Constraining viscous fluid models in $f(Q)$ gravity with data

This study utilizes combined cosmological datasets (BAO, CC, SNIa, and growth rate data) to constrain viscous fluid models within various $f(Q)$ gravity frameworks, finding that while exponential and logarithmic models are statistically rejected, the non-viscous power-law $f(Q)$ model remains a robust alternative to $\Lambda$CDM.

The Gist

Imagine the universe as a giant, expanding balloon. For decades, scientists have been trying to figure out exactly how that balloon is inflating and what's inside it. The current "gold standard" theory, called ΛCDM, suggests the balloon is being pushed apart by a mysterious, invisible force called Dark Energy.

But what if Dark Energy isn't a thing at all? What if the rules of gravity itself are slightly different than Einstein thought?

This paper is like a team of cosmic detectives testing a new set of "gravity rules" called f(Q) gravity. They wanted to see if these new rules could explain the universe's expansion without needing Dark Energy, and they added a twist: they asked, "What if the stuff inside the universe (like gas and dust) is a bit 'sticky'?"

Here is the breakdown of their investigation in simple terms:

1. The New Gravity Rules (f(Q) Gravity)

Think of Einstein's General Relativity as a perfectly smooth trampoline. If you put a bowling ball (a star) on it, it curves, and marbles (planets) roll toward it. That's gravity based on curvature.

The f(Q) gravity theory the authors are testing is like a trampoline made of a different material. Instead of bending, this material stretches or changes its shape in a way that isn't quite curvature. It's based on something called "non-metricity."

• The Analogy: Imagine walking on a floor. In Einstein's world, the floor is flat but bends under weight. In this new theory, the floor tiles themselves might be slightly warped or shifting as you walk, changing how you move without the floor actually "bending" down.

The authors tested three specific versions of these new rules:

• The Power-Law Model (f1CDM): Like a simple, straight-line rule.
• The Exponential Model (f2CDM): Like a rule that speeds up very quickly.
• The Logarithmic Model (f3CDM): Like a rule that slows down as it gets bigger.

2. The "Sticky" Twist (Bulk Viscosity)

The authors wondered: What if the cosmic fluid (the stuff filling the universe) isn't perfectly smooth? What if it has viscosity?

• The Analogy: Think of the difference between water and honey. Water flows easily (low viscosity). Honey is thick and resists movement (high viscosity).
• In the universe, "bulk viscosity" means the cosmic fluid resists the expansion a bit, like a thick syrup slowing down a stirrer. The authors asked: Does adding this "stickiness" help our new gravity rules fit the data better?

3. The Investigation (The Data)

To solve the case, they didn't just guess; they used a massive amount of real-world evidence, like a detective gathering clues:

• Cosmic Chronometers (CC): Measuring the "ages" of old galaxies to see how fast the universe was expanding in the past.
• BAO (Baryon Acoustic Oscillations): Looking at the "frozen sound waves" left over from the Big Bang, which act like a cosmic ruler.
• Supernovae (Pantheon+ SH0ES): Using exploding stars as "standard candles" to measure distances.
• Growth Rate (f & fσ8): Watching how fast galaxies clump together to form clusters.

They ran these clues through a super-computer simulation (using a tool called Kosmulator) to see which gravity model and which "stickiness" level matched reality best.

4. The Verdict (The Results)

After crunching the numbers, the team came to some very clear conclusions:

• The "Sticky" Idea Didn't Help: Adding viscosity (making the universe "honey-like") actually made the models worse. It was like trying to fix a leaky boat by adding more holes. Statistically, the "sticky" models were rejected because they didn't fit the data any better than the simple models, but they had more complicated math to explain them.
• The "Power-Law" Winner: Out of the three new gravity models, only the Power-Law model (f1CDM) survived. It was the only one that didn't get rejected by the data. It performed almost as well as the standard Dark Energy model (ΛCDM) but without needing the mysterious Dark Energy ingredient.
• The Other Two Lost: The Exponential and Logarithmic models were rejected. They just didn't match the cosmic clues.

5. Why This Matters

This paper is a significant step because it suggests that we might not need Dark Energy to explain why the universe is accelerating. Instead, we might just need to tweak the rules of gravity slightly (using the Power-Law f(Q) model).

However, the "sticky" fluid idea (viscosity) was a dead end. The universe seems to flow more like water than honey, at least when it comes to these specific gravity theories.

In a nutshell: The authors tried to replace the "Dark Energy" engine with a new type of gravity engine. They found one engine that works well, but they discovered that adding "thick syrup" to the fuel tank doesn't help the car run better. The universe is expanding, and maybe it's just the rules of the road that are different, not a mysterious invisible pusher.

Technical Summary

Here is a detailed technical summary of the paper "Constraining Viscous-Fluid Models in f(Q) Gravity with Data" by Sahlu et al.

1. Problem Statement

The standard $\Lambda$CDM model, while successful, faces significant challenges, including the $H_0$ tension (discrepancy between early and late-universe expansion rates) and the $S_8$ tension (discrepancy in matter clustering amplitude). Furthermore, the nature of dark energy remains unknown.

• Alternative Gravity: The authors investigate $f(Q)$ gravity, a modified theory based on non-metricity ($Q$) rather than curvature ($R$) or torsion ($T$). Unlike General Relativity (GR), $f(Q)$ offers a geometric explanation for cosmic acceleration without necessarily invoking a cosmological constant.
• Viscous Fluids: The study incorporates bulk viscosity into the cosmic fluid. In a homogeneous and isotropic universe, bulk viscosity modifies the effective pressure ($p_{eff} = p - \zeta H$), potentially driving acceleration and altering structure formation without dark energy.
• The Gap: While $f(Q)$ gravity and viscous fluids have been studied separately, there is a lack of comprehensive observational constraints on viscous $f(Q)$ models using modern large-scale structure data. This paper aims to test the viability of specific $f(Q)$ models with and without bulk viscosity against current cosmological datasets.

2. Methodology

Theoretical Framework

• Action: The theory is defined by the action $S = \int \sqrt{-g} [\frac{1}{2}f(Q) + \mathcal{L}_m] d^4x$, where $Q$ is the non-metricity scalar.
• Modified Friedmann Equations: Assuming a spatially flat FLRW metric and a viscous fluid, the authors derive modified Friedmann equations. The effective pressure includes a bulk viscosity term $\zeta(\rho)H$, where the viscosity coefficient is assumed to be linear ($\zeta = \text{const}$, i.e., $\delta=0$ in $\zeta \propto \rho^\delta$).
• Three Paradigmatic Models: Three specific functional forms for $f(Q)$ are tested:
  1. $f_1$CDM (Power-law): $F(Q) = \alpha Q^n$.
  2. $f_2$CDM (Exponential): $F(Q) = \beta Q_0 (1 - e^{-p\sqrt{Q/Q_0}})$.
  3. $f_3$CDM (Logarithmic): $F(Q) = \epsilon \ln(\Gamma Q/Q_0)$.
• Cosmological Perturbations: Using the 1+3 covariant formalism, the authors derive the evolution equations for density contrasts ($\delta_m$) and growth rates ($f(z)$) in the presence of non-metricity and bulk viscosity. They employ the quasi-static approximation to simplify the second-order differential equations governing structure growth.

Data and Statistical Analysis

• Datasets: The study utilizes a comprehensive set of observational data:
  • Cosmic Chronometers (CC): 31 data points for $H(z)$.
  • Baryon Acoustic Oscillations (BAO): From the DESI survey (isotropic and anisotropic).
  • Supernovae Type Ia (SNIa): Pantheon+ SH0ES sample (1701 light curves).
  • Growth Rate Data: Redshift-space distortions ($f\sigma_8$) and linear growth rate ($f$) from VIPERS and SDSS.
• Statistical Tools:
  • MCMC Simulations: Performed using the Kosmulator Python package (based on emcee and GetDist) to constrain parameters: $\Omega_m, H_0, r_d, M_{abs}, \zeta, \gamma, \sigma_8$, and model exponents ($n, p, \Gamma$).
  • Model Selection: The Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) are used to compare the $f(Q)$ models against the standard $\Lambda$CDM model. The Jeffreys scale is applied to interpret $\Delta$IC values (where $\Delta$IC $\ge$ 10 implies rejection).

3. Key Contributions

1. First Joint Analysis: This is the first study to simultaneously constrain bulk viscosity and three distinct $f(Q)$ models using a combination of background expansion data (CC, BAO, SNIa) and perturbation data ($f, f\sigma_8$).
2. Perturbation Formalism in $f(Q)$: The authors explicitly derive and implement the growth rate evolution equations for viscous fluids within the $f(Q)$ framework, extending previous work that focused primarily on background expansion.
3. Systematic Model Comparison: A rigorous statistical comparison of power-law, exponential, and logarithmic $f(Q)$ models, evaluating their performance both with and without the viscous term.

4. Results

Parameter Constraints

• Bulk Viscosity ($\zeta$): The inclusion of bulk viscosity generally leads to a slight increase in the matter density parameter ($\Omega_m$) and a decrease in the clustering amplitude ($\sigma_8$).
• $S_8$ Tension: In viscous models, the derived $S_8$ values tend to be lower. For instance, the viscous $f_1$CDM model yields $S_8 \approx 0.776$, which is closer to weak lensing measurements than the Planck $\Lambda$CDM value, though the viscous $f_2$CDM and $f_3$CDM models produce values that are statistically too low.
• $H_0$ Tension: The models do not significantly resolve the $H_0$ tension; the tension between BAO/CC and Pantheon+SH0ES persists across all models.

Statistical Viability (AIC/BIC)

• $f_1$CDM (Power-law):
  • Non-viscous ($\zeta=0$): This is the only model that is not outright rejected by any dataset. It achieves "substantial observational support" ($\Delta$AIC < 2) for four datasets and remains within the "less support" category for others.
  • Viscous ($\zeta \neq 0$): Adding viscosity increases $\Delta$AIC and $\Delta$BIC, leading to rejections in several datasets (specifically those involving Pantheon+SH0ES) based on BIC. The extra parameter $\zeta$ does not improve the fit enough to justify its inclusion.
• $f_2$CDM (Exponential) & $f_3$CDM (Logarithmic):
  • Both models are rejected by multiple datasets (particularly Pantheon+SH0ES) in both viscous and non-viscous cases.
  • The $\Delta$BIC values often exceed 10, indicating no observational support relative to $\Lambda$CDM.
  • The logarithmic model ($f_3$CDM) produces extreme variations in $\Omega_m$ (ranging from 0.19 to 0.39 depending on the dataset), rendering it unstable.

Growth Rate and Structure Formation

• The growth rate $f(z)$ and redshift-space distortion $f\sigma_8(z)$ predictions for the non-viscous $f_1$CDM model align closely with $\Lambda$CDM and observational data, especially at high redshifts.
• The inclusion of bulk viscosity introduces damping in the evolution of density perturbations, slightly altering the growth history but failing to provide a statistically superior fit.

5. Significance and Conclusion

• Viability of $f(Q)$ Gravity: The study concludes that among the three tested classes, the power-law $f_1$CDM model (without bulk viscosity) is the most viable alternative to $\Lambda$CDM. It provides robust parameter estimates and is not statistically rejected by current data.
• Role of Bulk Viscosity: The inclusion of bulk viscosity is not statistically justified in this context. It consistently increases the information criteria values ($\Delta$AIC, $\Delta$BIC), indicating that the data does not support the extra complexity introduced by the viscosity parameter.
• Implications: While viscous $f(Q)$ models do not currently offer a better fit than $\Lambda$CDM, the power-law $f(Q)$ model remains a strong candidate for explaining cosmic acceleration. Future work should explore other model selection criteria and incorporate CMB data to further constrain the parameter space, particularly regarding the $S_8$ tension.

In summary, the paper demonstrates that while $f(Q)$ gravity offers a promising geometric alternative to dark energy, the specific addition of bulk viscosity does not currently improve the model's fit to observational data, and the power-law variant remains the most robust candidate within this framework.