JBP Dark Energy: A Dynamical Model

• JBP is a two-parameter dark energy model that introduces a time-dependent equation of state, modifying ΛCDM with behavior peaking near z~1.
• It employs a quadratic ‘turn-on’ in its formulation to ensure bounded evolution at both low and high redshifts, avoiding divergences seen in other models.
• The model is rigorously tested through Bayesian inference and diagnostic tools, although its extra parameters often face statistical penalties compared to simpler ΛCDM.

The Jassal–Bagla–Padmanabhan (JBP) parameterization is a two-parameter family of dynamical dark energy models developed as a minimally complex extension of the cosmological constant for testing cosmic acceleration scenarios. In the JBP model, the dark energy equation of state (EoS) is constructed to vary only at low and intermediate redshift, reverting to its present-day value both as $z \to 0$ and $z \to \infty$, thereby producing an evolution in $w(z)$ that avoids divergence and is peaked near $z \sim 1$. This form is widely adopted in cosmological analyses to probe time-dependent deviations from $\Lambda$CDM, interrogate the robustness of inferred cosmic distances, and test correlations with key observables such as the Hubble constant, BAO scale, neutrino mass, and cosmic structure formation.

1. Mathematical Formulation and Physical Motivation

The canonical JBP equation of state is given by

$w(z) = w_0 + w_a \frac{z}{(1+z)^2}$

with $w_0$ as the present-day EoS parameter and $w_a$ describing the amplitude of its non-linear evolution. Alternatively, in terms of scale factor $a = 1/(1+z)$,

$w(a) = w_0 + w_a a(1 - a)$

Both limits $z \to \infty$0 ($z \to \infty$1) and $z \to \infty$2 ($z \to \infty$3) yield $z \to \infty$4, confining EoS variation to intermediate redshifts. This quadratic-only "turn-on" avoids the pathological behavior present in some alternative parameterizations (e.g., CPL) at high or low $z \to \infty$5.

The corresponding energy density evolves as

$z \to \infty$6

and the normalized expansion rate in a flat universe is

$z \to \infty$7

Adjustments for curvature (non-flat models) are straightforward: $z \to \infty$8

2. Cosmological Parameter Constraints and Model Comparison

Numerous analyses have tested the JBP form against geometric probes including SNe Ia, BAO, QSO standard rulers/candles, CMB distance priors, cosmic chronometers, and strong lensing systems (Staicova, 2022, Qi et al., 2016, Pan et al., 2010, Amante et al., 2019, Zheng et al., 2024, Zheng et al., 2021, Barua et al., 15 Jun 2025). Bayesian inference with large, contemporary datasets consistently reports best-fit parameters with $z \to \infty$9 in the range $w(z)$0 to $w(z)$1 and $w(z)$2 between $w(z)$3 and $w(z)$4, though the allowed region always encompasses $w(z)$5 (the cosmological constant). Marginalized constraints:

Dataset	$w(z)$6	$w(z)$7
DESI+Pantheon+QSO+CC (Barua et al., 15 Jun 2025)	$w(z)$8	$w(z)$9
DESI BAO “all” (Zheng et al., 2024)	$z \sim 1$0	$z \sim 1$1
Planck+Pantheon+DESI BAO (Rodrigues et al., 28 Feb 2025)	$z \sim 1$2	$z \sim 1$3
Strong lensing (SS3) (Amante et al., 2019)	$z \sim 1$4	$z \sim 1$5

Constraints from Bayesian evidence (Bayes factors, AIC/BIC penalties) show that JBP, while flexible, is consistently disfavored over $z \sim 1$6CDM after accounting for extra parameters, except in specific alternative gravity models where the fit may improve (see Section 5 below).

3. Diagnostic Tools for EoS Variation and Model Distinction

The JBP model has been thoroughly tested using geometrical diagnostics designed to distinguish dynamical dark energy:

• The $z \sim 1$7 diagnostic and its first derivative $z \sim 1$8 (Qi et al., 2016): JBP reproduces $z \sim 1$9 at $\Lambda$0 but fails in $\Lambda$1 at $\Lambda$2, where the measured expansion history diverges from JBP, indicating tension at high redshift.
• Statefinder hierarchy ($\Lambda$3) (Qi et al., 2015): Third and especially fourth derivatives of the scale factor can cleanly differentiate JBP from both $\Lambda$4CDM and alternative Padé and CPL forms at the present epoch. Growth-rate diagnostics add only marginal extra separation.
• Bayesian model selection (AIC, BIC, Bayes factor): JBP's additional freedom is not rewarded unless non-flat models or alternative gravity are considered.

4. Sensitivity to Systematic Errors and Observational Systematics

JBP is documented as highly sensitive to systematic errors in SN Ia calibration and progenitor evolution (Sharma et al., 11 Nov 2025). A 0.02 mag calibration offset shifts $\Lambda$5 and $\Lambda$6 by $\Lambda$7: $\Lambda$8, $\Lambda$9. Other systematics (dust, color scatter, $w(z) = w_0 + w_a \frac{z}{(1+z)^2}$0 error) move parameters more modestly but still at the $w(z) = w_0 + w_a \frac{z}{(1+z)^2}$1 level. The hierarchy of vulnerability to systematics is:

$w(z) = w_0 + w_a \frac{z}{(1+z)^2}$2

For precise future surveys, sub-millimagnitude calibration and careful modeling of progenitor properties are needed to achieve robust JBP constraints.

5. Extensions: Alternative Gravity, Accretion, and Quantum Consistency

The JBP ansatz is implemented in extended gravity frameworks (e.g., VCDM, 4D Einstein–Gauss–Bonnet, Horava–Lifshitz) as a testbed for the consistency of dynamical dark energy:

• In 4D EGB gravity, JBP fits current CC+SNe+BAO+CMB data with $w(z) = w_0 + w_a \frac{z}{(1+z)^2}$3, $w(z) = w_0 + w_a \frac{z}{(1+z)^2}$4 and is strongly favored by Bayesian evidence over $w(z) = w_0 + w_a \frac{z}{(1+z)^2}$5CDM (Mukherjee et al., 7 Jul 2025). The model realizes a transition from quintessence-like to phantom-like behavior in future cosmic evolution, with distinct implications for the mass evolution of black holes and wormholes. For black holes, mass increases until a late-time phantom regime ($w(z) = w_0 + w_a \frac{z}{(1+z)^2}$6), then decreases; for wormholes, the epochal trend is reversed.
• In VCDM, JBP admits stable phantom-crossing and fits Planck2018+DESI BAO DR2 slightly better than $w(z) = w_0 + w_a \frac{z}{(1+z)^2}$7CDM, remaining free of ghost instabilities (Arora et al., 5 Aug 2025).
• In Horava–Lifshitz gravity, accretion calculations with the JBP form indicate only modest BH mass growth ($w(z) = w_0 + w_a \frac{z}{(1+z)^2}$8–$w(z) = w_0 + w_a \frac{z}{(1+z)^2}$9 at $w_0$0), less sensitive than CPL or BA parameterizations, with possible mass loss triggered if $w_0$1 crosses the phantom divide (Biswas et al., 12 Jun 2025).
• Quantum gravity consistency: The Trans-Planckian Censorship Conjecture (TCC) imposes strong constraints on late-time acceleration. The JBP model can realize "quintom-B" crossing ($w_0$2), avoiding future eternal acceleration and remaining consistent with swampland bounds (Li et al., 10 Apr 2025).

6. Applications in Cosmological Inference and Implementation Methodologies

JBP is adapted into computational pipelines for cosmological parameter inference. The analytic form of $w_0$3 facilitates efficient ODE integration for Hubble rates. Physics-informed neural network (PINN) surrogates have been constructed to encode $w_0$4 for arbitrary $w_0$5 and have been validated for multi-run GPU-accelerated likelihood analyses, maintaining $w_0$6 bias in $w_0$7 and supporting full Pantheon+ SN cosmology pipelines (Verma et al., 16 Aug 2025).

For direct data-model mapping, "projection" procedures minimize $w_0$8 between quintessence microphysics and compressed $w_0$9 parameterizations; JBP is found to reproduce background observables to high accuracy and is equally robust to CPL, BA, and EXP forms (Wolf et al., 7 Feb 2025).

7. Tensions, Controversies, and Future Prospects

Despite the flexible dynamics conferred by JBP, most cosmological analyses find its best-fit parameter region consistent with $w_a$0CDM to within $w_a$1–$w_a$2, and Bayesian evidence generally disfavors JBP over simpler constant-$w_a$3 or polynomial models unless curvature or non-standard gravity is permitted (Staicova, 2022, Vazquez et al., 2012, Zheng et al., 2021, Zheng et al., 2024, Barua et al., 15 Jun 2025). Statistical measures (AIC/BIC) penalize the extra freedom unless observational evidence for dynamical EoS emerges distinctly.

The JBP form is notably useful in testing thawing/freezing dark energy scenarios, resolving neutrino mass hierarchy tension, and quantitatively probing phantom/quintessence transitions. Future improvements are contingent upon reducing systematics, improving calibration, and expanding high-$w_a$4 data coverage. The model's utility extends beyond standard cosmology into tests of modified gravity and primordial black hole evolution, with quantum consistency demands favoring forms, such as JBP, with controlled late-time EoS evolution.

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In summary, the Jassal–Bagla–Padmanabhan parameterization provides a well-behaved two-parameter framework for dynamical dark energy with bounded redshift evolution, enabling tests of departures from $w_a$5CDM, cosmic distance inference, quantum gravity consistency, and astrophysical object evolution. Its use in current research is characterized both by its analytic tractability and by nuanced sensitivities to observational systematics and model-selection penalties.