View source for Modified Newtonian dynamics

{{short description|Hypothesis proposing a modification of Newton's laws}}
{{Redirect|MOND||Mond (disambiguation){{!}}Mond}}
{{Use American English|date=August 2021}}
'''Modified Newtonian dynamics''' ('''MOND''') is a [[hypothesis]] that proposes a modification of [[Newton's second law]] to account for observed properties of [[galaxies]]. It is an alternative to the hypothesis of [[dark matter]] in terms of explaining why galaxies do not appear to obey the currently understood laws of physics.

Created in 1982 and first published in 1983 by Israeli physicist [[Mordehai Milgrom]],<ref name="Milgrom papers">{{Cite journal |last=Milgrom |first=M. |date=1983 |title=A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis |journal=Astrophysical Journal |volume=270 |pages=365–370 |doi=10.1086/161130 |bibcode=1983ApJ...270..365M|doi-access=free }}. {{Cite journal |last=Milgrom |first=M. |date=1983 |title=A modification of the Newtonian dynamics - Implications for galaxies |journal=Astrophysical Journal |volume=270 |pages=371–383 |doi=10.1086/161131 |bibcode=1983ApJ...270..371M}}. {{Cite journal |last=Milgrom |first=M. |date=1983 |title=A modification of the Newtonian dynamics - Implications for galaxy systems |journal=Astrophysical Journal |volume=270 |pages= 384|doi= 10.1086/161132 |bibcode=1983ApJ...270..384M}}.</ref> the hypothesis' original motivation was to explain why the velocities of stars in galaxies were observed to be larger than expected based on Newtonian mechanics. Milgrom noted that this discrepancy could be resolved if the [[gravitational force]] experienced by a star in the outer regions of a galaxy was proportional to the ''square'' of its centripetal [[acceleration]] (as opposed to the centripetal acceleration itself, as in [[Newton's second law]]) or alternatively, if gravitational force varied inversely with the radius directly (as opposed to with the [[inverse square]] of the radius, as in [[Newton's law of gravity]]). MOND departs from Newton's laws at extremely small accelerations that are characteristic of the outer regions of galaxies as well as the inter-galaxy forces within galaxy clusters, but which are far below anything encountered in the [[Solar System]] or on Earth.{{Dubious|reason=MOND could be observable in the outer solar system as orbits of KBOs should be influenced by the Galactic center. Source:  https://dx.doi.org/10.3847/1538-3881/acef1e|date=October 2023}}

{{unsolved|physics|
* What is the nature of [[dark matter]]? Is it a [[Elementary particle|particle]], or do the phenomena attributed to dark matter actually require a modification of the laws of gravity?
}}

MOND is an example of a class of theories known as [[modified models of gravity|modified gravity]], and is an alternative to the hypothesis that the dynamics of galaxies are determined by massive, invisible [[dark matter halo]]s. Since Milgrom's original proposal, proponents of MOND have claimed to successfully predict a variety of galactic phenomena that they state are difficult to understand as consequences of dark matter.<ref name="tale of two paradigms">{{Cite journal |last = McGaugh |first = S. |title = A Tale of Two Paradigms: the Mutual Incommensurability of LCDM and MOND |date=2015 |arxiv = 1404.7525 |bibcode = 2015CaJPh..93..250M |doi = 10.1139/cjp-2014-0203 |volume=93 |issue = 2 |journal=Canadian Journal of Physics |pages=250–259| s2cid = 51822163 }}</ref><ref>{{cite journal |author=Kroupa, P. |author2=Pawlowski, M. |author3=Milgrom, M. |year=2012 |title=The failures of the standard model of cosmology require a new paradigm |journal=International Journal of Modern Physics |volume=21 |issue=14 |page= 1230003 |doi=10.1142/S0218271812300030|arxiv=1301.3907 |bibcode= 2012IJMPD..2130003K|s2cid=118461811 }}</ref>

Though MOND explains the anomalously great rotational velocities of galaxies at their perimeters, it does not fully explain the velocity dispersions of individual galaxies within galaxy clusters. MOND reduces the discrepancy between the velocity dispersions and clusters' observed missing [[Baryon|baryonic]] mass from a factor of around 10 to a factor of about 2. However, the residual discrepancy cannot be accounted for by MOND, requiring that other explanations close the gap such as the presence of as-yet undetected missing baryonic matter.<ref name="Milgrom Scholarpedia">Mordehai, M. (2014) "The MOND paradigm of modified dynamics". [http://www.scholarpedia.org/article/The_MOND_paradigm_of_modified_dynamics#Galaxy_clusters ''Scholarpedia'', 9(6):31410.]</ref>

The accurate measurement of the speed of gravitational waves compared to the speed of light in 2017 ruled out a certain class of modified gravity theories but concluded that other MOND theories that dispense with the need for dark matter remained viable.<ref name="Boran">{{cite journal |last1=Oran |first1=Sibel |last2=Desai |first2=Santana |last3=Kaya |first3=Emre |last4=Woodard |first4=Richard |year=2018 |title=GW170817 Falsifies Dark Matter Emulators |journal=Physical Review D |volume=97 |issue=4 |pages=041501 |arxiv=1710.06168 |doi=10.1103/PhysRevD.97.041501 |bibcode=2018PhRvD..97d1501B |s2cid=119468128}}</ref> Two years later, theories put forth by Constantinos Skordis and Tom Zlosnik were consistent with gravitational waves that always travel at the speed of light. Later still in 2021, Skordis and Zlosnik developed a subclass of their theory called "RMOND", for "relativistic MOND", which had "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."<ref name="Milgrom Scholarpedia"/><ref name=RMOND>{{cite journal |title=New Relativistic Theory for Modified Newtonian Dynamics |author=Constantinos Skordis |author2=Tom Złośnik |name-list-style=amp |date=2021 |journal=Physical Review Letters |volume=127 |issue=16 |pages=161302 |doi=10.1103/PhysRevLett.127.161302 |pmid=34723619 |bibcode=2021PhRvL.127p1302S|arxiv = 2007.00082 |s2cid=220281053}}</ref>

== Overview ==
[[Image:M33 rotation curve HI.gif|thumb|upright 1.6|'''Figure 1''' – Comparison of the observed and expected rotation curves of the typical spiral galaxy [[Triangulum Galaxy|M33]]<ref>Data are from: {{cite journal |first1=E. |last1=Corbelli |first2=P. |last2=Salucci |year=2000 |title=The extended rotation curve and the dark matter halo of M33 |journal=[[Monthly Notices of the Royal Astronomical Society]] |volume=311 |issue=2 |pages=441–447 |doi=10.1046/j.1365-8711.2000.03075.x |arxiv=astro-ph/9909252 |bibcode = 2000MNRAS.311..441C|s2cid=10888599 }}</ref>]]

Several independent observations point to the fact that the visible mass in galaxies and galaxy clusters is insufficient to account for their dynamics, when analyzed using Newton's laws. This discrepancy – known as the "missing mass problem" – was first identified for clusters by Swiss astronomer [[Fritz Zwicky]] in 1933 (who studied the [[Coma cluster]]),<ref>{{cite journal |last=Zwicky |first=F. |date=1933 |title=Die Rotverschiebung von extragalaktischen Nebeln |journal=Helvetica Physica Acta |volume=6 |pages=110–127 |bibcode=1933AcHPh...6..110Z}}</ref><ref>{{cite journal |last=Zwicky |first=F. |title=On the masses of nebulae and of clusters of nebulae |date=1937 |journal=[[The Astrophysical Journal]] |volume=86 |page=217 |bibcode=1937ApJ....86..217Z |doi=10.1086/143864|doi-access=free }}</ref> and subsequently extended to include [[spiral galaxies]] by the 1939 work of [[Horace Babcock]] on [[Andromeda Galaxy|Andromeda]].<ref>{{cite journal |last=Babcock |first=H. |year=1939 |title=The rotation of the Andromeda Nebula |journal=Lick Observatory Bulletin |volume=498 |number=498|page=41 |doi=10.5479/ADS/bib/1939LicOB.19.41B |bibcode=1939LicOB..19...41B |doi-access=free }}</ref>

These early studies were augmented and brought to the attention of the astronomical community in the 1960s and 1970s by the work of [[Vera Rubin]] at the [[Carnegie Institution for Science|Carnegie Institute]] in Washington, who mapped in detail the rotation velocities of stars in a large sample of spirals. While Newton's Laws predict that stellar rotation velocities should decrease with distance from the galactic centre, Rubin and collaborators found instead that they remain almost constant<ref name="Rubin1970">{{cite journal |last1=Rubin |first1=Vera C. |author1-link=Vera Rubin |last2=Ford |first2=W. Kent Jr. |author2-link=Kent Ford (astronomer) |date=February 1970 |title=Rotation of the Andromeda Nebula from a spectroscopic survey of emission regions |journal=[[The Astrophysical Journal]] |volume=159 |pages=379–403 |bibcode=1970ApJ...159..379R |doi=10.1086/150317|s2cid=122756867 }}</ref> – the [[rotation curves]] are said to be "flat". This observation necessitates at least one of the following:
:{|
|- valign="top"
|(1) || There exists in galaxies large quantities of unseen matter which boosts the stars' velocities beyond what would be expected on the basis of the visible mass alone, or
|- valign="top"
|(2) || Newton's Laws do not apply to galaxies.
|}
Option&nbsp;(1) leads to the dark matter hypothesis; option&nbsp;(2) leads to MOND.

[[Image:Milgrom Mordechai.jpg|thumb|upright|MOND was proposed by Mordehai Milgrom in 1983]]

The basic premise of MOND is that while Newton's laws have been extensively tested in high-acceleration environments (in the Solar System and on Earth), they have not been verified for objects with extremely low acceleration, such as stars in the outer parts of galaxies. This led Milgrom to postulate a new effective gravitational force law (sometimes referred to as "'''Milgrom's law'''") that relates the true acceleration of an object to the acceleration that would be predicted for it on the basis of Newtonian mechanics.<ref name="Milgrom papers"/> This law, the keystone of MOND, is chosen to reproduce the Newtonian result at high acceleration but leads to different ("deep-MOND") behavior at low acceleration:

{{NumBlk|::|<math> F_\text{N} = m \, \mu\left( \frac{a}{a_0} \right)\, a ~. </math>|{{EquationRef|1}}}}

Here {{mvar|F}}{{sub|N}} is the Newtonian force, {{mvar|m}} is the object's (gravitational) [[mass]], {{mvar|a}} is its acceleration, {{mvar|μ}}({{mvar|x}}) is an as-yet unspecified function (called the ''interpolating function''), and {{mvar|a}}{{sub|0}} is a new fundamental constant which marks the transition between the Newtonian and deep-MOND regimes. Agreement with Newtonian mechanics requires

:<math>\begin{align} \mu(x) \longrightarrow 1 && \text{ for } x \gg 1 \end{align} ~,</math>

and consistency with astronomical observations requires

:<math>\begin{align} \mu(x) \longrightarrow x && \text{ for } x \ll 1 \end{align} ~.</math>

Beyond these limits, the interpolating function is not specified by the hypothesis, although it is possible to weakly constrain it empirically.<ref>{{cite journal |first1=G. |last1=Gentile |first2=B. |last2=Famaey |first3=W.J.G. |last3=de&nbsp;Blok |year=2011 |title=THINGS about MOND |journal=Astronomy & Astrophysics |volume=527 |issue=A76 |pages=A76 |doi=10.1051/0004-6361/201015283 |arxiv=1011.4148|bibcode=2011A&A...527A..76G |s2cid=73653467 }}</ref><ref>{{cite journal |first1=B. |last1=Famaey |first2=J. |last2=Binney |year=2005 |title=Modified Newtonian dynamics in the Milky Way |journal=[[Monthly Notices of the Royal Astronomical Society]] |volume=363 |issue=2 |pages=603–608 |doi=10.1111/j.1365-2966.2005.09474.x |arxiv=astro-ph/0506723|bibcode=2005MNRAS.363..603F |s2cid=150557 }}</ref> Two common choices are the "simple interpolating function":

:<math>\mu\left( \frac{a}{a_0} \right) = \frac{1}{1 + \frac{a_0}{a}} ~,</math>

and the "standard interpolating function":

:<math>\mu\left( \frac{a}{a_0} \right) = \sqrt{\frac{1}{1 + \left( \frac{a_0}{a} \right)^2}~} ~.</math>

Thus, in the deep-MOND regime ({{mvar|a}} ≪ {{mvar|a}}{{sub|0}}):

:<math> F_\text{N} = m \frac{\, a^2 \,}{\, a_0 \,} ~.</math>

Applying this to a star or other object of mass {{mvar|m}} in [[circular orbit]] around mass {{mvar|M}} (the total baryonic mass of the galaxy), produces

{{NumBlk|::|<math>\frac{G M m}{r^2} = m \frac{\left( \frac{v^2}{r} \right)^2}{a_0} \quad \Longrightarrow \quad v^4 = G M a_0 ~</math>|{{EquationRef|2}}}}

By fitting his law to rotation curve data, Milgrom found {{nowrap|''a''<sub>0</sub> ≈ 1.2&thinsp;×&thinsp;10<sup>&minus;10</sup>&nbsp;m/s<sup>2</sup>}} to be optimal.

MOND holds that for accelerations smaller than an ''a''<sub>0</sub> value of roughly {{nowrap|1.2&thinsp;×&thinsp;10<sup>&minus;10</sup>&nbsp;m/s<sup>2</sup>,}} accelerations increasingly depart from the standard {{nowrap|''M&thinsp;·&thinsp;G''&thinsp;/&thinsp;''r''&thinsp;<sup>2</sup>}} Newtonian relationship of mass and distance, wherein gravitational strength is proportional to mass and the inverse square of distance. Specifically, the theory holds that when gravity is well below the ''a''<sub>0</sub> value, its ''rate of change''—including the curvature of [[spacetime]]—increases with the ''square root of mass'' (rather than linearly as per Newtonian law) and decreases ''linearly with distance'' (rather than distance squared).

Whenever a small mass, ''m'' is near a much larger mass, ''M'', whether it be a star near the center of a galaxy or an object near or on Earth, MOND yields dynamics that are indistinguishably close to those of Newtonian gravity. This 1-to-1 correspondence between MOND and Newtonian dynamics is retained down to accelerations of about {{nowrap|1.2&thinsp;×&thinsp;10<sup>&minus;10</sup>&nbsp;m/s<sup>2</sup>}} (the ''a''<sub>0</sub> value); as accelerations decline below ''a''<sub>0</sub>, MOND's dynamics rapidly diverge from the Newtonian description of gravity. For instance, there is a certain distance from the center of any given galaxy at which its gravitational acceleration equals ''a''<sub>0</sub>; at ten times that distance, Newtonian gravity predicts a hundredfold decline in gravity whereas MOND predicts only a tenfold reduction.

It is important to note that the Newtonian component of MOND's dynamics remains active at accelerations well below the ''a''<sub>0</sub> value of {{nowrap|1.2&thinsp;×&thinsp;10<sup>&minus;10</sup>&nbsp;m/s<sup>2</sup>}}; the equations of MOND assert no minimum acceleration for the Newtonian component. However, because the residual Newtonian-like dynamics of MOND continue to decline as the inverse square of distance below ''a''<sub>0</sub>—just as they do above—they ''comparatively'' vanish as they become overwhelmed by the stronger “deep-MOND” linear dynamics of the theory.

MOND predicts stellar velocities that closely match observations for an extraordinarily wide range of distances from galactic centers of mass. The {{nowrap|1.2&thinsp;×&thinsp;10<sup>&minus;10</sup>}} magnitude of ''a''<sub>0</sub> establishes not only the distance from the center of the galaxy at which Newtonian and MOND dynamics diverge, but ''a''<sub>0</sub> also establishes the angle (when not plotted with [[Log–log plot|log/log scales]]) of the non-Newtonian linear slope on ''velocity/radius'' graphs like ''Fig.&nbsp;1''.

MOND-compliant gravity, which explains galactic-scale observations, was not previously detected closer to Earth, such as in national laboratories or the trajectories of interplanetary spacecraft, because the ''a''<sub>0</sub> acceleration, {{nowrap|1.2&thinsp;×&thinsp;10<sup>&minus;10</sup>&nbsp;m/s<sup>2</sup>}}, at which the dynamics of MOND begin diverging from Newtonian dynamics, is—as a practical matter—indistinguishably close to perfect [[weightlessness]]. Within the Solar System, the ''v''<sup>&thinsp;4</sup>&thinsp;=&thinsp;''GMa''<sub>0</sub> equation makes the effect of the ''a''<sub>0</sub> term virtually nonexistent; it is overwhelmed by the enormous—and highly Newtonian—gravitational influence of the Sun as well as the variability of Earth's surface gravity.

On Earth's surface—and in national laboratories when performing ultra-precise gravimetry—the ''a''<sub>0</sub> value is equal to 0.012&nbsp;[[Gal (unit)|microgal]] (μGal), which is only twelve-trillionths the strength of [[Gravity of Earth|Earth's gravity]]. A change in the laws of gravity below this acceleration is far too small to be resolved with even the most sensitive free-fall-style absolute [[Gravimetry|gravimeters]] available to national labs, like the FG5-X, which is accurate to just ±2&nbsp;μGal. When considering why MOND's effects aren't detectable with precision gravimetry on Earth, it is important to remember that ''a''<sub>0</sub> doesn't represent a spurious force; it is the gravitational strength at which MOND is theorized to significantly begin departing from the Newtonian dynamic. Moreover, the ''a''<sub>0</sub> ''strength'' is equivalent to the ''change'' in Earth's gravity brought about by an elevation difference of 0.04&nbsp;mm—the width of a fine human hair. Such subtle gravitational details, besides being unresolvable with current gravimeters, are overwhelmed by [[Earth tide|twice-daily distortions in Earth's shape]] due to lunar gravitational tides, which can cause local elevation changes nearly 10,000 times greater than 0.04&nbsp;mm. Such disturbances in local gravity due to tidal distortions are even detectable as variations in the [[Clock rate|rate]] of a [[Shortt–Synchronome clock|Shortt double-pendulum clock]], which was a national timekeeping standard in the late 1920s.

Even at the edge of the Solar System, the ''a''<sub>0</sub> point at which MOND dynamics significantly diverge from Newtonian dynamics is overwhelmed and masked by the much stronger gravitational fields from the Sun and planets, which follow Newtonian gravity. To give a sense of scale to ''a''<sub>0</sub>, a free-floating mass in space that was exposed for one hour to {{nowrap|1.2&thinsp;×&thinsp;10<sup>&minus;10</sup>&nbsp;m/s<sup>2</sup>}} would "fall" by just 0.8 millimeter—roughly the thickness of a credit card. An interplanetary spacecraft on a free-flying inertial path well above the Solar System's [[ecliptic plane]] (where it is isolated from the gravitational influence of individual planets) would, when at the same distance from the Sun as Neptune, experience a classic Newtonian gravitational strength that is 55,000 times stronger than ''a''<sub>0</sub>. For small Solar System asteroids, gravitational effects in the realm of ''a''<sub>0</sub> are comparable in magnitude to the [[Yarkovsky effect]], which subtly perturbs their orbits over long periods due to momentum transfer from the non-symmetric emission of thermal photons. The Sun's contribution to interstellar galactic gravity doesn't decline to the ''a''<sub>0</sub> threshold at which MOND's effects predominate until objects are 41&nbsp;[[Light-second#Use in astronomy|light-days]] from the Sun; this is 53 times further away from the Sun than [[Voyager 2]] was in November 2022, which has been in the [[interstellar medium]] since 2012.

Despite its vanishingly small and undetectable effects on bodies that are on Earth, within the Solar System, and even in proximity to the Solar System and other [[planetary system]]s, MOND successfully explains significant observed galactic-scale rotational effects without invoking the existence of as-yet undetected dark matter particles lying outside of the highly successful [[Standard Model]] of particle physics. This is in large part due to MOND holding that exceedingly weak galactic-scale gravity holding galaxies together near their perimeters declines as a very slow linear relationship to distance from the center of a galaxy rather than declining as the inverse square of distance.

Milgrom's law can be interpreted in two ways:
* One possibility is to treat it as a modification to [[Newton's second law]], so that the force on an object is not proportional to the particle's acceleration {{mvar|a}} but rather to <math display=inline>\mu\left( \frac{a}{a_0} \right) a.</math> In this case, the modified dynamics would apply not only to gravitational phenomena, but also those generated by other [[forces]], for example [[electromagnetism]].<ref>{{cite journal |first=M. |last=Milgrom |title=MOND – Particularly as modified inertia |journal=Acta Physica Polonica B |year=2011 |volume=42 |issue=11 |page=2175 |doi=10.5506/APhysPolB.42.2175 |arxiv=1111.1611|s2cid=119272458 }}</ref>
* Alternatively, Milgrom's law can be viewed as leaving [[Newton's Second Law]] intact and instead modifying the inverse-square law of gravity, so that the true gravitational force on an object of mass {{mvar|m}} due to another of mass {{mvar|M}} is roughly of the form <math display=inline>\frac{G M m}{\mu\left( \frac{a}{a_0} \right) r^2}.</math> In this interpretation, Milgrom's modification would apply exclusively to gravitational phenomena.

By itself, Milgrom's law is not a complete and self-contained [[physical theory]], but rather an ad hoc empirically motivated variant of one of the several equations that constitute classical mechanics. Its status within a coherent non-relativistic hypothesis of MOND is akin to [[Kepler's Third Law]] within Newtonian mechanics; it provides a succinct description of observational facts, but must itself be explained by more fundamental concepts situated within the underlying hypothesis. Several complete classical hypotheses have been proposed (typically along "modified gravity" as opposed to "modified inertia" lines), which generally yield Milgrom's law exactly in situations of high [[symmetry]] and otherwise deviate from it slightly. A subset of these non-relativistic hypotheses have been further embedded within relativistic theories, which are capable of making contact with non-classical phenomena (e.g., [[gravitational lensing]]) and [[cosmology]].<ref name="famaey mcgaugh">{{cite journal |first1=B. |last1=Famaey |first2=S. |last2=McGaugh |title=Modified Newtonian dynamics (MOND): Observational phenomenology and relativistic extensions |journal=Living Reviews in Relativity |year=2012 |volume=15 |issue=1 |page=10 |doi=10.12942/lrr-2012-10 |pmid=28163623 |pmc=5255531 |arxiv=1112.3960|bibcode=2012LRR....15...10F }}</ref> Distinguishing both theoretically and observationally between these alternatives is a subject of current research.

The majority of [[astronomers]], [[astrophysicists]], and [[cosmologists]] accept dark matter as the explanation for galactic rotation curves (based on general relativity, and hence Newtonian mechanics), and are committed to a dark matter solution of the missing-mass problem.<ref>{{cite journal|last1=Bertone|first1=Gianfranco|last2=Hooper|first2=Dan|journal=Reviews of Modern Physics|title=History of dark matter|year=2018|volume=90|issue=4|publisher=American Physical Society|pages=045002|doi=10.1103/RevModPhys.90.045002|arxiv=1605.04909 |bibcode=2018RvMP...90d5002B |s2cid=18596513 }}</ref> The primary difference between supporters of [[Lambda-CDM model|ΛCDM]] and MOND is in the observations for which they demand a robust, quantitative explanation, and those for which they are satisfied with a qualitative account, or are prepared to leave for future work. Proponents of MOND emphasize predictions made on galaxy scales (where MOND enjoys its most notable successes) and believe that a cosmological model consistent with galaxy dynamics has yet to be discovered. Proponents of ΛCDM require high levels of cosmological accuracy (which concordance cosmology provides) and argue that a resolution of galaxy-scale issues will follow from a better understanding of the complicated baryonic astrophysics underlying [[galaxy formation]].<ref name="tale of two paradigms"/><ref>{{cite journal |last=Sanders |first=R.H. |year=2014 |title=A historical perspective on modified Newtonian dynamics |journal=Canadian Journal of Physics |volume=93 |issue=2 |pages=126–138 |doi=10.1139/cjp-2014-0206 |arxiv=1404.0531 |bibcode=2015CaJPh..93..126S|s2cid=119240769 }}</ref>

== Observational evidence for MOND ==

Since MOND was specifically designed to produce flat rotation curves, these do not constitute evidence for the hypothesis, but every matching observation adds to support of the empirical law. Nevertheless, proponents claim that a broad range of astrophysical phenomena at the galactic scale are neatly accounted for within the MOND framework.<ref name="famaey mcgaugh"/><ref>{{Cite journal|arxiv = 1212.2568|doi = 10.1093/mnras/stt2066|title = MOND laws of galactic dynamics|year = 2014|last1 = Milgrom|first1 = Mordehai|journal = Monthly Notices of the Royal Astronomical Society|volume = 437|issue = 3|pages = 2531–2541|bibcode = 2014MNRAS.437.2531M|s2cid = 118840212}}</ref> Many of these came to light after the publication of Milgrom's original papers and are difficult to explain using the dark matter hypothesis. The most prominent are the following:

* In addition to demonstrating that rotation curves in MOND are flat, equation 2 provides a concrete relation between a galaxy's total baryonic mass (the sum of its mass in stars and gas) and its asymptotic rotation velocity. This predicted relation was called by Milgrom the [[mass-asymptotic speed relation]] (MASSR); its observational manifestation is known as the baryonic [[Tully–Fisher relation]] (BTFR),<ref>{{Cite journal|arxiv = astro-ph/0003001|doi = 10.1086/312628|title = The Baryonic Tully-Fisher Relation|year = 2000|last1 = McGaugh|first1 = S. S.|last2 = Schombert|first2 = J. M.|last3 = Bothun|first3 = G. D.|last4 = De Blok|first4 = W. J. G.|journal = The Astrophysical Journal|volume = 533|issue = 2|pages = L99–L102|pmid = 10770699|bibcode = 2000ApJ...533L..99M|s2cid = 103865}}</ref> and is found to conform quite closely to the MOND prediction.<ref>{{Cite journal|arxiv = 1107.2934|doi = 10.1088/0004-6256/143/2/40|title = The Baryonic Tully-Fisher Relation of Gas-Rich Galaxies as a Test of Λcdm and Mond|year = 2012|last1 = McGaugh|first1 = Stacy S.|journal = The Astronomical Journal|volume = 143|issue = 2|page = 40|bibcode = 2012AJ....143...40M|s2cid = 38472632}}</ref>
* Milgrom's law fully specifies the rotation curve of a galaxy given only the distribution of its baryonic mass. In particular, MOND predicts a far stronger correlation between features in the baryonic mass distribution and features in the rotation curve than does the dark matter hypothesis (since dark matter dominates the galaxy's mass budget and is conventionally assumed not to closely track the distribution of baryons). Such a tight correlation is claimed to be observed in several spiral galaxies, a fact which has been referred to as "Renzo's rule".<ref name = "famaey mcgaugh"/>
* Since MOND modifies Newtonian dynamics in an acceleration-dependent way, it predicts a specific relationship between the acceleration of a star at any radius from the centre of a galaxy and the amount of unseen (dark matter) mass within that radius that would be inferred in a Newtonian analysis. This is known as the mass discrepancy-acceleration relation, and has been measured observationally.<ref>R. Sanders, "Mass discrepancies in galaxies: dark matter and alternatives", The Astronomy and Astrophysics Review 1990, Volume 2, Issue 1, pp 1-28</ref><ref>{{Cite journal|arxiv = astro-ph/0403610|doi = 10.1086/421338|title = The Mass Discrepancy–Acceleration Relation: Disk Mass and the Dark Matter Distribution|year = 2004|last1 = McGaugh|first1 = Stacy S.|journal = The Astrophysical Journal|volume = 609|issue = 2|pages = 652–666|bibcode = 2004ApJ...609..652M|s2cid = 9544873}}</ref> One aspect of the MOND prediction is that the mass of the inferred dark matter goes to zero when the stellar centripetal acceleration becomes greater than ''a''<sub>0</sub>, where MOND reverts to Newtonian mechanics. In a dark matter hypothesis, it is a challenge to understand why this mass should correlate so closely with acceleration, and why there appears to be a critical acceleration above which dark matter is not required.<ref name = "tale of two paradigms"/>
* Both MOND and dark matter halos stabilize disk galaxies, helping them retain their rotation-supported structure and preventing their transformation into [[elliptical galaxy|elliptical galaxies]]. In MOND, this added stability is only available for regions of galaxies within the deep-MOND regime (i.e., with ''a'' < ''a''<sub>0</sub>), suggesting that spirals with ''a'' > ''a''<sub>0</sub> in their central regions should be prone to instabilities and hence less likely to survive to the present day.<ref>{{Cite arXiv|eprint = 1406.0537|last1 = Jiménez|first1 = M. A.|last2 = Hernandez|first2 = X.|title = Disk stability under MONDian gravity|year = 2014|class = astro-ph.GA}}</ref> This may explain the "[[Ken Freeman (astronomer)|Freeman]] limit" to the observed central surface mass density of spiral galaxies, which is roughly ''a''<sub>0</sub>/''G''.<ref>{{cite journal | last1 = McGaugh | first1 = S. | date = 1998 | title = Testing the Hypothesis of Modified Dynamics with Low Surface Brightness Galaxies and Other Evidence | journal = Astrophys J | volume = 499 | issue = 1| pages = 66–81 | doi=10.1086/305629 | bibcode=1998ApJ...499...66M|arxiv = astro-ph/9801102 | s2cid = 18901029 }}</ref> This scale must be put in by hand in dark matter-based galaxy formation models.<ref>{{cite journal | last1 = McGaugh | first1 = S. | date = 2005 | title = Balance of Dark and Luminous Mass in Rotating Galaxies | journal = Phys. Rev. Lett. | volume = 95 | issue = 17| page = 171302 | arxiv = astro-ph/0509305 | doi=10.1103/physrevlett.95.171302|bibcode = 2005PhRvL..95q1302M | pmid=16383816| s2cid = 1715002 }}</ref>
* Particularly massive galaxies are within the Newtonian regime (''a'' > ''a''<sub>0</sub>) out to radii enclosing the vast majority of their baryonic mass. At these radii, MOND predicts that the rotation curve should fall as 1/''r'', in accordance with [[Kepler's Laws]]. In contrast, from a dark matter perspective one would expect the halo to significantly boost the rotation velocity and cause it to asymptote to a constant value, as in less massive galaxies. Observations of high-mass ellipticals bear out the MOND prediction.<ref>{{cite journal | last1 = Romanowsky | first1 = A.J. | last2 = Douglas | first2 = N.G. | last3 = Arnaboldi | first3 = M. | last4 = Kuijken | first4 = K. | last5 = Merrifield | first5 = M.R. | last6 = Napolitano | first6 = N.R. | last7 = Capaccioli | first7 = M. | last8 = Freeman | first8 = K.C. | date = 2003 | title = A Dearth of Dark Matter in Ordinary Elliptical Galaxies | journal = Science | volume = 301 | issue = 5640| pages = 1696–1698 |arxiv=astro-ph/0308518 | doi=10.1126/science.1087441 | pmid=12947033|bibcode = 2003Sci...301.1696R | s2cid = 120137872 }}</</ref><ref>{{cite journal | last1 = Milgrom | first1 = M. | last2 = Sanders | first2 = R.H. | date = 2003 | title = Modified Newtonian Dynamics and the 'Dearth of Dark Matter in Ordinary Elliptical Galaxies' | journal = Astrophys J | volume = 599 | issue = 1| pages = 25–28 | arxiv = astro-ph/0309617 |bibcode = 2003ApJ...599L..25M |doi = 10.1086/381138 | s2cid = 14378227 }}</ref>
* In MOND, all gravitationally bound objects with ''a'' < ''a''<sub>0</sub> – regardless of their origin – should exhibit a mass discrepancy when analyzed using Newtonian mechanics, and should lie on the BTFR. Under the dark matter hypothesis, objects formed from baryonic material ejected during the merger or tidal interaction of two galaxies ("[[dwarf galaxy|tidal dwarf galaxies]]") are expected to be devoid of dark matter and hence show no mass discrepancy. Three objects unambiguously identified as Tidal Dwarf Galaxies appear to have mass discrepancies in close agreement with the MOND prediction.<ref>{{Cite journal|arxiv = 0705.1356|doi = 10.1126/science.1142114|title = Missing Mass in Collisional Debris from Galaxies|year = 2007|last1 = Bournaud|first1 = F.|last2 = Duc|first2 = P.-A.|last3 = Brinks|first3 = E.|last4 = Boquien|first4 = M.|last5 = Amram|first5 = P.|last6 = Lisenfeld|first6 = U.|last7 = Koribalski|first7 = B. S.|last8 = Walter|first8 = F.|last9 = Charmandaris|first9 = V.|journal = Science|volume = 316|issue = 5828|pages = 1166–1169|pmid = 17495138|bibcode = 2007Sci...316.1166B|s2cid = 20946839}}</ref><ref>{{Cite journal|arxiv = 0706.1976|doi = 10.1051/0004-6361:20078081|title = Tidal dwarf galaxies as a test of fundamental physics|year = 2007|last1 = Gentile|first1 = G.|last2 = Famaey|first2 = B.|last3 = Combes|first3 = F.|last4 = Kroupa|first4 = P.|last5 = Zhao|first5 = H. S.|last6 = Tiret|first6 = O.|journal = Astronomy & Astrophysics|volume = 472|issue = 2|pages = L25–L28|bibcode = 2007A&A...472L..25G|s2cid = 1288102}}</ref><ref>{{Cite journal|arxiv = 1204.2546|doi = 10.1071/AS12005|title = The Dark Matter Crisis: Falsification of the Current Standard Model of Cosmology|year = 2012|last1 = Kroupa|first1 = P.|journal = Publications of the Astronomical Society of Australia|volume = 29|issue = 4|pages = 395–433|bibcode = 2012PASA...29..395K|s2cid = 55470527}}</ref>
* Recent work has shown that many of the dwarf galaxies around the [[Milky Way]] and Andromeda are located preferentially in a single plane and have correlated motions. This suggests that they may have formed during a close encounter with another galaxy and hence be Tidal Dwarf Galaxies. If so, the presence of mass discrepancies in these systems constitutes evidence for MOND. In addition, it has been claimed that a gravitational force stronger than Newton's (such as Milgrom's) is required for these galaxies to retain their orbits over time.<ref>{{Cite book|arxiv = 1409.6302|doi = 10.1007/978-3-319-10614-4_28|chapter = Lessons from the Local Group (And Beyond) on Dark Matter|title = Lessons from the Local Group|year = 2015|last1 = Kroupa|first1 = Pavel|pages = 337–352|isbn = 978-3-319-10613-7|s2cid = 119114754}}</ref>
* In 2020, a group of astronomers analyzing data from the Spitzer Photometry and Accurate Rotation Curves (SPARC) sample together with estimates of the large-scale external gravitational field from an all-sky galaxy catalog, concluded that there was highly statistically significant evidence of violations of the [[strong equivalence principle]] in weak gravitational fields in the vicinity of rotationally supported galaxies.<ref name="Chae EFE">{{Cite journal|arxiv = 2009.11525|doi = 10.3847/1538-4357/abbb96|title = Testing the Strong Equivalence Principle: Detection of the External Field Effect in Rotationally Supported Galaxies|year = 2020|last1 = Chae|first1 = Kyu-Hyun|last2 = Lelli|first2 = Federico|last3 = Desmond|first3 = Harry|last4 = McGaugh|first4 = Stacy S.|last5 = Li|first5 = Pengfei|last6 = Schombert|first6 = James M.|journal = The Astrophysical Journal|volume = 904|issue = 1|page = 51|bibcode = 2020ApJ...904...51C|s2cid = 221879077 | doi-access=free }}</ref> They observed an effect consistent with the external field effect of modified Newtonian dynamics and inconsistent with tidal effects in the [[Lambda-CDM model]] paradigm commonly known as the Standard Model of Cosmology.
* In a 2022 published survey of dwarf galaxies from the Fornax Deep Survey (FDS) catalogue, a group of astronomers and physicists conclude that 'observed deformations of dwarf galaxies in the Fornax Cluster and the lack of low surface brightness dwarfs towards its centre are incompatible with ΛCDM expectations but well consistent with MOND.'<ref name="MNRAS">{{Cite journal|doi = 10.1093/mnras/stac1765 |title = The distribution and morphologies of Fornax Cluster dwarf galaxies suggest they lack dark matter|year = 2022|last1 = Asencio|first1 = Elena|last2 = Banik|first2 = Indranil|last3 = Mieske|first3 = Steffen|last4 = Venhola|first4 = Aku|last5 = Kroupa|first5 = Pavel|last6 = Zhao|first6 = Hongsheng|journal = Monthly Notices of the Royal Astronomical Society|volume = 515|issue = 2|pages = 2981–3013|arxiv = 2208.02265}}</ref>
* In 2022, Kroupa et al. published a study of open star clusters, arguing that asymmetry in the population of leading and trailing tidal tails, and the observed lifetime of these clusters, are inconsistent with Newtonian dynamics but consistent with MOND.<ref>{{cite journal |last1=Kroupa |first1=Pavel |last2=Jerabkova |first2=Tereza |last3=Thies |first3=Ingo |last4=Pflamm-Altenburg |first4=Jan |last5=Famaey |first5=Benoit |last6=Boffin |first6=Henri |last7=Dabringhausen |first7=Jörg |last8=Beccari |first8=Giacomo |last9=Prusti |first9=Timo |last10=Boily |first10=Christian |last11=Haghi |first11=Hosein |last12=Wu |first12=Xufen |last13=Haas |first13=Jaroslav |last14=Zonoozi |first14=Akram Hasani |last15=Thomas |first15=Guillaume |last16=Šubr |first16=Ladislav |last17=Aarseth |first17=Sverre J |title=Asymmetrical tidal tails of open star clusters: stars crossing their cluster's práh† challenge Newtonian gravitation |journal=Monthly Notices of the Royal Astronomical Society |date=26 October 2022 |volume=517 |issue=3 |pages=3613–3639 |doi=10.1093/mnras/stac2563 |url=https://academic.oup.com/mnras/article-abstract/517/3/3613/6773470 |access-date=2 November 2022|arxiv=2210.13472 }}</ref><ref>{{cite web |last1=University of Bonn |title=Astrophysicists make observations consistent with the predictions of an alternative theory of gravity |url=https://phys.org/news/2022-10-astrophysicists-alternative-theory-gravity.html |website=Phys.org |access-date=2 November 2022}}</ref>
* In 2023, a study claimed that cold dark matter cannot explain galactic rotation curves, while MOND can.<ref>{{cite journal|author=Kyu-Hyun Chae|title=Distinguishing Dark Matter, Modified Gravity, and Modified Inertia with the Inner and Outer Parts of Galactic Rotation Curves|journal=The Astrophysical Journal |date=18 October 2022|volume=941 |issue=1 |page=55 |doi=10.3847/1538-4357/ac93fc |arxiv=2207.11069 |doi-access=free |bibcode=2022ApJ...941...55C }}</ref>
* In 2023, a study measured the acceleration of 26,615 wide binaries within 200 parsecs. The study showed that those binaries with accelerations less than 1 nanometer per second squared systematically deviate from Newtonian dynamics, but conform to MOND predictions, specifically to [[AQUAL]].<ref>{{Cite journal|doi = 10.3847/1538-4357/ace101 |title = Breakdown of the Newton–Einstein Standard Gravity at Low Acceleration in Internal Dynamics of Wide Binary Stars|year = 2023|last1 = Kyu-Hyun|first1 = Chae|journal = The Astrophysical Journal|volume = 952|issue = 2| page=128 |doi-access = free|arxiv = 2305.04613| bibcode=2023ApJ...952..128C }}</ref> The results are disputed, with some authors arguing that the detection is caused by poor quality controls,<ref>{{cite journal | arxiv=2311.03436 | doi=10.1093/mnras/stad3393 | title=Strong constraints on the gravitational law from ''Gaia'' DR3 wide binaries | date=2024 | last1=Banik | first1=Indranil | last2=Pittordis | first2=Charalambos | last3=Sutherland | first3=Will | last4=Famaey | first4=Benoit | last5=Ibata | first5=Rodrigo | last6=Mieske | first6=Steffen | last7=Zhao | first7=Hongsheng | journal=Monthly Notices of the Royal Astronomical Society | volume=527 | issue=3 | pages=4573–4615 }}</ref> while the original authors claimed that the added quality controls do not significantly affect the results.<ref>{{Cite journal|doi = 10.3847/1538-4357/ad0ed5 |title = Robust Evidence for the Breakdown of Standard Gravity at Low Acceleration from Statistically Pure Binaries Free of Hidden Companions|year = 2024|last1 = Kyu-Hyun|first1 = Chae|journal = The Astrophysical Journal|volume = 960|issue = 2| page=114 |doi-access = free|arxiv = 2309.10404| bibcode=2024ApJ...960..114C }}</ref>

== Complete MOND hypotheses ==

Milgrom's law requires incorporation into a complete hypothesis if it is to satisfy [[conservation laws]] and provide a unique solution for the time evolution of any physical system. Each of the theories described here reduce to Milgrom's law in situations of high symmetry (and thus enjoy the successes described above), but produce different behavior in detail.

=== Nonrelativistic ===

The first hypothesis of MOND (dubbed [[AQUAL]]) was constructed in 1984 by Milgrom and [[Jacob Bekenstein]].<ref>{{cite journal|author=Jacob Bekenstein |author2=M. Milgrom |name-list-style=amp |title=Does the missing mass problem signal the breakdown of Newtonian gravity?|journal=Astrophys. J.|volume=286|pages=7–14|date=1984|bibcode = 1984ApJ...286....7B |doi = 10.1086/162570 }}</ref> AQUAL generates MONDian behavior by modifying the gravitational term in the classical [[lagrangian mechanics|Lagrangian]] from being quadratic in the gradient of the Newtonian potential to a more general function. (AQUAL is an acronym for A QUAdratic Lagrangian.) In formulae:

:<math>\begin{align}
\mathcal{L}_\text{Newton} &= - \frac{1}{8 \pi G} \cdot \|\nabla \phi\|^2 \\ [6pt]
\mathcal{L}_\text{AQUAL}  &= - \frac{1}{8 \pi G} \cdot a_0^2 F \left (\tfrac{\|\nabla \phi\|^2}{a_0^2} \right )
\end{align}</math>

where  <math>\phi</math> is the standard Newtonian gravitational potential and ''F'' is a new dimensionless function. Applying the [[Euler–Lagrange equations]] in the standard way then leads to a non-linear generalization of the [[Poisson's equation|Newton–Poisson equation]]:

:<math> \nabla\cdot\left[ \mu \left( \frac{\left\| \nabla\phi \right\|}{a_0} \right) \nabla\phi\right] = 4\pi G \rho </math>

This can be solved given suitable boundary conditions and choice of F to yield Milgrom's law (up to a [[Curl (mathematics)|curl]] field correction which vanishes in situations of high symmetry).

An alternative way to modify the gravitational term in the lagrangian is to introduce a distinction between the true (MONDian) acceleration field '''a''' and the Newtonian acceleration field '''a<sub>N</sub>'''. The Lagrangian may be constructed so that '''a<sub>N</sub>''' satisfies the usual Newton-Poisson equation, and is then used to find '''a''' via an additional algebraic but non-linear step, which is chosen to satisfy Milgrom's law. This is called the "quasi-linear formulation of MOND", or QUMOND,<ref>{{Cite journal|arxiv = 0911.5464|doi = 10.1111/j.1365-2966.2009.16184.x|title = Quasi-linear formulation of MOND|year = 2010|last1 = Milgrom|first1 = Mordehai|journal = Monthly Notices of the Royal Astronomical Society|volume = 403|issue = 2|pages = 886–895|bibcode = 2010MNRAS.403..886M|s2cid = 119305157}}</ref> and is particularly useful for calculating the distribution of "phantom" dark matter that would be inferred from a Newtonian analysis of a given physical situation.<ref name = "famaey mcgaugh"/>

Both AQUAL and QUMOND propose changes to the gravitational part of the classical matter action, and hence interpret Milgrom's law as a modification of Newtonian gravity as opposed to Newton's second law. The alternative is to turn the kinetic term of the action into a [[functional (mathematics)|functional]] depending on the trajectory of the particle. Such "modified inertia" theories, however, are difficult to use because they are time-nonlocal, require [[energy]] and [[momentum]] to be non-trivially redefined to be conserved, and have predictions that depend on the entirety of a particle's orbit.<ref name = "famaey mcgaugh"/>

=== Relativistic ===

In 2004, Jacob Bekenstein formulated [[TeVeS]], the first complete relativistic hypothesis using MONDian behaviour.<ref name=Bekenstein>{{cite journal |arxiv=astro-ph/0403694 |title=Relativistic gravitation theory for the MOND paradigm |author=Jacob D. Bekenstein |journal =Phys. Rev. |date=2004 |volume=D70 |issue=8 |pages=83509 |doi=10.1103/PhysRevD.70.083509|bibcode = 2004PhRvD..70h3509B }}</ref> TeVeS is constructed from a local Lagrangian (and hence respects conservation laws), and employs a unit [[vector field]], a dynamical and non-dynamical [[scalar field]], a free function and a non-Einsteinian [[Metric tensor (general relativity)|metric]] in order to yield AQUAL in the non-relativistic limit (low speeds and weak gravity). TeVeS has enjoyed some success in making contact with gravitational lensing and [[structure formation]] observations,<ref>{{Cite journal|arxiv = 1106.2476|doi = 10.1016/j.physrep.2012.01.001|title = Modified gravity and cosmology|year = 2012|last1 = Clifton|first1 = Timothy|last2 = Ferreira|first2 = Pedro G.|last3 = Padilla|first3 = Antonio|last4 = Skordis|first4 = Constantinos|journal = Physics Reports|volume = 513|issue = 1–3|pages = 1–189|bibcode = 2012PhR...513....1C|s2cid = 119258154}}</ref> but faces problems when confronted with data on the [[anisotropy]] of the [[cosmic microwave background]],<ref>{{Cite journal|arxiv = astro-ph/0508048|doi = 10.1103/PhysRevD.72.101301|title = Test of modified Newtonian dynamics with recent Boomerang data|year = 2005|last1 = Slosar|first1 = Anže|last2 = Melchiorri|first2 = Alessandro|last3 = Silk|first3 = Joseph I.|journal = Physical Review D|volume = 72|issue = 10|page = 101301|bibcode = 2005PhRvD..72j1301S}}</ref> the lifetime of compact objects,<ref name=Seifert2007>{{cite journal |last1=Seifert |first1=M. D. |date=2007 |title=Stability of spherically symmetric solutions in modified theories of gravity |journal=[[Physical Review D]] |volume=76 |issue=6 |pages=064002 |arxiv=gr-qc/0703060 |bibcode=2007PhRvD..76f4002S |doi=10.1103/PhysRevD.76.064002|s2cid=29014948 }}</ref> and the relationship between the lensing and matter overdensity potentials.<ref>{{cite journal |last1=Zhang |first1=P. |last2=Liguori |first2=M. |last3=Bean |first3=R. |last4=Dodelson |first4=S. |date=2007 |title=Probing Gravity at Cosmological Scales by Measurements which Test the Relationship between Gravitational Lensing and Matter Overdensity |journal=[[Physical Review Letters]] |volume=99 |issue=14 |pages=141302 |arxiv=0704.1932 |bibcode=2007PhRvL..99n1302Z |doi=10.1103/PhysRevLett.99.141302 |pmid=17930657|s2cid=119672184 }}</ref>

Several alternative relativistic generalizations of MOND exist, including BIMOND and generalized [[Einstein aether theory]].<ref name = "famaey mcgaugh"/> There is also a relativistic generalization of MOND that assumes a Lorentz-type invariance as the physical basis of MOND phenomenology.<ref name = "alzain">{{Cite journal|arxiv = 1708.05385|doi = 10.1007/s12036-017-9479-0|title = Modified Newtonian Dynamics (MOND) as a Modification of Newtonian Inertia|year = 2017|last1 = Alzain|first1 = Mohammed|journal = Journal of Astrophysics and Astronomy|volume = 38|issue = 4|page = 59|bibcode = 2017JApA...38...59A|s2cid = 119245210}}</ref>

== External field effect ==

In Newtonian mechanics, an object's acceleration can be found as the vector sum of the acceleration due to each of the individual forces acting on it. This means that a [[subsystem]] can be decoupled from the larger system in which it is embedded simply by referring the motion of its constituent particles to their centre of mass; in other words, the influence of the larger system is irrelevant for the internal dynamics of the subsystem. Since Milgrom's law is [[non-linear]] in acceleration, MONDian subsystems cannot be decoupled from their environment in this way, and in certain situations this leads to behaviour with no Newtonian parallel. This is known as the "external field effect" (EFE),<ref name = "Milgrom papers"/> for which there exists observational evidence.<ref name = "Chae EFE"/>

The external field effect is best described by classifying physical systems according to their relative values of ''a''<sub>in</sub> (the characteristic acceleration of one object within a subsystem due to the influence of another), ''a''<sub>ex</sub> (the acceleration of the entire subsystem due to forces exerted by objects outside of it), and ''a''<sub>0</sub>:

* <math> a_{\mathrm{in}} > a_0 </math> :  Newtonian regime
* <math> a_{\mathrm{ex}} < a_{\mathrm{in}} < a_0 </math> :  Deep-MOND regime
* <math> a_{\mathrm{in}} < a_0 < a_{\mathrm{ex}} </math> : The external field is dominant and the behavior of the system is Newtonian.
* <math> a_{\mathrm{in}} < a_{\mathrm{ex}} < a_0 </math> : The external field is larger than the internal acceleration of the system, but both are smaller than the critical value. In this case, dynamics is Newtonian but the effective value of ''G'' is enhanced by a factor of ''a''<sub>0</sub>/''a''<sub>ex</sub>.<ref>S. McGaugh, [http://www.astro.umd.edu/~ssm/mond/EFE.html The EFE in MOND] {{Webarchive|url=https://web.archive.org/web/20170716235250/http://www.astro.umd.edu/~ssm/mond/EFE.html |date=2017-07-16 }}</ref>

The external field effect implies a fundamental break with the [[strong equivalence principle]] (but not necessarily the [[weak equivalence principle]]). The effect was postulated by Milgrom in the first of his 1983 papers to explain why some [[open clusters]] were observed to have no mass discrepancy even though their internal accelerations were below a<sub>0</sub>. It has since come to be recognized as a crucial element of the MOND paradigm.

The dependence in MOND of the internal dynamics of a system on its external environment (in principle, the rest of the [[universe]]) is strongly reminiscent of [[Mach's principle]], and may hint towards a more fundamental structure underlying Milgrom's law. In this regard, Milgrom has commented:<ref>{{Cite arXiv|eprint = 0801.3133|last1 = Milgrom|first1 = Mordehai|title = The MOND paradigm|year = 2008|class = astro-ph}}</ref>

<blockquote>It has been long suspected that local dynamics is strongly influenced by the universe at large, ''a-la'' Mach's principle, but MOND seems to be the first to supply concrete evidence for such a connection. This may turn out to be the most fundamental implication of MOND, beyond its implied modification of Newtonian dynamics and general relativity, and beyond the elimination of dark matter.</blockquote>

Indeed, the potential link between MONDian dynamics and the universe as a whole (that is, cosmology) is augmented by the observation that the value of ''a''<sub>0</sub> (determined by fits to internal properties of galaxies) is within an order of magnitude of ''cH''<sub>0</sub>, where ''c'' is the [[speed of light]] and ''H''<sub>0</sub> is the [[Hubble constant]] (a measure of the present-day expansion rate of the universe).<ref name = "Milgrom papers"/> It is also close to the [[Accelerating universe|acceleration rate of the universe]], and hence the [[cosmological constant]]. Recent work on a transactional formulation of [[Entropic gravity]] by Schlatter and Kastner <ref>{{cite journal | doi=10.1088/2399-6528/acd6d7 | title=Gravity from transactions: Fulfilling the entropic gravity program | date=2023 | last1=Schlatter | first1=A. | last2=Kastner | first2=R. E. | journal=Journal of Physics Communications | volume=7 | issue=6 | page=065009 | s2cid=258791517 | doi-access=free | arxiv=2209.04025 | bibcode=2023JPhCo...7f5009S }}</ref> suggests a natural connection between ''a''<sub>0</sub>, ''H''<sub>0</sub>, and the [[cosmological constant]].

== Responses and criticism ==

=== Dark matter explanation ===

While acknowledging that Milgrom's law provides a succinct and accurate description of a range of galactic phenomena, many physicists reject the idea that classical dynamics itself needs to be modified and attempt instead to explain the law's success by reference to the behavior of dark matter. Some effort has gone towards establishing the presence of a characteristic acceleration scale as a natural consequence of the behavior of cold dark matter halos,<ref>{{Cite journal|arxiv = astro-ph/0107284|doi = 10.1086/340578|title = How Cold Dark Matter Theory Explains Milgrom's Law|year = 2002|last1 = Kaplinghat|first1 = Manoj|last2 = Turner|first2 = Michael|journal = The Astrophysical Journal|volume = 569|issue = 1|pages = L19–L22|bibcode = 2002ApJ...569L..19K|s2cid = 5679705}}</ref><ref>{{Cite journal|arxiv = 1310.6810|doi = 10.1093/mnras/stt2062|title = Using the topology of large-scale structure in the WiggleZ Dark Energy Survey as a cosmological standard ruler|year = 2014|last1 = Blake|first1 = Chris|last2 = James|first2 = J. Berian|last3 = Poole|first3 = Gregory B.|journal = Monthly Notices of the Royal Astronomical Society|volume = 437|issue = 3|pages = 2488–2506|bibcode = 2014MNRAS.437.2488B|s2cid = 56352810}}</ref> although Milgrom has argued that such arguments explain only a small subset of MOND [[wikt:phenomenon|phenomena]].<ref>{{Cite journal|arxiv = astro-ph/0110362|doi = 10.1086/341223|title = Do Modified Newtonian Dynamics Follow from the Cold Dark Matter Paradigm?|year = 2002|last1 = Milgrom|first1 = Mordehai|journal = The Astrophysical Journal|volume = 571|issue = 2|pages = L81–L83|bibcode = 2002ApJ...571L..81M| s2cid=120648795 }}</ref> An alternative proposal is to modify the properties of dark matter (e.g., to make it [[strongly interacting massive particle|interact strongly]] with itself or baryons) in order to induce the tight coupling between the baryonic and dark matter mass that the observations point to.<ref>J. Bullock (2014), [http://www.cfa.harvard.edu/events/2014/sackler/index/talks/Harvard2014_Bullock.pdf Self-Interacting Dark Matter]</ref> Finally, some researchers suggest that explaining the empirical success of Milgrom's law requires a more radical break with conventional assumptions about the nature of dark matter. One idea (dubbed "dipolar dark matter") is to make dark matter gravitationally [[Dipolar polarization|polarizable]] by ordinary matter and have this polarization enhance the gravitational attraction between baryons.<ref>{{Cite journal|arxiv = astro-ph/0605637|doi = 10.1088/0264-9381/24/14/001|title = Gravitational polarization and the phenomenology of MOND|year = 2007|last1 = Blanchet|first1 = Luc|journal = Classical and Quantum Gravity|volume = 24|issue = 14|pages = 3529–3539|bibcode = 2007CQGra..24.3529B|s2cid = 16832511}}</ref>

=== Outstanding problems for MOND ===

The most serious problem facing Milgrom's law is that it cannot eliminate the need for dark matter in all astrophysical systems: galaxy clusters show a residual mass discrepancy even when analyzed using MOND.<ref name = "tale of two paradigms"/> The fact that some form of unseen mass must exist in these systems detracts from the adequacy of MOND as a solution to the missing mass problem, although the amount of extra mass required is a fifth that of a Newtonian analysis, and there is no requirement that the missing mass be non-baryonic. It has been speculated that 2 eV neutrinos could account for the cluster observations in MOND while preserving the hypothesis's successes at the galaxy scale.<ref>{{Cite journal |last1=Angus |first1=Garry W. |last2=Shan |first2=Huan Yuan |last3=Zhao |first3=Hong Sheng |last4=Famaey |first4=Benoit |name-list-style=amp |date=2007 |title=On the Proof of Dark Matter, the Law of Gravity, and the Mass of Neutrinos |journal=The Astrophysical Journal Letters |volume=654 |issue=1 |pages=L13–L16 |doi=10.1086/510738 |arxiv=astro-ph/0609125 |bibcode=2007ApJ...654L..13A|s2cid=17977472 }}</ref><ref>{{Cite journal|author=R.H. Sanders|date=2007|title=Neutrinos as cluster dark matter|journal=[[Monthly Notices of the Royal Astronomical Society]]|arxiv=astro-ph/0703590 |bibcode = 2007MNRAS.380..331S |doi = 10.1111/j.1365-2966.2007.12073.x|volume=380|issue=1|pages=331–338 |s2cid=14237211}}</ref> Indeed, analysis of sharp lensing data for the galaxy cluster Abell 1689 shows that MOND only becomes distinctive at Mpc distance from the center, so that Zwicky's conundrum remains,<ref>{{Cite journal|title=How Zwicky already ruled out modified gravity theories without dark matter|first=Theodorus M.|last=Nieuwenhuizen |journal=Fortschritte der Physik|volume=65|issue=6–8|page=1600050 |doi=10.1002/prop.201600050|arxiv=1610.01543|date=2016 |s2cid=118676940}}</ref> and 1.8 eV neutrinos are needed in clusters.<ref>{{Cite journal|title=Dirac neutrino mass from a neutrino dark matter model for the galaxy cluster Abell 1689|first=Theodorus M.|last=Nieuwenhuizen |journal=Journal of Physics: Conference Series|volume=701|issue=1|page=012022(13pp)|date=2015 |doi=10.1088/1742-6596/701/1/012022|arxiv=1510.06958|bibcode=2016JPhCS.701a2022N |s2cid=3599969}}</ref>

The 2006 observation of a pair of colliding galaxy clusters known as the "[[Bullet Cluster]]",<ref>{{Cite journal |last1=Clowe |first1=Douglas |last2=Bradač |first2=Maruša |last3=Gonzalez |first3=Anthony H. |last4=Markevitch |first4=Maxim |last5=Randall |first5=Scott W. |last6=Jones |first6=Christine |last7=Zaritsky |first7=Dennis |name-list-style=amp |date=2006 |title=A Direct Empirical Proof of the Existence of Dark Matter |journal=The Astrophysical Journal Letters |volume=648 |issue=2 |pages=L109–L113 |doi=10.1086/508162 |arxiv=astro-ph/0608407 |bibcode=2006ApJ...648L.109C|s2cid=2897407 }}</ref> poses a significant challenge for all theories proposing a modified gravity solution to the missing mass problem, including MOND. Astronomers measured the distribution of stellar and gas mass in the clusters using [[visible light|visible]] and [[X-ray]] light, respectively, and in addition mapped the inferred dark matter density using gravitational lensing. In MOND, one would expect the "missing mass" to be centred on regions of visible mass which experience accelerations lower than a<sub>0</sub> (assuming the external field effect is negligible). In ΛCDM, on the other hand, one would expect the dark matter to be significantly offset from the visible mass because the halos of the two colliding clusters would pass through each other (assuming, as is conventional, that dark matter is collisionless), whilst the cluster gas would interact and end up at the centre. An offset is clearly seen in the observations. It has been suggested, however, that MOND-based models may be able to generate such an offset in strongly non-spherically symmetric systems, such as the Bullet Cluster.<ref>{{cite journal| author=G.W. Angus |author2=B. Famaey |author3=H. Zhao  |name-list-style=amp | title=Can MOND take a bullet? Analytical comparisons of three versions of MOND beyond spherical symmetry| journal=Mon. Not. R. Astron. Soc.| volume=371| issue=1| pages=138–146 |date=September 2006| doi= 10.1111/j.1365-2966.2006.10668.x| arxiv=astro-ph/0606216v1|bibcode = 2006MNRAS.371..138A |s2cid=15025801 }}</ref>

Some [[ultra diffuse galaxies]], such as [[NGC 1052-DF2]], appear to be free of dark matter. If this is indeed the case, it poses a problem for MOND because it cannot explain the rotation curves.{{efn|It is also a problem for standard cold dark matter, since it needs to demonstrate that it is capable of forming galaxies without dark matter.}} Efforts are under way to show that either MOND is capable of reproducing the rotation curves,<ref>{{Cite journal|last1=Haghi|first1=Hosein|last2=Kroupa|first2=Pavel|last3=Banik|first3=Indranil|last4=Wu|first4=Xufen|last5=Zonoozi|first5=Akram H.|last6=Javanmardi|first6=Behnam|last7=Ghari|first7=Amir|last8=Müller|first8=Oliver|last9=Dabringhausen|first9=Jörg|last10=Zhao|first10=Hongsheng|date=2019-08-14|title=A new formulation of the external field effect in MOND and numerical simulations of ultra-diffuse dwarf galaxies – application to NGC 1052-DF2 and NGC 1052-DF4|journal=Monthly Notices of the Royal Astronomical Society|volume=487|issue=2|pages=2441–2454|doi=10.1093/mnras/stz1465|arxiv=1906.03268|bibcode=|issn=0035-8711|doi-access=free}}</ref> or that the observations are incorrect.<ref>{{cite journal|title=The ultradiffuse galaxy AGC 114905 needs dark matter|author1=J. A. Sellwood|author2=R. H. Sanders|journal=Monthly Notices of the Royal Astronomical Society |date=June 2022|volume=514 |issue=3 |pages=4008–4017 |doi=10.1093/mnras/stac1604 |arxiv=2202.08678 }}</ref>

A significant piece of evidence in favor of standard dark matter is the observed anisotropies in the [[cosmic microwave background]].<ref>See [[Dark matter#Cosmic microwave background]].</ref> While ΛCDM is able to explain the observed angular power spectrum, MOND has a much harder time, although it is possible to construct relativistic generalizations of MOND that can fit the observations too.<ref name="RMOND"/> MOND also encounters difficulties explaining [[structure formation]], with density perturbations in MOND perhaps growing so rapidly that too much structure is formed by the present epoch.<ref>{{cite journal|last=McGaugh|first=Stacy |title=A tale of two paradigms: The mutual incommensurability of ΛCDM and MOND|journal=Canadian Journal of Physics|date=2015|volume=93|issue=2|pages=250–259 |doi=10.1139/cjp-2014-0203|arxiv=1404.7525v2|bibcode=2015CaJPh..93..250M |s2cid=51822163}}</ref> However, forming galaxies more rapidly than in ΛCDM can be a good thing to some extent.<ref name=Steinhardt>{{cite journal |title=The Impossibly Early Galaxy Problem |author=Charles L. Steinhardt |author2=Peter Capak |author3=Dan Masters |author4=Josh S. Speagle |name-list-style=amp |date=2016 |journal=The Astrophysical Journal |volume=824 |issue=1 |pages=21 |doi=10.3847/0004-637X/824/1/21 |bibcode=2016ApJ...824...21S|arxiv = 1506.01377 |s2cid=35183078 |doi-access=free }}</ref>

Several other studies have noted observational difficulties with MOND. For example, it has been claimed that MOND offers a poor fit to the velocity dispersion profile of [[globular clusters]] and the temperature profile of galaxy clusters,<ref name=Seife>{{cite book |title=Alpha and Omega |pages = [https://archive.org/details/alphaomegasearch0000seif/page/100 100]–101 |url=https://archive.org/details/alphaomegasearch0000seif |url-access=registration |quote=Modified Newtonian dynamics. |author=Charles Seife |isbn=0-14-200446-4 |publisher=Penguin Books |date=2004}}</ref><ref name=Aguirre>{{cite journal |title=Problems for Modified Newtonian Dynamics in Clusters and the Lyα Forest? |author=Anthony Aguirre |author2=Joop Schaye |author3=Eliot Quataert  |name-list-style=amp |date=2001 |journal=The Astrophysical Journal |volume=561 |issue=2 |pages=550–558 |doi=10.1086/323376 |bibcode=2001ApJ...561..550A|arxiv = astro-ph/0105184 |s2cid=119071058 }}</ref> that different values of a<sub>0</sub> are required for agreement with different galaxies' rotation curves,<ref>S. M. Kent, "Dark matter in spiral galaxies. II - Galaxies with H I rotation curves", 1987, AJ, 93, 816</ref> and that MOND is naturally unsuited to forming the basis of cosmology.<ref name = "fin du mond">{{Cite arXiv |last1=Scott |first1 = D. | last2 = White | first2 = M. | last3 = Cohn | first3 = J. D. | last4 = Pierpaoli | first4 = E. | date = 2001 | title = Cosmological Difficulties with Modified Newtonian Dynamics (or: La Fin du MOND?) | eprint = astro-ph/0104435}}</ref> Furthermore, many versions of MOND predict that the speed of light is different from the speed of gravity, but in [[GW170817|2017 the speed of gravitational waves was measured to be equal to the speed of light]] to high precision.<ref name="Boran"/> This is well understood in modern relativistic theories of MOND, with the constraint from gravitational waves actually helping by substantially restricting how a covariant theory might be constructed.<ref name=Skordis_2019>{{cite journal |title=New Relativistic Theory for Modified Newtonian Dynamics |author=Constantinos Skordis |author2=Tom Zlosnik |name-list-style=amp |date=2019 |journal=Physical Review D |volume=100 |issue=10 |pages=104013 |doi=10.1103/PhysRevD.100.104013 |bibcode=2019PhRvD.100j4013S|arxiv = 1905.09465 |s2cid=209924502}}</ref>

Besides these observational issues, MOND and its relativistic generalizations are plagued by theoretical difficulties.<ref name = "fin du mond"/><ref>{{Cite journal|arxiv = 0802.1215|doi = 10.1103/PhysRevD.78.044034|title = TeVeS gets caught on caustics|year = 2008|last1 = Contaldi|first1 = Carlo R.|last2 = Wiseman|first2 = Toby|last3 = Withers|first3 = Benjamin|journal = Physical Review D|volume = 78|issue = 4|page = 044034|bibcode = 2008PhRvD..78d4034C|s2cid = 119240967}}</ref> Several ad hoc and inelegant additions to general relativity are required to create a theory compatible with a non-Newtonian non-relativistic limit, though the predictions in this limit are rather clear. This is the case for the more commonly used modified gravity versions of MOND, but some formulations (most prominently those based on modified inertia) have long suffered from poor compatibility with cherished physical principles such as conservation laws. Researchers working on MOND generally do not interpret it as a modification of inertia, with only very limited work done on this area.

== Proposals for testing MOND ==

Several observational and experimental tests have been proposed to help distinguish<ref>{{cite journal|arxiv=0705.3408|title=Testing Gravity in the Outer Solar System: Results from Trans-Neptunian Objects|author=Wallin, John F.|author2=Dixon, David S.|author3=Page, Gary L.|date=23 May 2007|doi=10.1086/520528|volume=666|issue=2|journal=The Astrophysical Journal|pages=1296–1302|bibcode=2007ApJ...666.1296W|s2cid=18654075}}</ref> between MOND and dark matter-based models:

* The [[Dark matter#Detection of dark matter particles|detection]] of particles suitable for constituting cosmological dark matter would strongly suggest that ΛCDM is correct and no modification to Newton's laws is required.
* If MOND is taken as a theory of modified inertia, it predicts the existence of anomalous accelerations on the Earth at particular places and times of the year. These could be detected in a precision experiment. This prediction would not hold if MOND is taken as a theory of modified gravity, as the external field effect produced by the Earth would cancel MONDian effects at the Earth's surface.<ref>{{Cite journal|arxiv = 1408.3059|doi = 10.1139/cjp-2014-0164|title = Testing MOND on Earth|year = 2015|last1 = Ignatiev|first1 = A.Yu.|journal = Canadian Journal of Physics|volume = 93|issue = 2|pages = 166–168|bibcode = 2015CaJPh..93..166I| s2cid=119260352 }}</ref><ref>{{Cite journal|arxiv = 1002.2766|doi = 10.1051/0004-6361/200811520|title = Testing the Newton second law in the regime of small accelerations|year = 2009|last1 = De Lorenci|first1 = V. A.|last2 = Faúndez-Abans|first2 = M.|last3 = Pereira|first3 = J. P.|journal = Astronomy & Astrophysics|volume = 503|issue = 1|pages = L1–L4|bibcode = 2009A&A...503L...1D|s2cid = 53345722}}</ref>
* It has been suggested that MOND could be tested in the Solar System using the [[LISA Pathfinder]] mission (launched in 2015). In particular, it may be possible to detect the anomalous tidal stresses predicted by MOND to exist at the Earth-Sun saddlepoint of the Newtonian gravitational potential.<ref>{{Cite arXiv|eprint = 1001.1303|last1 = Trenkel|first1 = Christian|last2 = Kemble|first2 = Steve|last3 = Bevis|first3 = Neil|last4 = Magueijo|first4 = Joao|title = Testing MOND/TEVES with LISA Pathfinder|year = 2010|class = astro-ph.CO}}</ref> It may also be possible to measure MOND corrections to the [[perihelion precession]] of the planets in the Solar System,<ref>{{Cite arXiv|eprint = 1105.5815|last1 = Blanchet|first1 = Luc|last2 = Novak|first2 = Jerome|title = Testing MOND in the Solar System|year = 2011|class = astro-ph.CO}}</ref> or a purpose-built spacecraft.<ref>{{Cite journal|arxiv = gr-qc/0606063|doi = 10.1142/S0218271808012127|title = Apsis: An Artificial Planetary System in Space to Probe Extra-Dimensional Gravity and Mond|year = 2008|last1 = Sahni|first1 = Varun|last2 = Shtanov|first2 = Yuri|journal = International Journal of Modern Physics D|volume = 17|issue = 3n04|pages = 453–466|bibcode = 2008IJMPD..17..453S|s2cid = 6416355}}</ref>
* One potential astrophysical test of MOND is to investigate whether isolated galaxies behave differently from otherwise-identical galaxies that are under the influence of a strong external field. Another is to search for non-Newtonian behaviour in the motion of [[Binary star|binary star systems]] where the stars are sufficiently separated for their accelerations to be below a<sub>0</sub>.<ref>{{Cite journal|arxiv = 1105.1873|doi = 10.1140/epjc/s10052-012-1884-6|title = Wide binaries as a critical test of classical gravity|year = 2012|last1 = Hernandez|first1 = X.|last2 = Jiménez|first2 = M. A.|last3 = Allen|first3 = C.|journal = The European Physical Journal C|volume = 72|issue = 2|page = 1884|bibcode = 2012EPJC...72.1884H|s2cid = 119202534}}</ref>
* Testing MOND using the redshift-dependence of radial acceleration{{snd}} [[Sabine Hossenfelder]] and Tobias Mistele propose a parameter-free MOND model they call Covariant Emergent Gravity and suggest that as measurements of radial acceleration improve, various MOND models and particle dark matter might be distinguishable because MOND predicts a much smaller redshift-dependence.<ref>{{Cite journal|arxiv = 1803.08683|doi = 10.1142/S0218271818470107|title = The redshift-dependence of radial acceleration: Modified gravity versus particle dark matter|year = 2018|last1 = Hossenfelder|first1 = Sabine|last2 = Mistele|first2 = Tobias|journal = International Journal of Modern Physics D|volume = 27|issue = 14|bibcode = 2018IJMPD..2747010H|s2cid = 54663204}}</ref>

== See also ==
{{div col|colwidth=22em}}
* MOND researchers:
** {{annotated link|Mordehai Milgrom}}
** {{annotated link|Jacob Bekenstein}}
** {{annotated link|Stacy McGaugh}}
** {{annotated link|Pavel Kroupa}}
* {{annotated link|Alternatives to general relativity}}
** {{annotated link|Entropic gravity}}
* {{annotated link|AQUAL}}
* {{annotated link|Cold dark matter}}
* {{annotated link|Dark matter}}
* {{annotated link|Lambda-CDM model}}
* {{annotated link|Galaxy rotation curve}}
* {{annotated link|TeVeS}}
* {{annotated link|Tully–Fisher relation}}
{{div col end}}

==Notes==
{{reflist|group=lower-alpha}}

==References==
{{Reflist}}

==Further reading==

Technical:
*[[Merritt, David]] (2020). ''[[:openlibrary:works/OL27929169M/A Philosophical Approach to MOND|A Philosophical Approach to MOND: Assessing the Milgromian Research Program in Cosmology]]'' (Cambridge: [[Cambridge University Press]]), 282 pp. {{ISBN|9781108492690}} 
* {{cite journal |doi=10.4249/scholarpedia.31410|bibcode=2014SchpJ...931410M|title=The MOND paradigm of modified dynamics|year=2014|last1=Milgrom|first1=Mordehai|journal=Scholarpedia|volume=9|issue=6|pages=31410|doi-access=free}}
* {{cite journal |arxiv=1112.3960|doi=10.12942/lrr-2012-10|pmid=28163623|pmc=5255531|bibcode=2012LRR....15...10F|title=Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions|year=2012|last1=Famaey|first1=Benoît|last2=McGaugh|first2=Stacy S.|journal=Living Reviews in Relativity|volume=15|issue=1|pages=10}}
* {{cite journal |arxiv=1404.7525|doi=10.1139/cjp-2014-0203|bibcode=2015CaJPh..93..250M|title=A tale of two paradigms: The mutual incommensurability of ΛCDM and MOND|year=2015|last1=McGaugh|first1=Stacy S.|journal=Canadian Journal of Physics|volume=93|issue=2|pages=250–259|s2cid=51822163}}
* {{cite journal |arxiv=1404.7661|doi=10.1139/cjp-2014-0211|bibcode=2015CaJPh..93..107M|title=MOND theory|year=2015|last1=Milgrom|first1=Mordehai|journal=Canadian Journal of Physics|volume=93|issue=2|pages=107–118|s2cid=119183394}}
* {{cite journal |arxiv=1406.4860|doi=10.1139/cjp-2014-0179|bibcode=2015CaJPh..93..169K|title=Galaxies as simple dynamical systems: Observational data disfavor dark matter and stochastic star formation|year=2015|last1=Kroupa|first1=Pavel|journal=Canadian Journal of Physics|volume=93|issue=2|pages=169–202|s2cid=118479184}}
* {{cite journal |arxiv=1106.2476|doi=10.1016/j.physrep.2012.01.001|bibcode=2012PhR...513....1C|title=Modified gravity and cosmology|year=2012|last1=Clifton|first1=Timothy|last2=Ferreira|first2=Pedro G.|last3=Padilla|first3=Antonio|last4=Skordis|first4=Constantinos|journal=Physics Reports|volume=513|issue=1–3|pages=1–189|s2cid=119258154}}
* {{cite journal |arxiv=astro-ph/0505266|doi=10.1016/j.ppnp.2005.08.001|bibcode=2006PrPNP..56..340M|title=Alternatives to dark matter and dark energy|year=2006|last1=Mannheim|first1=P.|journal=Progress in Particle and Nuclear Physics|volume=56|issue=2|pages=340–445|s2cid=14024934}}

Popular:
* [https://aeon.co/essays/we-should-explore-alternatives-to-the-standard-model-of-cosmology A non-Standard model], David Merritt, Aeon Magazine, July 2021
* [https://arstechnica.com/science/2012/11/dark-matter-critics-focus-on-details-ignore-big-picture Dark matter critics focus on details, ignore big picture], Lee, 14 Nov 2012
* {{cite arXiv |eprint=0908.3842|last1=Milgrom|first1=Mordehai|title=MOND: Time for a change of mind?|class=astro-ph.CO|year=2009}}
* [http://www.world-science.net/othernews/070802_darkmatter.htm "Dark matter" doubters not silenced yet] {{Webarchive|url=https://web.archive.org/web/20160520115754/http://www.world-science.net/othernews/070802_darkmatter.htm |date=2016-05-20 }}, World Science, 2 Aug 2007
* [https://web.archive.org/web/20040919035917/https://www.astro.umd.edu/~ssm/mond/sad0802Milg6p.pdf Does Dark Matter Really Exist?], Milgrom, Scientific American, Aug 2002

==External links==
*{{Commons category-inline}}
{{wikiquote}}
* [http://astroweb.case.edu/ssm/mond/ The MOND pages], Stacy McGaugh
* [http://www.weizmann.ac.il/particle/milgrom/ Mordehai Milgrom's website]
* [http://www.scilogs.com/the-dark-matter-crisis/ "The Dark Matter Crisis" blog], Pavel Kroupa, Marcel Pawlowski
* [http://www.astro.uni-bonn.de/~pavel/kroupa_cosmology.html Pavel Kroupa's website]
* {{cite web
 | url=https://aeon.co/essays/is-dark-matter-subatomic-particles-a-superfluid-or-both
 | title=The superfluid Universe
 | last=Hossenfelder
 | first=Sabine
 | date=1 Feb 2016
 | access-date=2 Feb 2016 }} Superfluid dark matter may provide a more natural way to arrive at the MOND equation.

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