The $f(Q)$ gravity and affine EoS: Compatibility and observational constraints

This paper investigates the cosmological implications of an affine equation of state within General Relativity and $f(Q)$ gravity, demonstrating that while both frameworks can unify the universe's transition from decelerated to accelerated expansion, the affine model yields quintessence-like dark energy in General Relativity but phantom-like dark energy in $f(Q)$ gravity, with parameters constrained by Cosmic Chronometer and Supernovae type Ia data.

The Gist

Imagine the universe as a giant, expanding balloon. For a long time, scientists thought this balloon was inflating at a steady pace, or perhaps even slowing down due to gravity pulling everything together. But in the late 1990s, we discovered something shocking: the balloon isn't just inflating; it's speeding up. Something invisible is pushing the universe apart faster and faster. We call this mysterious pusher "Dark Energy."

This paper is a detective story where the authors try to figure out what this Dark Energy actually is, using two different rulebooks for how the universe works.

The Two Rulebooks

1. General Relativity (The Classic Rulebook): This is Einstein's famous theory. It's like the standard physics textbook we all learned. It says gravity is the curvature of space caused by mass.
2. f(Q) Gravity (The New Rulebook): This is a newer, more exotic theory. Instead of thinking about curved space, it looks at "non-metricity" (a fancy way of saying the rules for measuring distance change as the universe expands). Think of it as a new operating system for the universe that might explain things the old one can't.

The Mystery Ingredient: The "Affine" Fluid

The authors propose that the Dark Energy isn't just a random force. They suggest it behaves like a specific type of fluid with a special recipe called an "Affine Equation of State."

• The Analogy: Imagine a fluid that acts like a spring. If you squeeze it, it pushes back, but it also has a built-in "memory" or a constant baseline pressure that doesn't disappear.
• The Math: The formula they use is $p = \alpha\rho - \rho_0$.
  • $p$ is the pressure (the push).
  • $\rho$ is the energy density (how much "stuff" is there).
  • $\alpha$ and $\rho_0$ are constants (the recipe settings).

This fluid is special because it can act like normal matter in the past (slowing the universe down) and then switch gears to act like Dark Energy today (speeding the universe up). It's a unified scenario, meaning one single ingredient explains both the slow past and the fast present.

The Investigation: Testing the Theories

The authors took this "Affine Fluid" and ran it through both rulebooks (General Relativity and f(Q) Gravity) to see if it matches real-world observations. They used two main tools:

1. Cosmic Chronometers: Measuring the ages of old galaxies to see how fast the universe was expanding at different times.
2. Supernovae (Type Ia): Using exploding stars as "standard candles" to measure distances across the universe.

They used a statistical method (Bayesian analysis) to see which "recipe settings" for their fluid fit the data best.

The Results: Two Different Stories

Here is where the story gets interesting. The same fluid behaves very differently depending on which rulebook you use.

Story 1: General Relativity (Model I)

• The Vibe: This model is the "Goldilocks" zone. It's stable and sensible.
• The Dark Energy: It behaves like Quintessence. Think of Quintessence as a gentle, rolling hill. It pushes the universe apart, but it's a calm, controlled acceleration.
• Stability: The math says this model is stable. It won't break the laws of physics (like causality).
• The Future: As time goes on, this model starts to look more and more like the standard "Lambda-CDM" model (the current best guess of cosmology). It's a safe, reliable prediction.

Story 2: f(Q) Gravity (Model II)

• The Vibe: This model is wild and energetic.
• The Dark Energy: It behaves like Phantom Energy. Think of Phantom Energy as a runaway rocket. It pushes so hard that the acceleration gets crazier and crazier.
• Stability: The math says this model is unstable. In physics, "phantom" energy often leads to a "Big Rip," where the universe expands so fast it tears atoms apart. The authors found that for this model to fit the data, the fluid has to be in this unstable, "phantom" state.
• The Future: The universe would end in a dramatic, explosive expansion.

Can the Old Rulebook be the New Rulebook?

The authors asked a fascinating question: "Can the solution we found in the classic General Relativity (Model I) also be a solution in the new f(Q) gravity?"

They tried to translate the General Relativity solution into the language of f(Q) gravity. They found that to make it work, the new gravity theory would have to be a specific, simple function ($f(Q) = C\sqrt{Q}$). However, they discovered that in this specific case, the new gravity theory essentially vanishes. It becomes a "total derivative," meaning it doesn't actually change the physics at all.

The Conclusion: The solution that works in General Relativity cannot be a valid solution in f(Q) gravity unless the f(Q) gravity does nothing. They are fundamentally different paths.

The Final Verdict

• Both models fit the data: Both the "Classic" (General Relativity) and the "New" (f(Q)) models can explain the current speed of the universe's expansion.
• The Age of the Universe: Both models predict the universe is about 13.5 to 13.8 billion years old, which matches what we already know from other observations.
• The Winner?
  • If you want a stable, safe universe that behaves nicely and might eventually settle into a standard pattern, General Relativity with the Affine Fluid (Model I) is the winner.
  • If you are willing to accept a wild, unstable universe that might end in a "Big Rip," then f(Q) Gravity (Model II) is the path.

In simple terms: The paper shows that while we can explain the universe's acceleration with a special "affine fluid," the nature of that fluid depends entirely on which theory of gravity you choose. The classic theory gives us a calm, stable future, while the new theory suggests a chaotic, phantom-filled future. Both fit the current data, but they tell very different stories about the end of the universe.

Technical Summary

Here is a detailed technical summary of the paper "The f(Q) gravity and affine EoS: Compatibility and observational constraints" by Garg, Singh, and Singh.

1. Problem Statement

The paper addresses the challenge of explaining the late-time accelerated expansion of the universe without relying solely on the standard $\Lambda$CDM model, which faces issues like the fine-tuning problem and the cosmological constant problem. The authors investigate two specific theoretical frameworks:

1. General Relativity (GR): Using a dark fluid satisfying an Affine Equation of State (EoS), defined as $p_d = \alpha\rho_d - \rho_0$.
2. Modified Gravity ($f(Q)$ gravity): A theory based on non-metricity scalar $Q$, where the affine EoS fluid is analyzed to see if it can reproduce GR solutions or if it leads to distinct cosmological dynamics.

The core questions are:

• Can the affine EoS fluid unify the transition from decelerated to accelerated expansion in both GR and $f(Q)$ gravity?
• Are these models observationally compatible with current data (Cosmic Chronometers and Supernovae Type Ia)?
• Can the specific GR solution derived from the affine EoS be realized within the $f(Q)$ gravity framework?
• What is the nature of the dark energy (quintessence vs. phantom) in these models?

2. Methodology

Theoretical Framework

• Affine EoS: The authors assume a dark fluid with the equation of state $p_d = \alpha\rho_d - \rho_0$, where $\alpha$ and $\rho_0$ are constants. This form allows for hydro-dynamically stable propagation ($0 \le \alpha \le 1$) and can mimic various dark energy behaviors.
• Model I (GR): The authors derive the Hubble parameter $H(z)$ for a flat FLRW universe dominated by this dark fluid. The energy density evolves as $\rho_d = \rho_0 + \frac{\rho_{d0}(1+z)^{3(1+\alpha)}}{1+\alpha}$.
• Model II ($f(Q)$ Gravity): The authors consider a universe containing cold dark matter ($p_m=0$) and the dark fluid satisfying the affine EoS. They derive the corresponding Hubble parameter $H(z)$ for $f(Q)$ gravity. Crucially, they attempt to reverse-engineer the functional form of $f(Q)$ that would yield the same Hubble parameter as Model I.

Observational Analysis

• Datasets:
  • Cosmic Chronometers (CC): 31 data points measuring $H(z)$ via differential ages of galaxies ($0.07 \le z \le 1.965$).
  • Pantheon Sample: 1048 Type Ia Supernovae (SNIa) data points ($0.01 \le z \le 2.26$).
• Statistical Tools:
  • Bayesian Analysis: Implemented using the emcee Python library (MCMC).
  • $\chi^2$ Minimization: Used to constrain model parameters ($H_0, \alpha, \rho_0, \Omega_m$, etc.) by minimizing the difference between theoretical and observed values.
  • Joint Analysis: Combined CC and Pantheon datasets for tighter constraints.

Dynamical Analysis

The authors calculated key cosmographic parameters to characterize the universe's evolution:

• Deceleration Parameter ($q$): To identify the transition redshift ($z_t$) from deceleration to acceleration.
• Jerk ($j$) and Snap ($s$) Parameters: To describe higher-order derivatives of the scale factor.
• Equation of State ($\omega_e$): To determine if the dark energy is quintessence ($-1 < \omega < -1/3$), cosmological constant ($\omega = -1$), or phantom ($\omega < -1$).
• Age of the Universe ($t_0$): Calculated via numerical integration of $H(z)$.

3. Key Contributions and Results

A. Observational Constraints and Model Viability

• Model I (GR): The model is observationally compatible with both CC and joint datasets.
  • The parameter $\alpha$ is constrained to be slightly positive or near zero within $1\sigma$(e.g.,$0.032 \pm 0.054$for CC), satisfying the classical stability criterion ($0 \le \alpha \le 1$).
  • The transition redshift is found to be $z_t \approx 0.64$, consistent with standard cosmology.
  • The current age of the universe is calculated as $t_0 \approx 13.51$ Gyr (CC) and $13.41$ Gyr (Joint), consistent with Planck results.
• Model II ($f(Q)$ Gravity):
  • The model fits the data but reveals a critical instability. The parameter $\alpha$ is constrained to be negative (e.g., $\alpha \approx -1.08$ for joint data).
  • A negative $\alpha$ violates the classical stability criterion ($cs^2 > 0$), implying the model describes a phantom dark energy evolution that is classically unstable.
  • The current age is $t_0 \approx 13.76$ Gyr (CC) and $13.59$ Gyr (Joint).

B. The Equivalence Question (GR vs. $f(Q)$)

The authors investigated whether the Hubble parameter solution of Model I (GR) could be reproduced in $f(Q)$ gravity.

• By solving the field equations with the assumption that the dark fluid density in $f(Q)$ matches the GR density, they derived the functional form $f(Q) = C\sqrt{Q}$.
• Crucial Finding: The model $f(Q) = C\sqrt{Q}$ is a total derivative in the FLRW framework. Consequently, its contribution to the field equations vanishes identically.
• Conclusion: The specific solution derived in GR using the affine EoS cannot be realized as a non-trivial solution in $f(Q)$ gravity. The dynamics of Model I are unique to GR.

C. Nature of Dark Energy

• Model I (GR): The effective EoS parameter $\omega_e$ at present is $\approx -0.71$ (CC) and $\approx -0.72$ (Joint). This places the model in the quintessence regime ($-1 < \omega < -1/3$). As $z \to -1$ (future), the model approaches a de Sitter state ($\omega \to -1$), mimicking $\Lambda$CDM.
• Model II ($f(Q)$): The effective EoS parameter is $\approx -0.92$ (CC) and $\approx -0.73$ (Joint). However, due to the negative $\alpha$, the model exhibits phantom characteristics ($\omega < -1$) in certain evolutionary stages, leading to potential future singularities (Big Rip scenarios) and classical instability.

D. Cosmographic Parameters

• Deceleration ($q_0$): Both models show negative values ($q_0 \approx -0.56$ for Model I; $q_0 \approx -0.59$ to $-0.88$ for Model II), confirming current acceleration.
• Jerk ($j_0$): Model I approaches the $\Lambda$CDM limit ($j=1$) in the future, while Model II shows different behavior due to phantom dynamics.

4. Significance and Conclusion

1. Unified Scenario in GR: The paper successfully demonstrates that the affine EoS fluid in General Relativity provides a viable, observationally consistent unified scenario that transitions from matter-dominated deceleration to dark-energy-dominated acceleration.
2. Distinction between GR and $f(Q)$: The study highlights a fundamental difference between the two frameworks. While the affine EoS works well in GR, attempting to map this specific solution to $f(Q)$ gravity fails because the required functional form ($f(Q) \propto \sqrt{Q}$) is trivial in FLRW cosmology.
3. Stability Warning: The $f(Q)$ model with affine EoS and cold dark matter, while fitting data, requires a negative $\alpha$ parameter, rendering it classically unstable. This suggests that if $f(Q)$ gravity is to describe our universe with this specific fluid, it may require different constraints or fluid properties to ensure stability.
4. Dark Energy Nature: The analysis distinguishes between quintessence (Model I) and phantom (Model II) behaviors, suggesting that the choice of gravity theory significantly impacts the physical interpretation of dark energy stability and future evolution.

In summary, the paper validates the affine EoS as a robust tool for modeling cosmic acceleration in General Relativity but cautions against its direct application in $f(Q)$ gravity without addressing stability issues and the triviality of the resulting field equations.