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REVIEW 4 major objections 3 minor 1 cited by

Why do galaxies have extended flat rotation curves?

T0 review · 4 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Extended flat rotation curves imply that galaxies form already isothermal, not that outer halos relaxed slowly.

desk verdict A good question and a creative mechanism, but the argument rests on a false dichotomy and an internal factor-of-1.5 discrepancy, so the explanation as written does not hold. read the letter →

arxiv 2412.17869 v1 pith:LJUN3F3E submitted 2024-12-21 astro-ph.GA

Bruce Hoeneisen This is my paper · ORCID
classification astro-ph.GA PACS 95.35.+d98.62.Gq
keywords flatrotationcurvesisothermalspherewarmdarkmattergalaxyformationadiabaticinvariantcollisionsweakgravitationallensinghalos
topics Dark Matter
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

The paper addresses a tension: weak-lensing measurements show galaxy rotation curves staying flat out to roughly 1 megaparsec, far beyond the expected virial radius, while the outer halo has not had time to relax. It proposes that galaxies are born isothermal: warm dark matter particles with elastic collisions keep a Maxwell-Boltzmann velocity distribution with a common mean-square radial velocity at every radius, and the growing halo simply populates more of the tail of that distribution as the universe expands. In this picture the density run $\rho(r)\propto r^{-2}$ and the flat rotation curve are initial conditions rather than late-time equilibrium products. The paper also ties the core radius to a cosmological adiabatic invariant $v_{\rm hrms}(1)\approx 406\pm69$ m/s, so the 'warmness' of dark matter becomes a measurable parameter. If correct, extended flat rotation curves no longer require slow relaxation and become a direct probe of warm dark matter.

What carries the argument

The central object is the isothermal sphere in an expanding universe: a spherical halo with density $\rho(r)=\langle v_r^2\rangle/(2\pi G r^2)$ and velocity dispersion $\langle v_r^2\rangle$ independent of $r$, whose outer radius $r_{\rm max}$ grows as the universe expands. The load-bearing identity is $H_{\rm max}r_{\rm max}\approx\sqrt{4/3}\sqrt{\langle v_r^2\rangle}$ (equation 14 of the paper), which shows that particles captured at the growing boundary enter with the energy needed to populate the tail of the Maxwell-Boltzmann distribution, so the halo stays isothermal without relaxation. A second piece is the adiabatic invariant $v_{\rm hrms}(1)$, the comoving root-mean-square thermal velocity of the warm dark matter, which sets the core radius and is measured from dwarf galaxy rotation curves.

What would settle it

Use stellar kinematics or satellite dynamics in the outer halo of an isolated galaxy with a flat rotation curve to reconstruct the dark-matter velocity distribution: if the radial velocity dispersion is not constant with radius, or the velocity ellipsoid is anisotropic, the Maxwell-Boltzmann isothermal assumption fails and with it the growing-halo mechanism. A second concrete check is the lower envelope of $v'_{\rm hrms}(1)$ from dwarf galaxy cores: if it is not approximately $406\pm69$ m/s, the cosmological adiabatic invariant is not the quantity the model requires.

Watch

Extended reading notes

Core claim

The paper claims that the observed flat circular-velocity curves imply that the galaxy halo is in thermal equilibrium even at radii where particles had no time to relax, so galaxies must have assembled already in the isothermal state. The mechanism is that the halo radius grows with the expansion of the universe, capturing particles from the surrounding medium into the tail of the Maxwell-Boltzmann distribution; because the captured particles enter with expansion velocity $H_{\rm max}r_{\rm max}\approx\sqrt{4/3}\sqrt{\langle v_r^2\rangle}$, they land in a halo with $\langle v_r^2\rangle$ independent of radius. This yields $\rho(r)\propto r^{-2}$ and a flat rotation curve without a relaxation process. The warmness of the dark matter, $v_{\rm hrms}(1)$, is identified with a cosmological adiabatic invariant, measured at about $406\pm69$ m/s from dwarf galaxy cores, and the core velocity dispersion is the same invariant contracted adiabatically. With elastic collisions all species share the same mean-square radial velocity, making the object an 'iso-$\langle v_r^2\rangle$ sphere,' while inelastic baryons migrate inward and lower $\alpha\equiv\sqrt{\langle v_{rb}^2\rangle}/\sqrt{\langle v_{rh}^2\rangle}$ in elliptical galaxies.

Load-bearing premise

The load-bearing premise is that dark matter particles collide elastically and share one isotropic Maxwell-Boltzmann velocity distribution with a common mean-square radial velocity at every radius, including the outer halo where the dynamical time exceeds the age of the universe.

Editorial extensions

If this is right

  • Flat rotation curves extending to roughly 1 Mpc follow without slow relaxation: the growing halo captures particles into the tail of the Maxwell-Boltzmann distribution.
  • The dark-matter core radius is fixed by the cosmological adiabatic invariant $v_{\rm hrms}(1)\approx 406\pm69$ m/s, so core properties of dwarf, spiral, and elliptical galaxies are tied to one warmness parameter.
  • Because baryons have inelastic collisions and lower initial thermal velocities, $\alpha$ falls below 1 in elliptical galaxies; in the fully elastic limit all species share the same $\langle v_r^2\rangle$ and the halo is an iso-$\langle v_r^2\rangle$ sphere.
  • Halos grow until they meet voids or neighboring halos, so the universe becomes filled with galaxy halos and little intergalactic medium remains.
  • Warm-dark-matter simulations must include the thermal velocity to reproduce galaxy cores, and simulations used for Lyman-$\alpha$ forest studies must reproduce the extended flat halos.

Reading between the lines

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

  • A testable extension: the same formation mechanism predicts that flat rotation curves should already be present in high-redshift galaxies whose halos formed early, so deep weak-lensing or kinematic measurements at higher redshift could distinguish this isothermal-formation picture from collisionless cold-dark-matter assembly.
  • If overlapping isothermal halos leave little intergalactic space, quasar Lyman-$\alpha$ absorption at low redshift should trace halo outskirts rather than a smooth diffuse medium; this is a concrete prediction the paper hints at but does not develop.
  • The elastic-collision requirement could be probed by a future detection of two dark-matter components with different masses: in an iso-$\langle v_r^2\rangle$ sphere they must share the same velocity dispersion, so a measured difference would rule out the mechanism while a match would support it.
  • The value $v_{\rm hrms}(1)\approx 406$ m/s, taken as the lower envelope of measured $v'_{\rm hrms}(1)$, acts as a particle-physics cross-check: if laboratory or cosmological measurements of a warm dark matter particle mass predict a different comoving thermal velocity, the identification of the adiabatic invariant with the early-universe thermal velocity would need revision.
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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 / 3 minor

Summary. The paper argues that the recently observed extended flat rotation curves of galaxies, out to roughly 1 Mpc, imply that galaxy halos are isothermal spheres in thermal equilibrium at all radii, and that dark matter particles must undergo elastic collisions. The proposed explanation is that halo formation is approximately isothermal without relaxation: the galaxy halo radius grows with the expanding universe, and particles captured at the halo boundary populate the tail of the Maxwell-Boltzmann distribution, so the halo forms already in the isothermal state. The paper then connects the dark matter core radius to a cosmological adiabatic invariant vhrms(1), and discusses the role of baryons and warm dark matter in galaxy formation.

Significance. If the central claims were correct, the paper would be highly significant: it would imply that dark matter is collisional and warm, that galaxies form in an isothermal state rather than through hierarchical relaxation, and that the ubiquity of flat rotation curves follows from a single cosmological adiabatic invariant. The manuscript also benefits from explicit hydrostatic modeling and from comparisons to a wide range of galaxy observations, including dwarf, spiral, and elliptical galaxies. However, those empirical fits are drawn from the author's previous papers, and the present note's own derivation is short and contains load-bearing gaps and inconsistencies. The significance of the intended result is high, but the support provided here is not commensurate with the claim.

major comments (4)
  1. [Section 3, Eq. (14) and the following paragraph] The dichotomy between beta=1 (collisionless, radial velocities) and beta=3 (elastic collisions, isotropic velocities) is false. A collisionless system can have isotropic velocities: the phase-space density f proportional to exp(-E/sigma^2) is a stationary solution of the collisionless Boltzmann equation for the isothermal potential Phi = 2 sigma^2 ln(r/r_c), giving rho proportional to r^-2 and a flat rotation curve. Thus flat rotation curves do not, by themselves, establish that dark matter particles have elastic collisions or that the system is in thermodynamic equilibrium. This is load-bearing because the formation mechanism in Section 3 requires the thermalization implied by the Maxwell-Boltzmann tail, and the paper's own hydrostatic equations (2) do not require collisions to admit the isothermal solution (3).
  2. [Section 3, after Eq. (14)] Equation (14) gives H_max r_max approximately sqrt(4/3) sqrt(<v_r^2>), i.e. about 1.155 sigma with sigma = sqrt(<v_r^2>), while the immediately following text states that captured particles form a galaxy in thermal equilibrium if H_max r_max approximately sqrt(3 <v_r^2>), i.e. about 1.732 sigma. The factor is roughly 1.5, which is far larger than the stated 'approximately' due to inhomogeneity or neglecting the thermal velocity at a_max. This is a quantitative inconsistency in the central capture condition, and it needs to be resolved before the proposed formation mechanism can be accepted.
  3. [Section 3] The step from 'the expansion velocity at r_max approximately equals the thermal velocity' to 'captured particles populate the tail of the Maxwell-Boltzmann distribution and the halo forms in thermal equilibrium' is asserted without derivation. The paper does not show how infalling particles, which have not yet relaxed, acquire the same mean-square radial velocity <v_r^2> as the core, nor does it give a collision rate or relaxation timescale that would justify thermalization out to radii where the dynamical time exceeds the age of the universe. This missing derivation is the core of the paper's central claim that galaxies 'must have already formed in the isothermal state.'
  4. [Section 4] The argument for the cosmological origin of the adiabatic invariant is circular as presented. Equation (11) defines v'_hrms(1) from measured galaxy quantities, and Section 3 asserts that this equals the adiabatic invariant vhrms(1) if expansion and contraction are free of relaxation. But vhrms(1) = 406 +/- 69 m/s is itself obtained from fits to dwarf galaxy rotation curves in reference [6], i.e. from the same kind of data that Eq. (11) uses. Section 4 then uses the approximate equality |r_min| approximately r_c to conclude that the measured v'_hrms(1) 'is approximately equal to' vhrms(1) and hence is cosmological. As written, the equality is built in rather than tested, so the predicted core radii do not independently confirm the cosmological interpretation.
minor comments (3)
  1. [Section 2] The text contains raw LaTeX artifacts such as '/acute.ts1s' and 'greaterorsimilar'; these should be cleaned up before publication.
  2. [Section 6] The quantity r_eq is introduced as the 'break radius' and then used in Eqs. (20)-(21), but its definition is not explicit; please state clearly how r_eq is identified in the observed density run.
  3. [Throughout] The notation <v_r^2> is used both for the isothermal mean-square radial velocity and for measured quantities; a consistent notation distinguishing theoretical parameters from fitted values would improve readability.

Circularity Check

3 steps flagged · score 8.0 of 10

The isothermal-formation mechanism assumes the Maxwell-Boltzmann tail (Eq. 15 = Eq. 6) and the 'cosmological' adiabatic invariant is fitted to dwarf rotation curves, so the central prediction reduces to its inputs.

  1. self definitional [Section 3, after Eq. (14) and Eq. (15)]
    "The particles that are captured at rmax by the growing galaxy halo form a galaxy in thermal equilibrium if the expansion velocity Hmaxrmax ≈ sqrt(3⟨v_r^2⟩). These particles populate the tail end of the Boltzmann distribution. ... In conclusion, the halo formation is approximately isothermal without the need for relaxation: the galaxy halo radius grows populating the tail of the Maxwell-Boltzmann distribution (7)."

    The energy written in Eq. (15) is E = (1/2)m·3⟨v_r^2⟩ + 2m⟨v_r^2⟩ ln(r/r_c), which is exactly the assumed isothermal-sphere energy of Eq. (6) with β=3. The claim that captured particles populate the tail of the Maxwell-Boltzmann distribution therefore assumes the very isothermal distribution the paper sets out to explain. Moreover, Eq. (14) had just derived Hmaxrmax ≈ sqrt(4/3)√⟨v_r^2⟩, whereas the capture condition requires ≈ sqrt(3⟨v_r^2⟩); the gap is bridged only by asserting the desired condition. Thus 'formed already in the isothermal state' restates the ansatz rather than deriving it from expansion and infall.

  2. fitted input called prediction [Section 4, Eq. (16) and the following paragraph]
    "A dark matter particle orbit has a distance of closest approach to the galaxy center rmin that is obtained from rmax, the transverse thermal velocity ≈ ±vhrms(1)/a_max at rmax, ... and by conservation of angular momentum: rmin = ±rc [vhrms(1)/v'_hrms(1)] f(ρc/ρmax)^{1/3}. ... So, the core radius |rmin| ≈ rc implies that the measured v'_hrms(1) in the core of a galaxy is approximately equal to adiabatic invariant vhrms(1) defined in (10), and so is indeed of cosmological origin (as argued in section 3 and in [3], and as confirmed by measurements summarized in [7])."

    Section 3 states that 'Fits to dwarf galaxy rotation curves ... obtain vhrms(1) = 406 ± 69 m/s [6]', so vhrms(1) is a fitted input from dwarf-galaxy rotation curves, not an independent external constant. Eq. (16) puts that fitted vhrms(1) into the formula for rmin for a dwarf galaxy and then reads |rmin| ≈ rc as evidence that the measured v'_hrms(1) equals the same fitted vhrms(1). The 'prediction' of the core radius is therefore a consistency check on the fit, and the claimed cosmological origin is supported by the author's own summaries ([3], [7]) rather than by a parameter-free prediction.

1 more flagged steps
  1. self citation load bearing [Section 3, paragraph after Eq. (12)]
    "At r > rc the halo of the isolated galaxy approaches the density run ρ(r) ∝ r−2 until it reaches, in our example, the mean density of the expanding universe (9) (or a void, or the halo of a neighboring galaxy). This behavior can be seen by solving hydrodynamical equations [3]."

    Reference [3] is the author's own prior hydrostatic treatment in which √⟨v_rh^2⟩ is taken independent of r, i.e. the isothermal sphere is assumed. The growing-halo mechanism therefore takes the ρ(r) ∝ r−2 profile (the flat rotation curve it is meant to explain) as input from a self-cited model, and the conclusion that the halo forms isothermal inherits that assumption rather than deriving it from the expansion of the universe and infall of particles.

full rationale

The paper contains independent observational input (Mistele et al.'s flat rotation curves, Shajib et al.'s density profiles), and some algebraic relations (e.g. Eq. 14) are derived rather than assumed, so the circularity is not total. However, the central formation mechanism is circular: the Maxwell-Boltzmann tail that is claimed to make halos 'form isothermal' is written down in Eq. (15) with the same thermal-equilibrium energy assumed in Eq. (6), and the capture condition is asserted despite contradicting the paper's own Eq. (14). In addition, the core-radius 'prediction' of Section 4 uses the fitted vhrms(1) from dwarf rotation curves as an input and then reports agreement with that same fitted value, while the cosmological-origin conclusion leans on the author's self-cited [3] and [7]. These are specific reductions of output to input, not merely self-citation or lack of consensus. A separate logical gap exists in Section 2's inference that flat rotation curves require elastic collisions, since a collisionless isotropic distribution also yields ρ ∝ r^-2; that is a correctness concern rather than a self-definitional circularity, so it does not independently change the score.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

No new particles or forces are introduced; the paper instead assigns properties (warm, collisional, elastically scattering) to existing dark matter and uses the fitted adiabatic invariant vhrms(1) as a cosmological input.

free parameters (5)
  • vhrms(1) = 406 ± 69 m/s
    Warm dark matter adiabatic invariant measured from fits to dwarf galaxy rotation curves in reference [6]; used as an input to derive core radii in Section 4 and the cosmological interpretation in Section 3.
  • alpha = sqrt(<v_rb^2>)/sqrt(<v_rh^2>) = 0.4 to 2.5 (spirals), 0.5 to 0.7 (ellipticals)
    Ratio of baryon to dark matter rms thermal velocities fitted to galaxy density runs; used in Eqs (18) to (21) to derive baryon asymptotes and stellar mass relations.
  • hydrostatic boundary conditions = sqrt(<v_rb^2>), sqrt(<v_rh^2>), rho_b(rmin), rho_h(rmin)
    The four parameters varied to minimize chi-squared between numerical integration of the hydrostatic equations and the observed density run of galaxy J1313+4615 in Figure 1.
  • relaxation factor gamma = 1 to 3
    Empirical factor introduced in Eq (22) to absorb relaxation and rotation effects in relating v'_hrms(1) to vhrms(1).
  • beta = 1 or 3
    Modeling choice for whether dark matter is collisionless (beta=1) or collisional with isotropic velocities (beta=3); the paper selects beta=3 to match flat rotation curves.
assumptions (5)
  • standard math Hydrostatic equilibrium and Newtonian gravity apply to the galaxy halo.
    Invoked in Section 2, Eq (2), to derive the isothermal sphere density run rho proportional to r^{-2}.
  • ad hoc to paper Dark matter particles have elastic collisions and velocities become isotropic, at least in the core.
    Asserted in Section 2 based on the need for beta=3 to match flat rotation curves; there is no independent evidence for collisional dark matter on galactic scales.
  • domain assumption The universe is matter-dominated with a(t) proportional to t^{2/3}.
    Used in Section 3, Eq (9) and the following paragraph, to relate rho(a) to expansion and derive vhrms(a) proportional to 1/a.
  • ad hoc to paper Particles captured at rmax by the growing halo immediately populate the tail of the Maxwell-Boltzmann distribution and form the isothermal state.
    Central mechanism stated in Section 3, with no derivation; it is the load-bearing premise of the isothermal formation claim.
  • domain assumption The Press-Schechter formalism with a gaussian window function and a WDM free-streaming cutoff describes galaxy abundances.
    Invoked in Section 6, Eq (23), to connect stellar mass M* to primordial density perturbations; relies on the author's own reference [9] for validation.

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Cite this review

Pith. "Pith review of Why do galaxies have extended flat rotation curves?." pith.science (2026). https://pith.science/paper/LJUN3F3E

@misc{pith2026241217869,
  author       = {Pith},
  title        = {Pith review of: Why do galaxies have extended flat rotation curves?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LJUN3F3E}},
  note         = {Machine review of arXiv:2412.17869}
}
abstract

Recent observations by Mistele et al. show that the circular velocity curves of isolated galaxies remain flat out to the largest radii probed so far, i.e. $\approx 1$ Mpc. The velocity decline beyond the expected virial radius is not observed. These results imply that the galaxy halo is in thermal equilibrium even at large radii where particles did not have time to relax. The galaxies must have already formed in the isothermal state. How is this possible? In the present note we try to understand the formation of galaxies with warm dark matter in the expanding universe.

Figures

Figures reproduced from arXiv: 2412.17869 by the authors.

Figure 1
Figure 1. Observed [2] and calculated densities ρtot(r), ρb(r) and ρh(r) of galaxy J1313+4615. The fitted parameters are p hv 2 rbi, p hv 2 rhi, ρb(rmin) and ρh(rmin) (or vhrms(1)) [3]. Freeing a central black hole mass MBH = 0 does not change the fit significantly. The figure defines the core radius rc and density ρc, and the radius rmax with density ρmax ≡ ρ(rmax) = Ωmρcrit in this example (or a void, or the halo of a neigh… view at source ↗
Figure 1
Figure 1. Note that in the limit α = 1 we recover the “iso-hv 2 r i sphere”. Why is α < 1 in elliptical galaxies [4]? There are two reasons. For first generation galaxies, the “baryons”, mostly hydrogen and helium, become neutral and decouple from photons at redshift zeq = 1090, and so for first generation galaxies vbrms(1) ≈ 8 m/s ≪ vhrms(1) ≈ 406 m/s. Hydrodynamical equations show that the collapsing warmer dark matter deve… view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Testing CCC+TL Cosmology with Galaxy Rotation Curves

    astro-ph.CO 2025-09 reject novelty 5.0 of 10

    A covarying-coupling-constant cosmology with a locally varying parameter X is shown to reproduce the rotation curves of seven SPARC galaxies without dark matter, by fitting a single turn-off density parameter.

Reference graph

Works this paper leans on

12 extracted references · 10 canonical work pages · cited by 1 Pith paper

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