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On the initial conditions of the $\nu$HDM cosmological model

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A Bayesian fit of the MOND-based $\nu$HDM model to Planck CMB data gives $H_0\approx56$ km/s/Mpc, $\Omega_m\approx0.5$, and new initial conditions for MOND simulations.

desk verdict Honest first Bayesian fit of νHDM to Planck, but the ΛCDM-calibrated likelihood makes 'opt-νHDM' provisional, and the model is strongly disfavored by its own objective function. read the letter →

arxiv 2506.19196 v1 pith:LD67JTAK submitted 2025-06-23 astro-ph.CO

Nick Samaras , Sebastian Grandis , Pavel Kroupa This is my paper
classification astro-ph.CO
keywords νHDMMONDsterileneutrinocosmicmicrowavebackgroundinitialconditionsstructureformationHubbleconstanttensionBayesianparameterestimation
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 $\nu$HDM cosmological model—MOND gravity plus a hot, massive sterile neutrino in place of cold dark matter—can be Bayesian-fitted to the Planck CMB angular power spectrum, and this paper claims that the converged best-fit parameters, not the previous hand-tuned ones, are the proper initial conditions for MOND structure-formation simulations. The fit yields $H_0 \approx 55.6$ km/s/Mpc, $\Omega_m \approx 0.5$, $\omega_{\nu} \approx 0.1307$, $n_s \approx 0.872$, and a sterile neutrino mass of roughly 12.2 eV/$c^2$, values that deviate strongly from $\Lambda$CDM but that the authors argue can coexist with local distance measurements through a large-scale void. From these parameters the paper recomputes the CMB spectrum, matter power spectrum, and transfer function at $z=199$, and identifies an excess of power in the transfer function that is expected to alter how galaxy clusters form. Because the Planck foreground, lensing, integrated Sachs-Wolfe, and Sunyaev-Zeldovich corrections all assume $\Lambda$CDM, the authors present the result as the first step of an iterative procedure rather than a final cosmology.

What carries the argument

The load-bearing machinery is the Bayesian posterior sampling of the $\nu$HDM parameter set ($\omega_{\nu}$, $h$, $\tau$, $n_s$, $\ln(10^{10} A_s)$, $A_{Pl}$) against the Planck likelihood, using a modified Einstein-Boltzmann solver in which the cold dark matter density is held near zero and the energy budget is carried by baryons and a massive sterile neutrino. The central output is the converged posterior and its mean values, from which the CMB power spectrum, the matter power spectrum, and the transfer function at redshift $z=199$ are recomputed. The 'excess of power' in the absolute transfer function at $k<1\,h\,\mathrm{Mpc}^{-1}$ is the feature that is expected to change structure formation relative to the previous Wittenburg et al. (2023) initial conditions.

What would settle it

Run the same fit with a model-consistent Planck likelihood in which the five correction processes (non-integrated and integrated Sachs-Wolfe, Sunyaev-Zeldovich, weak lensing, and possible early foregrounds) are recomputed for the $\nu$HDM background and for MOND structure growth: if the best-fit parameters move significantly away from $H_0 \approx 56$ km/s/Mpc and $\Omega_m \approx 0.5$, the opt-$\nu$HDM initial conditions are an artifact of $\Lambda$CDM-calibrated data. A shorter check is to run a MOND hydrodynamical simulation from the proposed transfer function and compare the predicted galaxy mass function and massive-cluster abundance against the weak-lensing cluster catalog used by Wittenburg et al. (2023).

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Extended reading notes

Core claim

The central claim is that a fully converged Bayesian fit of the $\nu$HDM model to the Planck 2018 CMB temperature power spectrum defines a distinct 'opt-$\nu$HDM' cosmology: $H_0 = 55.64 \pm 0.32$ km/s/Mpc, $\Omega_m \approx 0.495$, $\omega_{\nu} = 0.1307 \pm 0.001$, $n_s = 0.872 \pm 0.0037$, $\tau \approx 0.0375$, and a sterile neutrino rest mass of about 12.2 eV/$c^2$, with the cold-dark-matter density held at a numerical floor. The fitted CMB spectrum reproduces the observed peak positions and amplitudes more closely than the manually tuned Wittenburg et al. (2023) model, and the paper argues that its matter power spectrum and transfer function at $z=199$ should therefore replace the older initial conditions in MOND hydrodynamical calculations. The transfer function is plotted in absolute value and shows an excess of power at $k<1\,h\,\mathrm{Mpc}^{-1}$, which the authors expect to enhance structure formation and possibly relieve the late emergence of the cosmic web seen in earlier $\nu$HDM simulations.

Load-bearing premise

The Planck likelihood used to constrain the $\nu$HDM model is computed with foreground, lensing, integrated Sachs-Wolfe, and Sunyaev-Zeldovich corrections that assume a $\Lambda$CDM universe, so the fitted parameters inherit the very model the paper is trying to test; the authors state this explicitly.

Editorial extensions

If this is right

  • Future MOND hydrodynamical simulations should be seeded with the opt-$\nu$HDM matter power spectrum and transfer function of Figs. 4 and 5 rather than the hand-tuned Wittenburg et al. (2023) initial conditions.
  • The increased sterile neutrino mass ($\approx 12.2$–$13$ eV) shortens the free-streaming length relative to the 11 eV case, so neutrinos cluster on larger scales and halo neutrino fractions must be recomputed.
  • The low $H_0 \approx 56$ km/s/Mpc gives an older universe ($t_0 \approx 14.9$ Gyr), which the authors point out is consistent with some recent stellar-age bounds and with explaining the local $H_0$ tension by a giant void.
  • Later reionization ($z \approx 5.9$) lengthens the dark ages, so JWST-era observations of galaxies at $z=10$–$20$ will provide a direct test of whether this cosmology forms bound objects early enough.
  • Because the Planck likelihood corrections assume $\Lambda$CDM, the fitted parameters are only a first step; a self-consistent $\nu$HDM fit requires recalculating the five foreground/lensing/ISW/SZ corrections and repeating the analysis until the parameters stabilize.

Reading between the lines

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

  • Editorial inference: the reported statistical uncertainties on $H_0$ and $\Omega_m$ are conditional on the $\Lambda$CDM-calibrated Planck likelihood, so a MOND-consistent likelihood could shift the best-fit values by more than the quoted error bars; the model comparison should not yet be read as a fair test.
  • Editorial inference: a direct way to stress-test the fit is to free the baryon density $\omega_b$ and widen the $\omega_{\nu}$ prior, since the current fit fixes $\omega_b$ and confines $\omega_{\nu}$ to $[0.1,0.15]$; if the posterior then moves, the claimed 12.2 eV mass and the 6-parameter setup are not robust.
  • Editorial inference: plotting $|T(k)|$ hides the sign of perturbation modes; the opt-$\nu$HDM initial conditions imply a specific pattern of overdense modes turning into underdensities (and vice versa) in the neutrino fluid, which is checkable directly in the initial velocity fields of a simulation built from these spectra.
  • Editorial inference: should the iterative CMB re-analysis converge and keep $H_0$ near 56, the model predicts a universe older than about 14 Gyr, making very old stellar populations in the local Universe a genuine falsifiable demographic prediction rather than a tension.
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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

3 major / 5 minor

Summary. This paper performs a Bayesian fit of the νHDM cosmological model—ΛCDM-like expansion with baryons plus a ~12 eV thermal sterile neutrino replacing CDM, and MOND for small-scale dynamics—to the Planck 2018 CMB angular power spectrum, using CosmoSIS with the mgcamb module. The authors obtain a 'best-fit' (posterior mean) cosmology with H0≈55.6 km/s/Mpc, Ωm≈0.5, ων=0.1307, and ns≈0.872, and they present the resulting matter power spectrum and transfer function as the initial conditions for future MOND hydrodynamical simulations. The paper explicitly acknowledges that the Planck likelihood it uses is calibrated within ΛCDM for foregrounds and for ISW, SZ, and lensing corrections, and frames the work as a first step in an iterative procedure.

Significance. If the fitted parameters and derived power spectra are taken at face value, the paper delivers a concrete, reproducible prescription for the initial conditions of νHDM simulations, which would directly update the hand-tuned ICs used in Wittenburg et al. (2023). The authors are commendably candid: the abstract and Section 5.5 clearly state that the assumed Planck initial conditions are only valid for ΛCDM, and the paper openly labels the exercise as a first iteration. The numerical pipeline (CosmoSIS plus mgcamb) is standard and the derived transfer functions are falsifiable inputs for simulation codes such as MUSIC and Phantom of RAMSES. These strengths are real. However, the headline claim of an 'optimized' fit is not yet supported because the objective function is a ΛCDM-calibrated likelihood, and the same data overwhelmingly disfavor the model on the authors' own likelihood comparison.

major comments (3)
  1. [§5.5 and Abstract] The central claim that opt-νHDM constitutes an 'optimized fit' to the Planck data is load-bearing and is not yet established. As the paper itself states, the Planck likelihood incorporates corrections for the non-integrated Sachs-Wolfe effect, the integrated Sachs-Wolfe effect, the Sunyaev-Zeldovich effect, weak lensing, and unmodeled foregrounds, all computed under ΛCDM structure growth. In νHDM with MOND, late-time growth differs substantially, so the likelihood used as the objective function is not the likelihood of the model being tested. The resulting parameter values H0≈55.6, Ωm≈0.5, and ων≈0.1307 are therefore the mode of a ΛCDM-calibrated likelihood, not of the true νHDM likelihood. The paper needs either to quantify the resulting bias (e.g., by a mock-based test or by recomputing the largest corrections in a MOND-compatible framework) or to explicitly rename the result as 'the best fit under ΛCDM-calibrated Planck assumptions' throughout, including the abstract and the name 'opt-νHDM'.
  2. [Table 1] The likelihood comparison in Table 1 is presented without statistical interpretation. The reported values give Δ(-2 ln L) = 768 between ΛCDM (-826) and opt-νHDM (-1210) with the same number of free parameters (six), which corresponds to a decisive rejection of the νHDM model by the Planck data at any conventional significance threshold. The paper does not quantify this discrepancy, nor does it discuss how the 'optimized' label relates to a model that the same data disfavor so strongly. This omission is load-bearing because the derived initial conditions are meant to be used for simulations claiming to represent the νHDM scenario; if the model is effectively ruled out by the CMB, the value of those ICs as a physical prediction is questionable. At minimum, the paper should report the Δχ² or an approximate p-value and discuss the implication.
  3. [§5.2 and Fig. 5] Section 5.2 interprets the 'excess of power' in the opt-νHDM transfer function as a distinct evolution scenario that might help solve the late-time emergence of the cosmic web in νHDM. This interpretive claim is premature because the transfer function is computed from parameters fitted to the same CMB data with a ΛCDM-calibrated likelihood; the excess is thus a consequence of the fitting assumptions, not an independent prediction of the νHDM model. The paper should either temper this statement or provide a demonstration that the excess is robust to the model-inconsistency of the likelihood.
minor comments (5)
  1. [Abstract and §5.1] The abstract quotes H0≈56 and Ωm0≈0.5, while Section 5.1 gives H0=55.64±0.32 and Ωm≈0.49; the rounding should be made consistent.
  2. [Table 2 and Table 4] The text states that the 'best-fit values of the posterior' are used to reevaluate the CMB, but Table 4 reports posterior means; please clarify which quantity is used for the power spectra and transfer functions.
  3. [Figure 2] The x-axis of the convergence plot is not labeled; please specify whether it shows step number, sample index, or something else, and report a convergence diagnostic such as Gelman-Rubin R̂.
  4. [§5.3] The three ad hoc values of the sound horizon (rd = 130.05, 145.05, 160.05 Mpc) are introduced without any physical justification; please state the reasoning for this ansatz and whether any of these values is compatible with the model's background expansion.
  5. [Throughout] There are several typographical and stylistic errors, including 'enlighten' instead of 'shed light on' and the L ATEX template header in the compiled PDF; a thorough proofread is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the best-fit parameters and derived transfer functions are fit outputs from external Planck data, not equations that reduce to their own inputs.

full rationale

The paper's central claim is a Bayesian best fit of the νHDM cosmological model to the Planck CMB power spectrum using CosmoSIS. The reported quantities (H0 ≈ 55.6 km/s/Mpc, Ωm ≈ 0.5, ων ≈ 0.1307, ns ≈ 0.872) are fit outputs from an external likelihood, not derivations from the model's own outputs. The matter power spectrum and transfer function of Section 5.2 are model outputs computed from those fitted parameters; they are not used as inputs to the fit, so the 'excess of power' is a consequence of the fitted model rather than a prediction that is forced by construction. The paper explicitly acknowledges that the Planck foreground, lensing, ISW, and SZ corrections assume a ΛCDM universe (Section 5.5) and warns that the 'Planck-derived CMB power spectrum might not be a valid constraint for non-ΛCDM models'; this is an honest model-consistency limitation, not a circular reduction. Self-citations to Wittenburg et al. (2023) and Haslbauer et al. (2020a) provide prior simulation context and are not load-bearing for the likelihood calculation, which uses external Planck data and the CosmoSIS mgcamb module. The comparisons to Pantheon+ and DESI are independent external checks. No equation-level or definitional circularity can be exhibited, so the appropriate finding is no significant circularity.

Assumptions & free parameters 6 free parameters · 5 assumptions · 1 invented entities

The central fit depends on six fitted cosmological parameters, several fixed fiducial values (Ωb, Ωk, Yp, Neff), and the assumption that a ΛCDM-calibrated Planck likelihood can be applied to a MOND-based model. The only invented entity is the sterile neutrino, which the paper inherits from prior νHDM work and adjusts to a slightly higher mass.

free parameters (6)
  • ων = Ων h² (sterile neutrino density) = 0.1307 ± 0.001 (posterior mean)
    Central model parameter replacing CDM density; fitted with uniform prior U(0.1, 0.15).
  • h (H0 / 100) = 0.5564 ± 0.0032
    Drives the low H0≈55.6 km/s/Mpc result; uniform prior initially [0.5, 0.8], enlarged to [0.3, 0.8] after MCMC runs.
  • ns (scalar spectral index) = 0.872 ± 0.0037
    Strongly red-tilted compared to ΛCDM 0.965; uniform prior U(0.7, 1.0).
  • τ (optical depth to reionization) = 0.0375 ± 0.0069
    Lower than ΛCDM; uniform prior U(0.01, 0.075).
  • ln(10^10 As) = 2.987 ± 0.015
    Primordial amplitude; uniform prior U(2.0, 3.1).
  • APl (Planck calibration) = 1.0 ± 0.0025
    Tight Gaussian prior N(1, 0.0025); acts as a calibration nuisance parameter.
assumptions (5)
  • domain assumption FLRW metric, adiabatic inflationary perturbations, and the Big Bang history apply to νHDM.
    Section 1 and 3.1 state the νHDM model keeps the standard FLRW background and inflationary seed spectrum.
  • ad hoc to paper MOND does not modify the background expansion, CMB anisotropies, or linear perturbation growth at high redshift.
    Section 2.1: 'by construction, MOND does not affect the description of the distance redshift relation, of the CMB and of the evolution of linear density fluctuations at high redshifts.' This is a model-specific assumption needed to use GR-based CMB fits.
  • ad hoc to paper The Planck 2018 CMB likelihood, including foreground and lensing corrections calibrated within ΛCDM, is applicable to νHDM.
    Abstract and Section 5.5 state the foreground corrections assume ΛCDM; the paper nevertheless uses this likelihood for the central fit.
  • domain assumption The mgcamb Boltzmann solver correctly evolves linear perturbations for the νHDM model.
    Section 3.1 uses mgcamb without an independent validation against another Boltzmann code for this nonstandard model.
  • domain assumption Fixed fiducial values for baryon density, curvature, helium fraction, and Neff are correct for νHDM.
    Section 3.2 fixes Ωb h²=0.0223828, Ωk=0, Yp=0.25, and Neff=4.048.
invented entities (1)
  • 11-13 eV sterile neutrino (νs), a hypothetical thermal relic replacing CDM in νHDM
    purpose: Supplies the dominant matter density and assists MOND on galaxy-cluster scales.
    No direct detection exists; KATRIN and STEREO only give bounds near ~10 eV, and the paper cites no positive laboratory or astrophysical detection. The fitted value shifts the mass to about 12.2-13 eV.

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Cite this review

Pith. "Pith review of On the initial conditions of the $\nu$HDM cosmological model." pith.science (2026). https://pith.science/paper/LD67JTAK

@misc{pith2026250619196,
  author       = {Pith},
  title        = {Pith review of: On the initial conditions of the $\nu$HDM cosmological model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LD67JTAK}},
  note         = {Machine review of arXiv:2506.19196}
}
abstract

The $\nu$HDM is the only cosmological model based on Milgromian Dynamics (MOND) with available structure formation simulations. While MOND accounts for galaxies, with a priori predictions for spirals and ellipticals, a light sterile neutrino of 11 eV can assist in recovering scaling relations on the galaxy-cluster scales. In order to perform MONDian cosmological simulations in this theoretical approach, initial conditions derived from a fit to the angular power spectrum of Cosmic Microwave Background (CMB) fluctuations are required. In this work, we employ CosmoSIS to perform a Bayesian study of the $\nu$HDM model. Using the best-fit values of the posterior, the CMB power spectrum is reevaluated. The excess of power in the transfer function implies a distinct evolution scenario, which can be used further as an input for a set of hydro-dynamical calculations. The resulting values H0 $\approx$ 56 km/s/Mpc and ${\Omega}_{m_{0}} \approx 0.5$ are far from agreement with respect to the best fit ones in the canonical Cold Dark Matter model, but may be significant in MONDian cosmology. The assumed Planck CMB initial conditions are only valid for the $\Lambda$CDM cosmology. This work constitutes a first step in an iterative procedure needed to disentangle the model dependence of the derived initial density and velocity fields.

Figures

Figures reproduced from arXiv: 2506.19196 by the authors.

Figure 1
Figure 1. Comparison of the different gravitational regimes as function of scale R and density ρ. At each scale, very low-density regions fall in the MOND regime (orange area), while high-density regions fall into the General Relativity (GR) regime (blue area). As a black line, we plot the critical density of the Universe as a function of the Hubble radius, showing the typical regime of the observable Universe. Color-coded th… view at source ↗
Figure 2
Figure 2. Likelihood convergence plot from CosmoSIS. Model ΛCDM νHDM opt-νHDM Likelihood -826 -5325 -1210 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. The Temperature Fluctuations power spectrum of the CMB for the 2 different cosmological νHDM models and the Planck 2018 data [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Matter power Spectrum for the 3 different Cosmological Models at z=199.0. MNRAS 000, 1–14 (2024) [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Transfer functions for the 3 different Cosmological Models at z=199.0. spectrum and over-plotting it with the data, shown in [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: Distance Modulus showing data points taken from Pantheon+SH0ES [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: DM. Three different green lines for the opt-νHDM, corresponding to opt-νHDM[1,2,3]->[130.05, 145.05, 160.05] Mpc. Data points are taken from DESI Collaboration (Adame et al. 2025) . (v) Hitherto not accommodated foreground sources of photons may exist from very early g…

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Pith tools

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