004 (Figs. 4, 5
there), [130]Tiret & Combes, 2009 (Fig. 3 there), [131]Wu & Kroupa, 2015 (Fig. 1
there), [132]McGaugh et al., 2016 (Fig. 3 there); and for early-type (elliptical)
galaxies by [133]Janz et al., 2016 and by [134]Tian & Ko, 2017.
An example of such a test is reproduced here in Figure [135]5. The important
lesson from Figure [136]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 ([137]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 ([138]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 [139]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 [140]Brouwer et
al. (2021), is shown in . 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 . 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 ([141]4) was a surprising, major prediction of MOND ([142]Milgrom, 1983b),
before it was found observationally that, indeed, all low-acceleration galaxies
show large mass discrepancies.
Figure 5: Test of the predicted MOND mass-discrepancy-acceleration relation (aka
radial-acceleration relation): The measured discrepancy: the ratio,
\(\eta=g/{g_N}\), at many radii, in 73 disc galaxies (courtesy of Stacy McGaugh).
In the upper panel, \(\eta\) is plotted against radius, where we see no
correlation. In the lower panel, it is plotted against \( {g_N}\). As predicted
by MOND ([143]Milgrom, 1983a), the discrepancy is a tight function of \( {g_N}\)
[the function \( {\nu}( {g_N}/ a_0)\) defined in eq. ([144]3) below], and departs
from 1 for accelerations smaller than \(\sim 10^{-8}~{\rm cm~s^{-2}}(=10^{-10}~
{\rm m~s^{-2}})\). The lines show the two asymptotic behaviors predicted by MOND:
the Newtonian limit, in blue, \(\eta=1\), and the deep-MOND limit, in red,
\(\eta=(a_0/g_N)^{1/2}\).
Figure 6: Test of the MOND MDAR with weak-lensing analysis ([145]Brouwer et al.
(2021)). The accelerations produced at many radii in many galaxies -- shown as
various data points, and measured by gravitational lensing -- are plotted against
the accelerations that would have been produced in Newtonian dynamics (with no
dark matter) by the observed (baryonic) matter. This is compared with the MOND
prediction in eq. ([146]3) below (from [147]Milgrom, 1983a, with an interpolating
function used by [148]McGaugh et al., 2016). Also shown are the data points
testing the same relation, from [149]McGaugh et al., 2016, using rotation curves
of the galaxies in the SPARC sample. We see that the lensing test reaches much
lower accelerations than can be probed with rotation curves. The inset at the
lower right shows the refined reanalysis of the same data by [150]Mistele et al.,
2024.
Figure 7: The MOND central-surface-densities relation (CSDR) ([151]Milgrom, 2016)
tested with the data of [152]Lelli et al. 2016. The thicker, blue line (full and
dashed) is the equality line (the Newtonian asymptote of the MOND prediction).
The thinner, red line (full and dashed) is the predicted, deep-MOND asymptote.
The thinnest, black line is the full MOND relation. For the data, the Toomre
surface density, \(\Sigma^0_T\), is taken as a proxy for \(\Sigma^0_D\), and the
proxy for \(\Sigma^0_B\) is the central, stellar surface density, \(\Sigma^0_*\).
The dotted line is the best-fit to the data in [153]Lelli et al., 2016, with some
3-parameter formula (not theoretically motivated). No fitting is involved in the
MOND curves. The values of the MOND surface density, \(\Sigma_M\) is marked.
Law ([154]9) is compared with the relevant data of [155]Lelli et al., 2016 in
Figure [156]7 (see more details in [157]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 [158]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 ([159]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 [160]5), and to compact dwarf galaxies ([161]Scarpa,
2005), but, on the face of it, does not hold for the cores of galaxy clusters
(see below).
Law ([162]14), Dictates and underlies correlations between properties of rotating
galaxies that involve their angular momentum [163]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 [164]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 `modified-gravity' 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
([165]Bekenstein & Milgrom, 1984; [166]Milgrom, 2010a). This was shown to also be
the case in some models of `modified-inertia' MOND ([167]Milgrom, 2022;
[168]Milgrom, 2023c) .
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)
([169]Milgrom, 1983a; [170]Bekenstein & Milgrom, 1984; [171]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 dependence on the specific MOND theory (e.g., [172]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
([173]Hernandez, Jimenez, and Allen, 2012; [174]Pittordis & Sutherland, 2018;
[175]Banik & Zhao, 2018; [176]Chae, 2023; [177]Hernandez, 2023; [178]Chae, 2023a;
[179]Hernandez et al., 2023; [180]Banik et al., 2024; [181]Hernandez and Chae,
2023; [182]Chae, 2024). These are expected to depart only a little from Newtonian
behavior; but exactly how little depends strongly on the MOND formulation.
[183]Milgrom, 2022 and [184]Milgrom, 2023c showed that in the framework of
`modified-inertia' formulations of MOND, the strength of the EFE, and, in
particular, its application to dynamics in the Galaxy, near the sun, can be quite
different from what is predicted in the framework of `modified-gravity'
formulations.
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 [185]Milgrom, 2011b
that this frame may be defined by the quantum vacuum.
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 [186]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 [187]Bekenstein &
Milgrom, 1984, and further elaborated on, for this theory, in [188]Milgrom, 1986.
The exact form in QUMOND was derived in [189]Milgrom, 2010a.
[190]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.
[191]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.
[192]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 [193]Milgrom, 2011b; [194]Milgrom, 2022; ([195]Milgrom, 2023c).
The EFE was implemented as part of the study of the observed internal dynamics of
dwarf satellites in [196]McGaugh & Milgrom, 2013b, and in [197]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 ([198]Milgrom, 1983a). [199]Haghi, et
al. 2016, and [200]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.)
[201]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.
[202]Thomas, et al. 2018 showed that the EFE of the Galaxy on the dynamics of the
globular cluster Palomar 5 can account for the observed asymmetry of its tidal
stream. [203]Kroupa, et al. 2022 and [204]Kroupa, et al. 2024 showed, using
detailed simulations, that the same MOND-specific effect naturally accounts for
the leading/trailing asymmetries observed in several open clusters, while such
asymmetries are not reproduced in their Newtonian simulations. Such asymmetries
appearing consistently in many clusters would be very hard to account for in the
dark-matter paradigm.
[205]Asencio, et al. 2022, showed that the distrubed appearance of a large
fraction of the dwarf galaxies in the Fornax Cluster argues against their being
protected by a dark-matter halo against tidal ripping and disruption. They showed
that the statistics of disturbed dwarfs is quite consistent with MOND, as the
cluster's EFE on the dwarfs reduces their internal acceleration field in just the
right amount to robe them of protection against disturbances.
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., [206]Milgrom, 1983b;
[207]Milgrom, 1983c; [208]Sanders, 1990; [209]McGaugh, 2004; [210]Scarpa, 2003;
[211]Tiret & Combes, 2009; [212]Milgrom, 2009b; [213]Famaey & McGaugh, 2012;
[214]Milgrom, 2014; [215]Trippe, 2014; [216]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 ([217]Milgrom, 1983a; [218]Milgrom, 1989;
[219]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.[220]^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) ([221]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 [222]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
[223]Milgrom, 1999; [224]Pikhitsa, 2010; [225]Li & Chang, 2011; [226]Kiselev &
Timofeev, 2011; [227]Klinkhamer & Kopp, 2011; [228]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 [229]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
([230]Milgrom, 1989; [231]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 ([232]Milgrom, 2017) found that the rotation curves of distant
disc galaxies presented by [233]Genzel et al. (2017) may already preclude large
variations of \(a_0\) with cosmic time.
Some "practical" consequences of this nearness in values, eq. ([234]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. This state of things is shown schematically
in Figure [235]8.
Figure 8: A schematic depiction of the different limits required from a MOND
theory, ilustrated for the dynamics at a distance \(R\) from a mass \(M\). We see
that the deep-MOND region (\(MG/R^2\ell_M\).
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 ([236]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 ([237]2), \(g_W\) remains above \(a_0\) for distances from
the source comparable with the Hubble distance ([238]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 ([239]Milgrom, 1994) a universal
algebraic relation between the Newtonian acceleration, \(g_N\), gotten from the
mass distribution, and the MOND acceleration, \(g=V^2(r)/r\) ([240]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. ([241]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.
([242]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 ([243]4)] to exhibit large mass
discrepancies everywhere in the disc, long before their dynamics were measured
([244]Milgrom, 1983b), [245]^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 ([246]Milgrom & Bekenstein, 1987; [247]Kent, 1987;
[248]Milgrom, 1988; [249]Begeman, Broeils & Sanders, 1991; [250]Sanders, 1996;
[251]Sanders & Verheijen, 1998; [252]de Blok & McGaugh, 1998; [253]Bottema, et
al., 2002; [254]Begum & Chengalur, 2004; [255]Gentile, et al., 2004;
[256]Gentile, et al., 2007a; [257]Corbelli & Salucci, 2007; [258]Barnes, et al.,
2007; [259]Sanders & Noordermeer, 2007; [260]Milgrom & Sanders, 2007;
[261]Swaters, Sanders, & McGaugh 2010; [262]Gentile, et al., 2011; [263]Famaey &
McGaugh, 2012; [264]Randriamampandry & Carignan, 2014; [265]Hees, et al., 2016;
[266]Haghi, et al., 2016; [267]Li, et al., 2018; [268]Sanders, 2019). The results
for a few galaxies of different mean accelerations are shown in Figure [269]9.
Figure 9: Observed rotation curves of five galaxies, (data points) compared with
the MOND predictions (solid lines going through the data points). The three
leftmost from [270]Begeman, Broeils, & Sanders, 1991, the lowest-right from
[271]Swaters, Sanders, & McGaugh, 2010, and the upper right from [272]Sanders,
2006. Other lines in the figures are the Newtonian curves for various baryonic
components.
Of particular note is the analysis of [273]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 [274]Sanders (2019) did not adjust these.) His results are shown in
Figure [275]10.
Figure 10: MOND predictions of rotation curves of gas-rich disc galaxies from
[276]Sanders, 2019. Each galaxy is represented in two panels. The upper panel
shows the surface densities of gas (dashed) and stars (dotted) as functions of
radius. The lower panel shows the observed rotation curve (points), the Newtonian
rotation curve for the baryonic components (long dashed curve), and MOND rotation
curve calculated from it. Because stars contibute very little to the baryonic
mass (as is seen in the upper panel for each galaxy), the exact conversion of
starlight to stellar mass (mass-to-light ratio) is practically immaterial. To
boot, the mass discrepancies are very large everywhere in these galaxies (as seen
in the lower panels: the observed velocities are much higher than the baryonic
ones). So the MOND predictions depend only very little on the exact form of the
interpolating function.
Rotation-curve tests of MOND differ conceptually from rotation-curve fits within
the dark-matter paradigm: In MOND, the observed baryon-mass distribution in a
given galaxy leads to a unique prediction of the rotation curve, which can be
compared with the observed curve. In the dark-matter paradigm, the relations
between the baryons and the total mass distribution strongly depend on the
unknowable formation and evolution history of the particular galaxy. It is thus
not possible to predict the rotation curve (mostly dominated by dark matter) from
the baryon distribution. At best one can fit a few-parameters dark halo to
reproduce the observed rotation curve.
Some observed rotation curves show features that are clearly traced back to
features in the baryonic mass distribution. Some examples are evident in Figure
[277]9. 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 [278]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 ([279]1)
is clearly seen to hold in the many observed rotation curves that go to large
enough radii (and is collectively subsumed in Figure [280]5).
Law ([281]7) ([282]Brada & Milgrom, 1999a) was tested in [283]Milgrom & Sanders,
2005 and [284]Milgrom, 2009b. Law ([285]8) was pointed out and tested in
[286]Milgrom, 2009b.
Pressure-supported systems
A pressure-supported system is a self gravitating system of masses in long-term
[287]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).[288]^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
([289]Milgrom, 1997; [290]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 ([291]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_{p2 dimensions.
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MOND Reviews
D. Merritt: [819]A Philosophical Approach to MOND: Assessing the Milgromian
Research Program in Cosmology, Cambridge University Press (2020) ([820]Google
books)
I. Banik and HS. Zhao: [821]From galactic bars to the Hubble tension - weighing
up the astrophysical evidence for Milgromian gravity , Symmetry (2021)
S. McGaugh: [822]Predictions and Outcomes for the Dynamics of Rotating Galaxies ,
Galaxies, vol. 8, issue 2, p. 35 (2020)
M. Milgrom: [823]MOND vs. dark matter in light of historical parallels, Studies
in History and Philosophy of Modern Physics (2020)
B. Famaey and S.S. McGaugh: [824]Modified Newtonian Dynamics (MOND):
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Relativity, 15, 10 (2012)
R.H. Sanders: [825]A historical perspective on modified Newtonian dynamics,
Canadian J. Phys. 93, 126 (2015)
Wikipedia: [826]Modified Newtonian Dynamics
R.H. Sanders, [827]"The dark matter problem: a historical perspective", Cambridge
U. Press, (2010) [828]Google Books
R.H. Sanders, [829]"Deconstructing Cosmology", Cambridge U. Press, (2016)
[830]Google Books
Useful links
[831]Papers with "MOND" in the abstract (from the Astrophysics Data System)
[832]Papers with "modified Newtonian dynamics" in the abstract (from the
Astrophysics Data System)
[833]Popular articles discussing MOND
[834]Milgrom's home page linking to "MOND: general resources"
Stacy McGaugh's [835]"The MOND pages"
Sponsored by: [836]Dr. Olivier Minazzoli, Université Côte d'Azur, Observatoire de
la Côte d'Azur, CNRS, Artemis, Nice, France AND Bureau des Affaires Spatiales,
Monaco
[837]Reviewed by: [838]Dr. Benoit Famaey, CNRS, Strasbourg, France
[839]Reviewed by: [840]Prof. Stacy McGaugh, Department of Astronomy - Case
Western Reserve, Cleveland, OH, United States of America
Accepted on: [841]2014-06-25 13:29:46 GMT
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