Observational constraints on viscous free-$γ$ fluid in $f(Q)$ gravity

1. Problem Statement

The standard $\Lambda$CDM model, while successful in fitting observational data, suffers from theoretical issues such as the cosmological constant problem (fine-tuning and the vacuum energy discrepancy) and the current $H_0$ tension. To address these, the authors explore two alternative frameworks:

1. Modified Gravity: Specifically, $f(Q)$ gravity, a theory based on non-metricity (symmetric teleparallel gravity) where the gravitational interaction is mediated by the non-metricity scalar $Q$ rather than curvature ($R$) or torsion ($T$). This theory offers second-order field equations, avoiding the pathologies often found in $f(R)$ gravity.
2. Dissipative Cosmology: The inclusion of bulk viscosity in the cosmic fluid. Standard cosmology often assumes a perfect fluid (dust), but dissipative processes (viscosity) can generate effective negative pressure, potentially driving late-time acceleration without a cosmological constant.

The paper aims to investigate the late-time cosmological dynamics of a spatially flat FLRW universe filled with a bulk viscous fluid characterized by a free equation-of-state (EoS) parameter $\gamma$ (where $p = (\gamma-1)\rho$), within the framework of $f(Q)$ gravity.

2. Methodology

Theoretical Framework

• Gravity Model: The authors adopt an exponential $f(Q)$ model:
  $$f(Q) = Q e^{\alpha Q_0 / Q}$$
  where $\alpha$ is a dimensionless constant and $Q_0 = 6H_0^2$. This model is designed to recover General Relativity (GR) at high redshifts (early times) while allowing deviations at late times.
• Matter Sector: The matter is modeled as a bulk viscous fluid. The effective pressure is defined as $p_{\text{eff}} = p - 3\zeta H$, where $\zeta$ is the bulk viscosity coefficient and $H$ is the Hubble parameter. The EoS is generalized as $p = (\gamma-1)\rho$, allowing $\gamma$ to vary freely (unlike the standard dust case where $\gamma=1$).
• Field Equations: Using the coincident gauge (where the affine connection vanishes), the authors derive the modified Friedmann equations for the $f(Q)$ framework coupled with the viscous fluid conservation equation.

Observational Data & Statistical Analysis

The model parameters are constrained using a Markov Chain Monte Carlo (MCMC) analysis (using the emcee package) against a combination of four independent datasets:

1. Cosmic Chronometers (CC): Differential age measurements of galaxies providing direct $H(z)$ data.
2. DESI DR2 BAO: Baryon Acoustic Oscillation measurements from the Dark Energy Spectroscopic Instrument (Data Release 2).
3. Gamma-Ray Bursts (GRBs): 162 long GRBs used as standardizable candles via the $E_{p,i}$–$E_{\text{iso}}$ correlation, extending the redshift range significantly.
4. Union3 Supernovae: A compilation of 2087 Type Ia supernovae with rigorous treatment of systematic uncertainties.

Diagnostics

To distinguish the proposed models from $\Lambda$CDM and other dark energy scenarios, the authors employ:

• Statefinder Parameters ($r, s$): Geometric diagnostics to track the evolution of the expansion.
• $Om(z)$ Diagnostic: A model-independent test to distinguish between dark energy and modified gravity.
• Deceleration Parameter ($q$) and Effective EoS ($w_{\text{eff}}$): To analyze the transition from deceleration to acceleration.
• Information Criteria: Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) are used to compare model performance relative to $\Lambda$CDM, penalizing for the number of free parameters.

3. Key Contributions

1. Generalized Viscous Model: The study extends previous work by treating the EoS parameter $\gamma$ as a free parameter rather than fixing it to the dust limit ($\gamma=1$). This allows for a broader phenomenological description of cosmic matter.
2. Combined Constraints: It provides the first comprehensive constraints on a viscous fluid within an exponential $f(Q)$ gravity model using the latest DESI DR2 data combined with high-redshift GRBs and Union3 supernovae.
3. Interplay of Geometry and Dissipation: The analysis explicitly quantifies the compensation mechanism between the modified gravity contribution (non-metricity) and bulk viscosity in driving cosmic acceleration.

4. Results

Parameter Constraints

• Hubble Constant ($H_0$): The models constrain $H_0$ to the range 65–68 km s$^{-1}$ Mpc$^{-1}$. The $f(Q)$ model generally favors slightly higher values than the pure viscous GR model, though both remain below local distance-ladder measurements (e.g., SH0ES), indicating the models do not fully resolve the $H_0$ tension.
• Viscosity ($\tilde{\zeta}$):
  • In pure GR (viscous only), a large viscosity coefficient is required ($\tilde{\zeta} \approx 1.6$ for $\gamma=1$ and $\approx 2.2$ for free $\gamma$) to drive acceleration.
  • In $f(Q)$ gravity, the required effective viscosity is significantly smaller ($\tilde{\zeta} \approx 0.3\text{--}0.5$). This demonstrates that the modified gravity sector efficiently drives acceleration, reducing the need for strong dissipative effects.
• Equation of State ($\gamma$): The free-$\gamma$ scenario yields best-fit values close to unity ($\gamma \approx 1.04\text{--}1.16$), consistent with dust-like matter but allowing for slight deviations.

Dynamical Evolution

• Acceleration: Both models successfully reproduce the transition from a decelerated early universe to the current accelerated phase. The transition redshift ($z_t$) is found around 0.5–0.7.
• Statefinder & $Om(z)$:
  • The trajectories in the $(r, s)$ plane approach the $\Lambda$CDM fixed point $(1, 0)$ at late times but evolve into the quintessence region ($s>0, r<1$) at higher redshifts.
  • The models remain clearly separated from the Chaplygin gas sector, indicating the acceleration is driven by effective quintessence-like dynamics rather than unified dark fluid behavior.
  • The $Om(z)$ diagnostic shows a mild redshift dependence, confirming the dynamical nature of the dark energy sector in these models.

Statistical Comparison

• Goodness of Fit: When using the full dataset (CC + DESI + GRB + Union3), the viscous model with free $\gamma$ shows a significant improvement in $\chi^2$ ($\Delta\chi^2 \approx -8.9$) compared to $\Lambda$CDM.
• Information Criteria:
  • The AIC favors the viscous free-$\gamma$ model and the $f(Q)$ free-$\gamma$ model (negative $\Delta$AIC), suggesting they are statistically preferred over $\Lambda$CDM given the improved fit.
  • The BIC is more conservative; while it penalizes the extra parameters, the viscous free-$\gamma$ model remains competitive ($\Delta$BIC $\approx -3.5$), whereas the $f(Q)$ free-$\gamma$ model is slightly disfavored ($\Delta$BIC $\approx +1.9$) due to the additional parameter freedom.

5. Significance

This work demonstrates that bulk viscosity within $f(Q)$ gravity provides a viable, observationally consistent alternative to the standard $\Lambda$CDM model. Key takeaways include:

• Viability of Modified Gravity: The $f(Q)$ framework can reproduce the late-time acceleration with a much weaker reliance on bulk viscosity compared to pure GR, suggesting a synergistic effect between geometric modifications and dissipative processes.
• Robustness of Free-$\gamma$: Allowing the EoS parameter to vary improves the fit to observational data, particularly when high-redshift GRB data is included, highlighting the importance of generalized matter descriptions in cosmology.
• Future Probes: The study underscores the power of combining low-redshift (CC, BAO) and high-redshift (GRB, SNe) probes to tightly constrain the degeneracy between modified gravity parameters and viscous coefficients.

The paper concludes that while these models do not fully resolve the $H_0$ tension, they offer a physically motivated and statistically competitive framework for understanding cosmic acceleration, warranting further investigation with future observational data.