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REVIEW 4 major objections 4 minor 70 references

The Galactic Pizza: Flat Rotation Curves in the Context of Cosmological Time-Energy Coupling

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Flat galaxy rotation curves may come from slow growth of disks over cosmic time, not dark matter.

desk verdict A new-looking mechanism for flat rotation curves, but the key constant a0 is put in by hand; worth discussing, not yet a real alternative to CDM or MOND. read the letter →

arxiv 2506.02045 v1 pith:DXGJ4NLL submitted 2025-05-31 physics.gen-ph

Artur Novais , André L. B. Ribeiro This is my paper
classification physics.gen-ph
keywords GravitationColdDarkMatterEnergySpiralGalaxiesHigh-redshiftModifiedNewtonianDynamicsrotationcurvesaccelerationscale
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that the flattening of spiral galaxy rotation curves, usually read as evidence for dark matter, can instead result from extremely slow growth of each galaxy's mass and radius over cosmic time. The claim is anchored to two cosmological conditions: the universe always has critical energy density, and its age is always the reciprocal of the Hubble parameter. Under those conditions, steady disk growth generates the transitional acceleration scale $a_0 \approx cH_0/2\pi \approx 10^{-10}$ m/s$^2$ at which rotation curves stop following the Keplerian fall-off, and it reproduces the observed flat outer velocities, the disk-mass versus rotation-speed scaling, and the radial acceleration relation. If right, it gives a third path beyond cold dark matter and modified Newtonian dynamics, with consequences for dark energy and for how much time early galaxies had to form.

What carries the argument

The load-bearing object is the ratio of the disk's mass-growth rate to its radius-growth rate, $\dot{M}_d/\dot{r}_d$. The paper assumes 'cosmological time-energy coupling' (CTEC) conditions for all epochs: the energy density equals the critical value $\rho = \rho_c$ and the Hubble parameter is the reciprocal of cosmic time, $H = 1/t$, so the internal energy of the universe, $U = \rho V$, grows linearly in $t$. For a galaxy written as $M = M_f + \dot{M}_d t$ and $r = r_f + \dot{r}_d t$ with $M_f/r_f = \dot{M}_d/\dot{r}_d$, polynomial division converts the Newtonian acceleration $GM/r^2$ into $G(\dot{M}_d/\dot{r}_d)/r$ once $r \gg r_f$. With $r \approx \dot{r}_d t_0 = \dot{r}_d/H_0$, the acceleration takes the form $a_c = G\dot{M}_d/\dot{r}_d^2 H(t)$; setting the coefficient equal to $c/2\pi$ fixes $a_0 = cH_0/2\pi$. Thus the galaxy's imperceptibly slow secular growth, rather than any unseen mass, is the mechanism that sets the empirical acceleration scale.

What would settle it

Measure $a_0(z)$ from rotation curves at redshifts $0.5 < z < 2$ using a sample of disk galaxies: this model predicts $a_0(z) = cH(z)/2\pi$, growing with redshift under the CTEC assumption $H = 1/t$, whereas MOND predicts a constant $a_0$; a measured $a_0$ that does not increase with redshift would falsify the proposed mechanism.

Watch

Extended reading notes

Core claim

The central discovery is that a Newtonian disk whose mass and radius each grow linearly with cosmic time, $M_d = \dot{M}_d t$ and $r_d = \dot{r}_d t$, carries a time-dependent critical acceleration $a_c(t) = G\dot{M}_d/\dot{r}_d^2 H(t)$. When the growth rates satisfy $G\dot{M}_d/\dot{r}_d^2 = c/2\pi$, this acceleration equals the observed scale $a_0 \approx cH_0/2\pi$ at the present epoch. Because the disk mass-to-radius ratio approaches $\dot{M}_d/\dot{r}_d$ in the outer regions, the centripetal acceleration shifts from an inverse-square to an inverse law, making circular velocities asymptotically flat. The same ratio is then used to derive the baryonic Tully-Fisher relation $M_d \propto v_f^4$, the radial acceleration relation, and the identification of the transition radius $r_d = v_f^2/a_0$. The paper interprets this galactic growth as the local manifestation of a cosmological time-energy coupling in which internal energy grows linearly with cosmic time while total energy, counting the negative gravitational contribution, remains zero.

Load-bearing premise

The growth rates of galactic mass and radius are free parameters, chosen so that $G\dot{M}_d/\dot{r}_d^2$ equals $c/2\pi$; if those rates cannot be independently measured or derived, the predicted $a_0$ is fitted rather than predicted and the central argument becomes circular.

Editorial extensions

If this is right

  • The critical acceleration is predicted to grow with cosmic time as $a_c(t) = cH(t)/2\pi$, so galaxies observed at higher redshift should show a larger transitional acceleration than local galaxies.
  • The baryonic Tully-Fisher relation follows with the coefficient $M_d = (2\pi/G c H_0)\, v_f^4$, tying the baryonic mass of a galaxy to the Hubble constant.
  • The equation of state of the cosmic fluid under CTEC is $\omega = -1/3$ with zero total energy, and the paper argues the supernova dimming can be reproduced without a cosmological constant.
  • Because the CTEC age-redshift relation gives $t = t_0/(1+z)$, high-redshift galaxies have up to several times more cosmic time to assemble, alleviating the problem of massive galaxies at $z \gtrsim 10$ while matching CDM behavior at $z \lesssim 6$.
  • A measurement of the local $a_0$ can be inverted to give the Hubble constant, $H_0 = 2\pi a_0/c$, providing an independent probe of the Hubble tension.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The predicted $a_0$ is fixed by the ratio $\dot{M}_d/\dot{r}_d^2$, and the paper does not independently measure those growth rates; checking them observationally is the sharpest test of whether the mechanism is predictive or merely fitted.
  • If the mechanism is real, precision astrometry of Milky Way disk stars over decades should eventually reveal a systematic outward radial drift of order $1$ km/s for a $10^{11}\,M_\odot$ disk, a signal accessible to next-generation astrometric surveys.
  • The same CTEC volume is much smaller than the standard observable universe, so estimates of cosmic baryon content and the 'missing baryon problem' would need to be recalculated under this model's geometry.
  • The predicted redshift growth of $a_0$ offers a clean way to discriminate this model from modified-Newtonian-dynamics theories with constant $a_0$ and from scale-invariant vacuum theories that predict a different $a_c(z)$.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper proposes that flat galaxy rotation curves and the MOND acceleration scale a0 ≈ cH0/2π can be explained by slow secular growth of galactic mass and radius, M = Mf + Mdot t and r = rf + rdot t, under CTEC assumptions (ρ = ρc and H = 1/t at all epochs). From this growth it derives acp ≈ G(Mdot/rdot^2)H(t), equates this to a0, and then uses the relation to interpret the Radial Acceleration Relation, the Baryonic Tully-Fisher Relation, the dark-energy budget, and the early appearance of massive galaxies. The central new physical claim is that a0 emerges from galactic growth rather than from dark matter or modified gravity.

Significance. If the growth rates Mdot and rdot were independently predicted and the RAR/BTFR fits were quantitative, this would be an interesting third path beyond CDM and MOND. The paper usefully collects the empirical evidence for a0 and connects it to the Hubble radius and mass, and it is candid about the limitations of CTEC-type models (Introduction). However, as written, the central step is a calibration, not a derivation: no independent physical mechanism fixes Mdot/rdot^2, and the cosmological framework is assumed rather than tested. The presentation is clear and the 'Galactic Pizza' analogy is effective, but the verification status is insufficient to support the abstract's claim.

major comments (4)
  1. [§3, Eq. (30)] The derivation of a0 is circular at its central step. Equation (12), Mdot ≈ c rdot^2/(2πG), is obtained by setting Eq. (11) equal to the empirical a0 of Eq. (2). This imposes the desired amplitude on the combination Mdot/rdot^2 rather than predicting it. The dimensional analysis in Eqs. (3)–(8) fixes only the time exponent (α − 2β = −1); the amplitude GA/B^2 remains free and is then fixed by the observed a0. Consequently, the abstract's claim that the approach 'elucidates the cosmological origins' of a0 is unsupported by Eqs. (7)–(12).
  2. [§2.3, Eq. (22)] The parameter choices in Section 3 are explicitly made to satisfy a0 = G Mdot/rdot^2 H0 = 1.1 × 10^−10 m/s^2, and the text admits that 'countless alternative combinations of Mf, Mdot, rf, and rdot' would fulfill the criteria. No independent measurement or physical mechanism fixes Mdot/rdot^2. Therefore, the agreement with the RAR data in Figure 1 is a consistency check on the assumed growth model with a calibrated constant, not an a priori prediction of a0 from galaxy growth or CTEC.
  3. [§4, Eqs. (31)–(32)] The radial-velocity profile ṙ = ṙ_d(1 − e^{−r/r_d}) is introduced without derivation; the 'feedback loop' argument preceding it is qualitative, and no dynamical equation is given that would lead to this specific exponential form. The RAR curve shown in Figure 1 therefore relies on an additional ad hoc choice beyond the growth functions of Eq. (13). In addition, the comparison with the binned data of Lelli et al. [23] is presented visually with no goodness-of-fit statistic or model error bars, which weakens the evidential weight of the fit.
  4. [§5, Eq. (40)] The CTEC premises ρ = ρc and H = 1/t at all epochs are assumed rather than derived, and the Introduction itself concedes that CTEC-type models do not yet reproduce the CMB fluctuations and structure-evolution successes of ΛCDM. Since Eq. (20) uses H = 1/t to generalize a0(t), the cosmological-origin claim rests on an unverified framework. The manuscript does not provide an independent test that would distinguish the CTEC assumption H = 1/t from the standard ΛCDM expansion history in the derivation of a0.
minor comments (4)
  1. [§5, Eq. (40)] The third expression in Eq. (40), a0 = (4/3)Gρc/(cH0), has incorrect dimensions: Gρc/(cH0) has units of inverse length, not acceleration. The same dimensional issue affects the following form with ρc RH/c^2. These expressions should be corrected or removed.
  2. [§4] Reference [55] appears in the text as 'Hoyle [ ? ]' rather than with a proper citation key and year; the placeholder should be completed.
  3. [§6] The phrase 'notorious agreement' (Section 6) should be replaced by, e.g., 'remarkable agreement' or 'notable agreement'; the current wording is not standard academic usage.
  4. [§2.4] The BTFR derivation in Eqs. (25)–(28) inserts acp = a0 into the Newtonian relation M = v^4/(G acp); this is an algebraic consequence of assuming a universal acceleration scale rather than a new prediction. If this point is intended to support the framework, it should be framed as a consistency check.

Circularity Check

3 steps flagged · score 8.0 of 10

The central a0 relation is imposed by choosing Mdot and rdot to satisfy Eq. (12), so the claimed 'emergence' of a0 is a calibration rather than a derivation.

  1. fitted input called prediction [Section 2.1, Eqs. (2), (11), (12)]
    "Thus, from this simple dimensional analysis, a Newtonian acceleration that is a function of the Hubble constant was obtained. As a final step of this exercise, one can even relate the rates of mass and radius growth by using Equations (2) and (11): ḍ_d ≈ c ḍ_d^2/(2πG)"

    Equation (2) is the empirically known MOND coincidence a0 ≈ cH0/2π, and Equation (11) is the paper's expression a0 = G Mdot/rdot^2 H0. Equating these two expressions and solving for Mdot does not derive a0; it merely imposes a0 by imposing a relation on the unmeasured growth rates Mdot and rdot. The later numerical value of a0 in Equation (30) is therefore a restatement of this imposed relation, not a prediction from first principles.

  2. fitted input called prediction [Section 3, Eq. (30)]
    "The values of M_f, ḍ_d, r_f, and Ḗ_d were chosen to comply with all the necessary conditions to make the peripheral orbital velocities flat and for the crossover acceleration to approach the observed order of magnitude a0 ∜ 10−10 ms−2. ... a0 = G ḍ_d/Ḗ_d^2 H0 = 1.1 × 10−10 m/s−2. ... Naturally, there exist countless alternative combinations of M_f, ḍ_d, r_f, and Ḗ_d that would fulfill those criteria."

    The paper explicitly states that the parameters were chosen so that the acceleration would match the observed order of magnitude, and it acknowledges that countless combinations of these parameters would work. No independent physical measurement or mechanism fixes the galaxy growth rates in the required combination. Consequently, the abstract's claim that the framework 'elucidates the cosmological origins' of a0 is unsupported: the value of a0 is inserted via the parameter choice, not derived from the growth phenomenon.

1 more flagged steps
  1. other [Section 2.4, Eqs. (27)–(29)]
    "This is the exact role that a0 plays by being a sole function of H0 and not varying locally from galaxy to galaxy. Thus, in the inner part of the disk, the high accelerations will dictate the dynamics, but in the outer regions of the spirals, the transition around a0 occurs, accounting for the flattening of the rotation curves and the BTFR, which can now be expressed in terms of the fundamental cosmological parameters: M_d = ... v^4_f"

    The BTFR section uses the already-calibrated constant a0 and the Newtonian identity M = v^4/(G a_cp) to recover the empirical M ∝ v^4 relation. Because a0 is an input obtained from Equations (2) and (12), this section does not provide an independent test of the proposed mechanism. It is a consistency exercise that propagates the calibrated value into a known empirical relation, adding no new support to the claim that a0 emerges from galaxy growth.

full rationale

The central derivation is circular in the sense of the fitted-input pattern. Equation (2) supplies the empirical target a0 ≈ cH0/2π; Equation (11) defines a0 in terms of the galaxy growth rates; Equation (12) then forces the growth rates to satisfy the target; and Equation (30) reports the target back as a derived value. The paper candidly states that the rates were 'chosen' and that many alternative combinations would work, so no independent measurement or mechanism fixes them. The CTEC thermodynamics of Section 4 is an independent derivation of U ∝ t from ρ = ρc and Ht = 1, but it does not determine the galactic Mdot and rdot; Section 2.2 only lists possible mechanisms. The RAR and BTFR sections likewise depend on the pre-assigned a0 and therefore do not add independent confirmation. The score is 8 because the paper's advertised central result, the emergence of a0 from galactic growth, reduces by construction to a calibration of Mdot and rdot.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The central claims rely on several free parameters (growth rates and initial conditions) that are calibrated to reproduce the target a0, plus strong cosmological assumptions (eternal flatness and H=1/t) that are asserted rather than proven. No new physical entities are introduced.

free parameters (7)
  • alpha, beta (power-law indices) = alpha = beta = 1
    Chosen so that alpha - 2 beta = -1, making the acceleration scale proportional to H(t) (Section 2.1, Eq. 8).
  • Mdot (galactic mass growth rate) = 4.2e23 kg/s
    Chosen so that G Mdot / rdot^2 H0 = 1.1e-10 m/s^2, reproducing a0 (Section 3, Eq. 30).
  • rdot (galactic radius growth rate) = 770 m/s
    Chosen together with Mdot to satisfy the empirical a0 relation; no independent measurement cited (Section 3, Eq. 30).
  • Mf (initial bulge mass) = 8.5e9 solar masses
    Chosen so that Mf/rf equals Mdot/rdot, satisfying the flat-rotation condition (Section 3, Eq. 30).
  • rf (initial disk radius) = 1 kpc
    Chosen together with Mf to satisfy the flat-rotation condition (Section 3, Eq. 30).
  • tau1, tau2 (growth spurt timescales) = tau1 = 500 Myr in Fig. 2
    Timescales in the evolution function Eq. (13) are chosen to match Illustris and observed high-z galaxies (Section 6, Fig. 2).
  • ti (start of galaxy evolution after Big Bang) = 200 Myr to 1 Gyr
    Varies to fit high-z galaxy formation curves (Section 6, Fig. 2).
assumptions (6)
  • standard math Newtonian gravity applies in the low-acceleration regime of galactic disks.
    Section 2 opening: 'At low accelerations, the field equations of General Relativity reduce to Newtonian dynamics.'
  • domain assumption The universe has always been spatially flat, with total density equal to critical density at all epochs.
    Section 4, Eq. (31): rho = rho_c = 3c^2 H^2 / 8 pi G is assumed to hold throughout expansion history.
  • domain assumption The cosmic age is always the reciprocal of the Hubble parameter, t = 1/H, because the Hubble radius grows at the speed of light.
    Section 4, Eq. (32): R_H = ct, H = 1/t, assumed independent of epoch.
  • ad hoc to paper Galaxy mass and radius follow the growth functions M = M_f (1 - exp(-t/tau1)) + Mdot t, r = r_f (1 - exp(-t/tau2)) + rdot t.
    Section 2.2, Eq. (13) proposes these functions without derivation from a physical model.
  • ad hoc to paper The disk boundary condition requires Mdot/rdot to become constant in the outer disk, causing flat rotation.
    Section 2.3: the feedback loop assumes this boundary condition rather than deriving it.
  • domain assumption The age-redshift relation t_source = t0/(1+z) holds for CTEC models.
    Section 6, Eq. (51), taken from ZEUS and Rh=ct models (Melia 2012; Novais & Ribeiro 2024).

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Cite this review

Pith. "Pith review of The Galactic Pizza: Flat Rotation Curves in the Context of Cosmological Time-Energy Coupling." pith.science (2026). https://pith.science/paper/DXGJ4NLL

@misc{pith2026250602045,
  author       = {Pith},
  title        = {Pith review of: The Galactic Pizza: Flat Rotation Curves in the Context of Cosmological Time-Energy Coupling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DXGJ4NLL}},
  note         = {Machine review of arXiv:2506.02045}
}
abstract

The phenomenon of augmented gravity on the scale of galaxies, conventionally attributed to dark matter halos, is shown to possibly result from the incremental growth of galactic masses and radii over time. This approach elucidates the cosmological origins of the acceleration scale $a_0\approx cH_0/2\pi\approx10^{-10}$ms$^{-2}$ at which galaxy rotation curves deviate from Keplerian behavior, with no need for new particles or modifications to the laws of gravity, i.e., it constitutes a new explanatory path beyond Cold Dark Matter (CDM) and Modified Newtonian Dynamics (MOND). Once one formally equates the energy density of the universe to the critical value ($\rho=\rho_c$) and the cosmic age to the reciprocal of the Hubble parameter ($t=H^{-1}$), independently of the epoch of observation, the result is the Zero-Energy condition for the cosmic fluid's equation of state, with key repercussions for the study of dark energy since the observables can be explained in the absence of a cosmological constant. Furthermore, this mass-energy evolution framework is able to reconcile the success of CDM models in describing structure assembly at $z\lesssim6$ with the unexpected discovery of massive objects at $z\gtrsim10$. Models that feature a strong coupling between cosmic time and energy are favored by this analysis.

Figures

Figures reproduced from arXiv: 2506.02045 by the authors.

Figure 1
Figure 1. Transition from the expected Keplerian regime into the flat rotation regime for a bulge-dominated spiral galaxy of Md = 1011M⊙. The left-hand side and central panels show, respectively, the radial and orbital velocities. The right-hand side panel shows the Radial Acceleration Relation (RAR) predicted by models that feature Cosmological Time-Energy Coupling (CTEC). The squares are the binned late-type galaxy data rep… view at source ↗
Figure 2
Figure 2. Galaxy mass evolution according to ZEUS, a CTEC-type model. The left-hand side panel shows M(t), as per Equation (13). For all curves Mf = 8.5 × 109M⊙, τ1 = 500 Myr, and M˙ d = 6.7M⊙/ yr. The only discerning parameter is ti : for the solid green curve, it takes 200 Myr for evolution to start after the big bang, changing to 500 Myr for the dotted line and 1 Gyr for the dash-dotted line. The central panel shows t(z) f… view at source ↗

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