Observationanal and Thermodynamic Analysis of Power Law form of $f (Q)$ gravity

Technical Summary: Observational and Thermodynamic Analysis of Power Law $f(Q)$ Gravity

Problem Statement
The paper addresses the accelerated expansion of the universe, a phenomenon currently explained either by exotic dark energy or by modifying General Relativity (GR). Specifically, the authors investigate the power law form of $f(Q)$ gravity, defined as $f(Q) = Q + mQ^n$, where $Q$ is the non-metricity scalar. While previous studies have constrained this model using energy conditions or constant sound speeds, this work aims to investigate the model without prior constraints, focusing on two specific domains:

1. Observational Viability: Testing the model against current cosmological data (Hubble parameter, Supernovae) to determine best-fit parameters and its ability to resolve the Hubble tension.
2. Thermodynamic Consistency: Analyzing the model through the lens of irreversible thermodynamics (Eckart theory) rather than traditional reversible thermodynamics. The study seeks to establish the validity of the Generalized Second Law of Thermodynamics (GSLT) and the conditions for Thermodynamic Equilibrium (TE) within this modified gravity framework.

Methodology
The study employs a dual-pronged approach combining statistical data analysis with theoretical thermodynamic derivation.

• Observational Analysis:

  • Datasets: The authors utilize three datasets: 31-point Hubble ($H(z)$) data from cosmic chronometers, and a combined dataset of 1721 points comprising Pantheon+ Type Ia Supernovae (SNIa) and SH0ES data.
  • Statistical Tool: A Markov Chain Monte Carlo (MCMC) method is used to maximize the likelihood function and minimize the total chi-square ($\chi^2$) to estimate the best-fit values for the Hubble constant ($H_0$), matter density ($\Omega_{m0}$), and the power-law index ($n$).
  • Model Comparison: The performance of the $f(Q)$ model is compared against the standard $\Lambda$CDM model using the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC).

• Thermodynamic Analysis:

  • Framework: The universe is treated as an isolated system bounded by an apparent horizon. The authors apply the Unified First Law of Thermodynamics to derive the horizon entropy ($S_A$) in $f(Q)$ gravity, which includes correction terms beyond the standard Bekenstein-Hawking area law.
  • Irreversibility: Unlike standard approaches, the study incorporates Eckart's first-order theory of irreversible thermodynamics. This involves calculating the entropy production density ($\sigma$) due to heat conduction, leading to a total entropy change equation ($dS_T/dt$) dependent on a non-equilibrium factor (thermal conductivity $\lambda$).
  • Conditions: The validity of the Generalized Second Law of Thermodynamics (GSLT) ($dS_T/dt \geq 0$) and Thermodynamic Equilibrium (TE) ($d^2S_T/dt^2 < 0$) are examined for both quintessence and phantom eras.

• Cosmographic Analysis:

  • The evolution of cosmographic parameters (deceleration $q$, jerk $j$, and snap $s$) is studied within the parameter ranges derived from the observational and thermodynamic analyses.
  • The $Om(z)$ diagnostic is employed to distinguish the model's behavior from $\Lambda$CDM.

Key Contributions and Results

1. Observational Constraints and Hubble Tension:

   • The combined analysis of Hubble, Pantheon+, and SH0ES data yields best-fit parameters of $H_0 \approx 72.67$ km s$^{-1}$ Mpc$^{-1}$ and $\Omega_{m0} \approx 0.229$ for the power law model with $n \approx 1.26$.
   • The model provides an excellent fit to the data ($\chi^2/dof \approx 1$) and is statistically consistent with $\Lambda$CDM based on AIC and BIC values.
   • Hubble Tension: The derived $H_0$ value lies significantly closer to local SH0ES measurements ($74.03 \pm 1.42$) than to the Planck CMB value ($67.4 \pm 0.5$), suggesting the model partially alleviates the Hubble tension.
   • The parameter $n$ is tightly constrained, with the combined dataset favoring values close to unity ($n \approx 1.26$).

2. Thermodynamic Validity:

   • Using the Unified Law and Eckart theory, the authors derived explicit conditions for GSLT and TE.
   • GSLT: The Generalized Second Law is found to be valid only when $n \geq 1$.
   • Thermodynamic Equilibrium: The study identifies a specific range, $1 \leq n \leq 2.4$, where GSLT holds ($dS_T/dt \geq 0$) but Thermodynamic Equilibrium is violated ($d^2S_T/dt^2 > 0$). The authors argue this is physically desirable, as the irreversible nature of heat flow in the universe implies that a state of maximum entropy (TE) should not be maintained.
   • The analysis confirms that for the best-fit observational value ($n \approx 1.26$), the thermodynamic requirements are satisfied.

3. Cosmographic Evolution:

   • In the far future ($z \to -1$), the $f(Q)$ model converges to the de Sitter solution ($q = -1$) and becomes indistinguishable from $\Lambda$CDM.
   • However, at the present epoch and in the near past, the models diverge. At the best-fit $n=1.26$, the deceleration parameter $q_0$ is approximately 30% higher than in $\Lambda$CDM, and the jerk parameter $j_0$ is roughly 60% lower.
   • The snap parameter ($s$) is identified as the most promising observational handle for distinguishing the two models, as it remains unambiguously higher than the $\Lambda$CDM prediction across the observable redshift range.
   • Transition Redshift ($z_t$): A notable discrepancy is found in the transition redshift. While $\Lambda$CDM predicts $z_t \approx 0.66$, the $f(Q)$ model with $n=1.26$ predicts a much earlier transition ($z_t \approx 2.3$). The authors note that increasing $n$ improves agreement with $\Lambda$CDM cosmographic values but further increases $z_t$, creating a tension between cosmographic fit and transition timing.

Significance and Claims
The paper claims to establish a robust correspondence between observational data and thermodynamic principles for the power law $f(Q)$ gravity model. Its primary significance lies in:

• Unifying Analysis: Successfully demonstrating that the parameter range favored by current observational data ($n \approx 1.26$) falls comfortably within the range required for the validity of the Generalized Second Law of Thermodynamics ($n \geq 1$) and the violation of Thermodynamic Equilibrium.
• Irreversible Framework: Highlighting the necessity of using irreversible thermodynamics (Eckart theory) to correctly describe the universe's evolution in modified gravity, rather than relying on reversible assumptions.
• Hubble Tension Mitigation: Proposing that this specific $f(Q)$ model offers a viable alternative to $\Lambda$CDM that naturally yields a higher Hubble constant, thereby reducing the tension between local and early-universe measurements.

The authors conclude modestly, acknowledging that while the thermodynamic and observational constraints are mutually consistent, the model still faces challenges regarding the transition redshift. They suggest that future work may require modifications to the power law form to resolve this specific discrepancy.