Bulk viscous cosmological models with a cosmological constant: Observational constraints

We investigate whether viscous cold dark matter (vCDM) in a $Λ$-dominated FLRW universe can alleviate the Hubble tension while satisfying thermodynamic constraints, examining both flat and curved geometries. We model vCDM with bulk viscosity $ζ= ζ_0,(Ω_{vc}/Ω_{vc0})^m$, where $m$ determines the viscosity evolution and $Ω_{vc}$ 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 ($\sim 1σ$ 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 $Ω{K0} > 0$ (at more than $2σ$), but this preference shifts toward flatness once the PPS+R22 prior are included. Notably, the current viscosity is constrained to $ζ_0 \sim 10^6$ Pa s in all scenarios, in agreement with the thermodynamic requirements. Although model selection via BIC and Bayesian evidence favors $Λ$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 $Λ$CDM model. To achieve more robust constraints, future analyses should incorporate CMB observations, which are expected to break parameter degeneracies involving $m$ and $\tildeζ_0$.

💡 Research Summary

This paper investigates whether introducing bulk viscosity into the cold dark matter sector (vCDM) within a Λ‑dominated Friedmann‑Lemaître‑Robertson‑Walker (FLRW) universe can alleviate the persistent Hubble tension while respecting thermodynamic constraints. The authors consider both spatially flat and curved geometries and model the bulk viscosity as ζ = ζ₀ (Ω_vc/Ω_vc0)ᵐ, where ζ₀ is the present‑day viscosity coefficient, Ω_vc is the density parameter of viscous CDM, and m governs the redshift evolution of the viscosity. Two specific cases are examined: (i) constant viscosity (m = 0) and (ii) variable viscosity with m treated as a free parameter. This yields four distinct cosmological models: flat ΛvCDM (m = 0), curved ΛvCDM + Ω_K (m = 0), flat ΛvCDM (m free), and curved ΛvCDM + Ω_K (m free). The standard ΛCDM model (ζ = 0) is used as a benchmark.

The theoretical framework adopts Eckart’s first‑order relativistic viscous fluid formalism, where the effective pressure is P_eff = p − 3Hζ. The continuity equation for vCDM becomes ρ̇_vc + 3Hρ_vc = 9ζH², which is recast in dimensionless form and solved numerically with the boundary condition Ω_vc(z = 0) = Ω_vc0. The Hubble function E(z) = H(z)/H₀ incorporates radiation, baryons, vCDM, curvature, and Λ, with the viscosity entering through the dimensionless parameter ζ̃₀ = 24πGc²ζ₀/H₀.

Parameter estimation is performed via Bayesian inference using the MultiNest sampler. Priors are chosen as uniform or Gaussian distributions: H₀∈U(30,99) km s⁻¹ Mpc⁻¹, Ω_b0h²∈N(0.022,0.01), Ω_vc0h²∈N(0.12,0.01), ζ̃₀∈U(0,1), m∈U(−2,2), and Ω_K0∈U(−0.4,0.4) for curved models. The data sets include: (a) the Pantheon+ Type Ia supernova compilation, both with the SH0ES calibration (PPS) and without (PP); (b) H(z) measurements from cosmic chronometers (CC) and baryon acoustic oscillations (BAO), with DESI DR2 included; and (c) a Gaussian prior on H₀ from the 2022 SH0ES result (R22, H₀ = 73.04 ± 1.04 km s⁻¹ Mpc⁻¹). The authors analyze the data both separately and in combined configurations (e.g., PPS + R22, PP + CC).

Model comparison employs the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Deviance Information Criterion (DIC), and Bayesian evidence (log‑Bayes factor). The Jeffreys scale is used to interpret the strength of evidence.

Key findings:

1. In the flat constant‑viscosity case (m = 0) with PPS + R22, the posterior H₀ = 71.05^{+0.62}_{-0.60} km s⁻¹ Mpc⁻¹, representing a ∼4 σ upward shift relative to the Planck ΛCDM value (≈67.4 km s⁻¹ Mpc⁻¹) but still leaving a ∼1 σ residual tension with the SH0ES measurement.
2. Curved models initially show a preference for positive curvature (Ω_K0 > 0) at >2 σ, but this signal weakens and becomes consistent with flatness once the PPS + R22 prior is added, indicating a strong degeneracy between curvature and the viscosity parameters.
3. The present‑day bulk viscosity is tightly constrained to ζ₀ ≈ 10⁶ Pa·s (ζ̃₀ ≈ 0.1–0.3) across all scenarios, satisfying the second law of thermodynamics (ζ > 0) and the near‑equilibrium condition |Π/p| ≪ 1.
4. Information criteria give mixed results: BIC and Bayesian evidence strongly favor ΛCDM (ΔBIC ≈ +6–10, ln B ≈ −3 to −5), whereas AIC and DIC show modest improvements for viscous models in some data combinations (ΔAIC ≈ −2, ΔDIC ≈ −3), reflecting the trade‑off between better fit (lower χ²) and additional parameters.
5. Overall, bulk viscous models can partially raise H₀ and improve χ², but they do not fully resolve the Hubble tension nor outperform ΛCDM in a statistically robust way.

The authors conclude that while bulk viscosity in the dark sector is a physically viable extension (consistent with thermodynamics) and can modestly shift H₀, current low‑redshift data alone cannot break the degeneracies involving ζ₀, m, and Ω_K0. Incorporating high‑precision early‑Universe observations such as the CMB power spectrum, as well as large‑scale structure probes (e.g., full‑shape BAO, redshift‑space distortions) and future standard‑sirens, will be essential to place tighter constraints on viscous parameters and assess whether such models can genuinely address the Hubble tension.