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The deep-MOND limit -- a study in Primary vs secondary predictions

Mordehai Milgrom

TL;DR

This work analyzes the deep-MOND limit (DML) to extract primary, theory-agnostic predictions that MOND would yield across broad classes of theories, even in the absence of a FUNDAMOND. By invoking spacetime scale invariance in the DML and exploring a one-parameter family of DML models, it derives universal mass–velocity relations (MASR) like $V_ ext{∞}^4=M\, ext{A}_0$ and investigates observables such as η_V and η_Σ that should hold (to within scatter) across diverse systems. The paper also clarifies the MG vs MI dichotomy, discusses how Newtonian limits constrain DML dynamics, and provides concrete predictions for binary circular orbits and for baryonic–dynamical surface-density correlations, offering robust tests for MOND without committing to a specific FUNDAMOND. It emphasizes observational realities—proxy measurements and systematics—and shows that the predicted primary relations remain testable over wide ranges (e.g., dwarfs versus groups), strengthening MOND as a paradigm if these predictions persist. Overall, it provides a framework of theory-agnostic tests grounded in SI-derived scaling, enabling meaningful confrontation of MOND with data while guiding theory development.

Abstract

In default of a fundamental MOND theory -- a FUNDAMOND -- I advocate that, alongside searching for one, we should try to identify predictions that follow from wide classes of MOND theories, if not necessarily from all. In particular, predictions that follow from only the basic tenets of MOND -- ``primary predictions'' -- are shared by all MOND theories, and are especially valuable. Such predictions permit us to test the MOND paradigm itself, or at least large parts of it, without yet having a FUNDAMOND. Concentrating on the deep-MOND limit, I discuss examples of either type of predictions. For some examples of primary predictions, I demonstrate how they follow from the basic tenets (which I first formulate). I emphasize that even predictions that pertain to the deep-MOND limit - namely, those that concern gravitating systems that have low accelerations everywhere -- require the full set of MOND tenets, including the existence of a Newtonian limit close to the deep-MOND regime. This is because Newtonian dynamics is a unique theory that all MOND theories must tend to in the limit of high accelerations, and it strongly constrains aspects of the deep-MOND regime, if the transition between the limits is fast enough, which is one of the MOND tenets.

The deep-MOND limit -- a study in Primary vs secondary predictions

TL;DR

This work analyzes the deep-MOND limit (DML) to extract primary, theory-agnostic predictions that MOND would yield across broad classes of theories, even in the absence of a FUNDAMOND. By invoking spacetime scale invariance in the DML and exploring a one-parameter family of DML models, it derives universal mass–velocity relations (MASR) like and investigates observables such as η_V and η_Σ that should hold (to within scatter) across diverse systems. The paper also clarifies the MG vs MI dichotomy, discusses how Newtonian limits constrain DML dynamics, and provides concrete predictions for binary circular orbits and for baryonic–dynamical surface-density correlations, offering robust tests for MOND without committing to a specific FUNDAMOND. It emphasizes observational realities—proxy measurements and systematics—and shows that the predicted primary relations remain testable over wide ranges (e.g., dwarfs versus groups), strengthening MOND as a paradigm if these predictions persist. Overall, it provides a framework of theory-agnostic tests grounded in SI-derived scaling, enabling meaningful confrontation of MOND with data while guiding theory development.

Abstract

In default of a fundamental MOND theory -- a FUNDAMOND -- I advocate that, alongside searching for one, we should try to identify predictions that follow from wide classes of MOND theories, if not necessarily from all. In particular, predictions that follow from only the basic tenets of MOND -- ``primary predictions'' -- are shared by all MOND theories, and are especially valuable. Such predictions permit us to test the MOND paradigm itself, or at least large parts of it, without yet having a FUNDAMOND. Concentrating on the deep-MOND limit, I discuss examples of either type of predictions. For some examples of primary predictions, I demonstrate how they follow from the basic tenets (which I first formulate). I emphasize that even predictions that pertain to the deep-MOND limit - namely, those that concern gravitating systems that have low accelerations everywhere -- require the full set of MOND tenets, including the existence of a Newtonian limit close to the deep-MOND regime. This is because Newtonian dynamics is a unique theory that all MOND theories must tend to in the limit of high accelerations, and it strongly constrains aspects of the deep-MOND regime, if the transition between the limits is fast enough, which is one of the MOND tenets.
Paper Structure (13 sections, 39 equations, 3 figures)

This paper contains 13 sections, 39 equations, 3 figures.

Table of Contents

  1. Introduction
  2. MOND -- basic tenets
  3. Between modified gravity and modified inertia
  4. General scaling relations of the DML
  5. Family of DML models
  6. Special cases
  7. Some consequences
  8. Mass-velocity relations
  9. Binary relative velocity -- circular orbits
  10. A "universal" DML $M-\sigma$ correlation
  11. Notes on comparison with observations
  12. Correlation between baryonic and dynamical surface densities
  13. Universal DML $\Sigma_{ D}-\Sigma_{ B}$ correlation

Figures (3)

  • Figure 1: From Ref. milgrom21: Predicted correction --in MG theories -- to the benchmark value of $\eta_{ V}=4/9$, plotted vs the polytropic index $(\gamma-1)/\gamma$, for DML, isotropic polytropes, when we use $V= \sqrt{3}\sigma_{ 0}$, where $\sigma_{ 0}$ is the density-weighted, central line-of-sight velocity dispersion of the polytrope ($\sqrt{3}\sigma_{ 0}$ is its 3-D equivalent).
  • Figure 2: From Ref. milgrom19: $L_{ K}/L_{ K,\odot}$ for the groups, and $\alpha L_{ V}/L_{ V,\odot}$ for dwarfs ($\alpha=2/0.7$), plotted vs. the line-of-sight velocity dispersion. The solid line is the prediction for $\eta_{ V}=4/9$ [see the paragraph containing Eq. (\ref{['kaoper']})] and taking $\sigma=\sqrt{3}\sigma_{ \|}$ ($\sigma_{ \|}$ being the literature value of the line-of-sight, global velocity dispersion)), for $M/L_{ K}=0.7s.u$ for the groups, and for the more appropriate V-band $M/L_{ V}=2s.u.$ for the dwarfs. The dashed lines are for 0.5 (upper) and 2 (lower) times these $M/L$ values, and the dotted lines are for 0.25 (upper) and 4 (lower) times these values. Details on the sources of the data, and further details are given in Ref. milgrom19 (caption of Fig. 6 there).
  • Figure 3: $\eta_{ \Sigma}$, predicted in AQUAL and QUMOND, constructed from the baryonic and dynamical central surface densities for anisotropic, DML polytropes [vs. $(\gamma-1)/\gamma$, $\gamma$ is the polytropic index], for anisotropy ratios (from bottom to top) $\beta=-0.5,~-0.3,~-0.1,~0,~0.1,~0.2$. Dots: isothermal spheres. $\gamma\rightarrow\infty$ corresponds to homogeneous spheres for all $\beta$ values.