Holographic Dark Energy

• Holographic dark energy is a framework that uses the holographic principle to relate cosmic horizon thermodynamics with quantum gravity in explaining accelerated expansion.
• The model derives dark energy density from entropy bounds on the horizon, forcing the bulk cosmological constant to vanish while aligning with observational data.
• This approach predicts a dynamic equation of state (w ≈ -0.9) and connects thermodynamic energy scaling with emergent gravitational effects on cosmological scales.

Holographic dark energy (HDE) is a theoretical framework that connects the observed late-time accelerated expansion of the universe to quantum gravity through the holographic principle. In this paradigm, the energy density responsible for acceleration is not a fundamental cosmological constant but arises dynamically from constraints imposed by holography on the number of degrees of freedom (DOF) in a given cosmological region. The central idea is that the vacuum energy density must not exceed the energy scale set by saturating the entropy bound defined on the causal or horizon surface, a concept rooted in black hole thermodynamics and entanglement entropy. This approach provides a mechanism for explaining why the cosmological constant should vanish while leaving a residual, nonzero, dynamical dark energy component that aligns well with cosmological observations.

1. Holographic Principle and Thermodynamic Foundation

The holographic principle posits that the maximal entropy (and thus the information content) within a spatial region is bounded by its boundary area, not its volume. This is described by the Bekenstein–Hawking entropy formula for a surface of area $A$: $S_{BH} = \frac{c^3 A}{4G\hbar}$ For a spherical cosmic causal horizon of radius $r$, the entropy saturating the Bekenstein bound can be written as: $S_h = \frac{\eta c^3 r^2}{G\hbar} \quad \text{(1)}$ where $\eta$ is a dimensionless parameter.

By associating a temperature with the horizon,

$$T_h = \frac{\varepsilon \hbar c}{k_B r} \quad \text{(2)}$$

($\varepsilon$ corresponds to the Gibbons–Hawking temperature when $\varepsilon = 1/2\pi$), and invoking a first-law thermodynamic relation applied to the boundary,

$dE_h = k_B T_h dS_h \quad \text{(3)},$

integration yields the total horizon energy: $$E_h = \frac{\eta \varepsilon c^4 r}{G} \quad \text{(4)}$$ This prescription demonstrates that the energy tied to the horizon scales linearly with $r$, a result distinct from local quantum field theory (QFT), which would yield a much larger scaling.

2. Holographic Dark Energy Density and Parameterization

To derive the corresponding energy density, the total horizon energy is divided by the enclosed volume ($\sim r^3$): $$\rho_h = \frac{E_h}{r^3} \sim \frac{\eta \varepsilon c^4}{G r^2}$$ This is recast in the standard HDE form: $$\rho_h = \frac{3 d^2 c^3 M_P^2}{\hbar r^2} \quad \text{(5)}$$ where $M_P = \sqrt{\hbar c / (8\pi G)}$ is the reduced Planck mass, and

$d = \sqrt{2 \eta \varepsilon} \quad \text{(6)}$

For the Bekenstein–Hawking bound and Gibbons–Hawking temperature ($\eta = \pi$, $\varepsilon = 1/2\pi$), one finds $d = 1$.

This dynamical energy component: $\rho_{DE} \propto \frac{1}{r^2} \propto H^2$ when the IR cutoff is taken to be the Hubble horizon, $r \sim H^{-1}$. Importantly, the bulk cosmological constant $\Lambda$ is forced to vanish because a constant $\Lambda$ would lead to bulk energy scaling as $r^3$, which eventually violates the holographic bound ($E_h \sim r$).

3. Equation of State and Observational Viability

The dark energy equation of state (EOS) in HDE models is parameterized as: $$w_0 = -\frac{1}{3} \left(1 + \frac{2\sqrt{\Omega_\ell^0}}{d}\right)$$ with $\Omega_\ell^0 \approx 0.73$ representing the current dark energy density fraction. Substituting $d=1$ yields $w_0 \approx -0.9$, well within the range favored by observations from SN Ia, BAO, and CMB data. The model also predicts a small but nonzero evolution in $w$, consistent with current data constraints.

The value $d=1$ emerges robustly from semiclassical holographic and thermodynamic assumptions, linking fundamental microphysics to cosmological phenomenology.

4. Relation to Quantum Vacuum Energy and Entropic Gravity

Conventional QFT predicts a vacuum energy density on the order of $M_P^4$, vastly exceeding the observed value by $\sim 120$ orders of magnitude (the cosmological constant problem). The holographic principle imposes a much stricter bound, limiting the effective degrees of freedom—not by the field modes in the bulk, but by the DOF on the horizon area.

As a result, the vacuum energy must scale as $M_P^2 / r^2$ (or $M_P^2 H^2$ for $r \sim H^{-1}$). This approach also finds a natural connection with entropic gravity frameworks: the horizon’s energy and the emergent gravitational force can be written in equipartition form,

$$E_h \sim N T_h, \quad F_h = \frac{dE_h}{dr} = \frac{c^4 \eta \varepsilon}{G}$$

where $N$ is proportional to the horizon area (the number of independent bits or DOF). Both gravity and dark energy are therefore seen as emergent quantum-informational phenomena, challenging the applicability of semiclassical QFT to cosmological scales.

5. Implications for the Cosmological Constant and Dynamics

The consistent application of the holographic principle and thermodynamics at the horizon leads to the following implications:

• The cosmological constant $\Lambda$ vanishes in the bulk, since its scaling with $r^3$ would eventually surpass the bound set by the horizon energy $E_h \sim r$.
• Dark energy is necessarily dynamical, arising from quantum fluctuations with a boundary-limited DOF.
• The EOS is determined by the parameter $d$, which is fixed by semiclassical considerations, and is in agreement with observed accelerated expansion.
• The reduction in effective DOF suggests the need for modification of QFT at large scales, pointing to emergent aspects of both gravitational and dark energy physics.

6. Synthesis and Broader Context

The holographic dark energy scenario provides a framework in which the observed dark energy is interpreted as a manifestation of boundary-confined quantum DOF, consistent with the Bekenstein–Hawking entropy bound and first-law thermodynamics applied to a cosmic horizon. This construction links quantum information, horizon thermodynamics, and large-scale cosmic acceleration, naturally producing a vanishing bulk cosmological constant and a viable, small, nonzero dark energy component.

Key features of the paradigm are summarized in the following table:

Quantity	Scaling in HDE	Interpretation
Entropy ($S_h$)	$\sim r^2$ (area law)	Horizon DOF bound
Energy ($E_h$)	$\sim r$	Thermodynamic energy on horizon
Density ($\rho_h$)	$\propto 1/r^2$	Dynamical dark energy
EOS ($w$)	$w_0 \approx -0.9$ (for $d=1$)	Matches observations

This approach also forges conceptual connections to entropic gravity and emphasizes the central role of the horizon as the relevant scale for quantum gravity corrections in cosmology, rather than the bulk volume. The framework thereby addresses both the cosmological constant problem and the origin of dark energy within a consistent holographic scheme, while highlighting the necessity of new physics at the intersection of gravity, thermodynamics, and quantum information on cosmological scales (Lee, 2010).