Bound Dark Energy (BDE) Overview

• Bound Dark Energy (BDE) is a family of dynamic dark-energy models in which cosmic acceleration is driven by bound state phenomena or bounded field dynamics instead of a fixed vacuum energy.
• BDE research includes distinct frameworks such as CKN-bound models based on UV/IR constraints, composite-meson quintessence from a dark SU(3) sector, and bounded quintessence with mirage cut-offs, each offering unique observational signatures.
• Observational tests of BDE highlight differences from ΛCDM, including modifications in matter density evolution and specific alterations in the matter power spectrum, providing insights into cosmic structure formation.

Bound Dark Energy (BDE) denotes a family of dynamical dark-energy proposals in which cosmic acceleration is tied to a bound, a bound state, or bounded scalar-field dynamics rather than to a strictly constant vacuum term. In current arXiv literature, the label covers at least three technically distinct constructions: dark energy driven by the Cohen–Kaplan–Nelson ultraviolet/infrared bound, a composite-meson quintessence field generated by a dark $SU(3)$ gauge sector, and “bounded dark energy” quintessence with a bounded hilltop-like potential stabilized by a mirage cut-off (Adolf et al., 2024, Almaraz et al., 2018, Borghetto et al., 14 Mar 2025). This indicates that the term is not standardized to a single framework.

1. Terminological scope and principal usages

The contemporary literature uses “Bound Dark Energy” or “Bounded Dark Energy” in several non-equivalent ways.

Usage	Defining mechanism	Representative papers
CKN-bound dark energy	UV/IR relation from the Cohen–Kaplan–Nelson bound, with $L_{\mathrm{IR}}=H^{-1}$	(Adolf et al., 2024, Adolf et al., 2024, Adolf et al., 21 Apr 2025)
Particle-physics BDE	lightest meson $\phi$ in a dark $SU(3)$ gauge sector with $V(\phi)=\Lambda_c^{4+2/3}\phi^{-2/3}$	(Almaraz et al., 2018, Macorra et al., 24 Mar 2025, Torres et al., 25 Jul 2025, Macorra et al., 13 Jan 2026)
“Bounded dark energy” quintessence	bounded hilltop potentials, mirage cut-off, symbolic-regression model building	(Borghetto et al., 14 Mar 2025)

This multiplicity matters because observational claims, parameter counts, and theoretical motivations differ sharply across these usages. In the CKN literature, the operative object is a vacuum-energy term scaling with $H^2(z)$. In the dark-$SU(3)$ literature, BDE is a composite scalar emerging at a condensation transition. In the mirage cut-off literature, the emphasis is on technically natural bounded quintessence potentials. Direct numerical comparisons across these literatures are therefore not model-independent.

2. Dark energy from the Cohen–Kaplan–Nelson bound

The CKN program starts from the claim that gravity constrains the regime of validity of QFT by tying the ultraviolet cutoff $\Lambda_{\mathrm{UV}}$ to an infrared scale $L$. In the form quoted in the recent cosmology papers, the QFT entropy satisfies

$$S_{\mathrm{QFT}}=\Lambda_{\mathrm{UV}}^3L^3\le \pi L^2 M_{\mathrm{Pl}}^2,$$

while the stronger Cohen–Kaplan–Nelson argument excludes states that would already describe black holes, yielding

$L_{\mathrm{IR}}=H^{-1}$0

Setting the infrared cutoff by the Hubble horizon, $L_{\mathrm{IR}}=H^{-1}$1, gives

$L_{\mathrm{IR}}=H^{-1}$2

and hence a vacuum-energy contribution proportional to $L_{\mathrm{IR}}=H^{-1}$3 rather than a constant (Adolf et al., 2024, Adolf et al., 2024).

The resulting dark-energy density is written in two closely related ways in the cited papers. One form is

$L_{\mathrm{IR}}=H^{-1}$4

with $L_{\mathrm{IR}}=H^{-1}$5 an order-one parameter. A more general implementation writes

$L_{\mathrm{IR}}=H^{-1}$6

where $L_{\mathrm{IR}}=H^{-1}$7 is a constant contribution and $L_{\mathrm{IR}}=H^{-1}$8 corresponds to the original CKN-motivated scaling. The literature calls the $L_{\mathrm{IR}}=H^{-1}$9 realization the CKN model and the generalized case the $\phi$0CKN model (Adolf et al., 21 Apr 2025).

At the homogeneous level, the model modifies the redshift dependence of matter and dark-energy fractions. The quoted expressions are

$\phi$1

$\phi$2

The same papers state that matter and dark energy are no longer conserved separately, so the framework is not merely a reparameterization of $\phi$3CDM but a coupled late-time background model (Adolf et al., 2024).

Conceptually, this approach interprets dark energy as a QFT–gravity interplay: the Hubble horizon acts as the relevant IR scale, and the allowed vacuum energy is limited by black-hole exclusion rather than by an unconstrained zero-point sum. A plausible implication is that the model is attractive precisely because its dynamical behavior follows from a cutoff argument rather than from an added scalar degree of freedom.

3. Composite-meson BDE from a dark $\phi$4 sector

A distinct and older BDE literature derives dark energy from particle physics. In this framework, dark energy is the lightest scalar meson $\phi$5 formed in a hidden supersymmetric dark gauge group, usually $\phi$6 with $\phi$7 massless flavors, unified with the Standard Model gauge couplings at the GUT scale and interacting with the visible sector only via gravity below $\phi$8 (Almaraz et al., 2018, Torres et al., 25 Jul 2025).

Above the condensation scale $\phi$9, the dark gauge group particles are massless and behave as extra radiation. Below $SU(3)$0, the coupling becomes strong and composite bound states form. The lightest neutral meson $SU(3)$1 then acts as the dark-energy field. Its scalar potential is the inverse-power-law form

$SU(3)$2

derived from non-perturbative gauge dynamics and identified in the papers with the Affleck–Dine–Seiberg mechanism. The condensation scale is written as

$SU(3)$3

with $SU(3)$4, and the condensation epoch satisfies

$SU(3)$5

while the 2018 analysis quotes $SU(3)$6 (Almaraz et al., 2018, Macorra et al., 24 Mar 2025).

The cosmological evolution is staged. Before condensation, the sector has $SU(3)$7. At condensation, the scalar enters a kinetic-dominated phase with $SU(3)$8 and $SU(3)$9. Later, Hubble friction drives the system toward a quintessence-like regime with $V(\phi)=\Lambda_c^{4+2/3}\phi^{-2/3}$0 and a present value near $V(\phi)=\Lambda_c^{4+2/3}\phi^{-2/3}$1. The 2018 complete analysis reported

$V(\phi)=\Lambda_c^{4+2/3}\phi^{-2/3}$2

$V(\phi)=\Lambda_c^{4+2/3}\phi^{-2/3}$3

$V(\phi)=\Lambda_c^{4+2/3}\phi^{-2/3}$4

and stated that these are in complete agreement with the theoretical prediction $V(\phi)=\Lambda_c^{4+2/3}\phi^{-2/3}$5 (Almaraz et al., 2018).

Later analyses sharpened the same picture. One paper reported $V(\phi)=\Lambda_c^{4+2/3}\phi^{-2/3}$6 eV and $V(\phi)=\Lambda_c^{4+2/3}\phi^{-2/3}$7, together with a present equation of state $V(\phi)=\Lambda_c^{4+2/3}\phi^{-2/3}$8 (Macorra et al., 24 Mar 2025). A later DR2-based analysis reported $V(\phi)=\Lambda_c^{4+2/3}\phi^{-2/3}$9 eV, $H^2(z)$0, and $H^2(z)$1, with $H^2(z)$2 maintained throughout cosmic history (Macorra et al., 13 Jan 2026). These papers emphasize that the dark-energy sector has no free parameters and that the model has one less parameter than $H^2(z)$3CDM and three less than $H^2(z)$4CDM (Torres et al., 25 Jul 2025).

4. Bounded quintessence and the mirage cut-off program

The 2025 paper “Bounded Dark Energy” introduced another framework, explicitly as a quintessence scenario. Here the defining feature is not a UV/IR vacuum-energy bound or a dark bound state, but a scalar potential that is flat near the origin and grows steeply at finite field values, thereby bounding field excursions. The paper describes this as a technically natural way to realize late-time dynamical dark energy and connects it to a bottom-up “mirage cut-off” construction (Borghetto et al., 14 Mar 2025).

Prototype potentials quoted in that work include

$H^2(z)$5

$H^2(z)$6

$H^2(z)$7

and the general series

$H^2(z)$8

The hierarchy

$H^2(z)$9

is used to argue that loop corrections do not spoil the desired shape (Borghetto et al., 14 Mar 2025).

The same work links the construction to swampland and de Sitter conjectures, and it combines analytical model building with symbolic regression through PySR. The quoted machine-learning-inspired potential is

$SU(3)$0

For cosmological testing, the paper uses Planck 2018, DESI DR1 BAO, full-shape clustering, and the Pantheon+, Union3, and DES-Y5 supernova samples, with CAMB, Cobaya, and a Scipy minimizer, and states that bounded dark energy provides a good fit to current observations (Borghetto et al., 14 Mar 2025).

This framework is therefore separate from the dark-$SU(3)$1 meson model despite the similar name. Its scalar degree of freedom is fundamental within the EFT description, its late-time evolution is thawing rather than condensation-driven, and its main theoretical novelty lies in technical naturalness and bounded field range.

5. Observational constraints and comparison with $SU(3)$2CDM

The CKN-bound literature confronts DESI BAO, supernova, and Hubble data directly. Using DESI year-1 BAO, DES-SN5YR or Pantheon+, and model-independent Hubble measurements, the 2024 analysis reported reduced $SU(3)$3 values $SU(3)$4 for DESY5 and $SU(3)$5 for Pantheon+, with $SU(3)$6 and $SU(3)$7, corresponding to a $SU(3)$8 preference for $SU(3)$9CKN over $\Lambda_{\mathrm{UV}}$0CDM in DESY5 and a $\Lambda_{\mathrm{UV}}$1 preference in Pantheon+, which the paper described as not significant (Adolf et al., 2024). The 2025 addendum using DESI BAO DR2 reported $\Lambda_{\mathrm{UV}}$2, $\Lambda_{\mathrm{UV}}$3 for DESY5 and $\Lambda_{\mathrm{UV}}$4 for Pantheon+, with a statistical preference of up to $\Lambda_{\mathrm{UV}}$5; it also noted that $\Lambda_{\mathrm{UV}}$6CDM and $\Lambda_{\mathrm{UV}}$7CDM sometimes fit even better (Adolf et al., 21 Apr 2025). Forecasts in the 2024 paper further stated that, if current trends persist, DESI-5Y plus Euclid-like improvements could raise the preference to $\Lambda_{\mathrm{UV}}$8 (Adolf et al., 2024).

The particle-physics BDE literature reports a different statistical profile. The 2018 analysis found that BDE improves the likelihood ratio of BAO measurements by 2.1 relative to $\Lambda_{\mathrm{UV}}$9CDM and has an equivalent fit for type Ia supernovae and the CMB (Almaraz et al., 2018). A 2025 paper combining DESI BAO, CMB, and DES-SN5YR stated that BDE achieves a $L$0 and $L$1 reduction in the reduced $L$2 compared to $L$3CDM and $L$4CDM, respectively, while keeping an equivalent fit for type Ia supernovae and the CMB; it also reported that the $L$5 contour is 10,000 times smaller than in $L$6CDM (Macorra et al., 24 Mar 2025). The 2026 DR2-based analysis quoted $L$7 and $L$8 relative to $L$9CDM for BAO+DESY5 and called this strong evidence favoring BDE-CDM (Macorra et al., 13 Jan 2026).

These numerical results are not interchangeable across the different BDE meanings. The CKN papers compare a vacuum-energy model with one extra amplitude parameter $$S_{\mathrm{QFT}}=\Lambda_{\mathrm{UV}}^3L^3\le \pi L^2 M_{\mathrm{Pl}}^2,$$0, while the dark-$$S_{\mathrm{QFT}}=\Lambda_{\mathrm{UV}}^3L^3\le \pi L^2 M_{\mathrm{Pl}}^2,$$1 papers compare a parameter-free dark-energy sector with fixed $$S_{\mathrm{QFT}}=\Lambda_{\mathrm{UV}}^3L^3\le \pi L^2 M_{\mathrm{Pl}}^2,$$2 and $$S_{\mathrm{QFT}}=\Lambda_{\mathrm{UV}}^3L^3\le \pi L^2 M_{\mathrm{Pl}}^2,$$3 predictions. This suggests that the empirical status of “BDE” is intrinsically model- and dataset-dependent, even when the same observational tendency—preference for time-varying dark energy over a strict cosmological constant—is emphasized.

6. Structure formation, consistency tests, and adjacent bound-based proposals

The nonlinear phenomenology of the particle-physics BDE model has been studied with $$S_{\mathrm{QFT}}=\Lambda_{\mathrm{UV}}^3L^3\le \pi L^2 M_{\mathrm{Pl}}^2,$$4-body simulations. The 2019 structure-formation analysis reported that before condensation the extra relativistic dark gauge sector enhances the Hubble rate by approximately $$S_{\mathrm{QFT}}=\Lambda_{\mathrm{UV}}^3L^3\le \pi L^2 M_{\mathrm{Pl}}^2,$$5, while linear theory gives an enhancement of matter perturbations on small scales for modes $$S_{\mathrm{QFT}}=\Lambda_{\mathrm{UV}}^3L^3\le \pi L^2 M_{\mathrm{Pl}}^2,$$6, reaching $$S_{\mathrm{QFT}}=\Lambda_{\mathrm{UV}}^3L^3\le \pi L^2 M_{\mathrm{Pl}}^2,$$7. At late times, however, nonlinear mode coupling washes out this small-scale enhancement before dark energy becomes dominant, leaving instead a $$S_{\mathrm{QFT}}=\Lambda_{\mathrm{UV}}^3L^3\le \pi L^2 M_{\mathrm{Pl}}^2,$$8 suppression of the matter power spectrum on the largest scales at $$S_{\mathrm{QFT}}=\Lambda_{\mathrm{UV}}^3L^3\le \pi L^2 M_{\mathrm{Pl}}^2,$$9 (Almaraz et al., 2019). The same work found more small halos, fewer large halos, and essentially unchanged halo concentration relative to $L_{\mathrm{IR}}=H^{-1}$00CDM. By contrast, the 2026 BDE-CDM analysis reported a $L_{\mathrm{IR}}=H^{-1}$01 enhancement in the matter power spectrum at $L_{\mathrm{IR}}=H^{-1}$02 (Macorra et al., 13 Jan 2026). Read together, these papers indicate that the status of small-scale signatures depends on whether one is discussing linear predictions or nonlinear late-time clustering.

Additional consistency studies exist. A 2019 analysis of cosmological-parameter and fundamental-constant evolution considered BDE models with $L_{\mathrm{IR}}=H^{-1}$03 and concluded that the resulting histories for $L_{\mathrm{IR}}=H^{-1}$04, $L_{\mathrm{IR}}=H^{-1}$05, $L_{\mathrm{IR}}=H^{-1}$06, and $L_{\mathrm{IR}}=H^{-1}$07 are consistent with observational constraints for sufficiently small couplings to the particle sector (Thompson, 2019).

The broader “bound” literature also contains adjacent, but non-identical, dark-energy proposals. One paper derived a self-gravitational upper bound $L_{\mathrm{IR}}=H^{-1}$08 for localized energy and used $L_{\mathrm{IR}}=H^{-1}$09 with a cosmological horizon scale to estimate a passable dark-energy density (Kauffmann, 2012). Another modeled dark energy as a Bose–Einstein condensate of cosmologically massive photons in de Sitter space, with $L_{\mathrm{IR}}=H^{-1}$10 and a photon mass tied to $L_{\mathrm{IR}}=H^{-1}$11 (Resca, 2020). Conversely, an entropy-bound study of McVittie spacetime argued that only the cosmological constant satisfies a D-bound–Bekenstein-bound identification for that particular black-hole setup, while quintessence and phantom fail (Hadi et al., 2019). That result was stated for a specific entropic criterion and spacetime, not as a universal theorem about all BDE frameworks.

The principal misconception surrounding BDE is therefore terminological: the same name labels several distinct dynamical dark-energy programs. Their shared denominator is opposition to a strictly constant $L_{\mathrm{IR}}=H^{-1}$12, but their microscopic ontology, EFT control, and phenomenological predictions are different. Any technical discussion of BDE must specify which construction is intended.