MDAR: Mass Discrepancy–Acceleration Relation

• MDAR is an empirical relation linking observed centripetal acceleration with baryonic-induced Newtonian acceleration in disk galaxies, highlighting its local and scale-dependent nature.
• It underpins the transition from Newtonian dynamics to MOND-like behavior, with distinct asymptotic regimes and implications for the baryonic Tully–Fisher relation.
• MDAR is reproduced in simulations under both ΛCDM and alternative dark matter frameworks, serving as a critical benchmark in debates over galaxy dynamics.

MDAR, usually the mass discrepancy–acceleration relation, is the empirical relation in disk galaxies between the observed local centripetal acceleration and the Newtonian acceleration generated by the observed baryons at the same radius. In standard rotation-curve notation,

$g(R)=\frac{V^2(R)}{R},$

while $g_N(R)$ denotes the Newtonian acceleration due to the baryonic mass distribution. The same relation can be written as a discrepancy ratio $D\equiv g/g_N$, or, in a spherical approximation, as $M_{\rm dyn}/M_{\rm bar}$ as a function of acceleration. In the recent literature it is often presented interchangeably with the radial acceleration relation (RAR), which uses $(g_{\rm bar},g_{\rm obs})$ rather than a discrepancy ratio. Its importance is conceptual as well as empirical: MOND treats the MDAR as a law-like consequence of a single acceleration scale $a_0$, whereas in $\Lambda$CDM it is generally treated as an emergent product of galaxy formation, halo structure, and baryonic physics (Milgrom, 2016, Mayer et al., 2022).

1. Definition and equivalent formulations

Historically, the MDAR is not a single plotting convention but a family of equivalent statements about the same regularity. One may plot $g/g_N$ against $g_N$, $g/g_N$ against $g_N(R)$0, or directly $g_N(R)$1 against $g_N(R)$2; these are all representations of the same underlying relation. In modern notation one also encounters $g_N(R)$3 and $g_N(R)$4, so that the RAR is simply the acceleration-space form of the MDAR. Milgrom emphasized that the 2016 SPARC-based presentation was an update of a relation that had already been “plotted and studied time and again,” rather than a conceptually new discovery (Milgrom, 2016).

The relation is local, not merely global. It is constructed radius by radius within a galaxy, rather than from a single characteristic speed or mass. That locality is what makes it more restrictive than the baryonic Tully–Fisher relation. In disk systems the discrepancy may be expressed as

$g_N(R)$5

and, in the spherical approximation used by several later analyses, as

$g_N(R)$6

or, for the “simple interpolation function” $g_N(R)$7,

$g_N(R)$8

This suggests that MDAR and RAR are best regarded as two coordinate choices for the same baryon–dynamics coupling (Mayer et al., 2022).

2. MOND interpretation and asymptotic structure

Within MOND, the MDAR is a direct consequence of three tenets: a single acceleration constant $g_N(R)$9, Newtonian correspondence at $D\equiv g/g_N$0, and scale invariance in the deep-MOND limit $D\equiv g/g_N$1. In the usual formulation,

$D\equiv g/g_N$2

with asymptotic behavior

$D\equiv g/g_N$3

and

$D\equiv g/g_N$4

Milgrom stressed that this is one of the major predicted “MOND laws,” not merely an empirical fit (Milgrom, 2016).

A widely used one-parameter fitting function for the updated SPARC relation was

$D\equiv g/g_N$5

with

$D\equiv g/g_N$6

Milgrom identified $D\equiv g/g_N$7 directly with $D\equiv g/g_N$8, and also noted that other MOND-compatible interpolation functions fit nearly as well, including

$D\equiv g/g_N$9

which corresponds to $M_{\rm dyn}/M_{\rm bar}$0. The critical content, in this view, lies not in a unique fitting formula but in the asymptotes and the single acceleration scale (Milgrom, 2016).

The low-acceleration branch is tightly linked to asymptotic flatness and the baryonic Tully–Fisher relation. For asymptotically flat rotation curves,

$M_{\rm dyn}/M_{\rm bar}$1

which implies

$M_{\rm dyn}/M_{\rm bar}$2

This is why MOND-oriented discussions treat the BTFR, asymptotically flat rotation curves, and the low-acceleration MDAR as different manifestations of the same dynamical structure (Milgrom, 2016).

3. Observational basis in disk galaxies

The principal observational benchmark in the modern discussion is the SPARC dataset. In one summary used in the MDAR debate, SPARC spans baryonic masses from

$M_{\rm dyn}/M_{\rm bar}$3

or about 4.5 orders of magnitude, and roughly 2.5 orders of magnitude in surface brightness. The low-surface-brightness and low-mass galaxies in that sample are especially important because many of them are low-acceleration systems throughout most of their extent, not merely in their outskirts. Those systems probe the deep-MOND branch globally rather than trivially through outer flat rotation curves (Milgrom, 2016).

This observational breadth is central to later methodological disputes. Milgrom argued that the most nontrivial content of the MDAR lies in the fact that low accelerations produced in very different dynamical circumstances nevertheless fall on the same $M_{\rm dyn}/M_{\rm bar}$4-versus-$M_{\rm dyn}/M_{\rm bar}$5 relation. One source of low-$M_{\rm dyn}/M_{\rm bar}$6 points is the outer, asymptotically flat part of rotation curves. Another is galaxies that are low-acceleration throughout their bulk. In his formulation, these two regimes “have nothing to do with each other, outside the framework of MOND,” which is why the agreement of both with the same relation is theoretically salient (Milgrom, 2016).

At the same time, the MDAR is a compressed diagnostic rather than a substitute for full rotation-curve analysis. Milgrom emphasized that it is a summary of the baryon–rotation-curve connection and therefore loses radial information. Distinct features in the baryonic mass distribution that generate corresponding features in $M_{\rm dyn}/M_{\rm bar}$7 can collapse into a back-and-forth excursion along the same locus in the $M_{\rm dyn}/M_{\rm bar}$8-$M_{\rm dyn}/M_{\rm bar}$9 plane. A plausible implication is that any interpretation of the MDAR alone must be read together with the richer information contained in full resolved rotation curves (Milgrom, 2016).

4. $(g_{\rm bar},g_{\rm obs})$0CDM accounts, simulations, and the question of explanation

Several $(g_{\rm bar},g_{\rm obs})$1CDM-based studies argue that the MDAR can emerge once realistic galaxy scaling relations and baryonic physics are included. A semi-empirical analysis built galaxies inside $(g_{\rm bar},g_{\rm obs})$2CDM haloes using abundance matching, observed structural relations, and either an NFW profile or a mass-dependent feedback-modified halo profile. In that framework the observed MDAR shape was reproduced, but the difference between halo prescriptions mattered: a mass-dependent density profile could account for the observed MDAR shape, whereas a universal NFW profile showed a discrepancy with dwarf galaxies with $(g_{\rm bar},g_{\rm obs})$3. The same framework reproduced the slope and normalization of the BTFR with 0.17 dex scatter (Cintio et al., 2015).

A complementary argument treats the MDAR as a consequence of self-similar CDM halo acceleration profiles plus the restricted halo mass range in which luminous disk galaxies form. On that view, the observed relation has two characteristic accelerations: $(g_{\rm bar},g_{\rm obs})$4, above which galaxies are baryon-dominated, and an effective minimum acceleration $(g_{\rm bar},g_{\rm obs})$5, below which isolated galaxies are seldom probed by kinematic tracers. The MDAR is then interpreted as a reflection of the self-similar nature of cold dark matter haloes and of the physical scales introduced by galaxy formation, rather than as evidence for a new force law (Navarro et al., 2016).

Hydrodynamical simulations push the same claim further. In the Magneticum simulations, disk galaxies at low redshift lie very close to MOND-like MDAR and RAR curves. Fitting the MDAR form

$(g_{\rm bar},g_{\rm obs})$6

at $(g_{\rm bar},g_{\rm obs})$7 gives

$(g_{\rm bar},g_{\rm obs})$8

while fitting the RAR form gives

$(g_{\rm bar},g_{\rm obs})$9

consistent with the SPARC-based value

$a_0$0

However, the same simulations infer an effective $a_0$1 that increases by a factor of approximately 3 from $a_0$2 to $a_0$3, which the authors propose as a sharper discriminator between MOND and $a_0$4CDM than the existence of the local MDAR alone (Mayer et al., 2022).

Against strong simulation-based claims, Milgrom’s rebuttal to Keller and Wadsley sharpened a methodological distinction between “consistency” and “accounting for” the MDAR. He argued that their sample contained only 18 high-mass galaxies, with baryonic masses in the range $a_0$5, overlapping only the highest decade of the observed SPARC mass range, and omitting the low-mass, globally low-acceleration galaxies that carry the most nontrivial information. He further argued that half of the simulated galaxies had unrealistic, steeply peaked rotation curves yet still lay on the high-acceleration branch, because baryon-dominated inner regions generically satisfy $a_0$6. In this criticism, lying on the MDAR is not a sufficient condition for having produced realistic galaxies or for having explained why the relation exists (Milgrom, 2016).

5. Beyond late-type disks: ellipticals, the Milky Way, and dwarfs

The MDAR has also been studied outside the canonical late-type disk context. In strong gravitational lensing, an analysis of 57 SLACS elliptical galaxies with Einstein rings found that the discrepancy between lensing mass and baryonic mass is larger when the baryonic Newtonian acceleration is smaller. The same systems show a surface-mass-density discrepancy relation, and the sample occupies the high-surface-density, low-discrepancy end of the broader acceleration relation. In that work, relativistic MOND was argued to account naturally for the lensing, dynamical, and stellar masses simultaneously (Tian et al., 2017).

Within the Milky Way, the local dynamics sharpen the problem because they supply both radial and vertical accelerations. A study of superfluid dark matter argued that a class of models designed to reproduce MOND-like or MDAR-like phenomenology can fit the radial rotation curve while overpredicting the local vertical acceleration, because the extra phonon-mediated force tracks the baryonic field and therefore boosts both directions together. Using rotation-curve data between 5 and 18 kpc and local K-dwarf tracers, the authors found a standard CDM halo preferred over the static superfluid model; their main point was not that the MDAR is false, but that any fundamental-force explanation of it must satisfy a more stringent directional test in the Milky Way (Lisanti et al., 2019).

A related Milky Way study compared MOND, Weyl conformal gravity, and the empirical MLS/RAR scaling law against inferred Galactic accelerations. In the RAR plane all three non-dark-matter descriptions performed comparably, while the authors argued that the halo acceleration relation

$a_0$7

provides a stronger test than the RAR alone. This suggests that subtractive diagnostics can separate theories that look similar in the usual $a_0$8-versus-$a_0$9 representation (Islam et al., 2019).

Dwarf-galaxy rotation curves remain especially contentious. One study argued that the MDAR, if interpreted as an exact rule mapping baryonic structure to total acceleration, is inconsistent with the observed trend between rotation-curve shape and the inner dynamical importance of baryons. In particular, the authors emphasized dwarf galaxies with slowly rising rotation curves where baryons are important in the inner regions, and concluded that either those trends arise from substantial observational systematics or the MDAR does not apply in that detailed, object-by-object sense (Santos-Santos et al., 2019).

6. Conceptual status, alternative mechanisms, and open problems

The MDAR occupies a distinctive methodological position in cosmology. Massimi treated it as one of the principal galaxy-scale phenomena underlying the “downscaling problem” for $\Lambda$0CDM: large-scale cosmology is highly successful, yet galaxy-scale regularities such as the MDAR and BTFR are difficult to explain, rather than merely retrieve, from simulations with context-sensitive feedback prescriptions. In that framework, MOND is strongest at the meso-scale of galaxies because it encodes the acceleration scale directly, whereas $\Lambda$1CDM is strongest at large scales; the central issue is therefore multi-scale model performance rather than a single decisive datum (Massimi, 2018).

Several alternative mechanisms attempt to reproduce the MDAR without modifying Newtonian gravity in the MOND sense. One line of work proposes baryon-interacting dark matter, treating the dark sector as a fluid heated or cooled through direct interactions with baryons. Under assumptions that the dark matter equation of state approximates an ideal gas, the relaxation time is of order the Jeans or dynamical time, and the heating rate scales inversely with the dark matter density, the hydrodynamical equations acquire an anisotropic scaling symmetry that reproduces the low-acceleration branch of the MDAR and related relations such as the BTFR and the central surface density relation (Famaey et al., 2017, Famaey et al., 2019). Another proposal, Refracted Gravity, replaces the acceleration threshold with a density threshold in a modified Poisson equation, yet still yields a MOND-like square-root force law in flattened low-density systems and is described as reproducing the MDAR and BTFR while giving a less exact account of the RAR (Cesare, 2023).

A different lesson comes from halo-profile studies that fit individual rotation curves. In a large SPARC-based comparison among NFW, DC14, Einasto, Burkert, and a fermionic maximum-entropy halo, the MDAR and RAR were reproduced by all tested halo models with comparable quality, whereas the detailed fits to individual rotation curves preferred cored or effectively flattened inner profiles over cuspy NFW. This suggests that the MDAR is robust observationally but comparatively weak as a discriminator of halo microphysics; the sharper discrimination comes from the full radial structure of rotation curves (Krut et al., 2023).

In current usage, the MDAR is therefore simultaneously an empirical regularity, a MOND benchmark, a simulation target, and a methodological test of what counts as explanation in galaxy dynamics. Its observational core is simple: the apparent mass discrepancy is tightly organized by baryonic acceleration. Its interpretive status remains contested: for MOND it is close to a built-in law, for $\Lambda$2CDM it is an emergent outcome that may or may not amount to a genuine explanation, and for alternative dark-sector models it is a target relation that constrains the permissible coupling between baryons and the unseen gravitating component.