Modified Newtonian dynamics (MOND) is a

hypothesis A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method requires that one can test it. Scientists generally base scientific hypotheses on previous obse ...

that proposes a modification of Newton's law of universal gravitation to account for observed properties of

galaxies A galaxy is a system of stars, stellar remnants, interstellar gas, dust, dark matter, bound together by gravity. The word is derived from the Greek ' (), literally 'milky', a reference to the Milky Way galaxy that contains the Solar System. ...

. It is an alternative to the hypothesis of

dark matter Dark matter is a hypothetical form of matter thought to account for approximately 85% of the matter in the universe. Dark matter is called "dark" because it does not appear to interact with the electromagnetic field, which means it does not ab ...

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 Mordehai "Moti" Milgrom is an Israeli physicist and professor in the department of Particle Physics and Astrophysics at the Weizmann Institute in Rehovot, Israel. Biography He received his B.Sc. degree from the Hebrew University of Jerusalem i ...

,. . . 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 In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...

(as opposed to the centripetal acceleration itself, as in

Newton's second law Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...

) or alternatively, if gravitational force came to vary inversely ''linearly'' with radius (as opposed to the inverse square of the radius, as in

Newton's law of gravity Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distan ...

). In MOND, violation of Newton's laws occurs at extremely small accelerations, characteristic of galaxies yet far below anything typically encountered in the

Solar System The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar S ...

or on Earth. MOND is an example of a class of theories known as

modified gravity Modified may refer to: * ''Modified'' (album), the second full-length album by Save Ferris *Modified racing, or "Modifieds", an American automobile racing genre See also * Modification (disambiguation) * Modifier (disambiguation) Modifier may re ...

, and is an alternative to the hypothesis that the dynamics of galaxies are determined by massive, invisible

dark matter halo According to modern models of physical cosmology, a dark matter halo is a basic unit of cosmological structure. It is a hypothetical region that has decoupled from cosmic expansion and contains gravitationally bound matter. A single dark matte ...

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

. However, MOND and its generalizations do not adequately account for observed properties of

galaxy clusters A galaxy cluster, or a cluster of galaxies, is a structure that consists of anywhere from hundreds to thousands of galaxies that are bound together by gravity, with typical masses ranging from 1014 to 1015 solar masses. They are the second-la ...

, and no satisfactory

cosmological model Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of f ...

has been constructed from the hypothesis. The accurate measurement of the speed of gravitational waves compared to the speed of light in 2017 ruled out many hypotheses which used modified gravity. However, both Milgrom's bi-metric formulation of MOND and nonlocal MOND are compatible with this measurement.

Overview

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 Fritz Zwicky (; ; February 14, 1898 – February 8, 1974) was a Swiss astronomer. He worked most of his life at the California Institute of Technology in the United States of America, where he made many important contributions in theoretical and ...

in 1933 (who studied the

Coma cluster The Coma Cluster (Abell 1656) is a large galaxy cluster, cluster of galaxies that contains over 1,000 identified galaxies. Along with the Leo Cluster (Abell 1367), it is one of the two major clusters comprising the Coma Supercluster. It is loc ...

), and subsequently extended to include

spiral galaxies Spiral galaxies form a class of galaxy originally described by Edwin Hubble in his 1936 work ''The Realm of the Nebulae''Horace Babcock on Andromeda. 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 Vera Florence Cooper Rubin (; July 23, 1928 – December 25, 2016) was an American astronomer who pioneered work on galaxy rotation rates. She uncovered the discrepancy between the predicted and observed angular motion of galaxies by studyi ...

at the 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 – the rotation curves are said to be "flat". This observation necessitates at least one of the following: : Option (1) leads to the dark matter hypothesis; option (2) leads to MOND. 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. 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: Here is the Newtonian force, is the object's (gravitational)

mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...

, is its acceleration, () is an as-yet unspecified function (called the ''interpolating function''), and is a new fundamental constant which marks the transition between the Newtonian and deep-MOND regimes. Agreement with Newtonian mechanics requires :

\begin \mu(x) \longrightarrow 1 && \text x \gg 1 \end ~,

and consistency with astronomical observations requires :

\begin \mu(x) \longrightarrow x && \text x \ll 1 \end ~.

Beyond these limits, the interpolating function is not specified by the hypothesis, although it is possible to weakly constrain it empirically. Two common choices are the "simple interpolating function": :

\mu\left( \frac \right) = \frac ~,

and the "standard interpolating function": :

\mu\left( \frac \right) = \sqrt ~.

Thus, in the deep-MOND regime ( ≪ ): :

F_\text = m \frac ~.

Applying this to an object of mass in

circular orbit A circular orbit is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle. Listed below is a circular orbit in astrodynamics or celestial mechanics under standard assumptions. Here the centripetal force is ...

around a point mass (a crude approximation for a star in the outer regions of a galaxy), we find: that is, the star's rotation velocity is independent of , its distance from the centre of the galaxy – the rotation curve is flat, as required. By fitting his law to rotation curve data, Milgrom found

a_0 \approx 1.2 \times 10^ \mathrm^

to be optimal. This simple law is sufficient to make predictions for a broad range of galactic phenomena. Milgrom's law can be interpreted in two different ways: * One possibility is to treat it as a modification to

, so that the force on an object is not proportional to the particle's acceleration but rather to

\mu\left( \frac \right) a.

In this case, the modified dynamics would apply not only to gravitational phenomena, but also those generated by other forces, for example

electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...

. * Alternatively, Milgrom's law can be viewed as leaving

Newton's Second Law Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...

intact and instead modifying the inverse-square law of gravity, so that the true gravitational force on an object of mass due to another of mass is roughly of the form

\frac.

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 Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...

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 A gravitational lens is a distribution of matter (such as a galaxy cluster, cluster of galaxies) between a distant light source and an observer that is capable of bending the light from the source as the light travels toward the observer. This ...

) and

cosmology Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount (lexicographer), Thomas Blount's ''Glossographia'', and in 1731 taken up in ...

. Distinguishing both theoretically and observationally between these alternatives is a subject of current research. The majority of astronomers,

astrophysicists The following is a list of astronomers, astrophysicists and other notable people who have made contributions to the field of astronomy. They may have won major prizes or awards, developed or invented widely used techniques or technologies within a ...

, 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. MOND, by contrast, is actively studied by only a handful of researchers. The primary difference between supporters of

ΛCDM The ΛCDM (Lambda cold dark matter) or Lambda-CDM model is a parameterization of the Big Bang cosmological model in which the universe contains three major components: first, a cosmological constant denoted by Lambda (Greek Λ) associated with d ...

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 The study of galaxy formation and evolution is concerned with the processes that formed a heterogeneous universe from a homogeneous beginning, the formation of the first galaxies, the way galaxies change over time, and the processes that have ge ...

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. 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 In astronomy, the Tully–Fisher relation (TFR) is an empirical relationship between the mass or intrinsic luminosity of a spiral galaxy and its asymptotic rotation velocity or emission line width. It was first published in 1977 by astronomer ...

(BTFR), and is found to conform quite closely to the MOND prediction. * 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". * 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. 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''₀, 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. * Both MOND and dark matter halos stabilize disk galaxies, helping them retain their rotation-supported structure and preventing their transformation into elliptical galaxies. In MOND, this added stability is only available for regions of galaxies within the deep-MOND regime (i.e., with ''a'' < ''a''₀), suggesting that spirals with ''a'' > ''a''₀ in their central regions should be prone to instabilities and hence less likely to survive to the present day. This may explain the " Freeman limit" to the observed central surface mass density of spiral galaxies, which is roughly ''a''₀/''G''. This scale must be put in by hand in dark matter-based galaxy formation models. * Particularly massive galaxies are within the Newtonian regime (''a'' > ''a''₀) 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. * In MOND, all gravitationally bound objects with ''a'' < ''a''₀ – 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 (" 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. * Recent work has shown that many of the dwarf galaxies around the

Milky Way The Milky Way is the galaxy that includes our Solar System, with the name describing the galaxy's appearance from Earth: a hazy band of light seen in the night sky formed from stars that cannot be individually distinguished by the naked eye ...

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 further 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. * 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 the theory of general relativity, the equivalence principle is the equivalence of gravitational and inertial mass, and Albert Einstein's observation that the gravitational "force" as experienced locally while standing on a massive body (suc ...

in weak gravitational fields in the vicinity of rotationally supported galaxies. They observed an effect consistent with the external field effect of Modified Newtonian dynamics and inconsistent with tidal effects in the

Lambda-CDM model The ΛCDM (Lambda cold dark matter) or Lambda-CDM model is a parameterization of the Big Bang cosmological model in which the universe contains three major components: first, a cosmological constant denoted by Lambda (Greek Λ) associated with d ...

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.' * 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.

Complete MOND hypotheses

Milgrom's law requires incorporation into a complete hypothesis if it is to satisfy

conservation laws In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, co ...

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 AQUAL is a theory of gravity based on Modified Newtonian Dynamics (MOND), but using a Lagrangian. It was developed by Jacob Bekenstein and Mordehai Milgrom in their 1984 paper, "Does the missing mass problem signal the breakdown of Newtonian grav ...

) was constructed in 1984 by Milgrom and

Jacob Bekenstein Jacob David Bekenstein ( he, יעקב בקנשטיין; May 1, 1947 – August 16, 2015) was an American and Israeli theoretical physicist who made fundamental contributions to the foundation of black hole thermodynamics and to other aspects of ...

. AQUAL generates MONDian behavior by modifying the gravitational term in the classical

Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...

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: :

\mathcal_\text &= - \frac \cdot a_0^2 F \left (\tfrac \right ) \end

where

\phi

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 Newton–Poisson equation: :

= 4\pi G \rho

This can be solved given suitable boundary conditions and choice of F to yield Milgrom's law (up to a

curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was fi ...

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_N. The Lagrangian may be constructed so that a_N 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, 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. 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 Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional sy ...

depending on the trajectory of the particle. Such "modified inertia" theories, however, are difficult to use because they are time-nonlocal, require

energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...

and

momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...

to be non-trivially redefined to be conserved, and have predictions that depend on the entirety of a particle's orbit.

Relativistic

In 2004, Jacob Bekenstein formulated TeVeS, the first complete relativistic hypothesis using MONDian behaviour. 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 In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...

, a free function and a non-Einsteinian

metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathema ...

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, but faces problems when confronted with data on the anisotropy of the

cosmic microwave background In Big Bang cosmology the cosmic microwave background (CMB, CMBR) is electromagnetic radiation that is a remnant from an early stage of the universe, also known as "relic radiation". The CMB is faint cosmic background radiation filling all spac ...

, the lifetime of compact objects, and the relationship between the lensing and matter overdensity potentials. Several alternative relativistic generalizations of MOND exist, including BIMOND and generalized Einstein-Aether theories. There is also a relativistic generalization of MOND that assumes a Lorentz-type invariance as the physical basis of MOND phenomenology.

The 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 mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...

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), for which there exists observational evidence. The external field effect is best described by classifying physical systems according to their relative values of ''a''_in (the characteristic acceleration of one object within a subsystem due to the influence of another), ''a''_ex (the acceleration of the entire subsystem due to forces exerted by objects outside of it), and ''a''₀: *

a_ > a_0

: Newtonian regime *

a_ < a_ < a_0

: Deep-MOND regime *

a_ < a_0 < a_

: The external field is dominant and the behavior of the system is Newtonian. *

a_ < a_ < a_0

: 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''₀/''a''_ex. The external field effect implies a fundamental break with the

(but not necessarily the

weak equivalence principle In the theory of general relativity, the equivalence principle is the equivalence of gravitational and inertial mass, and Albert Einstein's observation that the gravitational "force" as experienced locally while standing on a massive body (suc ...

). The effect was postulated by Milgrom in the first of his 1983 papers to explain why some

open clusters An open cluster is a type of star cluster made of up to a few thousand stars that were formed from the same giant molecular cloud and have roughly the same age. More than 1,100 open clusters have been discovered within the Milky Way galaxy, and ...

were observed to have no mass discrepancy even though their internal accelerations were below a₀. 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 The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. Acc ...

) 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:

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.

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''₀ (determined by fits to internal properties of galaxies) is within an order of magnitude of ''cH''₀, where ''c'' is the

speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...

and ''H''₀ is the

Hubble constant Hubble's law, also known as the Hubble–Lemaître law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther they are, the faster they are moving ...

(a measure of the present-day expansion rate of the universe). It is also close to the acceleration rate of the universe, and hence the cosmological constant. However, as yet no full hypothesis has been constructed which manifests these connections in a natural way.

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, although Milgrom has argued that such arguments explain only a small subset of MOND

phenomena A phenomenon ( : phenomena) is an observable event. The term came into its modern philosophical usage through Immanuel Kant, who contrasted it with the noumenon, which ''cannot'' be directly observed. Kant was heavily influenced by Gottfried W ...

. An alternative proposal is to modify the properties of dark matter (e.g., to make it 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. 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 polarizable by ordinary matter and have this polarization enhance the gravitational attraction between baryons.

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. 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. 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, and 1.8 eV neutrinos are needed in clusters. The 2006 observation of a pair of colliding galaxy clusters known as the " Bullet Cluster", 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 Visibility, in meteorology, is a measure of the distance at which an object or light can be seen. Visibility may also refer to: * A measure of turbidity in water quality control * Interferometric visibility, which quantifies interference contrast ...

and

X-ray An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10 picometers to 10 nanometers, corresponding to frequencies in the range 30&nb ...

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 a₀ (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. A significant piece of evidence in favor of standard dark matter is the observed anisotropies in the

. While ΛCDM is able to explain the observed angular power spectrum, MOND has a much harder time, though recently it has been shown that MOND can fit the observations too. 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. However, forming galaxies more rapidly than in ΛCDM can be a good thing to some extent. 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 A globular cluster is a spheroidal conglomeration of stars. Globular clusters are bound together by gravity, with a higher concentration of stars towards their centers. They can contain anywhere from tens of thousands to many millions of member ...

and the temperature profile of galaxy clusters, that different values of a₀ are required for agreement with different galaxies' rotation curves, and that MOND is naturally unsuited to forming the basis of cosmology. Furthermore, many versions of MOND predict that the speed of light is different to the speed of gravity, but in 2017 the speed of gravitational waves was measured to be equal to the speed of light to high precision. 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. Besides these observational issues, MOND and its relativistic generalizations are plagued by theoretical difficulties. 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 between MOND and dark matter-based models: * The 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. * It has been suggested that MOND could be tested in the Solar System using the

LISA Pathfinder LISA Pathfinder, formerly Small Missions for Advanced Research in Technology-2 (SMART-2), was an ESA spacecraft that was launched on 3 December 2015 on board Vega flight VV06. The mission tested technologies needed for the Laser Interferometer S ...

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. It may also be possible to measure MOND corrections to the

perihelion precession In celestial mechanics, apsidal precession (or apsidal advance) is the precession (gradual rotation) of the line connecting the apsides (line of apsides) of an astronomical body's orbit. The apsides are the orbital points closest (periapsi ...

of the planets in the Solar System, or a purpose-built spacecraft. * 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 systems where the stars are sufficiently separated for their accelerations to be below a₀. * Testing MOND using the redshift-dependence of radial acceleration

Sabine Hossenfelder Sabine Hossenfelder (born 1976) is a German theoretical physicist, author, musician and YouTuber. She is currently employed as a research fellow at the Frankfurt Institute for Advanced Studies. She is the author of ''Lost in Math: How Beauty Le ...

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.

References

External links

*
The MOND pages
Stacy McGaugh
Mordehai Milgrom's website

"The Dark Matter Crisis" blog
Pavel Kroupa, Marcel Pawlowski

* Superfluid dark matter may provide a more natural way to arrive at the MOND equation. {{Authority control Astrophysics Classical mechanics Theories of gravity Unsolved problems in astronomy Unsolved problems in physics Astronomical hypotheses Dark matter Physical cosmology Celestial mechanics Physics beyond the Standard Model

Matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic partic ...

Theoretical physics