Flat Galactic Rotation Curves Interpreted as Evidence for the Mach's Principle

The paper explores a Machian interpretation for the phenomena associated with asymptotically flat rotation curves in disk galaxies, providing an alternative to both the Lambda-Cold Dark Matter (ΛCDM) paradigm and MOdified Newtonian Dynamics (MOND). The authors argue that the MOND acceleration scale a0 is likely not a new fundamental constant, but rather a manifestation of the cosmological scales of the observable Universe (a0 ∼ c2 /Ru ∼ c4 /GMu ∼ c/tu). They rewrite the Baryonic Tully-Fisher Relation (BTFR) in a cosmological format, Mbar /Mu = (Vc /c) 4, suggesting that the transition to the non-Newtonian regime occurs when local accelerations become small enough to feel the global cosmic boundary, and showing that any Machian interpretation must fulfill a relation of this form. Furthermore, this framework naturally predicts a cosmological evolution of the BTFR normalization as the mass in the observable Universe evolves. The authors test these predictions against recent high-redshift observations (z ∼ 2 and z ∼ 5), finding that the data favor a flat, baryon-only, Λ-dominated quasi-de Sitter universe (omegam ≈ 0.05, omegaΛ ≈ 0.95) over standard MOND or ΛCDM parameters. These results suggest that flat galactic rotation curves may be interpreted as compelling evidence in favor of Mach’s Principle, in at least some of its versions.

The paper begins by noting that for half a century, since the discovery of asymptotically flat rotation curves in disk galaxies (Rubin et al. 1978), evidence has accumulated for mass-energy discrepancies in the Universe. The ΛCDM paradigm has emerged as the concordance model, but its success relies partially on extra assumptions such as the baryonic fraction, which behaves almost as a free parameter. Searches for WIMPs are not encouraging, and theorists have postulated ever more extreme properties for dark matter particles instead of searching for alternatives. MOND proposes that the law of inertia or gravity takes on a specific non-Newtonian form at accelerations well below a definite universal value a0 ≈ 1.2 × 10−10 m/s2 (Milgrom 1983). MOND works very well on the scale of galaxies, but its success appears to break down at the scales of clusters of galaxies, still requiring an extra factor of 2–3 of unobserved matter.

In §2, the authors analyze the implications of having a0 as a new universal constant. They perform dimensional analysis on a0, c, and ħ, arriving at a first set of MONDian units: TM = c/a0 = 2.5 × 10 18 s, LM = c 2/a0 = 7.5 × 10 26 m, MM = ħa0/c cubed = 4.7 × 10−70 kg, the first two being of cosmological significance (similar to the Hubble time and radius in the observable Universe) and the third one being too small to have any physical significance. Another possibility is to use Ga0, giving again a mixture of extremely small and cosmological scales. Only if a0/G, ħ, and c are chosen does it give units of the same (tiny) scale, but a0/G has a less motivated relevance. Therefore, it is very hard to argue in favor of replacing G with a0 in the list of most fundamental physical constants. On the other hand, if we choose a0, c, and G, we derive for TM, LM, and MM the fundamental scales of the observable Universe: TM = c/a0 ∼ tu, LM = c 2/a0 ∼ Ru, MM = c 4/(Ga0) ∼ Mu. In particular, for the mass MM = c 4/(Ga0) = 1.01 × 10 54 kg, the value is strikingly similar to the one required by the total mass-energy in the observable Universe to have a flat geometry (Mu ≈ 1.5 × 10 54 kg). These relations can be interpreted as a version of the coincidence a0 ≈ cH0, noticed as early as the development of MOND. The authors conclude that a0 is probably not a new fundamental constant with the same status as ħ and c, but the role of a0 (and MOND) can still be viewed as a useful prescription to describe the dynamics of galactic motions.

In §3, the authors rewrite the Baryonic Tully-Fisher relation in a cosmological form. Starting with the (deep-)MOND relation for low accelerations, gobs = sqrt(gbar · a0), when substituting gobs = V 2/R (centripetal acceleration) and gbar = GMbar/R squared, the R terms cancel out, leaving a Baryonic Tully-Fisher relation in the deep MOND limit: V 4 = GMbar a0. Assuming the link between the acceleration constant a0 and the total mass of the Universe Mu, i.e., Ga0 = c 4/Mu from Eq. 1c, leads to the simple expression: (Mbar/Mu) = (V/c) 4, which has a straightforward cosmological interpretation. Figure 1 shows the Baryonic Tully-Fisher relation rewritten in this cosmologically suggestive form, with red circles displaying the SPARC dataset of galaxies compiled by Lelli et al. (2016), and the blue curve using the standard MOND interpolating function, µ(x) = x/(1+x), expressed in terms of the acceleration ratio x = g/a0, translated into physical parameters by evaluating g at a characteristic transition radius Rt = 15 kpc. The black dashed line indicates the Newtonian regime (Mbar ∝ V 2) at high velocities, while the black dotted line represents the deep MOND limit (Mbar ∝ V 4) at low velocities. The authors note that under this interpretation, the Newtonian limit (M ∝ V 2) arises at high enough accelerations where local dynamics fully dominate over the cosmological boundary, while the deep MOND regime appears when motions are dominated by the cosmological boundary. In this framework, a0 is not a fundamental constant of nature like the speed of light; it is only a scaling constant that tells us when a local system has accelerations slow enough to finally ‘feel the edge of the Universe’.

In §3.1, the authors discuss examples of Machian theories that predict a Tully-Fisher relation that can be expressed like Eq. 2. They start with Sciama’s inertial induction, which proposed that a particle’s inertia is actually a gravitational inductive effect. In Sciama’s theory, the relationship between the gravitational constant G, the mass of the observable Universe Mu, and its radius Ru is defined by GMu/(Ru c 2) ≈ 1. If we consider the ‘Machian’ acceleration to be the characteristic gravitational acceleration produced by the mass of the Universe at its own boundary, the Hubble scale, we can define a ‘cosmic acceleration’: ac = GMu/Ru squared. When we substitute Sciama’s definition of G into this acceleration, the mass Mu cancels out, leaving ac = c 2/Ru, which is the relation fulfilled in Eq. 1b by the MONDian acceleration constant a0. Moreover, in the context of Sciama’s inertial induction it can be shown (Giné 2009) that this leads to the same MONDian interpolation function µ(x) = x/(1+x) used in Figure 1. Nevertheless, it is straightforward that Eq. 3 implies a cosmological variation for the constant of gravitation G, since Ru and Mu vary across cosmological time. Since modern measurements restrict G variations to a level of Ġ/G < 10−12, it is not worthy to discuss in further detail these Machian theories in which inertia is derived from gravity. Another set of derivations includes those related to possible interpretations of the cosmological constant Λ, for example, those that relate Unruh radiation to a minimum acceleration constant a0 (McCulloch 2007, 2013). According to the Unruh effect, an object accelerating at a given rate does not experience ‘empty’ space; instead, it perceives the vacuum as being filled with a thermal bath of radiation. At the same time, the Universe itself has a ‘temperature’ associated with its boundary, the cosmological horizon, and this Hawking radiation onto the cosmological horizon is called the Gibbons-Hawking effect. The basis of this process lies in Unruh radiation wavelengths truncated by the cosmological horizon, since at very low accelerations (a < a0), these waves become so long that they can be truncated by the cosmic horizon. To find the point where the new behavior begins, the authors look for the threshold where local thermal effects (from moving) are equal to global thermal effects (from the Universe itself). Setting the Unruh temperature TU = ħaU/(2πckB) equal to the Gibbons-Hawking temperature TGH = ħc/(2πkB Ru), most terms cancel out, particularly the quantum (ħ) and thermodynamic (kB) ones, leaving again aU = c 2/Ru, the relation fulfilled in Eq. 1b by the acceleration constant a0. The authors summarize that using two examples with completely different physics and assumptions, any Machian derivation should give a Baryonic Tully-Fisher relation like Eq. 2, as long as it relies on a minimum (or base) acceleration that is linked to some cosmological parameter (i.e., Ru or Mu).

In §4, the authors study the cosmological implications for the Baryonic Tully-Fisher relation. They conclude that regardless of the particular interpretation, any Machian derivation must fulfill a relation of the type a0(z) ≈ c 2/Ru(z). This predicts cosmological variations in the Baryonic Tully-Fisher relation, Eq. 2, through changes in the total mass of the Universe Mu as follows: Mu(z) = c 4/(Ga0(z)), which depends on the different density parameters (omegam, omegaΛ, omegak) as a0(z) = a0 sqrt(omegam(1+z) cubed + omegak(1+z) squared + omegaΛ), where omegam is the matter density parameter (baryons and/or dark matter), omegaΛ is the dark energy (cosmological constant) parameter, omegak = 1 − (omegam + omegaΛ) is the curvature parameter, and a0 = 1.2 × 10−10 m/s squared is the local value today. This directly affects the normalization in the Baryonic Tully-Fisher relation (i.e., Mu(z) c−4 = 1/Ga0(z) in Eq. 2), predicting that the relation must evolve on cosmological timescales. The authors note that Genzel et al. (2017, 2020), later extended by Nestor-Shachar et al. (2023), reported extreme cosmological variations of dark matter fractions, in particular, the existence of strongly baryon-dominated disk galaxies at z ∼ 2, where flat rotation curves are not typically present. Milgrom (2017) argued that the ‘falling’ rotation curves in Genzel et al. (2017) occur because MOND effects had not ‘taken over’ yet, since dynamical accelerations in such z ∼ 2 samples are higher compared to ones in the nearby Universe (i.e., accelerations at half-light radii, g(R1/2) = [3 − 11] a0). Milgrom (2017) also argued that this directly challenges the idea that a0 cosmologically scales upward (proportional to (1+z) 3/2), since it implies a0(z) ∼ 5 a0 at z ∼ 2, already in the acceleration regime of the Genzel et al. (2017) sample. However, the authors note that only in a matter-only flat universe (omegam = 1, omegaΛ = 0) does a0(z) scale exactly as (1+z) 3/2, as stated in Milgrom (2017). Since there is no observational evidence that currently favors such an omegam = 1 universe, a more realistic estimation would be to assume a flat Universe with a more standard omegam = 0.05 (due to baryonic matter alone) and omegaΛ = 1 − omegam = 0.95, which gives a0(z) ∼ 1.5 a0 at z ∼ 2, therefore compatible with the ‘falling’ rotation curves seen in the Genzel et al. (2017) sample.

The authors then perform a quantitative comparison of fitted normalizations in the Baryonic Tully-Fisher relation with the prediction from Eq. 5. Since the reported values are on a log scale, Eq. 2 should be rewritten as log10 Mbar = 4 log10(Vcirc/c) + log10 Mu(z); therefore, Eq. 5 predicts that any offset in mass at a fixed circular velocity (Vcirc) is given by: ∆log10 Mbar = log10(Mu(z)/Mu(0)) = −log10 sqrt(omegam(1+z) cubed + omegak(1+z) squared + omegaΛ). In Fig. 2, they compare the prediction from Eq. 6 for different cosmologies and fitted normalizations in the Baryonic Tully-Fisher relation at different redshifts. The zero-point mass offsets at a fixed circular velocity are denoted by square (Übler et al. 2017) and triangle (Gómez-Espinoza et al. 2026) symbols. A significant cosmological variation is indeed observed for the fitted normalizations in the Baryonic Tully-Fisher relation, something inconsistent with standard MOND (where it should be fixed equal to Ga0), but expected in a Machian interpretation with Mu(z). The blue curve is the prediction implemented in Milgrom (2017) (baryonic matter-only Universe, omegam = 1, omegaΛ = 0), the yellow line is the standard ΛCDM cosmology (omegam = 0.315, omegaΛ = 0.685), and the green one represents the cosmological parameters used in this work, both compatible with Planck’s flat Universe and the absence of dark matter in the Machian interpretation, thus close to a de Sitter universe (omegam = 0.05, omegaΛ = 0.95). The authors find that at higher redshifts (z ∼ 2 and 5), data points clearly favor the cosmology consistent with this work, i.e., a quasi-de Sitter universe without dark matter, with the lower redshift point (z = 0.9) being consistent only with the flat baryon-only universe (standard ΛCDM cosmology parameters are not consistent with any of the points). Nevertheless, z = 0.9 datapoint is also not consistent with the updated model (based on CDM) in Gómez-Espinoza et al. (2026), which is based on dark matter fraction (fDM) estimations determined in the RC100 sample (Nestor-Shachar et al. 2023); therefore, z = 0.9 datapoint is also not consistent with an independent measurement within the context of the CDM paradigm. This also illustrates how fDM operates nowadays almost as an extra free parameter, without any reference to why it should vary on cosmological timescales, as long as it is convenient to fit the data under study. In summary, the authors find that a significant cosmological variation is indeed observed for the fitted normalizations in the Baryonic Tully-Fisher relation, as predicted in the Machian interpretation. Also, higher-redshift (z ∼ 2 and 5) data points clearly favor the cosmology currently consistent with this Machian scenario (no dark matter in a Λ-dominated flat universe). Moreover, there is considerable evidence at higher redshifts for the existence of strongly baryon-dominated disk galaxies (Genzel et al. 2017, 2020; Nestor-Shachar et al. 2023), challenging CDM but in agreement with the Machian interpretation for the MOND paradigm (with accelerations still higher than a0(z)).

In §5, the authors discuss the results and their implications. They note that a critical achievement of this approach is the successful prediction of the cosmological evolution of the Baryonic Tully-Fisher relation’s zero-point. When evaluated against high-redshift data up to z ∼ 5, the observed offsets significantly favor a flat, baryon-only, Λ-dominated universe (omegam ≈ 0.05, omegaΛ ≈ 0.95). While this shifts the cosmic composition from the standard ΛCDM values (omegam ∼ 0.3), reducing the matter contribution to a negligible fraction (< 5%) transforms the Universe into a quasi-de Sitter state. This aligns closely with specific definitions of Mach’s Principle, where global geometry and mass distribution directly constrain local inertial states (e.g., omega is a definite number of order unity; Mach8 in Bondi & Samuel 1996). The empirical success of MOND as a predictive algorithm on galactic scales, combined with the observed evolution of the BTFR normalization at high redshifts, motivates a shift away from the dark matter paradigm. While ΛCDM relies heavily on the dark matter fraction (fDM) operating essentially as a free parameter to reconcile galactic dynamics across epochs and scales, which is critical to its success, the Machian interpretation offers a predictive, parameter-free explanation for the coupling between baryonic matter and cosmic scales. The authors also discuss that beyond individual galactic scales, this Machian framework offers a fresh perspective on larger cosmological structures, specifically regarding Baryon Acoustic Oscillations (BAO) and galaxy clusters. While global cosmological observables like the cosmic microwave background and BAO have historically presented severe challenges to dark-matter-free models, there are enough uncertainties that there is still room for the BAO second peak to be successfully reproduced within an effective MONDian gravity framework, especially without knowing the specific Machian process that operates. Within their interpretation, this would imply that the early Universe’s plasma dynamics were already operating under the influence of an effective acceleration threshold a0(z) scaled to the boundary conditions of the nascent cosmos. Another well-known breakdown of standard MOND …

Based on the paper, here are specific improvements for AI systems:

1. Cosmological-Aware Gravitational Modeling

• Improvement: Implement a dynamic acceleration threshold a0(z) that scales with cosmic epoch, rather than treating it as a fixed constant. The AI should compute a0(z) = c2/Ru(z) using cosmological parameters (Ωm, ΩΛ, Ωk).

• Capability: An AI astrophysics simulator can now predict galaxy rotation curves at any redshift (e.g., z=2, z=5) without invoking dark matter, by adjusting the MOND interpolation function based on the universe's expansion state. It can automatically transition from Newtonian to deep-MOND regimes based on local acceleration relative to the cosmic boundary.

2. Bayesian Model Selection with Machian Priors

• Improvement: Add a prior distribution favoring flat, baryon-only, Λ-dominated cosmologies (Ωm≈0.05, ΩΛ≈0.95) when fitting galaxy dynamics, instead of defaulting to ΛCDM parameters (Ωm≈0.3). The AI should compute likelihoods for both frameworks and report Bayes factors.

• Capability: An AI data-analysis tool can now objectively compare ΛCDM vs. Machian interpretations for any galaxy dataset (e.g., SPARC, high-z samples) and automatically flag when dark matter fractions are being used as free parameters to force a fit, providing a statistical warning about overfitting.

3. Predictive Scaling for Tully-Fisher Evolution

• Improvement: Encode the relation M bar/M u = (V c/c)4 as a time-evolving normalization, where M u(z) is derived from the Friedmann equation. The AI should output predicted zero-point offsets Δlog10M bar = -log10√(Ωm(1+z)3 + Ωk(1+z)2 + ΩΛ) for any given redshift.

• Capability: An AI observational planner can now predict exactly where the Baryonic Tully-Fisher relation should sit at any redshift, allowing telescopes to target galaxies that would best discriminate between Machian and dark-matter models, optimizing observation time.

4. Anomaly Detection for Free Parameter Abuse

• Improvement: Train the AI to detect when a model (e.g., ΛCDM) adjusts a parameter like dark matter fraction (f DM) to fit individual galaxies, and flag this as a potential sign of model inadequacy. The AI should compare the number of free parameters used vs. the Machian framework's zero-parameter prediction.

• Capability: An AI review system can automatically audit published galaxy rotation curve fits, identifying cases where the missing mass is being arbitrarily tuned, and highlight those as candidates for alternative interpretations.

5. Multi-Scale Boundary Condition Integrator

• Improvement: Implement a hierarchical acceleration cutoff where local dynamics (galaxy, cluster) are coupled to the cosmic horizon via a0. The AI should compute the effective acceleration at any radius, including contributions from the Unruh/Gibbons-Hawking temperature equivalence, to handle cluster-scale discrepancies (factor 2-3) without invoking extra matter.

• Capability: An AI cosmological simulation can now model galaxy clusters by treating the intracluster medium's thermodynamic state as a modifier of the local-to-global acceleration transition, potentially resolving the cluster-scale MOND failure without dark matter.

6. Redshift-Dependent Interpolation Function Selector

• Improvement: Replace the fixed MOND interpolation function µ(x) = x/(1+x) with a redshift-aware version, where x = g/a0(z). The AI should automatically switch between Newtonian and MOND regimes based on the epoch-specific a0.

• Capability: An AI galaxy formation simulator can now produce rotation curves for high-z galaxies (e.g., z=2) that naturally show falling curves (as observed by Genzel et al.) without requiring ad-hoc dark matter fractions, because the acceleration threshold is higher at earlier epochs.

7. Conceptual Reinterpretation Engine

• Improvement: Train the AI to reframe missing mass problems as boundary condition effects. Given any dataset showing anomalous dynamics, the AI should generate alternative explanations based on global cosmic constraints (e.g., a0 c2/Ru) before defaulting to exotic particle hypotheses.

• Capability: An AI research assistant can now propose novel, testable predictions for unexplained phenomena (e.g., cluster lensing, BAO peak positions) by applying the Machian boundary condition, potentially opening new avenues for empirical validation.

8. Automated Cosmological Parameter Estimator

• Improvement: Use the BTFR evolution as a cosmological probe. The AI should fit observed galaxy rotation curves at multiple redshifts to directly estimate Ωm and ΩΛ, bypassing CMB-based constraints.

• Capability: An AI data pipeline can now derive cosmological parameters from galaxy kinematics alone, providing an independent check on Planck results and potentially revealing tensions if the universe is indeed quasi-de Sitter (Ωm≈0.05).

We explore the possibility of a Machian interpretation for the phenomena associated with asymptotically flat rotation curves in disk galaxies, providing an alternative to both the Lambda-Cold Dark Matter (CDM) paradigm and MOdified Newtonian Dynamics (MOND). We argue that the MOND acceleration scale a 0 is likely not a new fundamental constant, but rather a manifestation of the cosmological scales of the observable Universe (a 0 about c 2/R u about c 4/GM u about c/t u). We rewrite the Baryonic Tully-Fisher Relation (BTFR) in a cosmological format, M bar over M u = (V over c) 4, suggesting that the transition to the non-Newtonian regime occurs when local accelerations become small enough to feel the global cosmic boundary, and showing that any Machian interpretation must fulfill a relation of this form. Furthermore, this framework naturally predicts a cosmological evolution of the BTFR normalization as the mass in the observable Universe evolves. We test these predictions against recent high-redshift observations (z about 2 and z about 5), finding that the data favor a flat, baryon-only, -dominated quasi-de Sitter universe (m about 0.05, about 0.95) over standard MOND or CDM parameters. These results suggest that flat galactic rotation curves may be interpreted as compelling evidence in favor of Mach's Principle, in at least some of its versions.