Bulk viscous cosmological models with a cosmological constant: Observational constraints

This study employs Bayesian inference on various cosmological datasets to demonstrate that bulk viscous cold dark matter models, while thermodynamically consistent and offering partial alleviation of the Hubble tension, ultimately fail to resolve the discrepancy or outperform the standard $\Lambda$CDM model.

The Gist

Imagine the universe as a giant, expanding balloon. For decades, scientists have had a very successful "instruction manual" for how this balloon inflates, called the Standard Model (or $\Lambda$CDM). It says the balloon is filled with invisible "dark matter" (which acts like glue holding galaxies together) and "dark energy" (which acts like a pump pushing the balloon to expand faster).

However, there's a problem. When we measure how fast the balloon is expanding right now (using local telescopes), we get one number. When we look at the "baby picture" of the universe (the Cosmic Microwave Background) and calculate how fast it should be expanding today, we get a different, slower number. This disagreement is known as the "Hubble Tension." It's like two mechanics looking at the same car engine and arguing about how many miles per hour it's actually going.

The New Idea: A Sticky Universe

In this paper, the authors ask: What if the "glue" (dark matter) isn't perfectly smooth and frictionless? What if it's a bit sticky?

In physics, this "stickiness" is called viscosity. Think of it like the difference between water and honey. Water flows easily (low viscosity), but honey resists flow (high viscosity). The authors propose that Dark Matter might be a bit like honey. As the universe expands, this "cosmic honey" creates internal friction (bulk viscosity), which generates heat and changes how the universe expands.

They tested two versions of this "sticky" universe:

1. Constant Stickiness: The honey is always the same thickness, no matter how much the universe expands.
2. Changing Stickiness: The honey gets thicker or thinner as the universe grows.

They also checked if the universe is perfectly flat (like a sheet of paper) or curved (like a saddle or a sphere).

How They Tested It

The team acted like cosmic detectives. They gathered all the latest clues from the universe:

• Supernovae: Exploding stars that act as "standard candles" to measure distances.
• Cosmic Chronometers: Measuring the ages of galaxies to see how fast time is passing relative to space.
• Sound Waves: Fossilized ripples from the early universe (Baryon Acoustic Oscillations).
• The "R22" Prior: A specific, very precise measurement of the current expansion rate from the SH0ES team (the "local" measurement).

They used powerful computer simulations (Bayesian inference) to see which model—the Standard Model or their new "Sticky" models—best fit these clues.

The Results: A Mixed Bag

Here is what they found, translated into plain English:

1. The "Hubble Tension" is still there, but it's less painful.
The sticky universe models didn't magically fix the disagreement between the "baby picture" and the "current picture." However, they did make the numbers come closer together. If the tension was a 5-point gap, the sticky models reduced it to a 1-point gap. It's not a cure, but it's a bandage.

2. The "Honey" amount is small but real.
They found that if dark matter is sticky, the "stickiness" (viscosity) is roughly $10^6$ Pascal-seconds. To put that in perspective, that's about a million times stickier than water, but still much less sticky than pitch. It's a tiny amount of friction, but enough to slightly tweak the expansion rate.

3. The Shape of the Universe.
At first, the data seemed to suggest the universe might be curved (like a saddle). But once they added the most precise local measurements (the R22 data), the universe looked much flatter again, just like the Standard Model predicts.

4. The "Best Fit" Verdict.
When the authors compared the models using statistical "scorecards" (like AIC, BIC, and Bayesian Evidence):

• The Standard Model ($\Lambda$CDM) still wins the popularity contest. It's the simplest explanation that fits the data well.
• The Sticky Models did a decent job and sometimes looked slightly better depending on which specific data set you looked at, but they didn't win decisively. In fact, some statistical tests strongly said, "Stick to the Standard Model; the extra complexity of the sticky fluid isn't worth it yet."

The Bottom Line

The authors conclude that while a "sticky" dark matter universe is a fascinating idea that partially helps explain why our measurements of the universe's speed are conflicting, it doesn't solve the mystery completely.

The "Hubble Tension" remains. The sticky models are a good "what if" scenario, but they aren't the final answer. To truly crack the case, the authors suggest we need to bring in more evidence, specifically from the Cosmic Microwave Background (CMB)—the oldest light in the universe. That data might be the missing key to finally tell us if the universe is made of frictionless water or cosmic honey.

In short: The universe might be a little sticky, but it's not sticky enough to fix the biggest argument in cosmology just yet. We need more data to be sure.

Technical Summary

1. Problem Statement

The paper addresses two major challenges in modern cosmology:

1. The Hubble Tension: The persistent $\sim 5\sigma$ discrepancy between the Hubble constant ($H_0$) derived from early-universe observations (Planck CMB, $H_0 \approx 67.4$ km s$^{-1}$ Mpc$^{-1}$) and late-time local measurements (SH0ES distance ladder, $H_0 \approx 73.0$ km s$^{-1}$ Mpc$^{-1}$).
2. The Nature of Dark Matter: The standard $\Lambda$CDM model treats Cold Dark Matter (CDM) as a non-relativistic, pressureless perfect fluid. However, physical arguments suggest cosmic matter may exhibit dissipative properties. The authors investigate whether introducing bulk viscosity into the dark matter sector can alleviate the Hubble tension while satisfying thermodynamic constraints, without abandoning the cosmological constant ($\Lambda$).

2. Methodology

The authors constructed and constrained four distinct cosmological models within a Friedmann-Lemaître-Robertson-Walker (FLRW) framework:

• Theoretical Framework:

  • Model: $\Lambda$vCDM (Lambda-viscous Cold Dark Matter).
  • Viscosity Ansatz: The bulk viscosity coefficient $\zeta$ follows a power-law dependence on the vCDM density parameter: $\zeta = \zeta_0 (\Omega_{vc}/\Omega_{vc0})^m$.
  • Scenarios:
    1. Constant Viscosity: $m = 0$ ($\zeta = \zeta_0$).
    2. Variable Viscosity: $m$ is a free parameter.
  • Geometry: Both spatially flat ($K=0$) and curved ($K \neq 0$) universes were analyzed.
  • Thermodynamics: The models utilize Eckart's first-order formalism, where the effective pressure is $P_{eff} = p - 3H\zeta$. The condition $\zeta > 0$ is enforced to satisfy the second law of thermodynamics (positive entropy production).

• Data & Statistical Analysis:

  • Datasets:
    • H(z): Cosmic Chronometers (CC) and Baryon Acoustic Oscillations (BAO) measurements (including independent BAO-based $H(z)$ and DESI DR2 data).
    • SNe Ia: Pantheon+ (PP) and Pantheon+ & SH0ES (PPS) samples.
    • Prior: A Gaussian prior on $H_0$ from the SH0ES R22 measurement ($73.04 \pm 1.04$ km s$^{-1}$ Mpc$^{-1}$).
  • Statistical Tools: Bayesian inference using MCMC (emcee).
  • Model Comparison: Evaluated using Akaike (AIC), Bayesian (BIC), and Deviance Information Criteria (DIC), alongside Bayesian Evidence (Jeffreys scale).

3. Key Contributions

• Comprehensive Parameter Space: The study systematically explores the interplay between bulk viscosity evolution ($m$), spatial curvature ($\Omega_{K0}$), and the Hubble constant using the latest high-precision datasets (DESI DR2, Pantheon+).
• Thermodynamic Consistency: It explicitly verifies that the constrained viscosity values satisfy the second law of thermodynamics and remain within bounds allowed by background expansion dynamics (distinct from tighter constraints imposed by linear perturbation theory).
• Model Selection Rigor: The paper provides a multi-criteria assessment (AIC, BIC, DIC, Bayes factors) to determine if the added complexity of viscous models is justified by the data compared to the standard $\Lambda$CDM.

4. Key Results

A. Hubble Tension

• Partial Alleviation: The inclusion of local measurements (R22 prior) in the viscous models reduces the tension between early and late-universe $H_0$ estimates to approximately $1\sigma$.
• Specific Values (Flat, $m=0$, Base 2 + CC + R22):
  • $H_0 = 71.05^{+0.62}_{-0.60}$ km s$^{-1}$ Mpc$^{-1}$.
  • While this is higher than Planck's value, it does not fully resolve the tension to the level of a single standard deviation across all datasets, but it significantly mitigates the discrepancy compared to standard $\Lambda$CDM fits without local priors.

B. Spatial Curvature ($\Omega_{K0}$)

• Initial Preference: Without local priors, some datasets (e.g., PPS alone) favored an open universe ($\Omega_{K0} > 0$) at $>2\sigma$ significance.
• Shift to Flatness: Once the PPS sample and R22 prior were included, the preference shifted strongly toward spatial flatness ($\Omega_{K0} \approx 0$).
• DESI Impact: The inclusion of DESI DR2 data further tightened constraints, confirming consistency with a flat universe ($\Omega_{K0} \approx 0$) with deviations $< 0.5\sigma$.

C. Viscosity Constraints

• Magnitude: The present-day bulk viscosity is consistently constrained to $\zeta_0 \sim 10^6$ Pa s across all scenarios (flat/curved, constant/variable).
• Thermodynamics: All best-fit values satisfy $\zeta_0 > 0$, ensuring non-negative entropy production.
• Evolution Parameter ($m$):
  • When using PPS alone, $m$ tends to be negative ($m < 0$), implying viscosity increases as the universe expands (density decreases).
  • When DESI data is included, the preference for $m < 0$ strengthens in both flat and curved scenarios.
  • However, $m$ remains weakly constrained, showing significant degeneracy with other parameters.

D. Model Selection (Evidence)

• Information Criteria:
  • AIC & DIC: Show mild support for viscous models in specific datasets (e.g., Base 1 + CC), suggesting they improve the fit quality enough to offset the penalty for extra parameters in some contexts.
  • BIC: Strongly favors the standard $\Lambda$CDM model ($\Delta$BIC < 0 for viscous models), penalizing the additional parameters ($m$, $\zeta_0$, $\Omega_{K0}$) too heavily.
• Bayesian Evidence:
  • Curved Models: Moderate to strong evidence against curved viscous models ($\ln B_{0i} > 3$ to $>5$), particularly when DESI data is included.
  • Flat Models: Evidence is inconclusive or weakly favors $\Lambda$CDM.
  • Conclusion: While viscous models offer a "better fit" in terms of $\chi^2$ for some datasets, they do not statistically outperform the simpler $\Lambda$CDM model according to Bayesian evidence and BIC.

5. Significance and Conclusion

• Resolution Status: Bulk viscous models do not resolve the Hubble tension completely; they only partially alleviate it (reducing it to $\sim 1\sigma$) when local priors are included.
• Parsimony: The standard $\Lambda$CDM model remains the most parsimonious description of the data. The addition of bulk viscosity does not provide a statistically significant improvement in model selection criteria (BIC/Evidence) to justify the extra complexity.
• Physical Viability: The study confirms that bulk viscosity of order $10^6$ Pa s is physically viable within the background expansion framework and consistent with thermodynamics, even if it is not the primary driver for solving current cosmological tensions.
• Future Directions: The authors conclude that current datasets (SNe Ia, H(z), BAO) cannot definitively break the degeneracies between $m$ and $\zeta_0$. Future analyses must incorporate Cosmic Microwave Background (CMB) observations to achieve robust constraints and potentially distinguish viscous scenarios from the standard model.