# The Multiple Determination Strategy to Solve the Dark Matter Problem

Simon Beyne

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# The Multiple Determination Strategy to Solve the Dark Matter Problem

Simon Beyne

Centre Gilles Gaston Granger, Aix Marseille Université.
Centre de Physique Théorique, Aix Marseille Université.

## Abstract

We propose to highlight two multiple determination strategies employed in the dark matter problem, that determining the matter density parameter $\Omega_m \approx 0,3$ of the standard model and that determining the acceleration parameter $a_0 \approx 1,2 \times 10^{-10}m/s^2$ of the modified gravity model MOND. The no-coincidence argument justifies each of these strategies: it would be unreasonable to believe that independent measurements converge to the same value by error or coincidence. The independent nature of these determination procedures is questioned in this article. For each of the two parameters, we emphasise that there is a procedure that is based on a "new/independent epistemic content" against the other procedures. In this sense, both strategies (coming a priori from competing paradigms) are a success and oblige us to admit that both parameters have an inevitable heuristic potential.

## Key words:

Dark matter, multiple determination, Standard Model matter density parameter, MOND model, theoretical independence.

*This version of the article is a preprint. The article was accepted by the European Journal for Philosophy of Science on 12/08/2025.*
The matter density parameter $\Omega_m$ is a robust element of the standard model of cosmology (ΛCDM) and astrophysical measurements. Dark matter, in this contemporary sense, emerges by noting that the density of matter known from particle physics (baryonic matter $\Omega_b$) is less than $\Omega_m$. Measurements of a high density of matter $\Omega_m$ lead us to consider the existence of this additional matter ($\Omega_{DM} = \Omega_m - \Omega_b$).

Our knowledge of the dark matter problem is consolidated by a multiple determination strategy. Each procedure for determining the value $\Omega_m \approx 0,3$ (greater than $\Omega_b \approx 0,05$) reinforces the dark matter hypothesis. Underlying this strategy is the no-coincidence argument: it would be unreasonable to believe that *independent* measurements would erroneously converge to the same value of $\Omega_m$.

Let's be clear in our designations against the ambiguity between “dark matter”, as a puzzling scientific problem, et “dark matter”, as a non-baryonic material substance. Here we mainly use the term "dark matter" as a scientific object formed from our knowledge about the *problem of dark matter*. In this scientific object called “dark matter”, there is this robust element $\Omega_m$ determined in multiple ways. This terminology allows us to broaden the framework for resolving the dark matter problem in order to include other robust elements than $\Omega_m$.

Advocates of a modified theory of gravity, which is often contrasted with the standard model based on the $\Omega_m$ parameter, have also developed a multiple determination strategy that has met with some success (Milgrom, 2020; Merritt, 2020). This concerns the value of the acceleration parameter $a_0$ of the MOND (MOdified Newtonian Dynamics) model. This model is used to describe the rotation curves of spiral galaxies, which are asymptotically flat (the velocity $V$ of galactic objects remains the same regardless of their position, instead of decreasing depending $1/\sqrt{radius}$ according to Newtonian laws and the distribution of luminous matter observed) without introducing a halo of non-baryonic matter. The parameter $a_0$ can be measured through various phenomena. The argument is the same: it would be unreasonable to believe that *independent* measurements would erroneously converge to the same value of $a_0$.
The crucial point we wish to highlight here in the study of these two multiple determination strategies is the following. In what sense are the different determination procedures independent of each other? What independence gives force to the no-coincidence argument underlying the two multiple determinations?

In particular, we consider that a determination procedure is independent of others if it is based on a "new epistemic content". An epistemic content is an element that supports the edifice of knowledge: a stated law, a model, etc. Each of these elements has this status because it has been empirically constrained. "New" here does not concern to temporality, it does not refer to a prediction of an as yet unobserved phenomenon. "New" indicate here an epistemic content that does not derive from the theoretical material used in the other procedures. We are on the same track as Elie Zahar: "a temporally new fact may have greater psychological impact than some known fact, but this, on its own, is irrelevant to the objective empirical support which it lends to a hypothesis. My re-definition of novelty amounts to the claim that in order to assess the relation between theories and empirical data within a research-programme, one has to take into account the way in which a theory is built and the problems it was designed to solve" (1973, p.103). We will therefore dissect the theoretical tools used in the different procedures for determining Ω<sub>m</sub> and a<sub>0</sub> in order to examine the link between them.

A new epistemic content is a new way, empirically consistent, of capturing a phenomenon. In this article, the equations of general relativity have a new epistemic content that captures the phenomenon of gravitational lensing (differently from Newtonian theory) and the MOND model has a new epistemic content that captures the dynamics of spiral galaxies (differently from Newtonian theory).

This paper does not break new ground in philosophical discussions regarding robustness, but uses the idea of "new epistemic content" to advance the ΛCDM/MOND debate from an epistemological point of view. The aim of this paper is to evaluate respectively the independence of the determinations of Ω<sub>m</sub> and a<sub>0</sub> through this idea, to highlight two decisive arguments. In an original way, we formulate this argument in favour of Ω<sub>m</sub> based on a "new epistemic content" and highlight what can equal it in the argument in favour of a<sub>0</sub>. In our view, it will lead to integrate Ω<sub>m</sub> and a<sub>0</sub> parameters
into a “trading zone”<fnref n="1" /> of the dark matter problem. By entering a trading zone where the flexibility of their paradigm is permitted, the physicists on one side would gain by seizing the parameter determined by the opposing side.

Our strategy is therefore not that of Patrick Duerr and William Wolf (2023), who show that the key ideas of major 20th century philosophers of science (Popper, Kuhn, Lakatos, and Laudan) finally serve the cause of ΛCDM rather than MOND. On the contrary, we wish to extract the crucial content provided by MOND and impose the success of its multiple determination strategy. Our strategy is not that of Siska De Baerdemaeker (2021) or Antonis Antoniou (2023) who anticipate the difficult multiple determination of the non-baryonic particle. Rather, our aim is to highlight a strong type of independence of the determinations of non-baryonic matter from their astrophysical dimensions.

Before doing so, we review the role of the multiple determination strategy identified in the literature on the history and philosophy of science through various case studies.

# 1. The Role of the Multiple Determination Strategy

The multiple determination of a phenomenon, object, result, etc. has always been present in the literature, but not in a central way. William Whewell and his "consilience of inductions" (1958) could give us a concrete illustration of this idea. Whewell asserts that Newton, faced with different types of phenomena (the orbits of Jupiter's satellites, planetary orbits, falling bodies, etc.) notices the same essence, the same cause (an inverse square force of attraction). Consilience, as a means of confirming our knowledge, would consist in achieving a causal unification of these different types of events.

Donald Campbell was perhaps the first to place multiple determination at the centre of his work (Campbell, 1958, 1996, 1969, 1977, Campbell & Fiske, 1959; Campbell & Cook, 1979). William Wimsatt (1981) then built on his work, preferring the term

<footnote n="1">¹ This expression refers to Galison (1997, 2010) who developed the concept of trading zone as a space where researchers from different fields make their paradigm more flexible in order to exchange and find compromises to overcome common objectives. We won't go into this concept in detail here, but simply borrow the expression, which is here opportune.</footnote>
robustness to multiple determination (here, we will not distinguish between the two terminologies).

The best known and most striking case of multiple determination is certainly that of Avogadro's number. Jean Perrin is said to have developed 13 independent procedures converging on $N_a$ (gas viscosity, Brownian motion, diffusion of dissolved bodies, disintegration of alpha particles, electron charge, etc.). The strength of this multiple determination can be seen in Perrin's conclusions:

> "I believe that it is impossible for a mind free of all prejudice to reflect on the extreme diversity of phenomena that converge towards the same result, without feeling a very strong impression, and I believe that it will henceforth be difficult to defend a hostile attitude towards molecular hypotheses with rational arguments. [...] One is seized with admiration at the miracle of such precise concordances, based on such different phenomena. That the same value can be found within each method, by varying the parameters of the experiment as much as possible, and that the numbers thus calculated unambiguously by these various methods coincide, gives molecular reality as much certainty as the principles of thermodynamics" (1913, p.111).

Based on such striking examples, multiple determination has been highlighted as a genuine strategy for scientists (Hacking, 1981, 1983; Cartwright, 1983, 1991; Franklin, 1986, 1998; Bechtel, 1990, 2002; Culp, 1994, 1995; Burian, 1997; Chalmers, 2003; Nederbragt, 2003; Staley, 2004; Weber, 2005; Woodward, 2006; Soler & al., 2012; Schickore & Coko, 2013; Kuorikoski & Marchionni, 2016; Basso, 2017; Schupbach, 2018), although it has been questioned (Hudson 1999, 2003, 2014; Stegenga 2009, 2012; Soler 2012; Boon 2012).

Catherine Allamel-Raffin and Jean-Luc Gangloff (2012), discussing a highly cited article in astronomy (Dame, & al, 2001), have shown that these physicists had to use this strategy to convince their peers. The results obtained (distribution of CO molecular clouds in the Milky Way) had to be corroborated by comparing them " with data obtained on the same regions by other teams of scientists using either radiotelescopes or other telescopes like optical, X-rays, gamma rays telescopes" (2012,
p.177). It is common in astrophysics to rely on different 'tracers'. This is the case when we trace the asymptotically flat rotation curves of galaxies (thanks to the Hα spectral lines of stars and HI spectral lines of atomic clouds), which are at the heart of the dark matter problem.

Multiple determination can be a driving force for extricating oneself from a highly entrenched paradigm. Emiliano Trizio (2012) described how the scientific community demanded electron microscopic corroboration of an unexpected biological process of endocytosis<fnref n="2" /> that had been highlighted by fluorescent microscopy. It is not surprising that the proponents of a modified gravity model, in the face of the Standard Model shared by the majority of physicists, point to the success of their multiple determination strategy.

The case studied by Alan Chalmers (2003) is a reminder of the possibility of a vicious circularity: the dislocations of molecular networks in crystals were discovered through the use of the electron microscope, whose underlying theories also remained hypothetical. This circularity was rendered non-vicious by a multitude of techniques and experimental arguments corroborating the discovery.

The question of vicious circularity also arises in the case of dark matter. When, at the beginning of the 20th century, Fritz Zwicky tried for the first time to quantify the quantity of dark matter on the scale of galaxy clusters using the virial theorem, he noted that

> "the virial theorem applied to clusters of nebulae provides a test of the validity of the inverse square law of gravitational forces. This is of fundamental interest because of the enormous distances separating the gravitational bodies whose movements are being studied" (Zwicky, 1937, p.234).

However, the validity of the dark matter hypothesis depends precisely on the validity of gravitational theory. William Vanderburgh calls this problem "the double bind of dark matter":

<footnote n="2">² Specifically, clathrin-mediated endocytosis with nutrients above 150 nm.</footnote>
"In order to evaluate the empirical adequacy of any gravitation theory at galactic and greater scales, the mass distribution in dynamical systems at those scales must first be known — but because of the astrophysical dynamical discrepancy the mass distribution is not known. In order to infer the mass distribution from the observed motions, a gravitational law must be assumed — but such a law cannot legitimately be assumed, since the very thing at issue is which gravitational law ought to be taken to apply at galactic and greater scales. " (Vanderburgh, 2003, p.824).

Our question will be whether multiple determination can be a solution for breaking a circularity that follows only one path.

Before we do that, let's complete our selection of case studies from the literature. Léna Soler's case study looks at the discovery of weak neutral currents in the 1970s, determined by a 'visual' and an 'electronic' experiment. The point is to recall the way in which one determination was motivated by the result of another:

> "For example, suppose that the visual experimenters think they have succeeded in deriving the existence of weak neutral currents from their visual experiment, and that they announce the news to the electronic experimenters. This will strongly encourage the latter to look for solutions that go in the same direction, to try hard to find conceptual interpretations and material modifications of their own electronic experiments that favor the convergence of the electronic conclusion with the visual conclusion. (...) We grasp here the possibility of a certain kind of influence that could potentially work as (what I would call a) 'convergence inducer' " (2012, p.32-33).

This "convergence inducer" would guide the search for other procedures for determining the result. However, this is not an argument that necessarily weakens the scientific credibility of the multiple determination strategy. It is normal and desirable for an isolated determination to motivate an independent determination. For example, Urbain Le Verrier's determination of the presence of Neptune *via* its gravitational effects acted as an inducer of convergence for the "visual" observation of Neptune by the astronomer Johann Galle. The particular orientation chosen by the LHC teams
towards an energy range around 125 GeV that led to the observation of the Higgs boson was influenced by the previous observations that led to the model determining the characteristics of the particles.

This does not necessarily weaken the strategy of multiple determination of the parameter Ω_m when physicists obtain it by adjusting the fit of the CMB temperature power spectrum. In fact, it is not an initially discordant result that is readjusted. A non-physicist or even a physicist who knows nothing about cosmology<fnref n="3" /> faced with the radar screen of the Planck telescope would not have a "raw value" of Ω_m which the physicist supporting the Standard Model would then adjust to Ω_m ≈ 0,3.

Recently, Klodian Coko has identified various 'dimensions' of multiple determination and assessed their epistemic force (2020). These dimensions strengthen or weaken the no-coincidence argument on which multiple determination is based. He lists: the number of determination procedures, their reliability, their plasticity (e.g. if there are free parameters), their clarity (e.g. for a visual result), their complexity (e.g. a coherent set of several values), their quality (e.g. according to the width of the uncertainties).

However, in our article, we focus on what we consider to be the decisive dimension: the independence of the determination procedures. It is the independence of the different determination procedures that underpins the effectiveness and legitimacy of this strategy. Jim Woodward writes:

> "It is common practice in many areas of science to take measurement robustness as grounds for increasing our confidence that the quantity has been measured accurately. The underlying rationale is usually taken to be something like this: The different measurement procedures are in some relevant sense or senses 'independent' of each other – they involve instruments of different design, operating according to different causal principles, they employ different assumptions to interpret the data they produce and so on. This suggests that while each procedure may be subject to various sources of error, they are unlikely to be subject to exactly the same

<footnote n="3">³ The first time they looked at their radar screen, Penzia and Wilson were unable to make sense of the observation. "When we first heard this inexplicable 'hum', we didn't understand what it meant, and we never imagined that it would be linked to the origins of the universe", said Arnau A. Penzias in a press release.</footnote>
kinds of error – the errors that may affect the operation of a mercury thermometer are likely to be at least somewhat different from the errors that may infect the electrical thermometer and so on." (2006, p.234).

To measure the epistemic strength of a multiple determination strategy, it is necessary to identify the nature of the independence of the determination procedures. In the case of the dark matter problem, we will mainly discuss the independence of the theoretical material used by the procedures.

## 2. Evaluation of the multiple determination of Ω$_m$

### *Independence of empirical material*

First of all, we would like to mention another type of independence, that of the empirical material with which the procedures are confronted. The robustness of the Standard Model of Cosmology is supported by the multiplicity of probes used to determine its parameters. There are several cosmological probes (and these are the most precise) that constrain the value of Ω$_m$ around 0.3, including the power spectrum of temperature fluctuations in the Cosmic Microwave Background (CMB), the power spectrum of matter fluctuations in the Large Scale Structure (LSS), the luminosity of Supernova Ia (SNIa), and cluster counts. Cosmologists juxtapose empirically constrained intervals of Ω$_m$ (e.g., Abbott, & al., 2018). These different procedures for determining Ω$_m$ are independent in a sense. The empirical constraint contained, for example, in the observation of one SNIa is independent of that contained in the observation of the CMB. The set of constraints allows a single value Ω$_m$ ≈ 0.3 whereas each of the probes could have excluded it. The multiple determination based on this independence must be given epistemic power. Indeed, it is unreasonable to believe that the different probes converge by error towards the value Ω$_m$ ≈ 0.3. This no-coincidence must mean something. If cosmologists finally change their model, the new model will have to account for the success of this convergence.

All these determination procedures are dependent on the validity of the hypotheses and gravitational theory on which the Standard Model is based. This is the
main argument of those who do not recognise the success of this multiple
determination.

Nevertheless, below cosmological dimension where we have only one solid
model, there are procedures for determining the matter density of astrophysical objects
that are not all based on common theoretical material. More precisely, the
determination procedures that depend on the validity of Einstein's gravitational theory
are based on a 'new epistemic content' compared with those that depend on the validity
of Newton's gravitational theory. This is why, to deploy our argument, we focus our
analysis on the 'astrophysical' (rather than 'cosmological') determination procedures
of $\Omega_m \approx 0.3$.

## Dependence on the law of Newtonian dynamics

We can count three determinations of $\Omega_m \approx 0.3$ from the study of galaxy clusters:

1. The velocity dispersion of galaxies in clusters can be related to the mass parameter using the virial theorem (the relationship between kinetic energy and potential energy).

2. The intensity of X-ray radiation from clusters can be related to the mass parameter with a procedure based on thermodynamic relationships (between the frequency/intensity of the radiation, the temperature/energy of the cluster and its density), the law of perfect gases and the fundamental law of hydrostatics.

3. The value of the gravitational lensing observed and produced by the clusters can be related to the mass parameter using the equations of general relativity.

(The asymptotically flat rotation curves of galaxies, based on Newton's law of dynamics, cannot be used to determine the cosmological parameter. $\Omega_m$.)

We wish to question the dependence (or independence) of the theoretical
material involved in these determination procedures. To do this, we need to "dissect"
the theoretical elements that play a role. We can see that Newton's law of dynamics is
"hidden" in the determination procedures (1) and (2).
The relationship between kinetic energy and potential energy described by the virial theorem is not in itself dependent on Newton's laws. However, Newton's laws do appear in the formulation of kinetic energy and potential energy. Gravitational potential energy $U = -G \frac{m \times M}{R}$ comes from Newton's law of attraction $F = G \frac{m \times M}{r^2}$ (by calculating the work done by the gravitational force from an infinite distance to the distance $R$ between two bodies: $W = \int_{\infty}^{R} F dr = \int_{\infty}^{R} G \frac{m \times M}{r^2} dr = \left[-\frac{GmM}{r}\right]_{\infty}^{R}$). Kinetic energy $\frac{1}{2}mv^2$ is defined in terms of the work done between 2 points in the Newtonian force field (we use the expression for the Newtonian force in the calculation of elementary work on an infinitesimal displacement $\delta W = \vec{F} \cdot d\vec{r} = m \frac{d\vec{v}}{dt} \cdot d\vec{r} = m \frac{d\vec{r}}{dt} d\vec{v} = m \vec{v} \cdot d\vec{v}$ to express this work between two positions A and B : $W_{A \to B} = \int_{p_A}^{p_B} m \vec{v} \cdot d\vec{v} = \left[\frac{1}{2}mv^2\right]_{p_A}^{p_B}$).

The thermodynamic relations in the determination procedure (2) were determined on the basis of empirical observations independently of the Newtonian paradigm. In contrast, Newton's law of dynamics is hidden in the fundamental law of hydrostatics $\vec{\nabla}P = \rho\vec{g}$. By defining pressure $P = \frac{F_n}{A}$ where $F_n$ is the normal component of the force exerted on a surface of area $A$, we then introduce the Newtonian expression $\vec{F} = m\vec{g}$. Thus $P = \frac{mg}{A} = \frac{\rho A h g}{A} = \rho h g$ and derive this formula with respect to the height $h$ to obtain $\vec{\nabla}P = \rho\vec{g}$.

Finally, like the argument for asymptotically flat rotation curves of galaxies, the determination procedures (1) and (2) are dependent on Newton's law of dynamics. Thus, if this law is not valid on large scales, as postulated by the MOND model, these procedures for determining $\Omega_m$ will be erroneous in the same way.

## *The new epistemic content of general relativity*

We will not dispute the link between Newton's theory and Einstein's theory: by expressing Einstein's equations in certain limiting cases (of weak gravitation and static
gravitation), we find what Newton's equations describe. However, this is not enough to
rule out the possibility that the determination procedures based on these two theories
are independent. We wish to show that the procedure for determining Ωm ≈ 0,3
according to gravitational lensing is based on a "new epistemic content" of general
relativity.

Let's recall how Einstein's theory of gravitation was constructed against the
Newtonian paradigm. His theory was motivated by the search for a formalism that did
not depend on the chosen reference frame (inertial or not), by respect for Lorentz
invariance<fnref n="4" /> and for Mach's principle<fnref n="5" /> (and perhaps by the principle of equivalence<fnref n="6" />).
Taking the Newtonian limit case as a starting point was part of Einstein's strategy
(Norton, 1984; Renn & Sauer, 2007). However, Einstein's attempts (like those of
Poincaré and Minkowski; Girard, 1981) to find a relativistic generalisation of
Newtonian laws proved unsuccessful. The failures of these attempts called for a
renewal of the tools and fundamental concepts of Newton's theory. Einstein relied in
particular on a relativistic extension of the Poisson field equation. The latter is a more
general formalism than Newton's<fnref n="7" /> and generates a different physical interpretation of
gravitation. In fact, Newton's and Einstein's theories are independent in several
respects: (i) the form of the equations, (ii) the general principles postulated (iii) and
the tools and concepts used. This independence leads general relativity to incorporate
a new epistemic content against Newton's theory, particularly in terms of the
components of the metric. Newtonian dynamics, although entirely contained
phenomenologically in general relativity, is exclusively linked to the temporal
component of the metric. However, gravitational lensing (which makes it possible to
determine Ωm ≈ 0,3 in another way) involves spatial components that play no role in
the dynamics. These spatial components of the metric constitute an epistemic content

<footnote n="4">⁴ A physical quantity is Lorentz invariant if it does not change under a Lorentz transformation, which can be thought of as the relativistic equivalent of the Galilean transformation.</footnote>

<footnote n="5">⁵ According to which all the masses in the cosmos determine the inertial behaviour of bodies.</footnote>

<footnote n="6">⁶ The equivalence between two properties that are in fact identical: inertial mass (the tendency of a body to resist any variation in its state of motion) and gravitational mass (the tendency of two bodies to attract each other).</footnote>

<footnote n="7">⁷ By setting certain conditions at infinity (by making the potential tend towards a constant) for the gravitational potential described by Poisson's equation, we can recover the Newtonian equation for the gravitational field.</footnote>
of general relativity that is independent of the temporal component of the metric (and
therefore of Newtonian dynamics).

This independent epistemic content of general relativity is clearly shown by the
differences in predictions under conditions of strong-field gravity. Indeed, in this
regime, while Newton's theory would indicate a value of Ωm significantly lower than Ωm
≈ 0,3 general relativity determines a value of Ωm ≈ 0,3 in agreement with that
determined from phenomena occurring in the classical regime. In this sense, the
determination of Ωm from phenomena occurring in the strong-field regime of gravity
by general relativity is independent of the determination of Ωm from phenomena
occurring in the classical regime by Newton's theory. This independence is decisive in
the strategy of multiple determination of Ωm.

It is important to note that these two independent determination procedures are
not based on incompatible theories. General relativity and Newtonian theory do not
provide the same physical interpretation of gravitational phenomena; nevertheless,
this does not make them incompatible theories. While it is true that in certain energy
and curvature regimes, the predictions of general relativity and Newtonian gravity
coincide, this occurs precisely because, in this domain of common validity, general
relativity reduces to Newton's law of gravitation. At that point, they are no longer
distinct theories.

But what interests us is the analysis of determinations of Ωm where the domains
of validity do not overlap: Newtonian dynamics and Einsteinian spatial distortion.
There can be no incompatibility between the validity of the Newtonian fundamental
principle of dynamics in the classical domain and the validity of the spatial component
of the metric of general relativity. Their validity concerns two distinct domains; these
theoretical elements are not "contradictory alternatives" (Cartwright, 1991).

Conclusion of the assessment

In the "classical regime", different procedures can be used to determine a density
of matter Ωm ≈ 0,3. However, we can imagine that the same error is contained in these
different procedures, since they all depend on the same epistemic element<fnref n="8" />, Newtonian

<footnote n="8">⁸ What Jacob Stegenga and Tarun Menon call "an inferential bottleneck" (Stegenga & Menon, 2017).</footnote>
dynamics. There remains the shadow of the most fundamental bias: the inapplicability
of Newton's theory to these scales.

In this regime, the equations of general relativity are equivalent to those of Newtonian theory. We know *a priori* that the theory of general relativity gives the same result as Newton's theory. This is why it would be completely irrelevant to engage an argument of no-coincidence by noting that general relativity makes it possible to determine the same Ωm in this framework. On the other hand, because of their independent foundations, their determination of Ωm in the strong-field diverge. The Newtonian approach determines a Ωm significantly lower than 0.3; whereas general relativity, which was not developed *ad hoc* (expressly to find a coherent Ωm), determines a Ωm consistent with the value determined in the weak field. The no-coincidence argument that can be formulated is therefore: it would be unreasonable to believe that the new content of Einstein's theory and the content of Newton's theory independently contain errors that make coincide their determinations of the value of Ωm in their respective fields of application.

### 3. Evaluation of the Multiple Determination of $a_0$

We have just evaluated the multiple determination of the matter density
parameter which is proving to be a powerful argument in favour of the non-baryonic
matter hypothesis. It is now very interesting to note that this strategy has also been
used by advocates of an alternative to the presence of non-baryonic matter. Indeed, the
physicists Mordehai Milgrom and David Merritt, members of a minority community in
favour of a modification of the theory of gravity, borrow this epistemological approach
of multiple determination (Milgrom, 2020; Merritt, 2020). They claim that there are
independent determination procedures of the value of the parameter $a_0$ essential to the
MOND model. The aim is not to repeat their discussion but to clearly extract what
seems to us to be crucial in their argument, justifying why the rest is less important.

In 1983, Milgrom presented<fnref n="9" /> the MOND model, which requires the Newtonian
law of dynamics to be "corrected" and makes it possible to describe the asymptotically
flat rotation curves of galaxies without relying on the presence of non-baryonic matter.

<footnote n="9">⁹ (Milgrom, 1983a).</footnote>
The "mondian" law of dynamics introduces an acceleration constant $a_0$ ($\approx 1,2 \times 10^{-10}m/s^2$) whose role emerges when the value of the acceleration involved approaches this constant (i.e. in the regime of weak accelerations, at the periphery of galaxies).

According to Milgrom and Merritt, the constant $a_0$ emerges from various phenomenological laws. They then present a historical parallel between their case and the convergence of phenomenological laws towards other well-established constants of nature. They list several cases: $N_\alpha$ Avogadro's number (as described in part 1), $h$ Planck's constant (determined from the blackbody spectrum, the photoelectric effect, the Balmer series, etc.), $c$ the speed of light (determined from length contraction/time dilation, the energy-impulse relation, the Schwarzschild metric); and add the mondian constant $a_0$. We will retain three phenomenological relations linked to $a_0$: *Baryonic Tully-Fisher Relation* (BTFR), *Mass Discrepancy Acceleration Relation* (MDAR) and *Central Surface Density Relation* (CSDR). From these we would obtain procedures for determining the value of the $a_0$ which converge towards $a_0 \approx 1,2 \times 10^{-10}m/s^2$ (see Merritt, 2020, p.207, for a table listing the various studies determining this value). According to this convergence, mondians "are therefore justified in considering $a_0$ as a new constant of nature" (Merritt, 2020, p.213) and there should be "new-physics principle underlying this convergence " (Milgrom, 2020, p.175) as was the case for the historical examples cited. The argument underlying these assertions would be the same as that in favour of $\Omega_{DM}$: it would be unreasonable to believe that these different relationships converge to the same value by coincidence.

We propose an evaluation of the multiple determination of $a_0$ through the question of the independence of determination procedures. It would have a strong epistemic power if a procedure were based on a "new epistemic content". We are therefore going to dissect the role of $a_0$ within the mondian theoretical structure and the phenomenological relations that highlight it.

## BTFR, Baryonic Tully-Fisher Relation

Following the first robust results of asymptotically flat rotation curves (Freeman, 1970; Rubin & Ford, 1970), Brent Tully and Richard Fischer observed in
1977¹⁰ a correlation for various galaxies between the value of their asymptotic velocity<fnref n="11" /> (determined by the spectral shift of a characteristic hydrogen line<fnref n="12" />) and their stellar mass (determined by the luminosity). Also verified by taking into account the mass of gas clouds, it became known as the Baryonic Tully Fischer Relation (BTFR): $V_∞^δ = M_{bar} × C_{BTFR}$ with $V_∞$ the asymptotic velocity, $δ$ the coefficient of the slope to be determined, $M_{bar}$ the baryonic mass and $C_{BTFR}$ a constant to be determined.

This correlation between baryonic mass and asymptotic velocity has led some physicists to envisage a dynamical model without non-baryonic matter<fnref n="13" />. In 1983¹⁴, Milgrom proposed correcting the Newtonian law of dynamics $F_i = ma$ with $a = \frac{v^2}{r}$ (he could also correct the law of attraction, $F_g = mg_N$ with $g_N = \frac{M×G}{r^2}$). G is the gravitational constant and r is the radial distance from the galactic centre. We also adopt these particular notations: a to refer to the acceleration due to inertia and $g_N$ the acceleration due to gravitation as described by Newton; in the MOND model, where there is only baryonic matter, $g_N = g_{N[bar]}$). Milgrom constrains the development of his model by the BTFR ($V_∞^δ = M_{bar} × C_{BTFR}$). Studies prior to 1983¹⁵,<fnref n="16" /> et <fnref n="17" /> "find values of $δ$ between 2.5 and 5" (1983b, p.375). We can imagine Milgrom's theoretical path, which made the bet that $δ=4$ in order to express an acceleration: $V_∞^4 = M_{bar} × C_{BTFR} \Leftrightarrow V_∞^2 = \sqrt{M_{bar} × C_{BTFR}} \Leftrightarrow \frac{v_∞^2}{r} = \sqrt{\frac{M_{bar}}{r^2} × C_{BTFR}} \Leftrightarrow a = \sqrt{g_{N[bar]} × \frac{C_{BTFR}}{G}}$. $\frac{C_{BTFR}}{G}$ is a constant of the dimension of an acceleration, the famous mondian constant that we note here $a_{02}$. In the MOND framework, the inertia of objects on the periphery of galaxies (with velocity $V_∞$) is therefore expressed as $g_{N[bar]} = \frac{a^2}{a_{02}}$ without non-baryonic matter.

More recent studies (McGaugh, 2005; Stark, McGaugh, & Swaters, 2009; McGaugh, 2011; Lelli, McGaugh, & Schombert, 2016) indeed indicate a slope $δ ≈$

<footnote n="10">¹⁰ (Tully & Fisher, 1977).</footnote>

<footnote n="11">¹¹ Or rather at the time: the characteristic speed width at 20% of the maximum flow.</footnote>

<footnote n="12">¹² The 21cm wavelength line.</footnote>

<footnote n="13">¹³ It should be noted that, to explain the great dispersion of velocities in clusters, there were early proposals to modify the Newtonian law of attraction (Finzi & Pirani, 1963) or to fix a negative cosmological constant (Forman, 1970; Jackson, 1970).</footnote>

<footnote n="14">¹⁴ (Milgrom, 1983a).</footnote>

<footnote n="15">¹⁵ (Aaronson, Huchra, & Mould, 1979).</footnote>

<footnote n="16">¹⁶ (Bottinelli, Gouguenheim, & Paturel, 1980).</footnote>

<footnote n="17">¹⁷ (Visvanathan, 1981).</footnote>
4. According to the relationship $V_\infty^4 = M_{bar} \times G \times a_{02}$ and the observation data ($V_\infty$ and $M_{bar}$), we can determine the value $a_{02} \approx 1,2 \times 10^{-10} m/s^2$ which holds for all the spiral galaxies tested. Note that this determination procedure is not based on any new epistemic content, since it was precisely on the basis of Tully and Fisher's empirical law that Milgrom developed MOND. The MOND model expresses the BTFR in its own terms. There is no no-coincidence argument to be made here.

## MDAR, Mass Discrepancy-Acceleration Relation

The BTFR and the mondian expression of inertia $g_{N[bar]} = \frac{a^2}{a_{02}}$ only concern the conditions at the periphery of galaxies, the "weak acceleration" regime. Milgrom's aim was to find a more general function to describe the radial acceleration as one move away from the galactic centre. He therefore suggested a function capable of making the transition between the classical regime where $g_{N[bar]} = a$ and the asymptotic regime of galaxies where $g_{N[bar]} = \frac{a^2}{a_{02}}$. He had to postulate the existence of a constant $a_{01}$ around which the transition between the two regimes occurs. In the classical regime where $a \gg a_{01}$, we must obtain $g_{N[bar]} = a$ and for the low acceleration regime where $a \ll a_{01}$, we must obtain $g_{N[bar]} = \frac{a^2}{a_{02}}$. The function $\mu$, which corrects the Newtonian law of dynamics, depends on the acceleration $a$ in relation to the transition acceleration $a_{01}$ (thus denoted $\mu(\frac{a}{a_{01}})$). Milgrom then made out a postulate on which his theory should be based: the transition acceleration threshold $a_{01}$ is equivalent to the acceleration constant $a_{02}$ derived from the BTFR. Given the form $\mu(x) = \frac{x}{1+x}$ we obtain a common law $g_{N[bar]} = a\mu(\frac{a}{a_{0}})$ consistent with the two regimes depending on the value of $a^{18}$. Thus, rather than describing galactic dynamics according to $g_{N[bar]} + g_{N[DM]} = a$,

<table><thead><tr><th>MOND general formula</th><th>Conditions</th><th>Value of a</th><th>Value of x = <sup>a</sup>&frasl;<sub>a<sub>0</sub></sub></th><th>Value of &mu;(x)</th><th>Particular formula</th></tr></thead><tbody><tr><td rowspan="2">F = m &times; a &times; &mu;(<sup>a</sup>&frasl;<sub>a<sub>0</sub></sub>)</td><td>Classics</td><td>a &gt;&gt; a<sub>0</sub></td><td>x &gt;&gt; 1</td><td>&mu;(x) = 1</td><td>F = m &times; a</td></tr><tr><td>Low acceleration</td><td>a &lt;&lt; a<sub>0</sub></td><td>|x| &lt;&lt; 1</td><td>&mu;(x) = x</td><td>F = m &times; <sup>a<sup>2</sup></sup>&frasl;<sub>a<sub>0</sub></sub></td></tr></tbody></table>

<footnote>18</footnote>
MOND describes it according to $g_{N[bar]} = a\mu(\frac{a}{a_0})$. Let's hear what MOND implies: *the difference described by Newton's theory between baryonic mass and 'dynamical mass'<fnref n="19" /> is a function of acceleration*. At the time, there was nothing to suggest such a relationship.

Before MOND was developed, astronomers simply knew that the asymptotic velocity $V_\infty$ was reached at a radius $r_d$. From this radius onwards, the ratio between the observed dynamics ($V$) and the dynamics imposed by the baryonic matter ($V_{bar}$) deviates from $1: (V/V_{bar})^2 \neq 1$. Attempts to modify Newton's law on the basis of a universal length scale were doomed to failure, because each galaxy has its own characteristic radius $r_d$ radius. If we measure $(V/V_{bar})$ at different radii along the rotation curve of a galaxy, make these measurements for hundreds of galaxies and plot all these measurements on a single graph (figure 1), nothing appears:

<figure><img src="image_2.png" /><figcaption>Figure 1. The absence of a characteristic radius common to all galaxies. $V_{bar}$ is the velocity attributable to the influence of the baryonic mass, $V$ is the observed velocity, and $r$ is the radius of the galaxy at which the measurements were made. The dots represent several hundred individual measurements along the rotation curves of nearly a hundred spiral galaxies. No relationship appears. Taken from (Famaey & McGaugh, 2012).</figcaption></figure>

In contrast, if we represent these same points not as a function of radius but as a function of acceleration $a = \frac{V^2}{r}$ we obtain (figure 2):

<footnote n="19">¹⁹ Mass determined from observed dynamics.</footnote>
<figure><img src="image_3.png" /><figcaption>Figure 2. The Mass Discrepancy-Acceleration Relation in spiral galaxies. a corresponds to the centripetal acceleration. The dots represent several hundred individual measurements along the rotation curves of nearly a hundred spiral galaxies. These plots form curves that can be predicted by the MOND model. Taken from (Famaey & McGaugh, 2012).</figcaption></figure>

MDAR shows that $\frac{V_{\infty}^2}{R_d} = a_0$ is the same for all spiral galaxies. As early as 1990, Robert Sanders "demonstrated that in galaxies whose rotation curves are well observed, the deviation apparently appears below a critical acceleration". He immediately concluded:

"this eliminates experimentally motivated modifications of Newton's law in which deviation from 1/r occurs beyond a fundamental length scale. [But not] Milgrom's suggestion that the force law essentially becomes 1/r below a critical acceleration (MOND)" (in the abstract).

MOND could predict how $(V/V_{bar})^2$ diverges from 1 on a scale common to all galaxies: acceleration. And this according to a single parameter: $a_0$.

The function $g_{N[bar]} = a\mu(\frac{gN[bar]}{a_0})$, constrained by the data, offers a new procedure for determining the value of $a_0 \approx 1,2 \times 10^{-10}m/s^2$ (Wu & Kroupa, 2015; McGaugh, Lelli, & Schombert, 2016; Lelli et al., 2017). This is based on a novel epistemic content which is consistent with the mondian interpretation of the BTFR: the MOND prediction of the MDAR.

## CSDR, Central Surface Density Relation

The mondian model can be used to express other phenomenological relationships. For example, it implies a limit to the gravitational influence exerted by dark matter in the competing model. Indeed, the acceleration produced by a dark
matter halo could not exceed $a_0$ (Milgrom, 2008, p.6). This is verified in most cases (Famaey & McGaugh, 2012, p.24 and 26).

Proponents of MOND present a number of phenomena like this that the model would have been able to predict. From its initial law, Milgrom (2016) is able to derive a version<fnref n="20" /> that offers a relationship between 'dynamical' and 'baryonic' 'surface densities' (an idealised situation of planar mass distributions) depending on $a_0$. This relationship is called the *Central Surface Density Relation* (CSDR). By measuring these surface densities, we can determine once again $a_0 \approx 1,2 \times 10^{-10} m/s^2$ (Donato & al., 2009; Lelli & al., 2016).

Milgrom thus describes numerous phenomenological laws in which $a_0$ appears in "independent roles" (2020, p.178), which would justify the need to involve this constant in a new theory. Merritt insists on the analogy:

> "This, perhaps, is a basis for the intuitive judgments of Perrin and Planck: namely that the convergence of the measured value of a 'constant of nature' implies a tight connection between facts that would otherwise not have been considered related" (2020, p.217).

We contest the turn their argument takes on one essential point: only the MDAR has provided a determination procedure that is independent in the strongest sense given here. Only the MDAR should count as a new epistemic content. All the other phenomenological laws, such as the CSDR, are not independent but derive from this relationship. Benoit Famaey, Justin Khoury, and Riccardo Penco (2018) state:

> "The BTFR, the variation in the shape of rotation curves with a given velocity scale, and their uniformity for a given baryonic surface density scale are all features that are captured by a single equation known as the *Mass Discrepancy-Acceleration Relation (MDAR)*" (p.4).

To make the no-coincidence argument in favour of $a_0$ striking, its multiple determination must be based on a determination procedure whose theoretical material does not derive from the theoretical material of the other procedures. The success of

<footnote n="20">²⁰ For a description of this version, see the chapter "Theory Variant T1: À Non-relativistic Lagrangian" in Merritt (2020).</footnote>
the mondian multiple determination strategy must be synthesised to the predicted
correlation between acceleration and the baryonic mass distribution profile of galaxies
(or the "conspiracy"<fnref n="21" /> between *a* and [*bar*]).

Conclusion of the assessment

We insist on the strong point of Merritt's argument: "the presence of the
Milgrom constant in the BTFR is independent of its appearance as a 'transition
acceleration' in the MDAR" (2020, p.212). We had expressly noted $a_{02}$ the constant
involved in the proportionality relation of the BTFR $V_\infty^4 \propto M_b \Rightarrow V_\infty^4 = M_b \times G \times a_{02}$;
while we called $a_{01}$ the constant of the function $\mu(\frac{a}{a_{01}})$ which ensures the transition
between the classical regime ($g_{N[bar]} = a$)) and the low acceleration regime ($g_{N[bar]} =
\frac{a^2}{a_{02}}$). Before the MDAR verification, nothing would have allowed us to assert that $a_{01} =
a_{02}$. Without MOND's audacious postulate, there would have been no reason to believe
that the constant ($a_{02}$) with respect to which the proportionality between the baryonic
mass of galaxies and their asymptotic velocity is determined is equivalent to a constant
($a_{01}$) above which the need to introduce dark matter would appear.

In other words, MOND implies a new epistemic content: a parameter common
to all galaxies capable of describing the difference between baryonic mass and
dynamical mass on the acceleration scale. This new epistemic content offers a
determination of $a_0$ (MDAR) in this sense independent of the measurement of the
BTFR constant. The no-coincidence argument that can be formulated is as follows. It
would be unreasonable to believe that the novel epistemic content of MOND is an error
that allows for an independent determination of the value of $a_0$ by coincidence.

On the other hand, determinations of $a_0$ on the basis of an epistemic content that
derives from the MDAR, as in the case of the CSDR, should not count as independent
determinations (at least in this sense of independence). Once the role of $a_0$ in the
MDAR admitted, we accept a priori the value and role of $a_0$ in the relationships arising

<footnote n="21">²¹ The term "conspiracy" between dark matter and baryonic matter is used in galactic studies literature to refer to this correlation.</footnote>
from the MDAR. These relations would remain unexplained only in a ΛCDM paradigm
that refused to include the success of this multiple determination, that is, that refused
to grant a property to dark matter that would make it possible to account for
phenomena as if $g_{N[bar]} = a\mu(\frac{a}{a_0})$. One would not need to multiply *ad hoc* hypotheses
on dark matter to account for BTFR, CSDR, etc. One would suffice: in galaxies, dark
matter behaves in such a way that its distribution is a function of $\frac{a}{a_0}$.

The strategy of multiple determination of $a_0$ is currently encountering
difficulties with objects other than galaxies. Determinations of $a_0$ at the scale of clusters
and at the cosmological scale are not convincing for a majority of physicists. Indeed,
while the no-coincidence argument in favour of $a_0$ is admittedly also based on
independence from epistemic content, it does not fit into as broad context as the no-
coincidence argument in favour of $\Omega_{DM}$.

We can thus highlight the scope for progress in the strategy of multiple
determination of $a_0$. This constant would have to play a role in a theory capable of
bringing cosmological observations into agreement<sup>22</sup>. It is in this perspective that the
supporters of MOND place their work. Constantinos Skordis and Tom Złośnik note:

> "What are the necessary phenomenological facts that any successful MOND
> theory should lead to? It must (i) return to GR (hence, Newtonian gravity)
> when $\nabla\vec{\Phi} \gg a_0$ in quasistatic situations while (ii) reproducing the MOND
> law (1) when $\nabla\vec{\Phi} \ll a_0$. It should also (iii) be in harmony with cosmological
> observations including the CMB and MPS<sup>23</sup>, (iv) reproduce the observed
> gravitational lensing of isolated objects without DM halos, and (v)
> propagate tensor mode gravitational waves (GWs) at the speed of light"
> (2021, p.1).

In this paper, Skordis and Złośnik manage to reproduce the CMB power
spectrum with a relativistic version of MOND including $a_0$. Guided by the concern to

<footnote n="22"><sup>22</sup> It should also be noted that the relationship between $a_0$ and the Hubble constant $H_0$ ($2\pi a_0 \approx cH_0$) motivates mondian to believe that $a_0$ can play a role on a cosmological scale (Bugg, 2013; Merritt, 2020, p.51; Milgrom, 2020, p.176).</footnote>

<footnote n="23"><sup>23</sup> Matter Power Spectrum.</footnote>
imitate the role of dark matter, this model would become an (extreme) variant of the ΛCDM model (Duerr & Wolf, 2023).

**Summary and Discussion**

Before assessing it for our case of dark matter, we noted that the role of the multiple determination strategy is essential, but underlying scientific practice. It needs to be more clearly explained and questioned. Multiple determination is used to ensure the robustness of a result, but it also has the capacity to convince in the face of a strong consensus or to reinforce the theoretical elements involved in the determinations. It can drive and guide empirical investigation. Its power is based on the argument of no-coincidence: it is unreasonable to believe that different determination procedures coincide on the same result from independent errors.

While it exists several "dimensions" of multiple determination that affect the quality of the no-coincidence argument (Coko, 2020), we have focused on the dimension that seems most relevant in our case: the quality of the independence of the determinations. In particular, we have analysed the independence of the epistemic contents on the basis of which the determinations are constructed.

If we consider only the procedures for determining matter density parameter $\Omega_m$ in the classical regime (velocity dispersion in clusters and hot radiation from clusters) we cannot claim independence from the underlying theoretical material. In contrast, the determination of $\Omega_m \approx 0,3$ using gravitational lensing is based on a new epistemic content against Newton's theory. In fact, this procedure is based on the spatial components of the metric of a gravitational theory with new foundations, whereas the procedures of the classical regime are based on the temporal component.

MOND's multiple determination strategy also offers different accesses to the parameter $a_0$. However, in our analysis we will only consider the independent determination of $a_0$ through the MDAR. Indeed, the mondian prediction of the MDAR implies a new epistemic content (the role of $a_{02}$ in the "conspiracy" between $a$ and $g_{bar}$) regarding the mondian interpretation of the BTFR (the role of $a_{01}$).

Thus, the multiple determination strategies of $\Omega_{DM}$ and $a_0$ are both successful.
On the one hand, it would be unreasonable to believe that the new content of Einstein's theory in the strong-field regime would make, based on errors, his determination of $\Omega_m$ via gravitational lenses coincide with the value of $\Omega_m$ which is determined by Newton's theory via phenomena taking place in the classical domain.

On the other hand, it would be unreasonable to believe that the MOND model predicted, based on errors, that the value of the BTFR proportionality constant would coincide with that of an MDAR transition constant.

Our type of no-coincidence argument can take the form of a no-miracle argument<sup>24</sup>. The success of these convergences cannot be attributed to a miracle: the multiple determinations of $\Omega_m$ and $a_0$ through ΛCDM and MOND capture something of reality. Although $\Omega_{DM}$ and $a_0$ belong to two competing theoretical frameworks that have incompatible physical interpretations, the success of the two multiple determinations is not necessarily contradictory. These theoretical frameworks and their multiple determinations are a means of capturing a parameter of what dark matter really is, but its exhaustive nature may resist them.

While these strategies do not allow us to decide on the correct physical interpretation of what dark matter is, they are nonetheless successful. They show that there is a epistemic authority to be given to both $\Omega_{DM}$ and $a_0$ beyond their current theoretical framework. These parameters have heuristic potential in the competing framework and should form part of a "trading zone" between communities seeking to make progress on the dark matter problem. The role of $a_0$ in MOND could in fact reveal a particular property of dark matter that concerns its interaction with baryons (e.g. in Famaey, Khoury et Penco, 2018). The role of $\Omega_{DM}$ in ΛCDM might actually reveal a property of space, a feature of a field, describable by a modified theory of gravity (e.g. in Skordis et Złośnik, 2021). Thus, $\Omega_{DM}$ and $a_0$ would refer to compatible dark matter realities.

<footnote n="24"><sup>24</sup> Presented in the literature as the first argument in favour of scientific realism: scientific realism is the only philosophy that does not attribute the success of science to a miracle (Putnam, 1975).</footnote>
It should be noted that the multiple determination of ΩDM covers the large scales (of clusters and cosmology); whereas the (current) limitation of the multiple determination of a₀ to the galactic scale makes it easier to "capture" mondian phenomenology by another model. In the ΛCDM model, it would suffice to grant a property to dark matter that manages to account for this relation to accommodate the entire mondian phenomenology (Famaey, Khoury & Penco, 2018; Paranjape & Sheth, 2021). The MOND paradigm is not currently capable of 'capturing' the phenomena of large scales. Duerr and Wolf (2023) argue that the mondian cosmology only has the possibility of a “parasitism” of the ΛCDM model, as “it lacks any heuristic force that may guide or inspire fellow researchers to emulate the MONDian solution to some gravitational puzzles (p.6-7)”. Nevertheless, the argument presented shows that the mondian success of the multiple determination of the a₀ parameter should oblige the competing framework to seek a role for this parameter in their lines of investigations, for it has managed to capture something of reality of dark matter.

Investigative leads granting a role to both parameters will be more likely to lead to a more satisfactory theoretical framework. Today, ΩDM and a₀ find meaning/interpretation in the ΛCDM and MOND frameworks. Tomorrow, if these frameworks evolve, or are replaced, and another meaning/interpretation is given to ΩDM and a₀, this does not make the ΛCDM and MOND grasp of dark matter wrong. On the contrary, they will have accomplished a fruitful multiple determination and will have been steps towards a more satisfactory framework. Here, the correctness of a theoretical framework is not measured by its ability to give the right physical interpretation, it is measured by its ability to have resulted in a heuristic approach, to have become a fruitful step in the history of science.
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