QCD-Induced Dynamical Dark Energy

• QCD-induced dynamical dark energy is defined as a framework where residual vacuum effects from non-perturbative QCD dynamics generate an energy density that scales with the Hubble rate.
• The mechanism modifies Friedmann equations to yield distinct equations of state that evolve from w ≈ -1/3 in radiation domination to w ≈ -1 in the late universe.
• This approach offers an economical alternative to ΛCDM by explaining cosmic acceleration without introducing new degrees of freedom, linking observed dark energy to established QCD scales.

Quantum Chromodynamics (QCD)-Induced Dynamical Dark Energy is a framework in which the cosmological dark-energy sector arises as a dynamical, residual vacuum effect from the non-perturbative topological and chiral properties of the QCD vacuum, rather than from a fundamental cosmological constant or an elementary scalar field. The central mechanism is the emergence of a vacuum energy density linked to the response of QCD to an expanding Friedmann-Lemaître-Robertson-Walker (FLRW) background, typically scaling as $\rho_{\rm DE}(t)\propto H(t)\Lambda_{\rm QCD}^3$, with $H$ the Hubble rate and $\Lambda_{\rm QCD}$ the QCD scale. This approach yields an effective equation of state that interpolates between $w\simeq -1/3$ in the radiation era, $w\simeq -1/2$ in matter domination, and $w\to -1$ at late times. Multiple variants exist, with distinctive predictions for cosmic acceleration, the onset of the phantom regime, and stability properties. No new propagating degrees of freedom or ad hoc mass scales are required; all scales and couplings are fixed by the Standard Model and General Relativity.

1. QCD Vacuum Structure and the Dynamical Vacuum Energy Mechanism

At the core of QCD-induced dark energy is the realization that the QCD vacuum is not unique but a superposition of topological sectors labeled by integer Chern-Simons numbers. Instanton-induced tunneling between these vacua gives rise to a non-perturbative vacuum energy (the topological susceptibility, $\chi_t$) that is formally large ($\sim \Lambda_{\rm QCD}^4$) in flat Minkowski space. However, in an expanding (curved) cosmological background, the relevant physical quantity is the difference in vacuum energy between FLRW and flat space,

$$\Delta\rho = \rho_{\rm vac}^{\rm FLRW} - \rho_{\rm vac}^{\rm Mink},$$

as originally advocated by Zeldovich. This subtraction renders the large flat-space vacuum energy unobservable, and only the residual, curvature-dependent piece (a function of $H$) remains. Multiple explicit computations (including Bogoliubov transformations for QCD ghosts in curved backgrounds and path-integral evaluations for topological sectors) demonstrate that this dynamical vacuum energy scales as

$H$0

where $H$1 encodes non-perturbative factors (Ohta, 2010, Waerbeke et al., 17 Jun 2025, Lee et al., 18 Jun 2026).

2. Effective Field Theoretic Realizations and Equations of State

The prototypical “ghost dark energy” model interprets the residual vacuum energy as sourced by the Veneziano ghost field, an unphysical degree of freedom introduced to resolve the $H$2 anomaly. While this ghost decouples in flat space, it yields a vacuum energy proportional to $H$3 in a curved cosmological background: $H$4 where $H$5 is fixed by QCD ($H$6). Generalizations allow for an $H$7 subleading term, or more complicated functionals (e.g., inclusion of $H$8 or quartic-in-$H$9 terms) capturing quantum corrections or IR physics (Cruz et al., 2022, Cai et al., 2012).

The effective equation of state (EoS) for the ghost sector is derived from the continuity equation: $\Lambda_{\rm QCD}$0 leading to

$\Lambda_{\rm QCD}$1

where $\Lambda_{\rm QCD}$2 is the EoS of matter. Thus, $\Lambda_{\rm QCD}$3 interpolates from $\Lambda_{\rm QCD}$4 in the early universe ($\Lambda_{\rm QCD}$5) to $\Lambda_{\rm QCD}$6 at late times ($\Lambda_{\rm QCD}$7) (Cruz et al., 2022). Models allowing derivative (e.g., $\Lambda_{\rm QCD}$8) dependence can yield $\Lambda_{\rm QCD}$9, i.e., realize a genuine phantom regime.

3. Modifications of FLRW Dynamics and Cosmological Solutions

The dynamical vacuum energy modifies the Friedmann equation: $w\simeq -1/3$0 for the simplest case, expanded in more general scenarios to

$w\simeq -1/3$1

where $w\simeq -1/3$2 encodes quantum gravity or RG-corrections (Cai et al., 2012, Ohta, 2010, Waerbeke et al., 17 Jun 2025). Solving this equation yields a cosmic evolution from matter (or radiation) domination to an asymptotic de Sitter phase $w\simeq -1/3$3, where the ghost-induced vacuum energy mimics a cosmological constant at late times.

Table: Typical cosmic regimes in the basic ghost model

Epoch	Dominant Component	$w\simeq -1/3$4	$w\simeq -1/3$5 Scaling
Radiation era	$w\simeq -1/3$6	$w\simeq -1/3$7	$w\simeq -1/3$8
Matter era	$w\simeq -1/3$9	$w\simeq -1/2$0	$w\simeq -1/2$1
DE era	$w\simeq -1/2$2	$w\simeq -1/2$3	$w\simeq -1/2$4

Generalizations can produce transient or even singular behaviors (e.g., adding $w\simeq -1/2$5 terms leads to Type III singularities with $w\simeq -1/2$6 at finite scale factor/time (Cruz et al., 2022)), or stable quartic-in-$w\simeq -1/2$7 constructions emulating a transition from phantom to de Sitter (Cruz et al., 2022).

4. Phenomenological Extensions and Stability

Several extensions of the basic scenario have been constructed:

• Holography-inspired modifications ($w\simeq -1/2$8): Allow for phantom behavior ($w\simeq -1/2$9) with solutions explicitly exhibiting $w\to -1$0 and possible future singularities. These models serve as local analogues of IR quantum-gravity or holographic cutoff effects (Cruz et al., 2022).
• Entropic-force motivated ($w\to -1$1) terms: Originate in higher-dimensional gravity or emergent entropic-force paradigms of DE, yielding a transient phantom-divide crossing (diverging $w\to -1$2) at the epoch of DE domination, but stable late-time attractor (Cruz et al., 2022).
• Invisible QCD (IQCD): Postulates a dark-sector copy of QCD with spontaneous chiral symmetry breaking, leading to a condensate of dark pions and dark gluons. The gauge-pion interaction energy dynamically mimics vacuum-like energy density with $w\to -1$3 and is cosmologically stable to perturbations (Alexander et al., 2016, Addazi et al., 2016).
• PNJL-inspired modifications: Introduce a Hubble-coupled term to the Polyakov-loop potential in the PNJL effective theory with a power $w\to -1$4 (i.e., $w\to -1$5), allowing for a range of late-time DE behaviors strongly constrained by cosmological data to $w\to -1$6, i.e., close to $w\to -1$7CDM (Rincón et al., 14 Jun 2025).

Most models possess a unique feature: no DE perturbations at the level of linear cosmological perturbation theory. The vacuum energy is genuinely global, tracking the (unperturbed) expansion, in contrast to scalar-field quintessence or coupled DE models (Lee et al., 18 Jun 2026).

5. Connection to Observables and Current Constraints

Direct comparison with cosmological data (Planck, DESI BAO, SNIa, DES-Dovekie SNe, CMB anisotropies) demonstrates that QCD-induced DE models with $w\to -1$8 ($w\to -1$9) fit current data at a level comparable to or slightly better than $\chi_t$0CDM, with moderate Bayesian evidence in favor (Lee et al., 18 Jun 2026, Waerbeke et al., 17 Jun 2025). The mild running of $\chi_t$1 (typically $\chi_t$2 today, phantom crossing at $\chi_t$3–$\chi_t$4, asymptoting to $\chi_t$5 in the future) is consistent with DESI and other large-scale structure measurements (Waerbeke et al., 17 Jun 2025, Lee et al., 18 Jun 2026).

Table: Representative best-fit cosmological parameters (Lee et al., 18 Jun 2026)

Model	$\chi_t$6 [km/s/Mpc]	$\chi_t$7	$\chi_t$8 [km/s/Mpc]	$\chi_t$9 (transition)
$\sim \Lambda_{\rm QCD}^4$0CDM	$\sim \Lambda_{\rm QCD}^4$1	$\sim \Lambda_{\rm QCD}^4$2	$\sim \Lambda_{\rm QCD}^4$3	–
QCD-DE (exp)	$\sim \Lambda_{\rm QCD}^4$4	$\sim \Lambda_{\rm QCD}^4$5	$\sim \Lambda_{\rm QCD}^4$6	$\sim \Lambda_{\rm QCD}^4$7
QCD-DE (tanh)	$\sim \Lambda_{\rm QCD}^4$8	$\sim \Lambda_{\rm QCD}^4$9	$$\Delta\rho = \rho_{\rm vac}^{\rm FLRW} - \rho_{\rm vac}^{\rm Mink},$$0	$$\Delta\rho = \rho_{\rm vac}^{\rm FLRW} - \rho_{\rm vac}^{\rm Mink},$$1

No significant anomalies are observed in growth rates or CMB peaks. The QCD-induced DE scenario can accommodate a small upwards shift in $$\Delta\rho = \rho_{\rm vac}^{\rm FLRW} - \rho_{\rm vac}^{\rm Mink},$$2 relative to $$\Delta\rho = \rho_{\rm vac}^{\rm FLRW} - \rho_{\rm vac}^{\rm Mink},$$3CDM, potentially addressing the $$\Delta\rho = \rho_{\rm vac}^{\rm FLRW} - \rho_{\rm vac}^{\rm Mink},$$4 tension (Waerbeke et al., 17 Jun 2025).

6. Theoretical Consistency, Vacuum Cancellation, and Open Issues

Theoretical consistency relies on exact cancellation of large UV vacuum energies (e.g., QCD instanton contributions, gluon condensate) in flat space. Semiclassical gravity provides a controlled expansion (graviton corrections) in which a small, residual term of order $$\Delta\rho = \rho_{\rm vac}^{\rm FLRW} - \rho_{\rm vac}^{\rm Mink},$$5 persists post-QCD phase transition due to incomplete cancellation (e.g., due to chiral symmetry breaking or Planck-suppressed $$\Delta\rho = \rho_{\rm vac}^{\rm FLRW} - \rho_{\rm vac}^{\rm Mink},$$6 breaking) (Pasechnik et al., 2013, Addazi et al., 2021). This result is numerically consistent with the observed vacuum energy density,

$$\Delta\rho = \rho_{\rm vac}^{\rm FLRW} - \rho_{\rm vac}^{\rm Mink},$$7

Within the standard instanton-liquid model, all time dependence in $$\Delta\rho = \rho_{\rm vac}^{\rm FLRW} - \rho_{\rm vac}^{\rm Mink},$$8 is suppressed by $$\Delta\rho = \rho_{\rm vac}^{\rm FLRW} - \rho_{\rm vac}^{\rm Mink},$$9, effectively reproducing $H$0CDM dynamics unless the scalar glueball is unphysically light (Musakhanov, 16 Jun 2025).

Possible loopholes for dynamical $H$1 include quantum anomalies, running vacuum models ($H$2 with $H$3 corrections), and non-trivial non-perturbative effects or IR modifications (Fritzsch et al., 2012). All such constructions must respect the precise cancellation mechanism in flat space and the rigid empirical constraints on time variation in $H$4 and $H$5 (Fritzsch et al., 2012).

7. Outlook, Distinguishing Predictions, and Future Probes

QCD-induced dynamical dark energy offers a parameter-economical, theoretically-motivated alternative to ad hoc scalar-field or modified-gravity models. Distinctive observational predictions include: a time-varying $H$6 with “safe” phantom crossing, correlated small deviations in $H$7 and growth at low redshift, absence of DE perturbations, and potential laboratory detection via topological Casimir effects in Maxwell theory (Lee et al., 18 Jun 2026, Waerbeke et al., 17 Jun 2025).

Current and future surveys (DESI, Euclid, LSST, CMB-S4) are expected to further constrain the allowed parameter space, test the hypothesis of a running vacuum proportional to $H$8, and potentially reveal signatures (e.g., BAO pattern, late-time ISW effects, mild $H$9 runnings) distinct from $H$00CDM and canonical quintessence (Lee et al., 18 Jun 2026, Waerbeke et al., 17 Jun 2025, Pasechnik, 2016).

This approach also opens novel connections between high-energy QCD/topological field theory, cosmology, and quantum gravity, providing a natural explanation for the coincidence between the dark-energy scale and the QCD scale, and demonstrating that cosmic acceleration may be sourced by known Standard Model physics in the non-perturbative regime.