Barboza-Alcaniz Dark Energy Model

• Barboza–Alcaniz (BA) is a two-parameter dark-energy equation-of-state parameterization characterized by its bounded behavior over redshift and clear asymptotic limits.
• The model provides analytic expressions for cosmic expansion, enhancing its application in FLRW-based analyses and facilitating comparisons with ΛCDM.
• Empirical investigations using SNe, BAO, CMB, and modified gravity frameworks highlight BA’s computational convenience and sensitivity to dynamical dark energy.

Barboza–Alcaniz (BA) is a two-parameter dark-energy equation-of-state parameterization used to describe redshift-dependent departures from the cosmological-constant limit within phenomenological late-time cosmology. In its standard redshift form,

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

it specifies the present-day value through $w(0)=w_0$ and a controlled evolution through the second coefficient $w_a$ (often denoted $w_1$ in older papers). BA is repeatedly used because it is bounded on $z\in[-1,\infty)$, remains finite in the asymptotic future, and admits simple analytic background expressions in many FLRW-based analyses (Verma et al., 16 Aug 2025, Wolf et al., 7 Feb 2025).

1. Definition and asymptotic structure

The BA parameterization is usually written either in redshift space,

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

or in scale-factor form,

$w(a)=w_0+w_a\frac{1-a}{a^2+(1-a)^2},$

with $a=(1+z)^{-1}$. The two forms are equivalent and are used interchangeably across the literature (Li et al., 27 Nov 2025).

Its limiting behavior is one of its defining properties:

$$w(0)=w_0,\qquad \lim_{z\to\infty}w(z)=w_0+w_a,\qquad \lim_{z\to-1}w(z)=w_0.$$

Several papers emphasize that this makes BA “well-behaved across the full redshift range of cosmological evolution” and “bounded for all $z\in[-1,\infty)$,” in contrast to parameterizations whose future or high-redshift behavior diverges (Verma et al., 16 Aug 2025). At low redshift, BA reduces to a linear form to first order, while at high redshift it approaches a finite asymptote rather than introducing an unbounded extrapolation (Biswas et al., 2018).

The parameter space is commonly interpreted in phase-language. One line of work notes that BA permits classification into quintessence, phantom, and phantom-crossing regions in the $w(0)=w_0$0 plane (Verma et al., 16 Aug 2025). A later DESI-era analysis instead organizes the plane with the lines $w(0)=w_0$1 and $w(0)=w_0$2, separating phantom, quintessence, Quintom-A, and Quintom-B sectors; in that convention BA frequently lands in the Quintom-B region, corresponding to quintessence-like behavior today and phantom-like behavior in the past (Li et al., 27 Nov 2025).

A thermodynamic follow-up to the original BA proposal quotes Barboza and Alcaniz’s early best fit as

$w(0)=w_0$3

at $w(0)=w_0$4, obtained from Supernova Type Ia (SNLS), BAO (SDSS), the WMAP shift parameter, and $w(0)=w_0$5 estimates from ages of high-$w(0)=w_0$6 galaxies (Mondal et al., 2018).

2. Background dynamics and analytic tractability

In standard flat late-time FRW analyses, BA is inserted through the continuity-equation factor

$w(0)=w_0$7

For the BA form, this integral is elementary and yields

$w(0)=w_0$8

In the flat late-time Friedmann equation,

$w(0)=w_0$9

this becomes

$w_a$0

This closed form is one reason BA is computationally convenient in supernova and distance-based inference pipelines (Verma et al., 16 Aug 2025).

Distance observables are then built in the usual way. In one Pantheon$w_a$1 implementation,

$w_a$2

with parameter vector

$w_a$3

and uniform priors restricted to the trained surrogate domain (Verma et al., 16 Aug 2025).

A related but distinct construction appears in analyses that work directly with the deceleration parameter. Using

$w_a$4

the BA form induces

$w_a$5

This version is used in thermodynamic and particle-creation studies rather than in standard late-time distance fitting (Mondal et al., 2018).

3. Constraints from low-redshift and mixed-probe analyses

Outside the DESI-driven dynamical-dark-energy literature, BA has been tested against several low-redshift probe combinations. With Pantheon, cosmic chronometers, and GWTC-1/GWTC-2 standard and dark sirens, BA remained consistent with the $w_a$6CDM point $w_a$7 within $w_a$8, but Bayesian evidence favored $w_a$9CDM over BA for the full Pantheon+CC+GW combination:

$w_1$0

so that $w_1$1 in favor of $w_1$2CDM. The same study found that current GW catalogs added very little statistical leverage to BA beyond SNe Ia and CC data (Escamilla-Rivera et al., 2021).

A different analysis based on the combined parameter $w_1$3 rather than separate $w_1$4 and $w_1$5 found that BA’s second coefficient could be comparatively tightly constrained. For BAO+CMB+SN+GRB it reported

$w_1$6

That study nevertheless concluded that BA could not satisfy the Planck-preferred values of both $w_1$7 and $w_1$8 simultaneously, and that $w_1$9CDM remained statistically preferred (Staicova, 2022).

In a supernova-only late-time analysis built around a physics-informed neural-network surrogate and Pantheon$z\in[-1,\infty)$0 with SH0ES calibration, BA remained fully compatible with the cosmological constant at 95% credibility. The reported $z\in[-1,\infty)$1 marginalized BA constraints were

$z\in[-1,\infty)$2

with no evidence that the data required dynamical dark energy (Verma et al., 16 Aug 2025).

4. DESI-era phenomenology and the status of dynamical dark energy

DESI-era analyses have made BA one of the principal test cases for parameterization dependence. The results are not uniform across data combinations, SN samples, or statistical criteria.

Study	Data	BA result
(Li et al., 27 Nov 2025)	CMB+DESI+DESY5	$z\in[-1,\infty)$3, $z\in[-1,\infty)$4, $z\in[-1,\infty)$5 preference for DDE
(Zheng et al., 2024)	full DESI BAO, flat BA	$z\in[-1,\infty)$6, $z\in[-1,\infty)$7, $z\in[-1,\infty)$8
(Rodrigues et al., 28 Feb 2025)	Planck 2018+Pantheon$z\in[-1,\infty)$9+DESI	$w(z)=w_0+w_a\frac{z(1+z)}{1+z^2},$0, $w(z)=w_0+w_a\frac{z(1+z)}{1+z^2},$1, $w(z)=w_0+w_a\frac{z(1+z)}{1+z^2},$2 eV (95%)
(Barua et al., 15 Jun 2025)	Pantheon$w(z)=w_0+w_a\frac{z(1+z)}{1+z^2},$3+QSO+DESI DR1+CC/MM	$w(z)=w_0+w_a\frac{z(1+z)}{1+z^2},$4 mildly shifted from $w(z)=w_0+w_a\frac{z(1+z)}{1+z^2},$5 with LRG1/LRG2; Bayes factors still strongly favor $w(z)=w_0+w_a\frac{z(1+z)}{1+z^2},$6CDM

The strongest pro-BA claim comes from a joint ACT+SPT+Planck CMB analysis combined with DESI DR2 and DESY5, PantheonPlus, or Union3 supernovae. In that study BA delivered some of the largest departures from the $w(z)=w_0+w_a\frac{z(1+z)}{1+z^2},$7CDM limit. For the headline CMB+DESI+DESY5 combination the result was

$w(z)=w_0+w_a\frac{z(1+z)}{1+z^2},$8

with a $w(z)=w_0+w_a\frac{z(1+z)}{1+z^2},$9 preference for dynamical dark energy, $w(a)=w_0+w_a\frac{1-a}{a^2+(1-a)^2},$0, and Bayesian evidence $w(a)=w_0+w_a\frac{1-a}{a^2+(1-a)^2},$1, classified there as moderate evidence in favor of BA over $w(a)=w_0+w_a\frac{1-a}{a^2+(1-a)^2},$2CDM. The same analysis interpreted BA as stably residing in the Quintom-B regime and found a reconstructed crossing of $w(a)=w_0+w_a\frac{1-a}{a^2+(1-a)^2},$3 near $w(a)=w_0+w_a\frac{1-a}{a^2+(1-a)^2},$4 (Li et al., 27 Nov 2025).

A more cautious DESI BAO 2024 study compared BA with CPL, JBP, and FSLL using full DESI BAO and a version with the LRG1/LRG2 points removed. In the flat BA model, full BAO alone pushed the fit to

$w(a)=w_0+w_a\frac{1-a}{a^2+(1-a)^2},$5

whereas removing LRG1/LRG2 relaxed the result to

$w(a)=w_0+w_a\frac{1-a}{a^2+(1-a)^2},$6

After adding SN and QSO, the preferred BA point moved much closer to $w(a)=w_0+w_a\frac{1-a}{a^2+(1-a)^2},$7CDM,

$w(a)=w_0+w_a\frac{1-a}{a^2+(1-a)^2},$8

and the authors concluded that the parameterization choice had little impact on the existence of the DESI deviation, while LRG1 and LRG2 were primarily responsible for driving it (Zheng et al., 2024).

BA has also been used to test whether DESI-era dark-energy freedom relaxes neutrino-mass bounds. With Planck 2018 TT/TE/EE+lensing + Pantheon$w(a)=w_0+w_a\frac{1-a}{a^2+(1-a)^2},$9 + DESI, BA gave

$a=(1+z)^{-1}$0

together with

$a=(1+z)^{-1}$1

substantially weaker than representative $a=(1+z)^{-1}$2CDM neutrino-mass bounds quoted in that paper. The same work emphasized that BA stayed more than $a=(1+z)^{-1}$3 away from the $a=(1+z)^{-1}$4CDM point in the $a=(1+z)^{-1}$5 plane when $a=(1+z)^{-1}$6 was free (Rodrigues et al., 28 Feb 2025).

By contrast, a late-Universe-only study using PantheonPlus, quasars, DESI DR1 BAO, and either cosmic chronometers or megamasers found only mild shifts of BA away from $a=(1+z)^{-1}$7CDM. Typical flat-BA fits with LRG1/LRG2 included were around

$a=(1+z)^{-1}$8

while removing the LRG pair drove $a=(1+z)^{-1}$9 closer to $$w(0)=w_0,\qquad \lim_{z\to\infty}w(z)=w_0+w_a,\qquad \lim_{z\to-1}w(z)=w_0.$$0 and kept $$w(0)=w_0,\qquad \lim_{z\to\infty}w(z)=w_0+w_a,\qquad \lim_{z\to-1}w(z)=w_0.$$1 statistically compatible with zero. In every such configuration, Bayesian evidence strongly favored $$w(0)=w_0,\qquad \lim_{z\to\infty}w(z)=w_0+w_a,\qquad \lim_{z\to-1}w(z)=w_0.$$2CDM over BA (Barua et al., 15 Jun 2025).

5. BA beyond standard GR and in model-to-parameter mappings

BA is frequently used outside standard GR because it supplies a smooth, closed two-parameter background history.

In power-law symmetric teleparallel gravity, $$w(0)=w_0,\qquad \lim_{z\to\infty}w(z)=w_0+w_a,\qquad \lim_{z\to-1}w(z)=w_0.$$3, BA is used as the closure relation for the effective dark-energy sector. The resulting background is analytic:

$$w(0)=w_0,\qquad \lim_{z\to\infty}w(z)=w_0+w_a,\qquad \lim_{z\to-1}w(z)=w_0.$$4

with derived

$$w(0)=w_0,\qquad \lim_{z\to\infty}w(z)=w_0+w_a,\qquad \lim_{z\to-1}w(z)=w_0.$$5

Using DESI DR2 BAO and previous BAO data, this framework yielded present acceleration and transition redshifts around $$w(0)=w_0,\qquad \lim_{z\to\infty}w(z)=w_0+w_a,\qquad \lim_{z\to-1}w(z)=w_0.$$6–$$w(0)=w_0,\qquad \lim_{z\to\infty}w(z)=w_0+w_a,\qquad \lim_{z\to-1}w(z)=w_0.$$7, while remaining competitive with $$w(0)=w_0,\qquad \lim_{z\to\infty}w(z)=w_0+w_a,\qquad \lim_{z\to-1}w(z)=w_0.$$8CDM in AIC/BIC, albeit usually slightly weaker than the CPL+$$w(0)=w_0,\qquad \lim_{z\to\infty}w(z)=w_0+w_a,\qquad \lim_{z\to-1}w(z)=w_0.$$9 alternative (Mazumdar et al., 8 Jul 2025).

In VCDM, a minimally modified gravity theory with no extra local physical degrees of freedom beyond the tensor modes, BA is embedded as a reconstructible dark-energy history. Planck 2018 + DESI DR2 gave

$z\in[-1,\infty)$0

with

$z\in[-1,\infty)$1

Within that analysis BA was one of the statistically strongest-performing dynamical parameterizations, but it did not alleviate the $z\in[-1,\infty)$2 or $z\in[-1,\infty)$3 tensions (Arora et al., 5 Aug 2025).

BA has likewise been inserted into 4D Einstein–Gauss–Bonnet cosmology and into length-preserving biconnection gravity. In the 4D EGB analysis the fitted BA values were

$z\in[-1,\infty)$4

and BA was interpreted there as strictly quintessence-like, with monotonic black-hole mass growth and monotonic wormhole mass loss under accretion (Mukherjee et al., 7 Jul 2025). In biconnection gravity, BA described the effective geometric dark-energy sector with best-fit

$z\in[-1,\infty)$5

and was statistically competitive with $z\in[-1,\infty)$6CDM under AICc and DIC, though penalized by BIC (Mhamdi et al., 9 May 2026).

A different use of BA is as a phenomenological image of concrete scalar-field models. A robustness study mapping minimally and non-minimally coupled thawing quintessence into BA, CPL, JBP, and EXP found that BA reproduces the observable phenomenology of these model classes to high accuracy. In that framework, the viability conclusions were insensitive to whether BA or another common two-parameter form was used: minimally coupled thawing quintessence remained disfavored, while non-minimally coupled thawing quintessence remained viable (Wolf et al., 7 Feb 2025).

6. Thermodynamic interpretations and related phenomenology

BA has also been examined as a thermodynamic model rather than only as a kinematic fit. In a flat FLRW universe bounded by the apparent horizon, one study derived the total entropy

$z\in[-1,\infty)$7

and found

$z\in[-1,\infty)$8

throughout cosmic evolution, so the generalized second law always holds, while

$z\in[-1,\infty)$9

at late times, implying thermodynamic equilibrium in the final stages. The same work used the BA-induced deceleration parameter

$w(0)=w_0$00

to derive a particle-creation rate

$w(0)=w_0$01

and concluded that BA reproduces a qualitatively reasonable pattern: strong creation in the early universe, suppression during deceleration, renewed importance during late acceleration, and further increase in the future (Mondal et al., 2018).

A more recent non-equilibrium thermodynamic study constrained the product $w(0)=w_0$02 of dark-energy chemical potential and number density using BA together with Pantheon$w(0)=w_0$03, DESI DR2, and Planck. The BA cosmological fit in that paper was

$w(0)=w_0$04

However, unlike CPL, BA never produced simultaneous compatibility between entropy positivity and the second law for any value of the particle-creation/destruction parameter $w(0)=w_0$05, so no BA posterior interval for $w(0)=w_0$06 could be extracted (Netto et al., 19 Nov 2025).

In compact-object and accretion applications, BA has been used as a time-varying ambient fluid rather than as a cosmological model of first resort. In an MCG-contaminated black-hole setup, BA produced a non-monotonic mass-ratio curve with a maximum near $w(0)=w_0$07 followed by a decline toward $w(0)=w_0$08, interpreted as transiently enhanced accretion followed by repulsion-dominated suppression (Biswas et al., 29 Oct 2025). This suggests that BA’s redshift structure can generate qualitatively distinct local-astrophysical signatures once it is reinterpreted as a background pressure history.

7. Methodological issues, controversies, and recurrent caveats

Despite its regularity advantages, BA remains a phenomenological ansatz rather than a unique microphysical theory. Several papers explicitly stress that alternative explanations of DESI-era anomalies remain possible, including interacting dark energy, modified gravity, or nonstandard matter sectors, so BA should be read as an effective description rather than a unique physical mechanism (Li et al., 27 Nov 2025).

A second recurring caveat is data dependence. Inferences about BA move substantially with the supernova sample, with the inclusion of ACT/SPT information, and with the treatment of the DESI LRG1/LRG2 points. Some analyses find BA among the most stable and best-constrained dynamical forms, while others find that removing LRG1/LRG2 drives the BA posterior back toward the $w(0)=w_0$09CDM point and restores clear Bayesian preference for $w(0)=w_0$10CDM (Zheng et al., 2024, Barua et al., 15 Jun 2025).

A third issue is expository nonuniformity. One Pantheon$w(0)=w_0$11 PINN study explicitly notes an internal ambiguity over whether BA’s dark-energy sector is actually learned by the PINN or inserted analytically in the final likelihood, and it points out a wording slip in which BA was described as approaching $w(0)=w_0$12 at both low and high redshift, even though the exact formula gives $w(0)=w_0$13 (Verma et al., 16 Aug 2025). More broadly, the papers summarized here do not describe future behavior in completely uniform language: many present BA as bounded on $w(0)=w_0$14, while one DESI-era study remarks more generally that polynomial-like parameterizations such as BA are useful mainly for past evolution (Li et al., 27 Nov 2025). The safest interpretation is therefore formula-first: BA’s exact analytic form controls its asymptotics, and individual narrative summaries should be checked against that form.

Taken together, the literature portrays BA as a technically convenient, globally regular, and empirically competitive two-parameter dark-energy parameterization. It often performs better than JBP and other less regular alternatives, frequently behaves similarly to CPL at low redshift while differing more strongly at intermediate and high redshift, and remains a central test bed for assessing whether present hints of dynamical dark energy are robust or parameterization-driven.