MOND and modified gravity

Milgrom, Mordehai.  “A Modification of the Newtonian Dynamics as a Possible Alternative to the Hidden Mass Hypothesis.”  ApJ 270:365–370.  1983.  A 1983 article, marked Paper I, that proposes changing Newtonian dynamics at very low accelerations as an alternative to hidden mass; it was received 1982 February 4 and accepted 1982 December 28.  It gives the modified equation and the constant a0, which the author compares with cH0, a quantity from cosmology.  The article uses it for the equation, for the value of a0, and for the author's reading of their coincidence through Mach's principle.

1. 1a.

   Pages 365–366:  “With some simplifying assumptions I was led to the form  m_gμ(a/a₀)a = F,  (1)  μ(x ≫ 1) ≈ 1,  μ(x ≪ 1) ≈ x, replacing m_ga = F.”  Page 366:  “it is thus possible that the dynamics need be modified only when gravitational forces are involved, in which case we will consider the modification one of gravity and not of the law of inertia.” ↩ ↩

2. 1b.

   Page 366:  “I determine, in Paper II, the acceleration constant, in a few independent ways, and find a₀ ≈ 2 × 10⁻⁸ cm s⁻² which turns out to be of the same order as cH₀ = 5 × 10⁻⁸ (H₀/50 km s⁻¹ Mpc⁻¹) cm s⁻².” ↩

3. 1c.

   Page 370:  “It is best to wait until we have a relativistic theory for the modified dynamics before its cosmological implications are discussed.  In view of the fact that a₀ ∼ cH₀, the universe is the only massive system which is both relativistic and involves accelerations not much larger than a₀.” ↩

4. 1d.

   Page 370:  “The possible connection between such numerical 'coincidences,' which relate parameters of 'local' physics to cosmological parameters and the Mach principle, is quite obvious.  If, for example, the inertia force is due to the interaction of the accelerated particle with an inertia field produced by totality of mass in the universe (see, for example, Sciama 1961) such that the inertia force is not proportional to the accelerations any more, the introduction of an acceleration constant into the local equations of dynamics is implied and cH₀ (perhaps up to a function of q₀) is the natural cosmological acceleration parameter which can play this role.”  The reference is D. W. Sciama, The Unity of the Universe (New York: Doubleday, 1961), p. 115. ↩

5. 1e.

   Page 365:  “The hidden mass hypothesis (HMH) explains the dynamics in galaxies and systems of galaxies by assuming that much of the mass in these systems is in, as yet, unobserved form… This hypothesis has not yet encountered any fatal objection.  However, in order to explain the observations in the framework of this idea, one finds it necessary to make a large number of ad hoc assumptions concerning the nature of the hidden mass and its distribution in space.” ↩

Cirelli, Marco et al.  Dark Matter.  Oct 31, 2025.  A 524-page review of dark matter.  Its chapter 11, "Alternatives to Dark Matter," covers MOND, including the 1984 Bekenstein and Milgrom version and a relativistic model by Skordis and Złośnik.  The article uses it for two points: that the bare MOND formula is not a consistent theory, and that the scalar field in a relativistic MOND model can play the role of dark matter in cosmology.

1. 2a.

   Page 400:  “note that the phenomenological MOND postulate, eq. (11.1), is not a consistent theory.  Such an empirical law, for instance, violates momentum conservation:  the forces between two different masses are not equal and opposite, given that the lighter mass is subject to a bigger acceleration, and therefore to a bigger MOND effect.” ↩

2. 2b.

   Page 404:  On Skordis and Złośnik's theory.  “The critical acceleration a_∗ depends on the scalar vacuum expectation values, and a_∗ ∼ H₀ is not a generic outcome.” ↩ ↩

3. 2c.

   Page 402:  On the Bekenstein-Milgrom 1984 Lagrangian.  “The above theory extends the MOND postulate beyond spherical or cylindrical symmetry, and respects momentum conservation, since its action is invariant under translations.  However, it is neither Galilean nor Lorentz invariant:  gravity is corrected in a way that is vaguely analogous to how a dielectric modifies equations of electromagnetism, so that the physics depends on the absolute acceleration with respect to the special 'rest' frame.  This gives the 'external field effect'… In contrast, Newtonian and Einstein gravity give rise to tidal forces only, related to spatial gradients of g_ext, and thus respect the strong equivalence principle.” ↩ ↩

4. 2d.

   Page 404:  “The cosmological time evolution of θ allows the scalar θ to play the role of DM in cosmology.” ↩

5. 2e.

   Page 404:  “The above set-up does not solve all the observational problems;  open issues remain.  For instance, it is not clear how the theory can explain the behaviour of the bullet cluster and of ordinary galaxy clusters… The CMB polarization spectra might be affected by U_µ.  Furthermore, the action for U_µ seems not to be gauge invariant, endangering the quantum consistency of the theory.” ↩

6. 2f.

   Page 401:  “The present situation is confusing;  while some authors that analysed the Gaia data claim that wide binaries agree with the Newtonian dynamics, and thus exclude MONDian effects with up to 19σ significance (see Banik et al. (2023)), others instead claim a discovery of MONDian anomalies with a significance of up to 10σ, and thus an exclusion of Newtonian dynamics (see Chae (2023)).  The main issue is that some wide binaries involve a third unseen object, that produces MONDian-like deviations from Newtonian gravity.” ↩

7. 2g.

   Page 401:  “In the solar system, the solar gravitational acceleration is about 10⁴ a_∗ around Pluto's orbit, and becomes smaller than a_∗ at distance r ≈ 0.1 lyr from the Sun.  Vokrouhlicky et al. (2024) claim that simulations of comets and planetesimals around the Oort cloud at distances 10⁻³ − 1 light years from the Sun prefer Newtonian dynamics over its MOND modification at a ≲ a_∗.  Full MOND theories where this modification is screened by the EFE are not disfavoured.  However, Desmond et al. (2024) claim that such MONDian theories predict a quadrupolar correction to the Sun gravitational potential, which was not observed in the precise Cassini measurements of Saturn's orbit.” ↩

McGaugh, Stacy S. et al.  “The Radial Acceleration Relation in Rotationally Supported Galaxies.”  Physical Review Letters 117:201101.  2016.  A 2016 letter that reports a tight relation, across 153 rotating galaxies, between the acceleration traced by their rotation and the acceleration predicted from the observed ordinary matter alone.  Its 2693 data points follow one relation with small scatter, which the authors say is largely caused by measurement uncertainty, and they call the relation a natural law for rotating galaxies.  The article uses it as the source of the best modern measurement of MOND's acceleration constant, and for its point that the scatter is about the size of the measurement errors.  Mirror:  arXiv

1. 3a.

   Section Abstract:  “We report a correlation between the radial acceleration traced by rotation curves and that predicted by the observed distribution of baryons.  The same relation is followed by 2693 points in 153 galaxies with very different morphologies, masses, sizes, and gas fractions.  The correlation persists even when dark matter dominates.  Consequently, the dark matter contribution is fully specified by that of the baryons.  The observed scatter is small and largely dominated by observational uncertainties.  This radial acceleration relation is tantamount to a natural law for rotating galaxies.” ↩

2. 3b.

   Page 4:  Preprint page numbers.  “Residuals from the fit are well described by a Gaussian of width 0.11 dex (Fig. 3).  The rms scatter is 0.13 dex owing to the inevitable outliers.  These are tiny numbers by the standards of extragalactic astronomy.”  The error budget table gives a total expected observational scatter of 0.12 dex (0.05 inclination, 0.08 distance, 0.06 mass-to-light, 0.01 HI calibration). ↩

3. 3c.

   Page 3:  Preprint page numbers.  “For our adopted Υ⋆, we find g^† = 1.20 ± 0.02 (random) ± 0.24 (systematic) × 10⁻¹⁰ m s⁻².” ↩

McGaugh, Stacy S.  “A Novel Test of the Modified Newtonian Dynamics with Gas Rich Galaxies.”  Physical Review Letters 106:121303.  2011.  A 2011 letter that compares MOND with 47 galaxies dominated by gas, and reports that the data fall where MOND predicts with zero free parameters.  The article uses it for the author's claim that the relation the data follow is unique to MOND, and for the author's own admission that galaxy clusters are a problem for the theory.  Mirror:  arXiv

1. 4a.

   Page 1:  Preprint page numbers.  “the centripetal acceleration required to keep a star in orbit in a galaxy is very much lower than that experienced by the planets orbiting the sun:  ∼ 10⁻¹⁰ m s⁻² vs. 6 × 10⁻³ m s⁻² for the Earth.” ↩

2. 4b.

   Page 2:  Preprint page numbers.  On the sample of 47 gas-dominated galaxies.  “The data fall precisely where MOND predicts.  This happens with no fitting whatsoever — there are zero free parameters in Fig. 2.  Computing χ² with the slope fixed to 4 and the normalization fixed at the previously determined value of a₀ gives χ² = 44.3 for 46 degrees of freedom for a reduced χ²_ν = 0.96.  If we treat a₀ as a fit parameter and minimize χ² with the slope fixed to 4, we find a₀ = 1.24 ± 0.14 × 10⁻¹⁰ m s⁻².  This is indistinguishable from the previous value, and χ² actually increases because we have added an unneeded degree of freedom:  χ²_ν = 0.99.  If we further treat the slope as an additional free parameter, we find 3.8 ± 0.2.  This does not differ significantly from the MOND prediction of 4, nor does it improve the fit:  χ²_ν = 0.98.” ↩ ↩ ↩

3. 4c.

   Page 2:  Preprint page numbers.  BTFR means the baryonic Tully–Fisher relation, which links a galaxy's ordinary-matter mass to how fast it rotates.  “The specific BTFR that the data follow is unique to MOND.  Indeed, to the best of my knowledge, MOND is the only theory to make a strong a priori prediction for the BTFR.  The dark matter paradigm makes no comparably iron-clad prediction.” ↩

4. 4d.

   Page 3:  Preprint page numbers.  “The most serious observational problem facing MOND is the dynamics of rich clusters of galaxies.  These appear to weigh more than can be accounted for with the observed baryons even with the modified dynamics.  This residual mass discrepancy is roughly a factor of two in mass.  On the one hand, this is very disturbing — a theory that seeks to eliminate the need for cosmic dark matter itself suffers a missing mass problem.”  The same page continues:  “While some of the mass in clusters appears to be dark, even in MOND, there is nothing that requires this unseen mass to be in some new form of non-baryonic particle.  Indeed, big bang nucleosynthesis implies the existence of considerably more baryons than have so far been detected.  If only a fraction of these missing baryons reside in clusters it would suffice to resolve the residual mass discrepancy suffered by MOND.” ↩ ↩ ↩

5. 4e.

   Page 3:  Preprint page numbers.  “A simple model motivated by MOND provided the only successful a priori prediction of the first-to-second peak amplitude ratio of the acoustic peaks of the cosmic background radiation:  A_1:2 = 2.4 predicted vs. 2.34 ± 0.09 measured… MOND also has its share of problems.  The same ansatz that correctly predicted the second acoustic peak amplitude also predicts a lower third peak than is observed.  This does not falsify MOND, but it does imply that a generally covariant parent theory should provide an effective forcing term.” ↩

Famaey, Benoît, and Stacy S. McGaugh.  “Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions.”  Living Reviews in Relativity 15:10.  2012.  A review article that surveys the observed evidence for MOND and for its relativistic versions.  It gives MOND's acceleration scale a0 as about 10 to the minus 10 metres per second per second, and notes that a0 is numerically close to cH0/2π (c is the speed of light and H0 is the expansion rate of the universe) and that a0 squared is of the same order as the cosmological constant Λ.  The article uses it for the authors' own verdict that whether these coincidences mean anything is an open question.

1. 5a.

   Section §4.1, page 12:  Page of the preprint.  “The value of this scale is a₀ ≃ 10⁻¹⁰ m s⁻², which yields in natural units, a₀ ∼ H₀ (or, more precisely, a₀ ≈ cH₀/2π).  It is perhaps even more meaningful to note that, in these same units:  a₀² ∼ Λ, where Λ is the currently favored value of the cosmological constant.  Whether these numerical coincidences are physically relevant or just true (insignificant) coincidences remains an open question, closely related to the nature of the dark sector… But, at this stage, it is in any case striking that the dark matter and dark energy sectors do have such a common scale.” ↩ ↩ ↩

Milgrom, Mordehai.  “A Modification of the Newtonian Dynamics: Implications for Galaxies.”  ApJ 270:371–383.  1983.  The second of Milgrom's three 1983 papers on modified Newtonian dynamics (MOND).  Its abstract lists the predictions the theory makes for galaxy rotation curves and gives its acceleration constant, a₀.  The article quotes those predictions as the tests the theory set for itself.

1. 6a.

   Section Abstract, page 371:  “The main predictions are:  1.  Rotation curves calculated on the basis of the observed mass distribution and the modified dynamics should agree with the observed velocity curves.  2.  The V_∞⁴ = a₀GM relation should hold exactly.  3.  An analog of the Oort discrepancy should exist in all galaxies and become more severe with increasing r in a predictable way.”  Result 5 of the same abstract:  “The value of the acceleration constant, a₀, determined in a few independent ways is approximately 2 × 10⁻⁸ (H₀/50 km s⁻¹ Mpc⁻¹)² cm s⁻², which is of the order of cH₀ = 5 × 10⁻⁸ (H₀/50 km s⁻¹ Mpc⁻¹) cm s⁻².” ↩

Milgrom, Mordehai.  “A Modification of the Newtonian Dynamics: Implications for Galaxy Systems.”  ApJ 270:384–389.  1983.  The third of Milgrom's three 1983 papers on modified Newtonian dynamics (MOND).  It re-derives the mass-to-light ratio (the mass a system has for each unit of light it gives off) of binary galaxies, small groups, clusters and the Virgo Supercluster from existing data, and finds the values are a few solar units.  The article uses it as the paper where the theory's later trouble with clusters was already visible.

1. 7a.

   Section Abstract, page 384:  “Masses and mass-to-light ratios are rederived, on the basis of existing data, for binary galaxies, small groups, clusters of galaxies, and the Virgo Supercluster.  For each type of galaxy system, the average M/L values come out to be a few solar units.  These results eliminate the need to assume large amounts of hidden mass in galaxy systems, if the modified dynamics applies.” ↩

2. 7b.

   Section §IV, page 387, table 3:  Table 3 is titled "Masses and M/L_V Values for Galaxy Clusters" and lists Coma with cz = 6821 km s⁻¹, σ = 975 km s⁻¹, M = 279 × 10¹² M⊙, two luminosities (9.4 and 30 × 10¹² L_V⊙) and correspondingly M/L_V = 30 or 9.3 in solar units.  The conventional values Milgrom is arguing against are given in the same section:  “With large uncertainties and scatter, the M/L_B values are typically between 100 and 300.” ↩

Sanders, Robert H., and Stacy S. McGaugh.  “Modified Newtonian Dynamics as an Alternative to Dark Matter.”  Annual Review of Astronomy and Astrophysics 40:263–317.  2002.  A review article on modified Newtonian dynamics (MOND) as an alternative to dark matter.  It tests how well the theory predicts galaxy rotation curves with one free parameter, the mass-to-light ratio of the visible disk, and it reports the galaxies where the fit fails.  The article uses it as the careful statement of the fits and of the failures.  Mirror:  arXiv

1. 8a.

   Page 12:  Preprint page, on the galaxy NGC 2841.  “At a distance of 15.6 Mpc, the MOND rotation curve of the galaxy still systematically deviates from the observed curve, and the implied M/L is 8.5;  thus, this galaxy remains the most difficult case for MOND.” ↩

2. 8b.

   Page 13:  Preprint page.  “Overall, the ability of MOND, as ad hoc algorithm, to predict galaxy rotation curves with only one free parameter, the M/L of the visible disk, is striking.  This implies that galaxy rotation curves are entirely determined by the distribution of observable matter.  Regardless of whether or not MOND is correct as a theory, it does constitute an observed phenomenology that demands explanation.” ↩ ↩

Lelli, Federico et al.  “The Small Scatter of the Baryonic Tully-Fisher Relation.”  ApJ Letters 816:L14.  2016.  A 2016 study of 118 disc galaxies, spiral and irregular, measured with extended H I (neutral hydrogen) rotation curves and Spitzer infrared photometry at 3.6 µm, which traces stellar mass.  It tests the baryonic Tully-Fisher relation, which links a galaxy's ordinary (baryonic) mass to the speed of its outer rotation, and finds that the relation's intrinsic scatter is about 0.1 dex, a logarithmic unit, below what ΛCDM predicts, which its authors call a fundamental issue for ΛCDM cosmology.  The article uses it as the sharper test of the relation, because the scatter it measures is smaller than ΛCDM expects.

1. 9a.

   Section Abstract:  “We study the BTFR using a sample of 118 disc galaxies (spirals and irregulars) with data of the highest quality:  extended H I rotation curves (tracing the outer velocity) and Spitzer photometry at 3.6 µm (tracing the stellar mass)… For any reasonable value of Υ_∗, the intrinsic scatter is ∼0.1 dex, below general ΛCDM expectations.  The residuals show no correlations with galaxy structural parameters (radius or surface brightness), contrary to the predictions from some semi-analytic models of galaxy formation.  These are fundamental issues for ΛCDM cosmology.” ↩ ↩

2. 9b.

   Section §1:  “Using a semi-analytic galaxy-formation model, Dutton (2012) predicts a minimum intrinsic scatter of ∼0.15 dex along the BTFR… The majority of this scatter comes from the mass-concentration relation of dark matter (DM) halos, which is largely independent of baryonic processes and well-constrained by cosmological DM-only simulations.” ↩

Bullock, James S., and Michael Boylan-Kolchin.  “Small-Scale Challenges to the ΛCDM Paradigm.”  Annual Review of Astronomy and Astrophysics 55:343–387.  2017.  A review of the small-scale problems facing ΛCDM, the standard model of cold dark matter, written by two authors.  The article cites it as an assessment from the dark-matter side rather than from MOND, and for what it says about the radial acceleration relation.

1. 10a.

   Figure 12, page 26:  The caption, in the preprint's page numbering.  “The tightness of the acceleration relation is remarkable (consistent with zero scatter given observational error, red cross), especially given the variation in central densities seen on the right.” ↩

2. 10b.

   Section §3.1.4 Explaining the radial acceleration relation, page 32:  Page of the preprint.  “Keller & Wadsley (2017) responded by demonstrating that a similar relation can be obtained using ΛCDM hydrodynamic simulations of disk galaxies.  Importantly, however, the systems simulated did not include low-mass galaxies, which are dark-matter-dominated throughout.  The smallest galaxies are the ones with low acceleration in their centers as well as in their outer parts, and they remain the most puzzling to explain.”  “It remains to be seen whether the absolute normalization and shape of the RAR in the low-acceleration regime can be reproduced in ΛCDM simulations that span the full range of galaxy types that are observed to obey the RAR.  As stated previously, these same simulations must also simultaneously reproduce the observed diversity of galaxies at fixed V_max that is seen in the data.” ↩

McGaugh, Stacy S.  “MOND Prediction for the Velocity Dispersion of the 'Feeble Giant' Crater II.”  ApJ Letters 832:L8.  2016.  A 2016 letter, posted as an arXiv preprint on 19 October 2016, that predicts the velocity spread of stars in the faint galaxy Crater II before anyone had measured it.  The prediction rests on MOND's external field effect, the pull of the Milky Way strong enough to change the galaxy's own dynamics, and it gives about 2 km/s where a galaxy isolated from the Milky Way would show about 4.  The article uses it as its clearest example of a prediction made in advance.  Mirror:  arXiv

1. 11a.

   Section Abstract:  “Crater II should be in the deep MOND regime (g_in ≈ 34 km² s⁻² kpc⁻¹ ≪ a₀ = 3700 km² s⁻² kpc⁻¹).  Despite its great distance (≈ 120 kpc) from the Milky Way, the external field of the host (g_ex ≈ 282 km² s⁻² kpc⁻¹) comfortably exceeds the internal field.  Consequently, Crater II should be subject to the external field effect, a feature unique to MOND.  This leads to the prediction of a very low velocity dispersion:  σ_efe = 2.1 +0.9 −0.6 km s⁻¹.”  Section 3:  Section 3 adds:  “For Crater II, M_∗ ≈ 3.3 × 10⁵ M⊙, so σ_iso ≈ 4 km s⁻¹.  This value for the velocity dispersion of 4 km s⁻¹ is what MOND would predict for a diffuse, isolated galaxy of this stellar mass.” ↩ ↩

2. 11b.

   Section 4.3, page 5:  Preprint page numbers.  “Barring a drastic failure of the input data and necessary assumptions, the velocity dispersion of Crater II must be σ_efe ≈ 2 km s⁻¹.  It should not be 4 km s⁻¹ as it would be if Crater II were isolated.  It certainly should not be significantly larger:  a high observed velocity dispersion would falsify MOND.” ↩

3. 11c.

   Section 4.3, page 5:  Preprint page numbers.  “There is no uniquely predicted value;  we simply infer the amount of dark matter that corresponds to the observed velocity dispersion.  ΛCDM can accommodate any result.  In MOND, the prediction is unique.” ↩

Caldwell, Nelson et al.  “Crater 2: An Extremely Cold Dark Matter Halo.”  ApJ 839:20.  2017.  A spectroscopic study of Crater 2, a satellite of the Milky Way, which identified about 62 members among 390 candidate stars.  It measured their velocity spread at 2.7 km/s, the coldest of any dwarf galaxy resolved so far, and was posted to arXiv on 19 December 2016, two months after McGaugh's prediction.  The article uses it as the measurement that fell inside MOND's predicted band.

1. 12a.

   Section Abstract:  “We present results from MMT/Hectochelle spectroscopy of 390 red giant candidate stars along the line of sight to the recently-discovered Galactic satellite Crater 2… we identify ∼62 members of Crater 2, for which we resolve line-of-sight velocity dispersion σ_vlos = 2.7 +0.3 −0.3 km s⁻¹… Despite Crater 2's relatively large size (projected halflight radius R_h ∼ 1 kpc) and intermediate luminosity (M_V ∼ −8), its velocity dispersion is the coldest that has been resolved for any dwarf galaxy.  These properties make Crater 2 the most extreme low-density outlier in dynamical as well as structural scaling relations among the Milky Way's dwarf spheroidals.” ↩ ↩ ↩

Keller, B. W., and J. W. Wadsley.  “ΛCDM is Consistent with SPARC Radial Acceleration Relation.”  ApJ Letters 835:L17.  2017.  A letter reporting simulations of 32 galaxies from the MUGS2 suite, each a rotating disc of stars forming in a standard ΛCDM cosmology (the model built on dark energy, Λ, and cold dark matter).  The simulated galaxies match the SPARC acceleration relation, with less scatter than the observed galaxies, and the authors conclude that the relation follows from ordinary matter collapsing and losing energy as it forms galaxies, not from exotic dark-sector physics or new laws of dynamics.  The article uses it as a simulation in standard cosmology that reproduces the relation.

1. 13a.

   Section Abstract:  “In this letter we show that 32 galaxies from the MUGS2 match the SPARC acceleration relation.  These cosmological simulations of star forming, rotationally supported discs were simulated with a WMAP3 ΛCDM cosmology, and match the SPARC acceleration relation with less scatter than the observational data.  These results show that this acceleration relation is a consequence of dissipative collapse of baryons, rather than being evidence for exotic dark-sector physics or new dynamical laws.” ↩

Ludlow, Aaron D. et al.  “Mass-Discrepancy Acceleration Relation: A Natural Outcome of Galaxy Formation in Cold Dark Matter Halos.”  Physical Review Letters 118:161103.  2017.  A short letter reporting computer simulations of galaxies that form inside cold dark matter halos, the dark matter clumps that surround galaxies.  The team started several runs from the same initial conditions, each with different rules for stellar and black-hole feedback, the processes that push gas back out of galaxies.  Every run followed the same link between total and ordinary-matter acceleration, though only some reproduced observed galaxy scaling relations.  The article cites it for the point that the same relation also appears in simulations with cold dark matter.  Mirror:  arXiv

1. 14a.

   Section Abstract:  “Our runs start from the same initial conditions but adopt different prescriptions for unresolved stellar and AGN feedback, resulting in diverse populations of galaxies by the present day.  Some of them reproduce observed galaxy scaling relations, while others do not.  However, regardless of the feedback implementation, all of our galaxies follow closely a simple relationship between the total and baryonic acceleration profiles, consistent with recent observations of rotationally supported galaxies.  The relation has small scatter:  different feedback implementations – which produce different galaxy populations – mainly shift galaxies along the relation, rather than perpendicular to it.” ↩

Sanders, Robert H.  “Clusters of galaxies with modified Newtonian dynamics.”  MNRAS 342:901–908.  2003.  A 2003 test of modified Newtonian dynamics (MOND) on galaxy clusters.  It models the observed X-ray brightness of real clusters and finds the mass discrepancy is three to four times smaller than under Newtonian dynamics, but not removed.  The article uses it to show that MOND still needs some dark mass in clusters.  Mirror:  arXiv

1. 15a.

   Section Abstract:  “These problems may be resolved by adding an additional dark mass component in the central regions;  a constant density sphere contained within two core radii and having a total mass of one to two times that in the gas… When the observed X-ray surface brightness distribution in actual clusters is modeled by such a two-component structure, the typical mass discrepancy is three to four times smaller than with Newtonian dynamics.  Thus while MOND significantly reduces the mass of the dark component in clusters it does not remove it completely.” ↩

Angus, Garry W. et al.  “X-ray group and cluster mass profiles in MOND: unexplained mass on the group scale.”  MNRAS 387:1470–1480.  2008.  A study of hot X-ray gas in 26 systems, with temperatures from 0.5 to 9 keV, under MOND.  It finds that MOND still needs dark matter in these systems, and that ordinary neutrinos of 2 eV cannot supply it in the central part of a system.  The article uses it to show that the missing-mass problem gets worse, not better, for smaller systems.

1. 16a.

   Section Abstract:  “Although very successful in explaining the observed conspiracy between the baryonic distribution and the gravitational field in spiral galaxies without resorting to dark matter (DM), the modified Newtonian dynamics (MOND) paradigm still requires DM in X-ray bright systems… Even by considering that the neutrino density stays constant and maximal within the central 100 or 150 kpc (which is the absolute upper limit of a possible neutrino contribution there), we show that these neutrinos can never account for the required DM within this region.  The natural corollary of this finding is that, whereas clusters (T ≳ 3 keV) might have most of their mass accounted for if ordinary neutrinos have a 2 eV mass, groups (T ≲ 2 keV) cannot be explained by a 2 eV neutrino contribution.”  The sample is:  “26 X-ray emitting systems, with temperatures ranging from 0.5 to 9 keV.” ↩ ↩

2. 16b.

   Pages 9–10:  Preprint page numbers.  The passage is on Milgrom's proposal of cluster baryonic dark matter.  “the total sky cover factor of such clouds in the core of the clusters would be of the order of only 10⁻⁴, so that they would only occult a minor fraction of the X-rays emitted by the hot gas… Of course, this speculative idea also raises a number of questions, the most serious one being how these clumps form and stabilize, and why they form only in clusters, groups and some ellipticals, but not in individual spiral galaxies.” ↩

Aker, M. et al.  Direct neutrino-mass measurement based on 259 days of KATRIN data.  Jun 19, 2024.  A 2024 report by the KATRIN Collaboration, the team that runs the KATRIN experiment, on its measurement of the neutrino's mass using tritium beta decay.  It combines 259 days of data from five measurement campaigns, from 2019 to 2021, finds no sign of a neutrino mass, and sets an upper limit of 0.45 eV (an electronvolt, a unit of energy used to express particle masses) at 90 percent confidence.  The article uses it to rule out the idea that neutrinos of about 2 eV could supply the extra mass that MOND needs in clusters.

1. 17a.

   Section Abstract:  “we report on the neutrino-mass results from five measurement campaigns conducted from 2019 to 2021… We find no indication for a non-zero neutrino mass and improve the upper limit by a factor of almost two to m_ν < 0.45 eV at 90 % confidence level.” ↩

Kelleher, Ruth, and Federico Lelli.  “Galaxy clusters in Milgromian dynamics: Missing matter, hydrostatic bias, and the external field effect.”  A&A 688:A78.  2024.  A 2024 study of galaxy clusters in Milgromian dynamics, the theory the article calls MOND, asking how much missing matter the clusters need.  For three relaxed clusters (A1795, A2029 and A2142), the ratio of missing mass to visible mass is about 1 to 5 at 200 to 300 kpc, falling to 0.4 to 1.1 at 2 to 3 Mpc, so the missing mass is no larger than the intracluster gas.  The article uses it as the most careful recent measurement of this question, and it comes from MOND's own side: it confirms that clusters still need missing matter under MOND, though far less than dark matter needs under Newtonian dynamics.

1. 18a.

   Section Abstract:  “We confirm that galaxy clusters require additional 'missing matter' in MOND, although the required amount is drastically reduced with respect to the non-baryonic dark matter in the context of Newtonian dynamics… The missing mass must be more centrally concentrated than the intracluster medium (ICM).  For relaxed clusters (A1795, A2029, A2142), the ratio of missing-to-visible mass is around 1−5 at R ≃ 200−300 kpc and decreases to 0.4−1.1 at R ≃ 2−3 Mpc, showing that the total amount of missing mass is smaller than or comparable to the ICM mass.” ↩

Clowe, Douglas et al.  “A Direct Empirical Proof of the Existence of Dark Matter.”  ApJ Letters 648:L109–L113.  2006.  A 2006 weak-lensing study of 1E0657-558, a pair of galaxy clusters that collided.  It maps the mass from the way it bends light, using ground-based and Hubble images, and finds that the mass sits with the galaxies rather than with the X-ray-emitting plasma, the largest share of ordinary matter in the system, an offset of 8σ.  The article uses it as the direct test that changing the gravity law cannot explain that offset.

1. 19a.

   Section Abstract:  “An 8σ significance spatial offset of the center of the total mass from the center of the baryonic mass peaks cannot be explained with an alteration of the gravitational force law, and thus proves that the majority of the matter in the system is unseen.” ↩

2. 19b.

   Section §4:  “Our κ map, however, has no evidence of any mass concentration between the two galaxy subclusters other than the small perturbations consistent with the gas mass contribution in Newtonian gravity… In 1E0657−558, however, the X-ray plasma, which would provide the central peak, lies north of the line connecting the two κ peaks.  Further, the absolute κ levels of the peaks observed in 1E0657−558 are in good agreement with those in systems with similar velocity dispersions and X-ray temperatures.”  “Any non-standard gravitational force that scales with baryonic mass will fail to reproduce these observations.” ↩ ↩

3. 19c.

   Section §4:  “While the existence of dark matter removes the primary motivation for alternative gravity models, it does not preclude non-standard gravity.” ↩ ↩

Angus, Garry W. et al.  “Can MOND take a bullet? Analytical comparisons of three versions of MOND beyond spherical symmetry.”  MNRAS 371:138–146.  2006.  An analytic paper that compares three versions of MOND in systems that are not spherical.  It shows that a multi-centred system of ordinary (baryonic) matter, with a bullet-shaped light distribution, can produce the lensing signal of the merging cluster 1E 0657-56.  The article cites it as MOND's own attempt to explain that cluster.

1. 20a.

   Section Abstract:  “In particular, we can generate a multi-centred baryonic system with a weak lensing signal resembling that of the merging galaxy cluster 1E 0657-56 with a bullet-like light distribution.” ↩

Springel, Volker, and Glennys R. Farrar.  “The speed of the 'bullet' in the merging galaxy cluster 1E0657−56.”  MNRAS 380:911–925.  2007.  A 2007 computer-simulation study of the 'bullet' subcluster in the merging galaxy cluster 1E0657−56.  It uses hydrodynamic models of the collision and asks how fast the subcluster itself is moving, as opposed to the shock it creates.  The article uses it to answer the argument, made by MOND supporters against ΛCDM, that the collision speed is too high.  Mirror:  arXiv

1. 21a.

   Section Abstract:  “The inferred shock velocity of ∼ 4700 km s⁻¹ has been commonly interpreted as the velocity of the 'bullet' subcluster itself.  This velocity is unexpectedly high in the ΛCDM cosmology… However, despite a shock speed of 4500 km s⁻¹, the subcluster's mass centroid is moving only with ∼ 2600 km s⁻¹ in the rest frame of the system.  The difference arises in part due to a gravitationally induced inflow velocity of the gas ahead of the shock towards the bullet, which amounts to ∼ 1100 km s⁻¹ for our assumed 10 : 1 mass ratio of the merger.” ↩

McGaugh, Stacy S.  “Distinguishing Between Cold Dark Matter and Modified Newtonian Dynamics: Predictions for the Microwave Background.”  ApJ Letters 523:L99.  1999.  A 1999 letter that works out what the microwave background, the faint afterglow of the early universe, should look like in a universe with no cold dark matter, and compares it with the standard cold dark matter prediction.  Its key number is the amplitude ratio of the first and second peaks, which it gives as about 2.4 for MOND and about 1.7 for cold dark matter.  The article uses it as an advance prediction, made before the peaks were measured well.  Mirror:  arXiv

1. 22a.

   Section 2, page 4:  Preprint page numbers.  “The ratio of the absolute amplitudes of the peaks can also distinguish the two cases, but require comparable accuracy.  In CDM, (C_ℓ,1/C_ℓ,2)_abs ≈ 1.7, while in MOND (C_ℓ,1/C_ℓ,2)_abs ≈ 2.4.” ↩

Page, Lyman et al.  “First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Interpretation of the TT and TE Angular Power Spectrum Peaks.”  ApJ Supplement 148:233–241.  2003.  Written by the WMAP Science Team from the first year of data from the Wilkinson Microwave Anisotropy Probe satellite.  It reports the amplitudes of the first two peaks in the cosmic microwave background (the glow left over from the early universe) and of the trough between them, with their uncertainties.  The article uses its measured peak heights to check a prediction that models without cold dark matter made before the data were in.  Mirror:  arXiv

1. 23a.

   Section Abstract:  “From the WMAP data we find the first peak in the temperature spectrum at ℓ = 220.1 ± 0.8 with an amplitude of 74.7 ± 0.5 µK;  the first trough at ℓ = 411.7 ± 3.5 with an amplitude of 41.0 ± 0.5 µK;  and the second peak at ℓ = 546 ± 10 with an amplitude of 48.8 ± 0.9 µK.  All uncertainties are 1σ and include calibration and beam errors.”  The squared amplitude ratio (74.7/48.8)² = 2.34 ± 0.09 was computed by the article from those figures; Page et al. do not print it. ↩

McGaugh, Stacy S.  “Confrontation of Modified Newtonian Dynamics Predictions with WMAP First Year Data.”  ApJ 611:26–39.  2004.  A 2004 journal article that checks MOND's predictions about the microwave background against the first year of data from the WMAP satellite.  It reports that the predicted first-to-second peak amplitude ratio, about 2.4, matches the measured 2.34 ± 0.09.  It also calls the third peak an outstanding test, because experiments that probe high-ℓ appear to find a larger third peak than the model predicts.  The article uses it for both halves of the story, the second-peak match and the author's own warning about the third peak.  Mirror:  arXiv

1. 24a.

   Section Abstract:  “An a priori prediction of such no-CDM models was a first-to-second peak amplitude ratio A_1:2 ≈ 2.4.  WMAP measures A_1:2 = 2.34 ± 0.09.  The baryon content is the dominant factor in fixing this ratio;  no-CDM models which are consistent with the WMAP data are also consistent with constraints on the baryon density from the primordial abundances of ²H, ⁴He, and ⁷Li.  However, in order to match the modest width of the acoustic peaks observed by WMAP, a substantial neutrino mass is implied:  m_ν ≈ 1 eV… An outstanding test is in the amplitude of the third acoustic peak.  Experiments which probe high-ℓ appear to favor a third peak which is larger than predicted by the no-CDM model.” ↩ ↩

Dodelson, Scott.  “The Real Problem with MOND.”  International Journal of Modern Physics D 20:2749–2753.  2011.  A short essay by Scott Dodelson, a physicist at Fermilab and the University of Chicago.  It grants that MOND fits galaxy rotation curves well, and then names the most severe challenge to it: the shape of the matter power spectrum, which measures how clumpy matter is at each scale.  The article uses it for two failures: the third acoustic peak in the microwave background, and the power spectrum.

1. 25a.

   Page 4:  Page of the preprint.  “The biggest challenge facing MOND today is the shape of the matter power spectrum.”  Figure 1:  The caption to figure 1 says, for a model with no dark matter:  “the spectrum has pronounced Baryon Acoustic Oscillations, in violent disagreement with the data.”  Page 4:  On the peaks:  “The problem is that the third peak should be very small in a baryon-dominated model that lacks the extra gravitational forcing usually supplied by dark matter.  The observed third peak is therefore very difficult to accommodate in the framework of MOND.” ↩ ↩

Angus, Garry W.  “Is an 11 eV sterile neutrino consistent with clusters, the cosmic microwave background and modified Newtonian dynamics?”  MNRAS 394:527–532.  2009.  A theory paper asking whether one sterile neutrino of about 11 eV could solve MOND's outstanding problems, on the scale of clusters and on the scale of the microwave background.  Its model is flat, with the neutrino density parameter Ω at about 0.23, and has the same expansion history as the standard model, ΛCDM.  The article uses it as the MOND cosmology that matches the microwave background, by adding a hot dark matter component.

1. 26a.

   Section Abstract:  “If a single sterile neutrino exists such that m_νs ∼ 11 eV, it can serendipitously solve all outstanding issues of the Modified Newtonian Dynamics.  With it one can explain the dark matter of galaxy clusters without influencing individual galaxies, match the angular power spectrum of the cosmic microwave background and potentially fit the matter power spectrum.  This model is flat with Ω_νs ∼ 0.23 and the usual baryonic and dark energy components, thus the Universe has the same expansion history as the ΛCDM model and only differs at the galactic scale where the Modified Dynamics outperforms ΛCDM significantly.” ↩ ↩

Angus, Garry W., and Antonaldo Diaferio.  “The abundance of galaxy clusters in modified Newtonian dynamics: cosmological simulations with massive neutrinos.”  MNRAS 417:941–949.  2011.  Computer simulations of how galaxy clusters form under MOND in a universe with massive neutrinos.  The authors find that MOND does form clusters, and that the simulated X-ray clusters are about the right order of magnitude, but that there are too many very rich clusters and too few lower-mass ones.  The article uses it to show that structure formation still goes wrong even with the neutrino fix.

1. 27a.

   Section Abstract:  “We clearly demonstrate the necessity of MOND for developing the large scale structure in a hot dark matter cosmology and contradict the naive expectation that MOND cannot form galaxy clusters.  We find that the correct order of magnitude of X-ray clusters (with T_X > 4.5 keV) can be formed, but that we overpredict the number of very rich clusters and seriously underpredict the number of lower mass clusters.  The latter is a shortcoming of the resolution of our simulations, whereas we suggest that the over production of very rich clusters might be prevented by incorporating a MOND acceleration constant that varies with redshift and an expansion history that cannot be described by the usual Friedmann models.” ↩

Bekenstein, Jacob D.  “Relativistic gravitation theory for the MOND paradigm.”  Physical Review D 70:083509.  2004.  A theory paper that builds a relativistic version of MOND, one that fits Einstein's theory of relativity, in which gravity is carried by a metric, a scalar field and a four-vector field, all three dynamical.  The author says it resolves the problems that earlier relativistic MOND theories had, and an erratum followed in Physical Review D 71 (2005): 069901.  The article calls it the flagship attempt to give MOND a relativistic form, under the name TeVeS.

1. 28a.

   Section Abstract:  “So far relativistic gravitation theories proposed to underpin MOND have either clashed with the post-Newtonian tests of general relativity, or failed to provide significant gravitational lensing, or violated hallowed principles by exhibiting superluminal scalar waves or an a priori vector field.  We develop a relativistic MOND inspired theory which resolves these problems.  In it gravitation is mediated by metric, a scalar field and a 4-vector field, all three dynamical.” ↩

Skordis, Constantinos et al.  “Large Scale Structure in Bekenstein's Theory of Relativistic Modified Newtonian Dynamics.”  Physical Review Letters 96:011301.  2006.  A 2006 paper that works out how Bekenstein's relativistic version of modified Newtonian dynamics would affect the distribution of galaxies and the cosmic microwave background.  Its abstract says the theory may be able to reproduce both sets of observations.  The article cites it in its account of Bekenstein's theory, TeVeS.  Mirror:  arXiv

1. 29a.

   Section Abstract:  “The impact on galaxy distributions and the cosmic microwave background is calculated in detail.  We show that it may be possible to reproduce observations of the cosmic microwave background and galaxy distributions with Bekenstein's theory of MOND.” ↩

Boran, Sibel et al.  “GW170817 falsifies dark matter emulators.”  Physical Review D 97:041501.  2018.  A theory paper on a class of modified-gravity theories called dark matter emulators, which make light and gravitational waves follow different paths and so predict a delay between them (article 041501(R) in Physical Review D).  The authors estimate the delay from dark matter at the order of 400 days, and conclude that the simultaneous detection of GW170817 falsifies these theories.  The article uses it against TeVeS, which the authors name as one of these emulators, and notes that it does not touch Milgrom's bi-metric MOND or nonlocal MOND.

1. 30a.

   Section §II:  Section of the preprint.  “The null geodesics of GWs are sourced by only the baryonic matter, however, photons/neutrinos propagate on null geodesics sourced by baryonic matter and the much larger pools of dark matter which would be required if GR were correct.  Therefore, the differential Shapiro delay between GWs and photons/neutrinos is due to the gravitational potential of only the dark matter.  Some examples of these Dark Matter (DM) emulator theories include Bekenstein's TeVeS theory and Moffat's Scalar-Tensor-Vector gravity theory.” ↩

2. 30b.

   Section §III:  Section of the preprint.  “we estimate the total Shapiro delay due to the dark matter component of the order of 400 days.” ↩

3. 30c.

   Section §III:  Section of the preprint.  “Because GR predicts coincident arrival times for photons and gravitational radiation, whereas DM emulators predict delays of over a year, the simultaneous optical detection of GW170817 immediately and decisively falsifies DM emulator models.” ↩

4. 30d.

   Section §II:  Section of the preprint.  “It is important to understand that dark matter emulators constitute a special class of modified gravity theories which attempt to dispense with dark matter.  Many modifications of gravity do not fall within this class, including Milgrom's bi-metric formulation of MOND and nonlocal MOND.  Nor does it include hybrid theories such as dipolar dark matter or superfluid dark matter.” ↩

Abbott, B. P. et al.  “Gravitational Waves and Gamma-Rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A.”  ApJ Letters 848:L13.  2017.  A joint report by the LIGO and Virgo collaborations, with the Fermi GBM and INTEGRAL teams, on one merger of two neutron stars.  The gravitational waves and the gamma-ray burst from that event arrived 1.74 seconds apart, and the authors used the gap to limit how far the speed of gravitational waves can differ from the speed of light.  The article uses the event to test whether gravity and light travel at the same speed.  Mirror:  arXiv

1. 31a.

   Section Abstract and §2:  “the observed time delay of (+1.74 ± 0.05) s between GRB 170817A and GW170817.” ↩

2. 31b.

   Section §4.1:  The limit uses a conservative travel distance of 26 Mpc and allows the gamma-ray burst to have been emitted up to 10 s after the merger.  “The resulting constraint on the fractional speed difference is −3 × 10⁻¹⁵ ≤ Δv/v_EM ≤ +7 × 10⁻¹⁶.” ↩

Skordis, Constantinos, and Tom Złośnik.  “New Relativistic Theory for Modified Newtonian Dynamics.”  Physical Review Letters 127:161302.  2021.  A 2021 paper that proposes a relativistic gravity theory giving modified Newtonian dynamics (MOND) in galaxies, called aether scalar tensor theory (AeST).  Its abstract says it agrees with the cosmic microwave background and matter power spectra on large scales, and that its action is free of ghost instabilities at second order.  The article calls it the current best attempt.  Mirror:  arXiv

1. 32a.

   Section Abstract:  “We propose a relativistic gravitational theory leading to modified Newtonian dynamics, a paradigm that explains the observed universal galactic acceleration scale and related phenomenology.  We discuss phenomenological requirements leading to its construction and demonstrate its agreement with the observed cosmic microwave background and matter power spectra on linear cosmological scales.  We show that its action expanded to second order is free of ghost instabilities.” ↩

2. 32b.

   Section Cosmology, page 4:  Preprint page.  “For a wide range of parameters, this relativistic MOND theory is consistent with the CMB measurements from Planck.  This happens because c²_ad and w are small enough so that Π → 0 and we get dustlike evolution.” ↩

Mistele, Tobias et al.  “Aether scalar tensor theory confronted with weak lensing data at small accelerations.”  A&A 676:A100.  2023.  A 2023 test of aether scalar tensor theory (AeST), a relativistic gravity theory that reproduces MOND only up to a certain galactocentric radius (distance from the galaxy's centre).  The authors test it against weak gravitational lensing, the slight bending of light by mass, at small accelerations.  The article uses it to show that AeST is not established.  Mirror:  arXiv

1. 33a.

   Section Abstract:  “the AeST model reproduces MOND only up to a certain maximum galactocentric radius.  Since MOND is known to fit very well to observations at these scales, this raises the question of whether the AeST model comes into tension with data… Results.  The AeST model shows some deviations from MOND at the radii probed by weak gravitational lensing.  The data show no clear indication of these predicted deviations.” ↩

Pittordis, Charalambos et al.  “Wide Binaries from Gaia DR3: testing GR vs MOND with realistic triple modelling.”  The Open Journal of Astrophysics 8.  2025.  A 2025 test of gravity using wide binaries, pairs of stars that orbit each other at great distances, with data from the Gaia DR3 star catalogue.  It models the triple-star systems that can mimic a wide binary, and compares the observed speed differences with Newtonian and MOND predictions.  The article uses it as a 2025 repeat of an earlier wide-binary test, with tighter cuts and better masses.  Mirror:  arXiv

1. 34a.

   Section §1:  “MOND-like models predict increased accelerations leading to faster relative velocities at scales where the internal accelerations are below the MOND characteristic acceleration scale a₀ ≃ 1.2 × 10⁻¹⁰ m s⁻².  This acceleration occurs at separation ≃ 7 kAU for a binary of 1 M⊙ total mass.” ↩

2. 34b.

   Section Abstract:  “We fit histograms of observed vs Newtonian velocity differences to a flexible mixture of binary + triple populations with realistic eccentricity distributions, plus unbound flyby and random-chance populations.  We find as before that Newtonian models provide a significantly better fit than MOND, though improved understanding of the triple population is necessary to make this fully decisive.” ↩

Banik, Indranil et al.  “Strong constraints on the gravitational law from Gaia DR3 wide binaries.”  MNRAS 527:4573–4615.  2024.  A statistical test of wide binaries, pairs of stars that orbit each other 2 to 30 thousand astronomical units (kAU) apart.  It uses 8,611 such pairs within 250 pc of the Sun from the Gaia DR3 data, and finds Newtonian gravity preferred at 19σ, with MOND excluded at 16σ.  The article uses it as a test where dark matter makes no prediction, but Newton and MOND disagree.

1. 35a.

   Section Abstract:  “We test Milgromian dynamics (MOND) using wide binary stars (WBs) with separations of 2 − 30 kAU.  Locally, the WB orbital velocity in MOND should exceed the Newtonian prediction by ≈ 20% at asymptotically large separations given the Galactic external field effect (EFE).  We investigate this with a detailed statistical analysis of Gaia DR3 data on 8611 WBs within 250 pc of the Sun… Directly comparing the best Newtonian and Milgromian models reveals that Newtonian dynamics is preferred at 19σ confidence.  Using a complementary Markov Chain Monte Carlo analysis, we find that α_grav = −0.021 +0.065 −0.045, which is fully consistent with Newtonian gravity but excludes MOND at 16σ confidence.” ↩ ↩

2. 35b.

   Section Abstract:  “We show that although our best-fitting model does not fully reproduce the observations, an overwhelmingly strong preference for Newtonian gravity remains in a considerable range of variations to our analysis.  Adapting the MOND interpolating function to explain this result would cause tension with rotation curve constraints.” ↩

Hernandez, Xavier.  “Internal kinematics of Gaia DR3 wide binaries: anomalous behaviour in the low acceleration regime.”  MNRAS 525:1401–1415.  2023.  A 2023 analysis of wide binaries, which are pairs of stars that orbit each other at large distances, using Gaia DR3, the third data release of the Gaia space telescope's star catalogue.  Using Gaia's own stellar masses, each star's chance of belonging to a closer pair, and more radial velocities to remove contaminating pairs, it finds that the velocity difference (ΔV) between the two stars of each pair matches Newton's gravity at high acceleration but not at low, which it says confirms a non-Newtonian low-acceleration effect.  The article uses it as the 2023 result behind its claim that low-acceleration gravity departs from Newton's.

1. 36a.

   Section Abstract:  “Internally determined Gaia stellar masses and estimates of binary probabilities for each star using spectroscopic information, together with a larger sample of radial velocities, allow for a significant improvement in the analysis and careful exclusion of possible kinematic contaminants.  Resulting ΔV scalings accurately tracing Newtonian expectations for the high acceleration regime, but markedly inconsistent with these expectations in the low acceleration one, are obtained.  A non-Newtonian low acceleration phenomenology is thus confirmed.” ↩

Chae, Kyu-Hyun.  “Breakdown of the Newton-Einstein Standard Gravity at Low Acceleration in Internal Dynamics of Wide Binary Stars.”  ApJ 952:128.  2023.  A paper that measures the orbits of wide binaries, pairs of stars orbiting each other slowly, at low acceleration, using 26,615 such pairs within 200 pc of the Sun.  It reports departures from Newtonian gravity at 10σ significance, and says the size of the departure matches the boost that the AQUAL version of MOND predicts.  The article uses it for the measured ratio of 1.43 at the lower acceleration point, and for the claim that this matches MOND.

1. 37a.

   Section Abstract:  “A gravitational anomaly parameter δ_obs−newt between the observed acceleration at g_N and the Newtonian prediction is measured to be:  δ_obs−newt = 0.034 ± 0.007 and 0.109 ± 0.013 at g_N ≈ 10^−8.91 and 10^−10.15 m s⁻², from the main sample of 26,615 wide binaries within 200 pc.  These two deviations in the same direction represent a 10σ significance… At g_N = 10^−10.15 m s⁻², the observed to Newton-predicted acceleration ratio is g_obs/g_pred = 10^{√2δ~obs−newt~} = 1.43 ± 0.06.  This systematic deviation agrees with the boost factor that the AQUAL theory predicts for kinematic accelerations in circular orbits under the Galactic external field.” ↩

Hernandez, Xavier et al.  “A critical review of recent Gaia wide binary gravity tests.”  MNRAS 533:729–742.  2024.  A 2024 critical review of recent tests of gravity with wide binaries, which are pairs of stars that orbit each other at large distances, in Gaia data.  It concludes that the low-acceleration binaries fit MOND if gravity is boosted by about 1.5 times, plus or minus 0.2, and it faults two earlier fits for ignoring observational errors and for finding hidden companion fractions that are too large.  The article uses it for that combined position, and for the dispute over how to calibrate such fits.

1. 38a.

   Section §6 Conclusions, page 15:  Page of the preprint.  “For the low acceleration 2-20 kau region, again full accord is found in all of the above studies, but finding observations of Gaia wide binaries to be consistent with MOND predictions, for elliptical orbits within a 1.5 boost to the gravitational constant, G → (1.5 ± 0.2)G.” ↩

2. 38b.

   Section §6 Conclusions, page 15:  Page of the preprint.  “It is important to note that in both PS23 and B24 the existence of observational errors is neglected from the statistical fitting procedure.  Also, no deep Newtonian region is included to calibrate any of the parameters being fitted… In the cases of parameters for which previous and/or direct observational determinations exist, the best fit values found by B24 are inconsistent, finding excessively large hidden tertiary fractions, excessively large α ellipticity parameters and excessively small numbers of line-of-sight contaminants.  As we have shown explicitly, working within excessively large hidden binary fractions will necessarily result in reduced effective gravity inferences.” ↩ ↩

Chae, Kyu-Hyun, and Youngsub Yoon.  Revisiting Data Quality Control and Multiple-star Modeling in Wide Binary Gravity Tests: Confirmation of MOND-type Gravitational Anomaly at Low Acceleration.  Jul 16, 2026.  A paper posted to arXiv on 16 July 2026 that argues the low-acceleration anomaly in wide binaries survives better data quality control and multiple-star modeling.  Its authors say that studies finding no anomaly did so through a bypassed calibration, a bias in data quality control, or too few stars in the low-acceleration regime.  The article presents it as the authors' direct reply to the quality-control criticism.

1. 39a.

   Section Abstract:  “we show that proper data quality control or reasonable variation in multiple-star modeling cannot remove the low-acceleration gravitational anomaly but confirms the MOND-type gravitational anomaly, particularly consistent with recent realistic MOND solutions of wide binary orbits.  We find that studies claiming no evidence for the low-acceleration gravitational anomaly are consequences of bypassed calibration of the fraction of multiple-star systems using the Newtonian-regime data, bias-introduction in data quality control that is not taken into account in gravity tests, or insufficient statistics in the low-acceleration regime.” ↩

Cookson, Stephen A. et al.  A Quality Framework for Testing Gravity with Wide Binaries: No Evidence for MOND.  Feb 27, 2026.  A 2026 preprint, submitted to the journal MNRAS, on how to test gravity with wide binaries, which are pairs of stars that orbit each other at large distances.  It builds a high-purity sample of wide binary pairs from the third data release of the Gaia space telescope's star catalogue, within 130 parsecs of the Sun, finds no 20 percent MOND boost in their velocities, and reports that earlier MOND signals shrink as the methods improve.  The article uses it as a recent case of a MOND signal disappearing once the data are cleaned.

1. 40a.

   Section Abstract:  “We show that undetected close binaries, chance alignments, and improper treatment of projection effects can mimic MOND-like signals… Applying this framework to Gaia DR3 data, we compile a high-purity sample of WBs within 130 pc with projected separations of 1 − 30 kAU, spanning the transition between the Newtonian and MOND regimes.  We find that the scaled relative velocity distribution of wide binaries does not exhibit the 20% enhancement expected from MOND and is consistent with Newtonian gravity across all separations.  A meta-analysis of previous WBTs shows that apparent MOND signals diminish as methodological rigour improves.” ↩ ↩

Makarov, Valeri V.  “Distributions of Wide Binary Stars in Theory and in Gaia Data. III. Orbital Momenta, Masses, and Manifestations of MOND.”  The Astronomical Journal 171:79.  2026.  The third part of a series on wide binary stars, which are pairs that orbit each other at large separations.  The author takes 103169 resolved binary stars from the Gaia DR3 catalogue, measures a new quantity for each pair called the projected orbital momentum, and checks whether the MOND effect expected for them appears.  It does not, and the article cites it with Cookson and colleagues' 2026 result as two results that push the same way.  Mirror:  arXiv

1. 41a.

   Section Abstract:  “Using the censored catalog of 103169 resolved Gaia DR3 binary stars with accurate astrometric data for each component, a new observable, object-specific parameter is computed for each pair:  the projected orbital momentum… The anticipated manifestation of MOND is computed in the investigated parameter space {separation, momentum}.  This effect is absent in the given data.” ↩

Chae, Kyu-Hyun et al.  “Testing the Strong Equivalence Principle: Detection of the External Field Effect in Rotationally Supported Galaxies.”  ApJ 904:51.  2020.  A statistical study of 153 rotating galaxies from the SPARC sample, testing for the external field effect that MOND predicts, which is the influence of a galaxy's surroundings on how it rotates.  The authors report a detection at more than 4σ, and say the result points to a breakdown of the strong equivalence principle, a basic assumption of Einstein's gravity, and so supports modified gravity theories.  The article calls this the statistical claim for the external field effect.

1. 42a.

   Section Abstract:  “Our detection is threefold:  (1) the EFE is individually detected at 8σ to 11σ in 'golden' galaxies subjected to exceptionally strong external fields, while it is not detected in exceptionally isolated galaxies, (2) the EFE is statistically detected at more than 4σ from a blind test of 153 SPARC rotating galaxies, giving a mean value of the external field consistent with an independent estimate from the galaxies' environments, and (3) we detect a systematic downward trend in the weak gravity part of the radial acceleration relation at the right acceleration predicted by the EFE of the MOND modified gravity.  Tidal effects from neighboring galaxies in the ΛCDM context are not strong enough to explain these phenomena… Our results point to a breakdown of the SEP, supporting modified gravity theories beyond General Relativity.” ↩ ↩

Paranjape, Aseem, and Ravi K. Sheth.  “The phenomenology of the external field effect in cold dark matter models.”  MNRAS 517:130–139.  2022.  A theory paper that re-examines a claimed detection of the external field effect (EFE), the claim that a galaxy's surroundings change how it behaves inside.  It argues that large uncertainties in rotation-curve analyses and inaccurate simulation estimates undermine the claimed detection, and that a similar signal is expected in the standard cosmological model (ΛCDM) without breaking the equivalence principle.  The article uses it as the rebuttal to the claimed EFE detection.  Mirror:  arXiv

1. 43a.

   Section Abstract:  “We show that large uncertainties in rotation curve analyses and inaccuracies in published simulation-based external field estimates compromise the significance of the claimed EFE detection.  We further show analytically that a qualitatively similar statistical signal is, in fact, expected in a Λ-cold dark matter (ΛCDM) universe without any violation of the strong equivalence principle.  Rather, such a signal arises simply because of the inherent correlations between galaxy clustering strength and intrinsic galaxy properties… Although the detection of an EFE-like signal is not, by itself, evidence for physics beyond GR, our work shows that the sign of the EFE-like correlation between the external field strength and the shape of the radial acceleration relation can be used to probe new physics:  e.g., in MOND, the predicted sign is opposite to that in our ΛCDM mocks.” ↩ ↩

Rodrigues, Davi C. et al.  “Absence of a fundamental acceleration scale in galaxies.”  Nature Astronomy 2:668–672.  2018.  A 2018 Bayesian statistical test (a method that weighs how well a hypothesis fits the data) of 193 disk galaxies from the SPARC and THINGS databases.  It asks whether one acceleration scale fits all of them and answers no, with a p-value below 10⁻²⁰, which the authors say rejects a common scale at more than 10σ (ten standard deviations, a measure of how unlikely a result is to be chance).  The article quotes this claim, that MOND is ruled out at more than 10σ.  Mirror:  arXiv

1. 44a.

   Section Abstract:  “Here, we consider 193 disk galaxies from the SPARC and THINGS databases and, using Bayesian inference, we show that the probability of existence of a fundamental acceleration that is common to all the galaxies is essentially zero:  the p-value is smaller than 10⁻²⁰ or, equivalently, the null hypothesis is rejected at more than 10σ.  We conclude that the acceleration scale unveiled by the Radial Acceleration Relation is of emergent nature, possibly caused by a complex interplay between baryons and dark matter.  In particular, the MOND theory, or any other theory that behaves like it at galactic scales, is ruled out as a fundamental theory for galaxies at more than 10σ.” ↩

Rodrigues, Davi C. et al.  “Reply to "Presence of a fundamental acceleration scale in galaxies" and "A common Milgromian acceleration scale in nature".”  Nature Astronomy 2:927–929.  2018.  The same authors' reply to two critics of their 2018 paper, McGaugh and colleagues and Kroupa and colleagues.  It defends their priors (the starting assumptions about each parameter) and their quality cuts (the rules for which galaxies to drop), and says the result stays above 10σ under every cut it tried.  The article cites it for the 10σ floor and for the authors' main analysis, which gave about 50σ.  Mirror:  arXiv

1. 45a.

   Figure 2, page 4:  The caption of Figure 2, on the preprint's page 4.  “b. As in Figure 1 of R18, but with an additional quality cut:  galaxies with relative error on D larger than 20% are excluded.  This leaves 59 galaxies and a fundamental acceleration is rejected at 26σ.  c. Similar to the previous case, but excluding all the galaxies with distance inferred from the Hubble flow.  This leaves 48 galaxies and a fundamental acceleration is rejected at 22σ.  d. R18 always excludes galaxies with inclination (i) less than 30°, but one can consider a stronger quality cut, such that galaxies with i − 2σ_i ≤ 40° are excluded, where σ_i is the SPARC observational error.  This leaves 76 galaxies and a fundamental acceleration is rejected at 41σ.” ↩

2. 45b.

   Page 3:  Preprint page, answering Kroupa et al.  “After a sequence of cuts on the sample from which the RAR is derived, 81% of the RAR sample is removed and still 17% of the remaining galaxies are incompatible with MOND, according to their criteria.”  A superscript reference marker after "RAR" is omitted. ↩

3. 45c.

   Page 3:  Preprint page.  “One can use an unlimited number of nuisance parameters to describe a galaxy, we selected the most relevant ones and found a result with high level of statistical significance (for all our quality cuts, it was always beyond 10σ, and for our main analysis, we find ∼ 50σ).” ↩

Kroupa, Pavel et al.  “A common Milgromian acceleration scale in nature.”  Nature Astronomy 2:925–926.  2018.  A short reply in Nature Astronomy by Kroupa and eight colleagues to Rodrigues and colleagues, who had claimed that MOND is ruled out at more than 10σ.  It argues that this rests on galaxies with very uncertain distances or nearly edge-on orientations, where dust obscures the light, and that even the few outliers left after quality cuts can be explained with Milgromian gravitation.  The article uses it as the rebuttal to the claim that MOND is ruled out.

1. 46a.

   Page 1:  Page of the preprint.  “Rodrigues et al. argue that Milgromian gravitation, which solves the flat rotation curve problem without the need for dark matter particles, is ruled out at > 10σ significance.  To a large extent, this conclusion relies on galaxies with very uncertain distances and/or nearly edge-on orientations, where dust obscuration often becomes significant.  Applying appropriate quality cuts to the data leaves only a handful of outliers to the predictions of Milgromian gravitation according to the analysis of Rodrigues et al., but even these outliers can be explained with Milgromian gravitation.” ↩

Li, Pengfei et al.  “Fitting the radial acceleration relation to individual SPARC galaxies.”  Astronomy & Astrophysics 615:A3.  2018.  A 2018 journal article that fitted the radial acceleration relation to 175 individual galaxies in the SPARC database, using a sampling method called Markov chain Monte Carlo.  It let the critical acceleration scale vary from galaxy to galaxy and found that this extra freedom did not improve the fit, so there was no credible sign that the scale varies.  The article uses it for that answer, which it sets against the claim by Rodrigues and colleagues that a single MOND constant is ruled out.  Mirror:  arXiv

1. 47a.

   Section Abstract:  “We use the Markov chain Monte Carlo method to fit the mean RAR to 175 individual galaxies in the SPARC database, marginalizing over stellar mass-to-light ratio (Υ_∗), galaxy distance, and disk inclination… We further consider a generalized version of the RAR that, unlike MOND, permits galaxy-to-galaxy variation in the critical acceleration scale.  The fits are not improved with this additional freedom:  there is no credible indication of variation in the critical acceleration scale.” ↩

Marra, Valerio et al.  “A fundamental test for MOND.”  MNRAS 494:2875–2885.  2020.  A 2020 journal paper that repeats a test of whether galaxies share one fundamental acceleration constant, a0, and finds that a shared constant is incompatible with the data at more than 5σ.  Rodrigues, a co-author here, led the 2018 version, and this paper replaces its Gaussian approximations with a Monte Carlo simulation, a method that builds the statistics from many random trials.  The article uses it because the same sample gives 26σ, 14σ and 5.3σ under three statistical treatments, which shows how much the result depends on method.  Mirror:  arXiv

1. 48a.

   Page 8:  Preprint page numbers.  “Under this approximation one can compute the confidence of rejecting the hypothesis that there is a fundamental acceleration scale using the χ² distribution.  For the sample S₂ one finds 14σ.  If we approximate the χ² distribution itself as a Gaussian, as done in R18, the confidence is 26σ.  This is due to the fact that the tail of the χ² distribution is longer than the one of the corresponding Gaussian.”  R18 is Rodrigues et al. 2018. ↩

2. 48b.

   Figure 2, page 8:  Preprint page numbers.  “The hypothesis that there is a fundamental acceleration scale is ruled out with the confidence of > 5.7σ for S₁ and of 5.3σ for S₂.”  S₁ is the 153-galaxy sample passing the quality cut used to build the radial acceleration relation; S₂ is the 91 galaxies that also pass a stronger statistical cut. ↩

3. 48c.

   Section Conclusions, page 9:  Preprint page numbers.  “In this way, we have confirmed that there is a strong evidence against a common a₀ value:  a fundamental acceleration scale a₀ is rejected at more than 5σ.” ↩

Chang, Zhe, and Yong Zhou.  “Is there a fundamental acceleration scale in galaxies?”  MNRAS 486:1658–1666.  2019.  A reanalysis, by two researchers at the Institute of High Energy Physics in Beijing, of 175 galaxy rotation curves from the SPARC sample.  It uses the same Bayesian method as a 2018 study by Rodrigues and colleagues, which had put the probability of a fundamental acceleration scale at essentially zero, and finds the conflict greatly reduced.  The authors still say the evidence rejects MOND as a fundamental theory, and they call for more accurate observations.  The article uses it for the finding that swapping flat priors for Gaussian ones halved the 2018 result.

1. 49a.

   Page 6:  Preprint page.  The result is for the RAR-inspired interpolating function, on the 100 galaxies that passed the authors' 5σ quality criteria.  “The confidence level in rejecting MOND with Gaussian priors is nearly 25σ, about half of 50σ in R18.”  R18 is Rodrigues et al. 2018. ↩

2. 49b.

   Section Conclusions, page 6:  Preprint page.  “More accurate observations for galaxy rotation curve and baryonic mass profile are needed to exclude or confirm the existence of a fundamental acceleration scale in galaxies.”  Section Abstract:  The abstract states the verdict itself:  “However, there still exists evidence that rejects MOND as a fundamental theory.” ↩