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 (from QFT) and the actual observed dark energy density: $\rho_{Planck}/ \rho_{\Lambda}$. 

$$\begin{equation} \rho_{\Lambda} = \frac{\Lambda c^2}{8 \pi G} \ \end{equation}$$ $$\begin{equation} \rho_{Planck} = \frac{c^5}{\hbar G^2} \end{equation}$$   The QFT result comes from quantising a 'particle in a box'. However, in 1998 the CKN bound was proposed. CKN realised that if you put particles in a box and heat them, you can only increase their energy so much before the box collapses into a black hole. The number of high energy states is proportional to the area of the box, not the volume, aka UV-IR mixing. This is also 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. 

 

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

That is, 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 $\rho_{Planck}/ \rho_{\Lambda}$. Now, the holographic dark energy model by Li used the CNK bound as a starting point. As surface area of a sphere is:

So, with $r=l_{\Lambda}=2l_{Planck}$ 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! Now, the mass-energy of the cosmic event horizon $m_{CEH}$ divided by the spherical volume is: $$\begin{equation} \rho_{\Lambda} \ c^2 = \frac{ 3m_{CEH} \ c^2}{4\pi l_{\Lambda}^3 }    = \frac{c^4 \Lambda} {8 \pi G} \end{equation}$$

We see that the CCP is overcome! A key outcome of the holographic dark energy conjecture, via the CKN bound.       

 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.