New generalization of the Barboza-Alcaniz parametrization of Dark energy

Technical Summary: New Generalization of the Barboza-Alcaniz Parametrization of Dark Energy

Problem Statement
The standard $\Lambda$CDM model, while successful in confronting observational data, faces fundamental theoretical challenges, including the cosmological constant problem (the discrepancy between observed vacuum energy and quantum field theory estimates) and current cosmological tensions (e.g., Hubble and $S_8$ tensions). Consequently, dynamical dark energy (DE) models are often explored as alternatives. Among phenomenological parametrizations, the Barboza-Alcaniz (BA) model is notable for ensuring the equation of state (EoS) parameter, $w_{de}$, remains finite at early times, the present, and the far future ($z \to -1$). However, the BA model possesses a specific shortcoming: its behavior at $z < 0$ (the future) is a duplicate of its behavior at $z > 0$ (the past). Specifically, $\lim_{z \to 0, -1} w_{de} = w_0$, implying the DE EoS loops back to its current value regardless of dynamics. This "duplication" is an artificial constraint that fixes the DE transition time and peak location, lacking physical justification.

Methodology
The authors propose a new three-parameter generalization of the BA model, termed BAn, to resolve the future-time shortcomings of the original parametrization.

1. Model Formulation:
   The new EoS parameter is defined as:
   $$w_{de}(z) = w_0 + \frac{n}{2} w_a \frac{(1+z)|z|^{n-1}}{1+|z|^n}$$
   where $w_0$ and $w_a$ are standard DE parameters, and $n$ is a new free parameter. To ensure real-valued results for non-integer $n$, the absolute value $|z|$ is utilized.

   • For $n=2$, the model reduces to the original BA model.
   • For odd integer $n$, the limit as $z \to -1$ differs from the present value ($w_{de}(z=0)$), breaking the artificial symmetry of the BA model.
   • The DE energy density $\rho_{de}$ is derived via the conservation equation, yielding a dimensionless form $X(z)$.

2. Theoretical Origin:
   The paper demonstrates that the BAn parametrization can be derived from a modified gravity framework involving a non-standard matter Lagrangian, $L_m = f(\rho, P)$. Specifically, by defining $f(\rho, P) = P - B(\rho)$, where $B(\rho)$ corresponds to the energy density of the BAn model, the field equations reproduce the proposed dynamics. The authors note that while this works at the background level, energy-momentum conservation may not hold at the perturbative level, implying interactions between dark matter and dark energy.

3. Statistical Analysis:
   The model was constrained using Markov Chain Monte Carlo (MCMC) analysis against four independent datasets:

   • Cosmic Chronometers (CC): 31 data points for $H(z)$.
   • Pantheon+: $\sim$1500 Type Ia Supernovae.
   • DESI DR2 BAO: Baryon Acoustic Oscillation data covering $0.2 \lesssim z \lesssim 2.4$.
   • CMB Distance Priors: Compressed geometric data from Planck.

   The analysis compared the BAn model against the standard $\Lambda$CDM and the original BA model using Bayesian evidence (Jeffreys scale) and reduced chi-squared ($\chi^2_{red}$) statistics.

Key Results

• Parameter Constraints: The best-fit value for the new parameter is found to be $n \approx 1.865$ (with $1\sigma$ uncertainty) when using the full dataset (CC + Pantheon+ + BAO + CMB). This value is close to, but distinct from, the original BA choice of $n=2$.
• Model Comparison:
  • Both BA and BAn models provide substantially improved fits over $\Lambda$CDM.
  • Bayesian evidence indicates strong evidence favoring BAn over $\Lambda$CDM and moderate evidence favoring BA over $\Lambda$CDM.
  • When comparing BAn directly to BA, the reduction in $\chi^2$ is modest and not statistically significant via the $\chi^2$ difference test. However, Bayesian evidence shows weak support for BAn over BA, suggesting the additional parameter $n$ improves overall performance without being decisively required by current data.
• Correlations: The parameter $n$ shows moderate correlation with $w_0$ and $w_a$, indicating that $n$ should be treated as a free parameter rather than fixed a priori.

Cosmological Implications and Cosmography

• Future Behavior: Unlike the BA model, the BAn model allows the future EoS ($z \to -1$) to differ from the present value.
• Acceleration Rate: The BAn model predicts a slightly smaller acceleration rate at present and in the future compared to both the BA model and $\Lambda$CDM.
• Transition Epochs:
  • The deceleration-to-acceleration transition redshift is $z \approx 0.76$ for BAn, compared to $z \approx 0.747$ for BA and $z \approx 0.636$ for $\Lambda$CDM. This implies the accelerated era is "younger" in $\Lambda$CDM.
  • The phantom-to-quintessence crossing occurs at $z \approx 0.42$ for BAn.
• Dynamical Evolution: The BAn model exhibits stronger phantom behavior at early times and weaker quintessence behavior at late times compared to BA. The jerk ($j$) and snap ($s$) parameters indicate that the slope of the Hubble diagram is lower for BAn at late times compared to BA and $\Lambda$CDM.
• Shape Functions: Analysis of shape functions ($S_0, S_1, S_2, S_3$) confirms that while BA and BAn behave similarly, the BAn model exhibits a "smoother" evolution in time, with the EoS crossing the $\Lambda$CDM line later than the BA model.

Significance and Claims
The paper claims that the BAn parametrization is a viable, observationally motivated extension of the BA model that resolves the artificial symmetry of the original model's future behavior. While the current data does not provide decisive evidence to rule out the original BA model (as $n \approx 1.865$ is close to 2), the introduction of the free parameter $n$ makes the BAn model the most favorable among the three tested models according to Bayesian evidence. The authors conclude that this generalization offers a flexible, bounded form for the DE equation of state that remains well-behaved near $z \le 0$ and provides a more nuanced description of cosmic acceleration, particularly regarding the acceleration rate and future evolution, without invoking complex microscopic physics.