Does a(0) Evolve Over Time?

The radical acceleration relation (RAR) which is implied by MOND but isn't necessary caused by MOND, holds true for all low-z observations (i.e. nearby galaxies). But this study concludes that while the RAR still holds in intermediate age galaxies (i.e. those that are farther away), that Milgrom's constant a(0) for these galaxies has a numerical value that is a factor of two greater than what it is for low-z galaxies.

The radial acceleration relation (RAR) is a tight empirical correlation between the observed radial acceleration (a_tot) and the baryonic radial acceleration (a_bar) measured across galaxy radii: these two accelerations start to deviate significantly from each other below a characteristic acceleration scale, a0. So far, observational studies of the RAR have predominantly focused on galaxies in the local Universe, leaving its evolution with cosmic time largely unexplored. 

Using high signal-to-noise data from the MUSE Hubble Ultra Deep Field survey, we investigate the RAR with a sample of 79 star-forming galaxies (complete above M* >10^8.8 Msun) at intermediate redshifts (0.33 < z <1.44). We estimate the observed intrinsic acceleration and the baryonic acceleration from a disk-halo decomposition that incorporates stellar, gas, and dark matter components, with corrections for pressure support, using 3D forward modelling. 

We find a RAR in our intermediate-z sample offset from the local relation, with a higher characteristic acceleration scale, a0(z~1) = 2.38+/-0.1* 10^-10 m/s^2, and a larger intrinsic scatter (~0.17 dex). Dividing the sample into redshift bins and refitting the RAR in each bin, we find a characteristic acceleration scale that systematically increases with z. Parametrizing the z-dependence as a0(z)= a0(0) + a1 * z, we obtain a1 = 1.59 +/- 0.1 * 10^-10 m/s^2, providing evidence for a z-evolution. 

We find similar results using various dark matter halo profiles as well as the Modified Newtonian Dynamics framework in our 3D forward modelling. Our results show that the RAR persists at intermediate redshift, with statistically significant redshift evolution of the characteristic acceleration, pointing to a possible evolution of the baryon-missing mass connection over cosmic time.

B. I. Ciocan, N. F. Bouché, J. Fensch, D. Krajnović, J. Freundlich, H. Desmond, B. Famaey, R. Techi, "MUSE-DARK III: The evolution of the radial acceleration relation at intermediate redshifts" arXiv:2604.22613 (April 24, 2026) (Accepted in A&A).

For reference z=0.33 is about 3.7 to 3.8 billion years ago, z=1 is about 7.7 to 8 billion years ago, and z=1.44 is about 9 to 10 billion years ago. The universe is about 13.8 billion years old. A variation of 0.17 dex is about ± 48%. The intrinsic scatter in the recent time SPARC galaxy sample is about ± 8% (0.034 dex), which is about is small as possible given the precision of the astronomy instrumentation involved. Milgrom's constant is about a(0) ≈ 1.2 × 10^−10 m/s^2.

Ciocan (2026), above, and the cluster data, both point to something very like MOND, except that a(0) evolves under certain circumstances to higher values. 

Missing baryonic matter (i.e. matter made up of ordinary atoms) is, at least, a partial explanation and one that could evolve other time. Indeed, it should evolve over time, because over time more baryonic matter ends up in stars, which are easy for astronomers to see, rather than interstellar gas and dust, which are hard for astronomers to see (and hence often called "missing" when it isn't seen and couldn't be seen even if it was there with current instrumentation). Still, missing baryonic matter may not be the entire explanation, because the magnitude of the change in a(0) may not be big enough, and changes in the naively measured value of Milgrom's constant shouldn't be very uniform since some galaxies are forming starts more actively than others (although this may be reflected in the greater dispersion of Milgrom's constant measurements in older samples).

Deur (who bibliography is linked in the sidebar) argues that the missing piece for cluster scale phenomena is the geometry of the mass distributions, by an appealing analogy to similar phenomena in QCD (which is attractive theoretically because in many respects gravity behaves like QCD squared). (QCD stands for quantum chromodynamics which is the Standard Model theory of the strong force that holds hadrons together and indirectly through hadron mediated forces accounts for the nuclear binding force that binds atomic nuclei together.)

Stacy McGaugh at Triton Station has another post about MOND v. dark matter particles (DM) and why the evidence favors something like MOND but the sociology of astrophysics favors dark matter particles.

The search for a final explanation of dark matter phenomena continues, and while toy-model MOND isn't the final solution, it does a remarkably good job over a very wide range of masses. McGaugh is surely right that the final solution looks a lot more like MOND than it does like most DM models, because for DM to describe the universe we see, we need a theory that explains how DM particles consistently form in a way entirely predicted by baryonic mass distribution, which contrary to protests that it has, it hasn't.

Even if a(0) changes over time, it provides a vastly smaller degree of freedom in how galaxy dynamics can vary than DM, especially if the variation is systemic between galaxies and galaxy clusters, or between galaxies over billions of years of time, and not just random.