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Hints of noncold dark matter? Observational constraints on barotropic dark matter with a constant equation of state parameter

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Current cosmological data give a marginal 95% credible hint that dark matter has a small positive pressure, a property that would suppress small-scale structure and could dissolve the S8 tension.

desk verdict A competent, transparent update of Mueller's Lambda wDM constraints whose claimed detection of non-cold dark matter rests entirely on a linear-theory S8 prior and a one-sided prior, so the hint is not yet supported. read the letter →

arxiv 2507.00478 v1 pith:7QW7VGHF submitted 2025-07-01 astro-ph.CO

Yan-Hong Yao , Jian-Qi Liu This is my paper
classification astro-ph.CO PACS 98.80.-k95.35.+d
keywords darkmatterequationofstatebarotropicwDMmodelS8tensioncosmologicalparametersweaklensingredshift-spacedistortionsCMB
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 tests a simple extension of the standard cosmological model in which dark matter is a barotropic fluid with a constant equation of state parameter $w_{\rm dm}$ rather than being perfectly cold. Combining Planck CMB, BAO (SDSS or DESI), Pantheon+ supernovae, redshift-space distortions, and a weak-lensing prior, the authors find a marginal preference for a positive value, $w_{\rm dm}=2.7^{+2.0}_{-1.9}\times10^{-7}$ (95% credible interval) for the SDSS-based dataset combination, and $w_{\rm dm}=2.29^{+1.9}_{-2.0}\times10^{-7}$ for the DESI-based combination. In this $\Lambda w$DM model, the long-standing discrepancy between early- and late-universe measurements of the $S_8$ parameter drops from above $3\sigma$ to below $1\sigma$. If the result holds, it would mean dark matter is not cold but has a tiny pressure and sound speed, slightly raising the Jeans scale and damping the growth of small-scale structure.

What carries the argument

The central object is the barotropic dark-matter fluid with a constant equation of state $w_{\rm dm}$. In the linear perturbation equations, the square of the sound speed equals $w_{\rm dm}$ (since the non-adiabatic sound speed is set to zero), so a small positive $w_{\rm dm}$ increases the effective pressure support of dark matter. This slightly raises the Jeans wavelength and suppresses the matter power spectrum at small scales, which lowers $\sigma_8$ and $S_8$ without appreciably altering the background expansion. The paper also notes an equivalent background description: the model can be recast as cold dark matter plus a dynamical dark-energy component that behaves like a cosmological constant at low redshift and like quintessence at high redshift when $w_{\rm dm}>0$.

What would settle it

Run the same parameter estimation using the full KiDS-1000 weak-lensing likelihood with a nonlinear halo model adapted to a dark-matter sound speed $c_s^2=w_{\rm dm}$, rather than the $S_8$ prior. If the posterior on $w_{\rm dm}$ then includes zero at 95% credibility while the $S_8$ tension remains, the claimed detection and tension reduction would be falsified.

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Extended reading notes

Core claim

The central claim is that the combined observational data prefer a small positive dark-matter equation of state parameter, at roughly 95% confidence, and that this single additional parameter resolves the $S_8$ tension. In the $\Lambda w$DM model, the dark matter fluid has $c_{s}^2 = w_{\rm dm}$, so a positive $w_{\rm dm}$ produces a positive sound speed that suppresses the matter power spectrum on small scales via an increased Jeans wavelength. The authors show that this mechanism reduces the tension between the CMB-based $S_8$ and the late-universe RSD+WL-based $S_8$ from $T_1=3.42$ in $\Lambda$CDM to $T_1=0.16$ in $\Lambda w$DM. They also find that the model is close to being positively preferred over $\Lambda$CDM by AIC for the SDSS-based joint dataset, though not decisively.

Load-bearing premise

The load-bearing assumption is that a Gaussian prior on $S_8$ from KiDS-1000, computed from the linear matter power spectrum, adequately represents the constraints the full weak-lensing likelihood would place on the $\Lambda w$DM model.

Editorial extensions

If this is right

  • If $w_{\rm dm}$ is genuinely positive at the level indicated, dark matter would be a barotropic fluid with a tiny pressure, and structure formation would be suppressed on small scales, potentially easing small-scale challenges like the core-cusp or missing-satellite problems.
  • The $S_8$ tension between early- and late-universe probes would be reduced from more than $3\sigma$ to below $1\sigma$ without invoking new interactions or modified gravity.
  • The model makes a testable prediction for the nonlinear matter power spectrum: a specific suppression at high $k$ that future weak-lensing and galaxy-clustering surveys could detect.
  • Since a positive $w_{\rm dm}$ implies a high-redshift quintessence-like dark energy component, the model's viability is directly tied to the dark-energy equation of state inferred from DESI BAO data; the authors find that including DESI lowers the preferred $w_{\rm dm}$ only slightly.
  • The AIC comparison shows that, for at least one dataset combination, $\Lambda w$DM is close to being preferred over $\Lambda$CDM, so the model is a viable competitor that future data could distinguish.

Reading between the lines

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

  • The authors' choice of a flat prior $w_{\rm dm}\in[0,100]\times10^{-7}$ forbids negative values, so the claimed 'non-zero' signal is one-sided; a prior allowing negative $w_{\rm dm}$ would be needed to assess whether the data truly exclude $w_{\rm dm}=0$ or merely prefer positive pressure over negative imaginary sound speed.
  • A direct test of the claim would be to replace the Gaussian $S_8$ prior with the full KiDS-1000 weak-lensing likelihood, including a halo model adapted to $\Lambda w$DM; if that full analysis drives $w_{\rm dm}$ back to zero, the reported signal would likely be an artifact of the approximate likelihood.
  • The equivalence to a dynamical dark-energy component suggests that the same data could be reinterpreted as a constraint on early dark energy or quintessence; combining future DESI BAO data with tomographic weak lensing may sharpen this discriminant.
  • The 95% credible interval for $w_{\rm dm}$ still includes values near zero, so the paper's conclusion is a hint rather than a detection; confirmation would require independent probes of the small-scale matter power spectrum, such as Lyman-$\alpha$ forest or 21-cm measurements.
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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 / 4 minor

Summary. The paper constrains a model (ΛwDM) in which dark matter is a barotropic fluid with a constant equation-of-state parameter wdm, so that its sound speed squared equals wdm. Using Planck CMB, SDSS and DESI BAO, Pantheon+ SNe, RSD, and a Gaussian prior on S8 from KiDS-1000 (computed from the linear matter power spectrum), the authors report a marginal 95% credible detection of positive wdm ≈ 2.7×10^-7 (SDSS case) or 2.29×10^-7 (DESI case) when the weak lensing prior is included, and a reduction of the S8 tension from >3σ in ΛCDM to <1σ in ΛwDM. Model comparison via AIC does not show a decisive preference for ΛwDM. The paper explicitly acknowledges that the full KiDS-1000 likelihood is replaced by a one-number S8 prior because nonlinear modeling for ΛwDM is unavailable.

Significance. If the central result held, it would constitute an interesting hint of non-cold dark matter and an economical resolution of the S8 tension with a single extra parameter. The statistical machinery is standard (MontePython, CLASS, GetDist), and the paper is transparent about its main assumption and limitations, which is a strength. However, the significance is currently limited because the reported hint and the tension reduction both depend critically on an unvalidated approximation—the use of a linear-theory Gaussian S8 prior in place of the full weak lensing likelihood. The paper does not overclaim in its conclusions, and the AIC analysis honestly shows no strong model preference.

major comments (3)
  1. [Section III (Weak Lensing) and Table II] The claimed 95% positive lower bound on wdm appears only when the Gaussian S8 prior from KiDS-1000 is added: for CMB+SDSS+PP+RSD, the 95% interval for 10^7 wdm is [0, 4.04] (mean 2.1, lower error -2.1), while after adding the WL prior it becomes [0.8, 4.7]. The paper states in Section III that the S8 prior is assumed to adequately represent the full KiDS-1000 likelihood, but no validation is provided. Because a positive wdm suppresses small-scale power in a scale-dependent way, the full shear correlation functions carry shape information that a one-number prior discards, and the prior itself is calibrated within ΛCDM. The central hint is therefore contingent on an equivalence that has not been demonstrated; the paper should either implement the full likelihood (even approximately) or clearly reframe the result as conditional on this untested assumption.
  2. [Table I: prior on wdm] The flat prior wdm ∈ [0, 100]×10^-6 truncates the parameter space at zero. Although a negative wdm would correspond to an imaginary adiabatic sound speed (c_s^2 = wdm), the hard boundary makes the reported detection one-sided. The 95% lower bounds quoted for the WL-inclusive datasets are computed on a truncated posterior, and the posterior mass is pushed away from the boundary. The authors should assess the sensitivity to this truncation, for example by running a follow-up with a prior that extends to negative wdm (even if treated as unphysical) to quantify how much of the "positive detection" is driven by the boundary, or by reporting the posterior percentile relative to a two-sided prior and discussing the implications.
  3. [Section IV: S8 tension estimators T1 and T2] The tension reduction from >3σ to <1σ is computed using the RSD+WL dataset as the low-redshift leg, where the WL component is the same approximate Gaussian S8 prior. This is not an independent low-redshift constraint: the model is being fit to the same prior that is used to claim the tension is resolved. The value of S8 in the RSD+WL run (0.765±0.021 for ΛwDM) is very close to the input prior mean of 0.759, so the reduction in T1/T2 largely reflects the fact that the model can lower the CMB-predicted S8 by increasing wdm, while the low-redshift S8 is essentially predetermined by the prior. A more convincing assessment would compare the model to the actual KiDS-1000 shear data (with nonlinear modeling) or at least use a tension metric that does not rely on the same approximate likelihood in both legs.
minor comments (4)
  1. [Abstract and Section V] The notation "2.7+2.0−1.9 ×10−7( at 95% confidence level)" has an extra parenthesis and missing spacing; the same issue appears in the concluding remarks. Please fix these typographical errors.
  2. [Table I vs Tables II-VI] The prior in Table I is labeled "10^6 wdm" with range [0,100], while all results are quoted as "10^7 wdm". This unit inconsistency is confusing; please standardize to a single unit (e.g., 10^7 wdm) throughout the paper.
  3. [Tables II-VI] The error format such as "3.64+0.54−3.64" is ambiguous when the lower error exceeds the mean; the lower bound is not transparent. Reporting the actual credible intervals (e.g., [0, 4.18] at 68%) would improve readability.
  4. [Section IV, Eq. (13) discussion] The statement that the dynamical-DE interpretation "is in tension with recently released DESI data since the latter support phantom behavior for DE at high redshifts" is asserted without a reference or quantitative support; either add a citation to the DESI results or soften the claim.

Circularity Check

2 steps flagged · score 4.0 of 10

The 95% hint of positive wdm and the claimed S8-tension relief are both driven by the KiDS-1000 S8 prior fed into the fit; the rest of the derivation is standard and transparently sourced from Mueller (2005).

  1. fitted input called prediction [Section III (Weak Lensing paragraph) and Section IV (Table II, CMB+SDSS+PP+RSD+WL row)]
    "we incorporate a prior on S8, i.e. S8 = 0.759+0.024−0.021 [42], which is based on the measurements from KiDS1000. (For the Λ wDM model, utilizing the full WL likelihood necessitates a thorough consideration of nonlinear effects. Due to the unavailability of these tools, we confine our analysis to the linear power spectrum and assume that including the S8 prior adequately represents the constraints imposed by the KiDS1000 likelihood on the Λ wDM model). ... we finally find a statistically significant signal for a positive DM parameter wdm = 2.70+2.0−1.9 × 10−7 (at the 95% confidence level)."

    Table II shows the 95% interval for 10^7 wdm contains zero for CMB, CMB+SDSS+PP, and CMB+SDSS+PP+RSD; it excludes zero (lower bound 0.8) only in the CMB+SDSS+PP+RSD+WL row. The paper itself states that S8 and wdm are negatively correlated, so a Gaussian prior on S8 centered at 0.759 (below the CMB-determined value ~0.83) statistically forces wdm upward in the fit. The 'hint of non-cold dark matter' is therefore the input low-S8 measurement re-expressed in wdm coordinates, under the explicitly flagged assumption that a linear-theory Gaussian S8 prior reproduces the full KiDS-1000 likelihood for ΛwDM. Because the prior is a ΛCDM-derived compressed summary of the same data class that defines the S8 tension, the detection is not independent of the input it claims to explain.

  2. fitted input called prediction [Section IV (S8-tension paragraph, T1/T2 definitions and values)]
    "Now, we assess the Λ wDM model's ability to relieve the S8 tension by using the following two quantities T1 = xCMB+SDSS+PP − xRSD+WL / sqrt(σ2CMB+SDSS+PP + σ2RSD+WL), T2 = xCMB+DESI+PP − xRSD+WL / sqrt(σ2CMB+DESI+PP + σ2RSD+WL), where x = S8. For the Λ wDM model, we have T1 = 0.16 and T2 = 0.10; for the ΛCDM model, we have T1 = 3.42 and T2 = 3.12, therefore, we can see that the Λ wDM model can reduce the S8 tension from beyond 3σ to below 1σ."

    The low-redshift leg xRSD+WL for ΛwDM is 0.765 ± 0.021 (Table VI), essentially the input KiDS-1000 S8 prior (0.759) that is a component of the fitted likelihood. The high-redshift leg xCMB+SDSS+PP = 0.770 is a posterior mean pushed down by the one-sided prior wdm ≥ 0, which truncates the wdm posterior at the boundary and inflates its mean. The claimed reduction from 3.42σ to 0.16σ is thus largely a consistency restatement of the input S8 prior plus the truncated wdm prior, rather than an independent prediction, although the fit's Δχ2 = 8.4 shows the low-S8 preference is real under the assumed likelihood.

full rationale

The ΛwDM model and its perturbation equations are transparently taken from the literature: the paper states 'it has been proposed and discussed in Ref. [21] for about two decades' and uses the GDM equations from Ref. [24]; no ansatz is smuggled via citation, and no uniqueness theorem is invoked. The authors' self-citations ([27]-[29]) are peripheral GDM follow-up references and are not load-bearing. The remaining concern is structural rather than derivational: the paper's headline results — the 95% exclusion of wdm = 0 and the S8-tension reduction to <1σ — are both produced by the KiDS-1000 S8 prior fed into the MCMC, combined with the one-sided flat prior wdm ∈ [0,100]×10^-6. The paper explicitly flags this: 'we can only use the model-dependent S8 prior to replace the complete WL likelihood to constrain Λ wDM. This may introduce some bias into the fitting results,' and in the concluding remarks lists 'the absence of applying the full WL likelihood for ΛwDM' as a critical limitation. Because the low-S8 prior is a compressed summary of the same data class that defines the S8 tension, the 'hint of non-cold dark matter' and the 'tension relief' are substantially the input measurement re-expressed in model coordinates. Nevertheless, the fit has independent multi-probe content: CMB, BAO, SNe, and RSD genuinely tolerate wdm ~ 2-3 × 10^-7, the χ2 improvement relative to ΛCDM is a real statistical preference under the stated likelihood, and the AIC comparison honestly reports no decisive model preference. Score 4 reflects this partial circularity: the central numerical claims reduce largely to the input S8 prior and the prior floor, while the model, equations, and remaining derivation are self-contained and externally sourced.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The central claim rests on one new fitted parameter (wdm) and several model assumptions: the barotropic fluid closure, the replacement of the WL likelihood by an S8 prior, and the fixed-k RSD treatment. No new particles or forces are introduced.

free parameters (1)
  • wdm = 2.7e-7 (95% CI 0.8 to 4.7e-7) for CMB+SDSS+PP+RSD+WL; 2.29e-7 for CMB+DESI+PP+RSD+WL
    Dark matter equation of state parameter, the only new parameter, fitted with a flat prior 0 to 100e-6. The positive detection is the central claim.
assumptions (3)
  • domain assumption The dark matter fluid is barotropic with zero non-adiabatic sound speed, so c_s^2 = wdm (Eqs. 5-8).
    This closes the perturbation equations; it is the defining phenomenological assumption of the model.
  • ad hoc to paper The Gaussian S8 prior from KiDS-1000, combined with the linear power spectrum, adequately represents the full weak lensing likelihood for Lambda wDM.
    Introduced in Section III because nonlinear tools are unavailable. Directly determines the wdm posterior and the S8 tension result.
  • domain assumption RSD measurements can be analyzed with the growth rate f evaluated at a fixed k = 0.1 h/Mpc even though f is scale-dependent in Lambda wDM.
    The paper states this is consistent with the effective wavenumber of the data, but it is an approximation.

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Cite this review

Pith. "Pith review of Hints of noncold dark matter? Observational constraints on barotropic dark matter with a constant equation of state parameter." pith.science (2026). https://pith.science/paper/7QW7VGHF

@misc{pith2026250700478,
  author       = {Pith},
  title        = {Pith review of: Hints of noncold dark matter? Observational constraints on barotropic dark matter with a constant equation of state parameter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7QW7VGHF}},
  note         = {Machine review of arXiv:2507.00478}
}
abstract

This study investigates the potential of a cosmological model termed $\Lambda w$DM, in which a cosmological constant play the role of dark energy and dark matter is barotropic and has a constant equation of state parameter ($w_{\rm dm}$), to address the $S_8$ tension between early- and late- universe observations. By incorporating the latest cosmological datasets -- including Planck Cosmic Microwave Background (CMB), Baryon Acoustic Oscillation (BAO), Ia supernovae (SNe Ia), Redshift Space Distortions (RSD), and weak lensing (WL) -- we constrain the $\Lambda w$DM compared to $\Lambda$CDM. Our analysis reveals a marginal preference for a non-zero $w_{\rm dm}=2.7^{+2.0}_{-1.9}\times10^{-7}$( at 95\% confidence level) when combining CMB, SDSS BAO, SNe Ia, RSD, and WL data, and a marginal preference for a non-zero $w_{\rm dm} = 2.29^{+1.9}_{-2.0} \times 10^{-7}$( at 95\% confidence level) when combining CMB, DESI Y1 BAO, SNe Ia, RSD, and WL data. In addition, we find that, compared to $\Lambda$CDM, $\Lambda w$DM can alleviate the $S_8$ tension from $>3\sigma$ to $<1\sigma$. Furthermore, we find that, for CMB+SDSS+PP+RSD+WL datasets, the $\Lambda w$DM model is close to being positively preferred over the $\Lambda$CDM model.

Figures

Figures reproduced from arXiv: 2507.00478 by the authors.

Figure 1
Figure 1. FIG. 1. One dimensional posterior distributions and two dimensional joint contours at 68% and 95% CL for the most relevant [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. One dimensional posterior distributions and two dimensional joint contours at 68% and 95% CL for the most relevant [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. One dimensional posterior distributions and two dimensional joint contours at 68% and 95% CL for the most relevant [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. One dimensional posterior distributions and two dimensional joint contours at 68% and 95% CL for the most relevant [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. One dimensional posterior distributions and two dimensional joint contours at 68% and 95% CL for the most relevant [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

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Forward citations

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