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The Cosmological Constant Problem Revisited

 

Holography, Horizons, and Thermodynamic Equilibrium

In a previous post we mentioned how the discrepancy in the classical and quantum estimates of the stiffness of space-time was another version of the cosmological constant problem (CCP). You will find some people claiming this is a non-problem, however, the CCP is actually one of the two great naturalness problems in modern physics. 

$$\Lambda \simeq 1.3 \times 10^{-52}\ \mathrm{m}^{-2}$$

yet naïve quantum field theory (QFT) estimates of vacuum energy overshoot the observed value by an astonishing factor of order $10^{121}$.

This discrepancy is often expressed as the ratio between the Planck energy density and the observed dark energy density:

$$\frac{\rho_{\text{Planck}}}{\rho_\Lambda} \sim 10^{121}$$

where

$$\rho_\Lambda = \frac{\Lambda c^2}{8\pi G}, \qquad \rho_{\text{Planck}} = \frac{c^5}{\hbar G^2}$$

 At face value, this looks like a total failure of theoretical physics. However, this interpretation rests on an assumption that turns out to be wrong: that vacuum degrees of freedom scale with volume. 

Where the naive QFT estimate comes from

In local QFT, the vacuum energy is estimated by summing zero-point energies of field modes in a box, aka: quantising a 'particle in a box'.

$$\rho_{\rm vac} \sim \int^{\Lambda_{\rm UV}} \frac{d^3 k}{(2\pi)^3} \frac{1}{2}\hbar \omega_k \sim \hbar c \Lambda_{\rm UV}^4.$$

Choosing the ultraviolet cutoff at the Planck scale,

$$\Lambda_{\rm UV} \sim L_{\rm Pl}^{-1},$$

yields the Planck density $\rho_{\text{Planck}}$. Implicit in this estimate is the assumption that every Planck-sized cell in the bulk carries an independent degree of freedom. Gravity, however, places a fundamental limit on this assumption.

The CKN bound and UV–IR mixing

In 1998, Cohen, Kaplan, and Nelson (CKN) pointed out that a QFT placed in a region of size $L$ cannot be arbitrarily excited. If the total energy exceeds that of a black hole of the same size, the region collapses gravitationally.

Requiring gravitational stability implies

$$L^3 \Lambda_{\rm UV}^4 \lesssim \frac{c^4}{G} L,$$

or equivalently,

$$\Lambda_{\rm UV}^4 \lesssim \frac{c^4}{G L^2}.$$

This is a striking result: the ultraviolet cutoff depends on the infrared scale. This UV–IR mixing means that the number of physically accessible states does not scale with volume.  The number of high energy states (DOF) is proportional to the area of the box, not the volume, aka UV-IR mixing. 

Holography and the counting of degrees of freedom

The CKN bound is equivalent to the holographic principle, which states that the maximum entropy (and hence the number of degrees of freedom) contained in a region scales with its boundary area, not its volume:

$$S_{\max} = \frac{A}{4 L_{\rm Pl}^2}.$$

In this picture, all information about the bulk is encoded on the horizon. For an observer inside, that horizon information is inaccessible and therefore manifests as entropy. The cosmological constant problem can thus be rephrased as follows:

Local QFT dramatically overcounts the degrees of freedom of the vacuum by treating spacetime as a box of independent Planck cells. 

While the holographic principle equates the actual number of degrees of freedom of a bulk to its surface area not its volume. 
 
The holographic principle: all the information-energy about the volume is printed on the horizon of the can. Inside the can, the horizon information-energy is unavailable (can't read it, can't use it), so is, by definition, entropy $S_{dS}$   

De Sitter space and the cosmic event horizon

Our universe, both during inflation and in its current accelerated phase, is well approximated by a quasi–de Sitter spacetime with Hubble parameter

$$H_\Lambda = \sqrt{\frac{\Lambda}{3}},$$

and a cosmic event horizon of radius

$$\ell_\Lambda = \frac{1}{H_\Lambda}.$$

This horizon is not merely a causal boundary—it is a thermodynamic object with well-defined temperature and entropy:

$$T_{dS} = \frac{\hbar H_\Lambda}{2\pi k_B}, \qquad S_{dS} = \frac{A_H}{4 L_{\rm Pl}^2}.$$

Horizon energy and thermodynamic consistency

Using quasilocal energy methods (Brown–York), the total energy associated with the de Sitter horizon satisfies the horizon entanglement identity

$$E_0 = 2 k_B T_{dS} S_{dS}.$$

The gravitating energy inside the horizon (the Misner–Sharp mass) is half of this total:

$$E_H = \frac{1}{2} E_0.$$

Dividing this bulk energy by the horizon volume ($V = \frac{4}{3}\pi \ell_\Lambda^3$), one obtains

$$\rho_\Lambda c^2 = \frac{E_H}{V} = \frac{3 c^4}{8\pi G \ell_\Lambda^2}.$$

Using $\ell_\Lambda^{-2} = \Lambda/3$, this becomes

$$\rho_\Lambda = \frac{\Lambda c^2}{8\pi G},$$

which is precisely the observed dark energy density.

No Planck-scale vacuum density ever appears.

What really resolves the cosmological constant problem

The CCP is not solved by cancelling enormous numbers. It is resolved by recognizing that:

  • Vacuum energy is not a bulk sum over Planck cells.
  • Gravity enforces a holographic bound on degrees of freedom.
  • De Sitter space is a thermodynamic equilibrium state with finite entropy.
  • The cosmological constant is a state variable, not a dynamical field.

In equilibrium, the horizon entropy is extremized, implying a constant $\Lambda$. Away from exact de Sitter (as in inflation or today's matter–dark energy mixture), small departures from equilibrium are allowed, leading to only tiny, observationally constrained variations.

Emergent spacetime

Finally, holography suggests that spacetime itself is emergent : one may associate roughly one degree of freedom with each Planck area on the cosmic horizon. Bulk geometry, dynamics, and even gravity arise collectively from these boundary degrees of freedom.

In this sense, the cosmological constant does not introduce a new fundamental energy scale. Instead, it reflects the thermodynamic equilibrium of spacetime itself.

Dark energy can be exactly constant in value while being emergent in origin.



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