Jassal-Bagla-Padmanabhan (JBP) Dark Energy Model

• JBP is a two-parameter phenomenological dark energy model defined by w(z)=w0 + wa*z/(1+z)^2, capturing low-to-intermediate redshift deviations from ΛCDM.
• It features a localized departure where w(z=0)=w0 and w(z≫1)→w0, distinguishing it from models like the CPL parametrization which retain high-redshift deviations.
• JBP is widely employed in cosmological inference pipelines, with diverse datasets (e.g., supernovae, BAO, QSOs) yielding varying constraints and highlighting sensitivity to systematics.

Searching arXiv for JBP and related dark-energy parametrization papers to ground the article in relevant literature. Searching arXiv for "Jassal Bagla Padmanabhan dark energy equation of state". The Jassal–Bagla–Padmanabhan (JBP) parametrization is a two-parameter phenomenological model for the dark-energy equation of state, written as $w(z)=w_0+w_a z/(1+z)^2$. Its defining feature is that $w(z=0)=w_0$ and $w(z\gg 1)\sim w_0$, so the deviation from a constant equation of state is localized at low-to-intermediate redshift rather than persisting into the distant past. In the dark-energy literature, JBP is therefore used both as a compact benchmark against $\Lambda$CDM and as a foil to the Chevallier–Polarski–Linder (CPL) form when testing whether cosmological data require time dependence in $w(z)$ (Vazquez et al., 2012, Gao et al., 2010).

1. Definition and formal properties

The JBP equation-of-state ansatz is

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

Several papers use the equivalent notation $w_1$ in place of $w_a$, but the role of the second parameter is the same: it controls the redshift-dependent departure from the present-day value $w_0$ (Kumari et al., 7 Apr 2026).

A standard equivalent form is obtained in the scale factor $a=1/(1+z)$,

$w(z=0)=w_0$0

which makes explicit that the time-dependent term vanishes at $w(z=0)=w_0$1 and $w(z=0)=w_0$2. This is the sense in which JBP is commonly described as a localized or low-redshift evolution model (Qi et al., 2015).

Relative to CPL, the JBP correction is more strongly suppressed at high redshift. In CPL, the time-dependent contribution approaches a constant in the distant past, whereas in JBP it dies away, so the asymptote is again $w(z=0)=w_0$3. The 2012 Bayesian reconstruction study states this explicitly and uses it to contrast JBP with more flexible node-based reconstructions that can accommodate turnovers or phantom-divide crossings (Vazquez et al., 2012). A curved-universe analysis similarly emphasizes that JBP is designed to allow present-day deviations from $w(z=0)=w_0$4CDM while keeping the early-time behavior more controlled (Pan et al., 2010).

This formal structure has an important interpretive consequence: JBP is not intended to encode sustained early-time running of dark energy. It is instead an effective two-parameter compression of late-time evolution, especially when one wishes to test whether the data prefer a bounded, intermediate-redshift departure from $w(z=0)=w_0$5.

2. Background expansion and derived observables

In the standard FLRW treatment, inserting JBP into the continuity equation gives the familiar dark-energy density scaling

$w(z=0)=w_0$6

For a curved FRW cosmology, one representative expression is

$w(z=0)=w_0$7

with

$w(z=0)=w_0$8

These formulae are the backbone of JBP likelihood analyses in both flat and non-flat settings (Pan et al., 2010).

In flat models the same structure reduces to

$w(z=0)=w_0$9

which is the version repeatedly used in supernova, quasar, GRB, and strong-lensing studies (Zheng et al., 2021).

Distance observables are then built from the usual line-of-sight integral. For luminosity distances, one recurring expression is

$w(z\gg 1)\sim w_0$0

used in GRB and supernova analyses (Gao et al., 2010). In strong-lensing applications, the same $w(z\gg 1)\sim w_0$1 enters the distance ratio

$w(z\gg 1)\sim w_0$2

which is then compared with the observed Einstein-radius-based ratio (Amante et al., 2019).

Because the JBP modification is concentrated at intermediate redshift, its effect on observables is correspondingly localized. This is why it is often described as a low-redshift effective description rather than a model of the full asymptotic history of dark energy.

3. Role in observational inference pipelines

JBP is frequently deployed as a benchmark model rather than as the primary reconstruction target. In the 2012 Bayesian study of dark-energy reconstruction, it appears alongside CPL and Felice–Nesseris–Tsujikawa (FNT) as a standard two-parameter comparator for node-based reconstructions. That analysis used WMAP 7-year temperature and polarization data, ACT 148 GHz power spectra, Union2 supernovae, BAO distance measurements, BBN baryon-density information, and an HST Gaussian prior on $w(z\gg 1)\sim w_0$3, with perturbations evolved in a modified CAMB using the PPF prescription and posteriors sampled using CosmoMC plus MultiNest (Vazquez et al., 2012).

The same parametrization has also been used in cosmography-based GRB analyses. One such study calibrated five GRB luminosity relations from the Union SN Ia set, derived distance moduli for 42 GRBs at $w(z\gg 1)\sim w_0$4, and fitted JBP with a total chi-square

$w(z\gg 1)\sim w_0$5

In that framework, JBP served as one of three phenomenological dark-energy parametrizations tested against high-redshift GRB data supplemented by CMB and BAO (Gao et al., 2010).

Quasar-based analyses provide a third use case. A 2021 study calibrated ultraviolet/X-ray and compact-radio-quasar relations using Gaussian-process reconstructions from 31 cosmic-chronometer $w(z\gg 1)\sim w_0$6 measurements and then fitted JBP to BAO-only and BAO+QSO combinations under a flat-universe assumption (Zheng et al., 2021). A separate quasar cosmology analysis combined Pantheon supernovae, a 2036-object QSO Hubble diagram up to $w(z\gg 1)\sim w_0$7, and BAO, with JBP treated as one of several flat evolving-$w(z\gg 1)\sim w_0$8 models extending the Hubble diagram beyond the supernova regime (Bargiacchi et al., 2021).

More recent late-time analyses have moved to DESI-era datasets and information-criterion comparisons. A 2026 study fitted JBP to cosmic chronometers, DESI DR2 BAO, and Pantheon+ using MCMC with the emcee package and model comparison via AIC and BIC rather than Bayesian evidence (Kumari et al., 7 Apr 2026). This methodological spread shows that JBP is not tied to a single inference philosophy: it appears in evidence-based Bayesian model selection, standard MCMC posterior inference, and compressed-likelihood late-time analyses.

4. Empirical constraints and model-comparison results

Early combined-probe studies generally found JBP consistent with $w(z\gg 1)\sim w_0$9CDM within broad uncertainties. In the GRB+CMB+BAO analysis, the best-fit values were $\Lambda$0, $\Lambda$1 for one cosmographic calibration and $\Lambda$2, $\Lambda$3 for another, with the paper concluding that all tested dark-energy parametrizations, including JBP, were compatible with $\Lambda$4CDM within $\Lambda$5 (Gao et al., 2010). A curved-universe analysis based on Constitution SNe Ia, BAO, WMAP5, and $\Lambda$6 reported marginalized constraints $\Lambda$7, $\Lambda$8, $\Lambda$9, and $w(z)$0, again finding flat $w(z)$1CDM consistent with the data at the $w(z)$2 level (Pan et al., 2010).

The 2012 Bayesian reconstruction analysis delivered a sharper model-selection statement. With flat priors $w(z)$3 and $w(z)$4, it found

$w(z)$5

and a Bayes factor relative to the cosmological constant

$w(z)$6

which the authors interpreted as strong disfavouring of JBP relative to $w(z)$7CDM. JBP was also disfavoured relative to the two-internal-node reconstruction, with $w(z)$8. The same paper showed substantial prior sensitivity: alternative prior ranges weakened the evidence penalty, and a narrow prior excluding the exact cosmological-constant value $w(z)$9 yielded $w(z)=w_0+w_a\frac{z}{(1+z)^2}.$0 (Vazquez et al., 2012).

High-redshift and DESI-era analyses have produced a more mixed picture. In a BAO+QSO study, JBP remained close to $w(z)=w_0+w_a\frac{z}{(1+z)^2}.$1CDM in the sense that the BAO+QSO best fit was $w(z)=w_0+w_a\frac{z}{(1+z)^2}.$2 and $w(z)=w_0+w_a\frac{z}{(1+z)^2}.$3, but it was substantially penalized by information criteria, with $w(z)=w_0+w_a\frac{z}{(1+z)^2}.$4 and $w(z)=w_0+w_a\frac{z}{(1+z)^2}.$5 relative to the preferred polynomial parametrization (Zheng et al., 2021). By contrast, the combined SNe+QSO+BAO analysis found $w(z)=w_0+w_a\frac{z}{(1+z)^2}.$6, $w(z)=w_0+w_a\frac{z}{(1+z)^2}.$7, and $w(z)=w_0+w_a\frac{z}{(1+z)^2}.$8, and described the result as roughly $w(z)=w_0+w_a\frac{z}{(1+z)^2}.$9–$w_1$0 away from the flat $w_1$1CDM reference point, with the preference driven mainly by the SNe+QSO Hubble diagram rather than BAO alone (Bargiacchi et al., 2021).

DESI-focused late-time studies continue to disagree on how strongly JBP is supported. An analysis using DESI BAO 2024, Pantheon+, and quasars characterized flat JBP as the most conservative of the tested parametrizations and still consistent with $w_1$2CDM at the $w_1$3 level, even though the full BAO set including LRG1 and LRG2 shifted the fit toward $w_1$4 and $w_1$5 (Zheng et al., 2024). A separate late-Universe probe analysis using PantheonPlus, quasars, and DESI DR1 BAO with cosmic chronometers or megamasers found typical JBP central values $w_1$6 to $w_1$7 and $w_1$8 to $w_1$9, but concluded that Bayesian evidence still strongly favored ACDM over JBP (Barua et al., 15 Jun 2025). In contrast, the 2026 DESI DR2 + Pantheon+ + cosmic-chronometer study reported

$w_a$0

with $w_a$1 versus $w_a$2, corresponding to $w_a$3, $w_a$4, and $w_a$5 in that paper’s convention $w_a$6 (Kumari et al., 7 Apr 2026).

Taken together, these results suggest that JBP is empirically viable but not stably preferred. Its status depends strongly on the dataset, the treatment of high-redshift tracers, the comparison metric, and the prior volume.

5. Diagnostics, degeneracies, and robustness issues

Several papers emphasize that JBP is hard to distinguish from neighboring dark-energy models using only low-order background observables. In a Statefinder analysis, JBP was found to be highly degenerate with other dark-energy models in $w_a$7 and $w_a$8, but clearly separable from $w_a$9CDM in the Statefinder hierarchy, especially through $w_0$0, which the authors judged more powerful than $w_0$1. The growth-rate diagnostic $w_0$2 was reported not to play a significant role in improving discrimination (Qi et al., 2015).

A differential-age study reached a complementary conclusion. It found that $w_0$3 could not distinguish JBP from $w_0$4CDM, CPL, and FSLL, whereas the derivative $w_0$5 was more sensitive at low redshift. In that framework, JBP remained broadly compatible with the reconstruction mainly for $w_0$6, while CPL appeared somewhat more closely aligned at low redshift (Rani et al., 2016).

Parameterization-robustness studies sharpen this point further. A 2025 analysis mapping minimally and non-minimally coupled quintessence models into CPL, JBP, BA, and EXP parameter spaces concluded that the broad physical inferences were independent of the chosen parametrization. JBP reproduced the observables well, often with $w_0$7, and performed especially well for the hilltop branch of thawing quintessence, but it was typically the worst-performing of the four forms in full-data $w_0$8 (Wolf et al., 7 Feb 2025). A related neutrino-mass study found that allowing JBP dynamics relaxed the bound to $w_0$9 from Planck 2018 + Pantheon+ + DESI, weaker than in $a=1/(1+z)$0CDM but tighter than in the BA case, illustrating the degeneracy between dark-energy evolution and the neutrino sector (Rodrigues et al., 28 Feb 2025).

Systematics analyses add another caution. A simulated SN-Ia study comparing CPL, JBP, LOG, and GEN found JBP to be the most systematics-sensitive of the time-varying equation-of-state models tested. In that analysis, a calibration offset $a=1/(1+z)$1 induced $a=1/(1+z)$2 and $a=1/(1+z)$3, while progenitor evolution in stretch produced $a=1/(1+z)$4 and $a=1/(1+z)$5 (Sharma et al., 11 Nov 2025).

A common misconception is that a successful JBP fit would by itself establish a specific microphysical model of dark energy. The literature does not support that reading. JBP is best understood as a compact phenomenological ansatz whose utility lies in diagnosing bounded late-time deviations from $a=1/(1+z)$6CDM and in testing the robustness of inferences to the assumed shape of $a=1/(1+z)$7.

6. Extensions beyond standard late-time phenomenology

JBP has also been embedded in broader theoretical and phenomenological settings. In a Trans-Planckian Censorship Conjecture analysis, the model was written as $a=1/(1+z)$8 and treated as a quintom-B candidate. The paper summarized the corresponding viability conditions as $a=1/(1+z)$9, $w(z=0)=w_0$00, and $w(z=0)=w_0$01, with the broader conclusion that viable late-time cosmologies must asymptotically avoid eternal acceleration (Li et al., 10 Apr 2025).

Within VCDM, JBP has been promoted from a purely phenomenological background fit to a realizable background history in a minimally modified gravity theory. A Planck 2018 + DESI DR2 analysis in that framework obtained

$w(z=0)=w_0$02

with $w(z=0)=w_0$03 and $w(z=0)=w_0$04 relative to $w(z=0)=w_0$05CDM. The authors interpreted this as quintessence-like behavior today, phantom-like behavior at earlier times, and only a marginal statistical improvement, with no resolution of the $w(z=0)=w_0$06 or $w(z=0)=w_0$07 tensions (Arora et al., 5 Aug 2025).

JBP has likewise been transplanted into modified-gravity and compact-object accretion studies. In a 4D Einstein–Gauss–Bonnet analysis, the observationally constrained JBP parameters were reported as $w(z=0)=w_0$08 and $w(z=0)=w_0$09, with a black-hole accretion transition from quintessence-like behavior for $w(z=0)=w_0$10 to phantom-like behavior for $w(z=0)=w_0$11, and the opposite trend for wormholes (Mukherjee et al., 7 Jul 2025). Strong-lensing constraints provide a different kind of extension: in a 204-system sample, JBP fits were found to be highly sample-dependent, and the preferred fiducial subset $w(z=0)=w_0$12 yielded $w(z=0)=w_0$13 and $w(z=0)=w_0$14, although CPL was favored overall by AIC/BIC/FoM (Amante et al., 2019).

These extensions do not establish JBP as a fundamental theory. They instead show how widely the parametrization is used as an effective input sector: once a bounded, low-redshift deformation of $w(z=0)=w_0$15 is desired, JBP provides a mathematically simple template that can be inserted into standard FLRW likelihoods, modified-gravity reconstructions, diagnostic hierarchies, and compact-object accretion models alike.