Bulk viscous cosmological models with a cosmological constant: Observational constraints
Pith reviewed 2026-05-18 22:26 UTC · model grok-4.3
The pith
Bulk viscous dark matter models ease the Hubble tension to about one sigma but do not outperform the standard Lambda CDM cosmology.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The analysis finds that viscous cold dark matter models in flat and curved geometries produce Hubble constant values near 71 km per second per megaparsec when local measurements are included, bringing the tension down to roughly one sigma. The current viscosity is constrained to approximately 10 to the 6 Pascal seconds in all scenarios, satisfying thermodynamic requirements. Although some information criteria indicate mild support for the viscous models on particular data sets, the Bayesian information criterion and Bayesian evidence still favor the standard Lambda CDM model over the extensions.
What carries the argument
The bulk viscosity form zeta equals zeta sub zero times the ratio of the viscous cold dark matter density parameter to its present value raised to the power m, which governs the evolution of the viscosity and is constrained through Bayesian inference on multiple cosmological data sets.
If this is right
- Adding the local Hubble prior shifts curved models toward spatial flatness.
- The viscosity today remains tightly bounded near 10 to the 6 Pascal seconds regardless of the data combination.
- Information criteria give mixed results with BIC and evidence favoring Lambda CDM while AIC and DIC sometimes mildly support viscous cases.
- Future work incorporating cosmic microwave background data is required to resolve degeneracies in the viscosity parameters m and zeta sub zero.
Where Pith is reading between the lines
- Allowing the viscosity to depend on the expansion rate in addition to density could open new parameter space that might further alleviate the tension.
- A full CMB analysis would likely exclude or confirm the mild preferences seen in the current data sets.
- Similar viscosity parametrizations could be tested in models without a cosmological constant to check consistency with other tension resolutions.
- Independent probes such as the growth of structure might reveal whether the required viscosity affects perturbation evolution in observable ways.
Load-bearing premise
The chosen power-law form for how bulk viscosity scales with the viscous dark matter density, combined with the exclusion of cosmic microwave background data that would otherwise break degeneracies.
What would settle it
Adding Planck or other CMB likelihoods to the fit and finding that the best-fit viscosity parameters either drive the Hubble constant back to early-universe values or produce unphysical negative viscosities would falsify the prospect of meaningful alleviation.
read the original abstract
We investigate whether viscous cold dark matter (vCDM) in a $\Lambda$-dominated FLRW universe can alleviate the Hubble tension while satisfying thermodynamic constraints, examining both flat and curved geometries. We model vCDM with bulk viscosity $\zeta = \zeta_0\,(\Omega_{vc}/\Omega_{vc0})^m$, where $m$ determines the viscosity evolution and $\Omega_{vc}$ is the density parameter of vCDM. We explore two particular scenarios: constant viscosity ($m=0$), and variable viscosity ($m$ free). Using Bayesian inference, we constrain these models with the latest datasets: the Pantheon+ SN Ia sample (both with SH0ES calibration, PPS, and without it, PP), $H(z)$ measurements from CC and BAO as separate datasets, and a Gaussian prior on $H_0$ from 2022 SH0ES baseline, $H_0=73.04 \pm 1.04$ km/s/Mpc (R22 prior). We compare the models via information criteria such as AIC, BIC, DIC, and Bayesian evidence. Our results reveal that the Hubble tension persists, although it shows partial alleviation ($\sim 1\sigma$ tension) in all investigated scenarios when local measurements are included. For the flat $m=0$ case, the joint analysis yields $H_0 = 71.05^{+0.62}_{-0.60}$ km/s/Mpc. Curved model initially favors $\Omega_{K0} > 0$ (at more than $2\sigma$), but this preference shifts toward flatness once the PPS+R22 prior are included. Notably, the current viscosity is constrained to $\zeta_0 \sim 10^6$ Pa s in all scenarios, in agreement with the thermodynamic requirements. Although model selection via BIC and Bayesian evidence favors $\Lambda$CDM, AIC and DIC show mild support for viscous models in some datasets. Bulk viscous models moderately improve fits but neither resolve the Hubble tension nor outperform the $\Lambda$CDM model. To achieve more robust constraints, future analyses should incorporate CMB observations, which are expected to break parameter degeneracies involving $m$ and $\tilde{\zeta}_0$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper investigates bulk viscous cold dark matter (vCDM) in a Λ-dominated FLRW universe, modeling bulk viscosity as ζ = ζ₀ (Ω_vc / Ω_vc0)^m for both constant (m=0) and variable (m free) cases in flat and curved geometries. Using Bayesian inference, it constrains the models with Pantheon+ SN Ia data (PPS with SH0ES calibration and PP without), H(z) from cosmic chronometers and BAO, and a Gaussian R22 H0 prior. Model comparison employs AIC, BIC, DIC, and Bayesian evidence. Results show the Hubble tension persists with only partial (~1σ) alleviation when local H0 data are included (e.g., H0 = 71.05^{+0.62}_{-0.60} km/s/Mpc for flat m=0 joint analysis), viscosity constrained to ζ₀ ∼ 10^6 Pa s consistent with thermodynamics at z=0, curved models initially favoring Ω_K0 > 0 but shifting to flatness with PPS+R22, and mixed information criteria where BIC/evidence favor ΛCDM while AIC/DIC show mild support for viscous models in some cases. The paper concludes viscous models offer moderate fit improvements but neither resolve the tension nor outperform ΛCDM, recommending future CMB inclusion to break m-ζ₀ degeneracies.
Significance. If the results hold, the work supplies concrete observational bounds on a phenomenological viscous dark matter component, demonstrating limited impact on the Hubble tension and no decisive preference over ΛCDM by standard model-selection metrics. The explicit acknowledgment of CMB data needs and the reported ζ₀ value provide a clear benchmark for subsequent viscous cosmology studies.
major comments (2)
- [Abstract] Abstract: the central claim of only partial (~1σ) Hubble tension alleviation in all scenarios when local measurements are included rests on fits that omit CMB data; the paper itself states that CMB observations are required to break degeneracies involving m and ζ̃₀, leaving the reported ζ₀ ∼ 10^6 Pa s and tension reduction potentially sensitive to additional constraints.
- [Model description] Model section (viscosity parametrization): thermodynamic consistency is asserted to hold for the constrained ζ₀ ∼ 10^6 Pa s, but the check is performed only at z=0 rather than verifying the entropy production condition integrated over the full expansion history probed by the datasets.
minor comments (2)
- [Abstract] Abstract: notation mixes ζ₀ and ζ̃₀; consistent use of one symbol would improve clarity.
- [Results] Results: the mixed AIC/DIC versus BIC/evidence outcomes are reported but the text could more explicitly state the weighting given to each criterion when concluding that viscous models do not outperform ΛCDM.
Simulated Author's Rebuttal
We thank the referee for the thoughtful and constructive report. We address the major comments point by point below, indicating where revisions will be made to the manuscript.
read point-by-point responses
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Referee: [Abstract] Abstract: the central claim of only partial (~1σ) Hubble tension alleviation in all scenarios when local measurements are included rests on fits that omit CMB data; the paper itself states that CMB observations are required to break degeneracies involving m and ζ̃₀, leaving the reported ζ₀ ∼ 10^6 Pa s and tension reduction potentially sensitive to additional constraints.
Authors: We agree that the abstract's statement on partial alleviation should be presented with appropriate caution given the absence of CMB data. The manuscript already notes in the abstract, results, and conclusions that CMB observations are needed to break degeneracies involving m and ζ̃₀. The reported ζ₀ ∼ 10^6 Pa s and the ~1σ tension level are the outcomes of the specific datasets employed (Pantheon+, H(z) from CC and BAO, and the R22 prior). These constitute valid constraints from late-universe probes. To improve clarity, we will revise the abstract to explicitly state that the tension alleviation and viscosity bounds are based on current data and may be refined by future CMB inclusion. revision: yes
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Referee: [Model description] Model section (viscosity parametrization): thermodynamic consistency is asserted to hold for the constrained ζ₀ ∼ 10^6 Pa s, but the check is performed only at z=0 rather than verifying the entropy production condition integrated over the full expansion history probed by the datasets.
Authors: The referee is correct that the thermodynamic consistency check in the current version is performed at z=0 using the constrained present-day value of ζ₀. This choice reflects that ζ₀ is the parameter directly constrained by the data and that the entropy production condition is evaluated at the epoch where the viscosity is reported. We acknowledge that verifying the condition integrated over the full redshift range of the datasets would provide a more complete assessment. We will add a short discussion or supplementary calculation in the revised manuscript confirming that the entropy production remains positive throughout the relevant expansion history for the best-fit parameters. revision: yes
Circularity Check
No circularity: constraints derived from external datasets via Bayesian fitting
full rationale
The paper sets up a phenomenological bulk-viscosity ansatz ζ = ζ0 (Ωvc/Ωvc0)^m and performs standard Bayesian parameter estimation against independent observational catalogs (Pantheon+, CC, BAO, R22 prior). The reported posteriors on H0, ζ0, m and the model-comparison statistics (AIC/BIC/DIC/evidence) are direct outputs of the likelihood evaluation on those external data; none of the central claims (persistence of Hubble tension, moderate fit improvement, thermodynamic consistency at z=0) reduce by the paper’s own equations to a quantity defined solely in terms of the fitted parameters. The acknowledged m–ζ0 degeneracy is stated explicitly and does not alter the fact that the numerical results are data-driven rather than tautological.
Axiom & Free-Parameter Ledger
free parameters (2)
- ζ₀ =
~10^6 Pa s
- m =
0 or free
axioms (2)
- standard math Universe geometry described by FLRW metric (flat or curved)
- domain assumption Bulk viscosity must satisfy thermodynamic inequalities
invented entities (1)
-
viscous cold dark matter (vCDM)
no independent evidence
discussion (0)
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