Dark Energy Quintessence Model

• Dark Energy Quintessence Model is a framework where a canonical scalar field mimics QCD ghost dark energy, linking its energy density and equation of state.
• The model explicitly reconstructs the scalar field potential and dynamics by mapping ghost dark energy through Friedmann equations and energy conservation.
• It explains cosmic acceleration via a dynamic transition from matter to dark energy domination while alleviating fine-tuning challenges seen in the cosmological constant.

A dark energy quintessence model, in the context of the referenced work, is a theoretical framework wherein the observed cosmic acceleration is attributed to the dynamics of a canonical scalar field (the “quintessence” field), whose evolution reproduces the phenomenology of ghost dark energy (GDE) sourced by nonperturbative effects in quantum chromodynamics (QCD). A detailed correspondence is established such that the energy density and equation of state of the quintessence field precisely track those of the GDE, enabling explicit reconstruction of the potential $V(\phi)$ and field dynamics from first principles.

1. Ghost Dark Energy: Physical Origin and Definition

The ghost dark energy model is rooted in the anomalous properties of the Veneziano ghost in low-energy QCD, which, in flat Minkowski space, does not contribute to vacuum energy but leaves a non-trivial remnant in a time-dependent (curved) background. The resulting vacuum energy density is linearly proportional to the Hubble parameter,

$\rho_D = \alpha H,$

where $\alpha \sim \Lambda_{QCD}^3$ and $\Lambda_{QCD}$ is the QCD mass scale. This proportionality offers a natural explanation for the observed magnitude of dark energy without fine-tuning, yielding $\rho_D \sim (10^{-3}~\mathrm{eV})^4$ for $H_0 \sim 10^{-33}~\mathrm{eV}$.

2. Quintessence Correspondence: Formulation and Mapping

By postulate, the quintessence field $\phi$ is constructed so that

$\rho_\phi = \rho_{D}, \qquad w_\phi = w_{D},$

with canonical kinetic and potential contributions: $$\rho_\phi = \frac{1}{2}\dot{\phi}^2 + V(\phi), \qquad p_\phi = \frac{1}{2}\dot{\phi}^2 - V(\phi), \qquad w_\phi = \frac{p_\phi}{\rho_\phi}.$$ Given the GDE density and its equation-of-state evolution, this mapping yields unique functional expressions for both $V(\phi)$ and $\dot{\phi}^2$ at each epoch, realizing the field’s cosmological trajectory.

3. Dynamical Equations and Explicit Reconstruction

The cosmological background is described by the flat Friedmann equation: $$H^2 = \frac{1}{3M_p^2}\left(\rho_m + \rho_D\right),$$ and the GDE equation of state $w_D$ arises from the energy conservation equation by differentiating $\rho_D=\alpha H$: $\dot{\rho}_D + 3H\rho_D(1 + w_D) = 0.$ Solving for $w_D$ yields

$w_D = -\frac{1}{2 - \Omega_D},$

where $\Omega_D = \rho_D/\rho_{cr}$ and $\rho_{cr} = 3 H^2 M_p^2$. Substituting $\rho_D$ and $w_D$ into the scalar field relationships gives

$$V(\phi) = \frac{1-w_D}{2}\rho_D,\qquad \dot{\phi}^2 = (1+w_D)\rho_D.$$

All relevant quantities are now functions of $\Omega_D(a)$.

An explicit differential equation for the field evolution is obtained: $$\frac{d\phi}{d\ln a} = \sqrt{3} M_p \sqrt{\frac{\Omega_D (1-\Omega_D)}{2-\Omega_D}},$$ while the effective potential is reconstructed as

$$V(\phi) = \frac{\alpha^2}{6 M_p^2}\frac{3 - \Omega_D}{\Omega_D (2 - \Omega_D)}.$$

Numerical integration of these expressions yields the scalar’s rolling trajectory and the (in general non-analytical) form of $V(\phi)$.

4. Cosmological Evolution and Late-Time Acceleration

The evolution of $w_D$ connects the physical behavior of GDE and the reconstructed quintessence:

• For $\Omega_D \ll 1$ (early times), $w_D \to -1/2$, distinct from a cosmological constant and yielding subdominant negative pressure.
• For late times ($\Omega_D \to 1$), $w_D \to -1$, recovering cosmological-constant-like behavior and de Sitter expansion.

This dynamical transition naturally explains the observed late-time acceleration without an explicit fine-tuned cosmological constant. In scenarios with suitable dark energy–dark matter coupling, $w_D$ may cross the phantom divide ($w < -1$), a regime inaccessible to standard single-field models without instabilities.

5. Comparison to Other Frameworks and Fine-Tuning Alleviation

Unlike conventional quintessence, which often employs ad hoc potentials, the potential here is derived from a field-theoretic ghost mechanism in QCD. Notably:

• The energy scale is set by hadronic physics, avoiding the extreme fine-tuning ($\sim 10^{-120}$) endemic to the cosmological constant problem.
• Only known degrees of freedom are invoked (no need for new particles or fields beyond the SM/Veneziano ghost).
• The scalar field potential arises from matching to quantum vacuum properties, granting a predictive linkage between particle physics and cosmology.

An explicit table relates the correspondence:

Model	Energy Density	Equation of State $w$	Potential $V(\phi)$
Ghost Dark Energy	$\rho_D = \alpha H$	$-\frac{1}{2-\Omega_D}$	--
Quintessence	$\rho_\phi$ from GDE	$w_\phi = w_D$	$$(\alpha^2/6M_p^2)\frac{3 - \Omega_D }{\Omega_D (2- \Omega_D)}$$

6. Implications, Limitations, and Extensions

The GDE-quintessence correspondence offers several avenues for cosmological modeling:

• The model anticipates and accommodates a transition from matter to dark energy domination, mimics $\Lambda$CDM at late times, and may contribute underpinnings to features such as the cosmic coincidence problem.
• When extended to include direct interaction terms with matter, the scalar field sector allows for a flexible range of late-time dynamical dark energy phenomena, including possible $w < -1$ evolution.
• Limitations include the lack of closed-form analytical potential $V(\phi)$ and sensitivity to the precise QCD scale via $\alpha$.
• The framework is not sensitive to initial conditions for the scalar field, as the evolution is driven predominantly by the QCD ghost contribution fixed by cosmological expansion.

7. Summary and Significance

In this model, ghost dark energy arising nonperturbatively from QCD is mapped precisely onto a scalar quintessence field through a one-to-one correspondence of energy density and equation of state. The resulting potential and field evolution are explicitly reconstructed (numerically), with the key output that late-time cosmic acceleration can be obtained from quantum vacuum effects tied to known physics, rather than invoking a fundamental cosmological constant or arbitrary inflation of field-theoretic parameter space. Extensions with dark sector interactions further enhance the dynamical range and potential observational signatures, solidifying this mapping as a promising interface between QCD vacuum structure and cosmological acceleration.