Testing $f(Q)$ Gravity with Logarithmic Equation of State Using Latest Cosmological Data

Technical Summary: Testing f(Q) Gravity with Logarithmic Equation of State Using Latest Cosmological Data

Problem Statement
The standard $\Lambda$CDM model, while successful, faces theoretical challenges regarding the nature and origin of the cosmological constant and growing tensions in measurements of the current expansion rate of the Universe. These limitations have motivated the exploration of alternative frameworks, including modified theories of gravity. Specifically, this paper investigates symmetric teleparallel gravity, where gravity is described by the non-metricity scalar $Q$ rather than spacetime curvature or torsion. The authors focus on a power-law form of $f(Q)$ gravity, $f(Q) = Q + 6\gamma H_0^2 (Q/Q_0)^n$, to explain the late-time accelerated expansion of the Universe without invoking an explicit dark energy component. A key challenge addressed is characterizing the effective dark energy sector emerging from this modified gravity with a flexible equation of state (EoS) that allows for deviations from the constant $\omega = -1$ of the standard cosmological constant.

Methodology
The study employs a theoretical and observational approach combining analytical derivation with statistical data analysis:

1. Theoretical Framework:

   • The authors adopt the coincident gauge within the symmetric teleparallel equivalent of General Relativity (STEGR), where the connection is flat and torsion-free.
   • They propose a specific power-law model for the modification function: $F(Q) = 6\gamma H_0^2 (Q/Q_0)^n$, where $Q_0 = 6H_0^2$.
   • To model the effective dark energy arising from this geometry, they introduce a logarithmic parametrization for the equation of state: $\omega_{de}(z) = \omega_0 + \omega_1 \ln(1+z)$. This allows for a smooth deviation from $\omega = -1$ across the redshift range.
   • An analytical expression for the Hubble parameter $H(z)$ is derived by solving the modified Friedmann equations with this logarithmic EoS.

2. Observational Data and Statistical Analysis:

   • The model is constrained using a Markov Chain Monte Carlo (MCMC) analysis against three independent datasets:
     • Pantheon+: 1701 Type Ia supernovae (SNIa) data points.
     • DESI DR2: 12 Baryon Acoustic Oscillation (BAO) data points from the Dark Energy Spectroscopic Instrument.
     • Cosmic Chronometers (CC): 32 direct measurements of the Hubble parameter $H(z)$.
   • The parameter space includes the Hubble constant $h$, matter density $\Omega_{m0}$, the EoS parameters $\omega_0$ and $\omega_1$, and the power-law index $n$.
   • Model performance is evaluated using the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) to compare the $f(Q)$ model against the standard $\Lambda$CDM model, accounting for the number of free parameters.

3. Cosmographic Diagnostics:

   • The dynamical behavior is probed using the deceleration ($q$), jerk ($j$), and snap ($s$) parameters.
   • Additional diagnostics include the evolution of density parameters ($\Omega_m, \Omega_{de}$), the $Om(z)$ diagnostic, and the statefinder parameters $\{s, j\}$.

Key Results

• Parameter Constraints: The MCMC analysis yields tightly constrained values for the Hubble parameter ($h \approx 0.732 - 0.736$) and matter density ($\Omega_{m0}$), which are consistent with $\Lambda$CDM predictions and SH0ES measurements. The logarithmic EoS parameters are constrained to $\omega_0 \approx -0.9$ and $\omega_1$ taking small values (positive or negative depending on the dataset combination), indicating a mild deviation from a cosmological constant.
• Model Comparison:
  • AIC: The power-law $f(Q)$ model shows comparable performance to $\Lambda$CDM, with $\Delta AIC \approx 0.91$ (Pantheon+ + DESI) and $1.06$ (Pantheon+ + DESI + CC), suggesting the models are statistically indistinguishable or only slightly favored.
  • BIC: The BIC yields $\Delta BIC \approx 17.3 - 17.4$, indicating a strong penalty for the $f(Q)$ model due to its additional free parameters, thus favoring the simpler $\Lambda$CDM model under this criterion.
• Dynamical Evolution:
  • The model successfully reproduces the transition from a matter-dominated decelerated phase to a late-time accelerated expansion.
  • The deceleration parameter $q_0$ is found to be approximately $-0.45$, and the jerk parameter $j_0$ is close to unity ($0.70 - 0.81$), consistent with standard cosmological scenarios.
  • The $Om(z)$ diagnostic lies below the constant $\Lambda$CDM value with a negative slope, suggesting an effective phantom-like behavior.
  • Statefinder trajectories in the $\{s, j\}$ plane show the model evolving from a phantom region, through quintessence, approaching the $\Lambda$CDM fixed point $(0, 1)$, and returning to the phantom region in the future.

Significance and Claims
The paper claims that the logarithmic parametrization within the power-law $f(Q)$ framework serves as a valid and more flexible alternative to standard dark energy models. Its primary significance lies in providing a richer phenomenology at low redshifts, allowing for a smooth, controlled departure from the constant equation of state ($\omega = -1$) without violating current observational constraints.

The authors conclude that while the standard $\Lambda$CDM model remains statistically preferred by information criteria due to its parsimony, the proposed $f(Q)$ model offers a geometric alternative to dark energy that is phenomenologically viable. It successfully reproduces the observed expansion history and demonstrates that non-metricity can play a central role in shaping cosmic dynamics. The study suggests that this framework is capable of mitigating cosmological tensions and warrants further investigation, particularly regarding perturbations and extensions to other observational probes.