[Paper Review] A general parametric model for the dynamic dark energy

Stefano Sello|arXiv (Cornell University)|Aug 2, 2013

Cosmology and Gravitation Theories5 references3 citations

TL;DR

This paper proposes two generalized parametric models—ξ MZp and DFp—for the equation of state (EoS) of dark energy, extending the standard CPL model to avoid unphysical divergences in the future cosmic evolution (z < 0). By incorporating logarithmic and arctangent functions, the models ensure finite EoS values at z → -1 and z → ∞, enabling consistent exploration of dark energy dynamics across past and future cosmic epochs without mathematical singularities.

ABSTRACT

In the present work we suggest new and more generalized parameterizations for the Equation of State, EoS, of dark energy, maintaining the basic structure of two-parameters CPL-model, but covering both the past and the future of the cosmic history, without divergences and consistently with the current observational data. We propose two generalizations, starting from the extended MZp-model by Ma and Zhang, 2011, the $ξ$MZp-model and the DFp-model. The potential advantages of using these new formulations is their extended range of validity, mainly in the future, to determine possible future scenarios of the cosmic evolution.

Motivation & Objective

• To address the unphysical divergence of the standard CPL model at z → -1, which limits its applicability in future cosmic evolution.
• To generalize the Ma and Zhang (2011) MZp model to ensure finite EoS values across both past (z > 0) and future (z < 0) cosmic epochs.
• To develop new parametric forms of the dark energy EoS that maintain simplicity while improving mathematical consistency and physical plausibility.
• To enable more robust future cosmological modeling by allowing reliable extrapolation of dark energy behavior beyond the present epoch.
• To provide a foundation for tighter constraints on dark energy parameters using current observational data (SNe Ia, CMB, BAO) in extended parameter space.

Proposed method

• Proposes the ξ MZp-model by introducing a shift parameter ξ to delay the divergence of the EoS to z = -(ξ + 1), extending its validity into the future.
• Introduces the DFp-model, a more general formulation using logarithmic and inverse hyperbolic tangent functions to ensure finite EoS values at both z → ∞ and z → -1.
• Employs L’Hôpital’s rule to analytically verify finite limiting values of the EoS in extreme redshift regimes.
• Derives exact analytical expressions for the integral ∫(1 + w(z'))/(1 + z') dz' from 0 to z, valid for both z > 0 and -1 < z < 0, essential for computing the Hubble parameter H(z).
• Ensures consistency at z = 0 by including normalization terms (±ln√2) so that w(0) = w₀, preserving the physical interpretation of w₀ as the present-day EoS.
• Uses the resulting EoS parameterizations in cosmological models to compute H(z), enabling consistent modeling of cosmic expansion history across all redshifts.

Experimental results

Research questions

• RQ1 Can the standard CPL model's unphysical divergence at z → -1 be avoided while preserving its simplicity and two-parameter structure?
• RQ2 How can the Ma and Zhang (2011) MZp model be generalized to ensure finite EoS values in both the far past (z → ∞) and far future (z → -1)?
• RQ3 What mathematical forms allow a continuous, non-divergent dark energy EoS across the entire redshift range, including negative redshifts?
• RQ4 Can the new parameterizations yield tighter observational constraints on w₀ and wₐ when fitted to current cosmological data (SNe Ia, CMB, BAO) compared to the CPL model?
• RQ5 What are the implications of these new EoS models for predicting future cosmic evolution scenarios, such as the Big Rip or de Sitter-like asymptotic behavior?

Key findings

• The ξ MZp-model extends the validity of the MZp model by shifting the divergence point to z = -(ξ + 1), allowing exploration of future cosmic evolution up to that redshift.
• The DFp-model achieves full non-divergence for all z > -1, with finite EoS values at both z → ∞ (w → w₀ + wₐ ln√2) and z → -1 (w → w₀ - wₐ(1/2 + ln√2)).
• The models ensure w(0) = w₀ through normalization terms ±ln√2, preserving the physical interpretation of w₀ as the present-day EoS.
• Analytical expressions for the critical integral ∫(1 + w(z'))/(1 + z') dz' are derived in closed form for both z > 0 and -1 < z < 0, enabling exact Hubble parameter computation.
• The new parameterizations avoid unphysical behavior in the future while maintaining the two-parameter simplicity of the CPL model, enabling consistent cosmological modeling.
• The models are mathematically consistent and suitable for future data fitting, with potential to yield tighter constraints on dark energy parameters when tested against current observational datasets.