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A cosmological constant solution via the holographic principle

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. 

From observations, the cosmological constant $\Lambda \sim 1.3\times10^{-52} \ m^{-2}$. The so-called $10^{121}$ crisis can be expressed as the ratio of the Planck density and the actual observed dark energy density: $\rho_{Planck}/ \rho_{\Lambda}$. 

\begin{equation}
\rho_{\Lambda} = \frac{\Lambda c^2}{8 \pi G} \
\end{equation}\begin{equation}
\rho_{Planck} = \frac{c^5}{\hbar G^2}
\end{equation} Equivalently, the cosmological constant problem can be expressed as observing that the number of degrees of freedom of dark energy in quantum field theory (QFT) are much too large to explain the observational data. The QFT approach relates bulk degrees of freedom $N_B$ to a sphere volume V.  

 As we know: \begin{equation}
\hbar = \frac{L_{Planck}^2 c^3}{G}
\end{equation} \begin{equation}
\frac{\rho_{Planck}}{\rho_{\Lambda}} = \frac{32 \pi}{3} \ = N_B
\end{equation}

\begin{equation}
N_B = V = \frac{4}{3} \pi r^3
\end{equation} In another previous post we showed how the radius of the CEH before inflation was $l_{\Lambda}=2l_{Planck}$.

\begin{equation}
V = \frac{32 \pi L_{Planck}^3}{3}
\end{equation} Divide this result by the unit volume $v=L_{Planck}^3$ and you have the same result as $\rho_{Planck}/ \rho_{\Lambda}$. However, a long time ago, Li applied the holographic principle, which equates the actual number of degrees of freedom of a bulk to its surface area not its volume, to our Universe. 

\begin{equation}
A = 4 \pi r^2
\end{equation} An application of  UV-IR mixing, this approach provides a solution to the CCP. 

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 $S_{dS}$  Image credit: Stable diffusion AI + SR Anderson

So, we could expect that the actual bulk degrees of freedom (as a dimensionless ratio) are $N_s=16 \pi$, aka the cosmic event horizon surface area before inflation.

 

To show this result, firstly, from the holographic principle, the equipartition mass-energy $m_{CEH}$ of the cosmic event horizon surface is equal to the bulk gravitating energy $m_{bulk}$ so: $N_s=N_B$. 

Our next trick is to connect the cosmic event horizon energy $E_H$ and Hawking's black hole temperature $T_{BH}$  to the degrees of freedom $N_s$ via the equipartition theorem: \begin{equation}
\notag
E_{H}=\frac{1}{2}N_{s}k_{B}T_{BH}
\end{equation}\begin{equation}
\notag
 N_s=4S_{ds}=\frac{A_{H}}{L_{Planck}^2} = 16 \pi
\end{equation} This is the expected energy density ratio, and so, the CCP is overcome; a key outcome of the holographic dark energy conjecture.       

 BH entropy $S_{dS}= \frac{A_{H}}{4L_{Planck}^2}$ Credit: Scholarpedia

We can also associate one degree of freedom with one Planck area at the surface of the past cosmic event horizon, which defines a theory of emergent space-time.

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