We also showed a diagram similar to Figure 1 below. Except here, we are once again thinking about the future dS state.
Figure 1. For an observer at O inside the cosmic event horizon (CEH) with radius $l_{\Lambda}$, the universe can be divided into two sub-vacuums, $(A)$ inside the CEH, and $(B)$, outside. The horizon surface $\Sigma$ has entanglement entropy $S_{dS}$ and rest energy $E_H$
Now, a comoving volume partition of the
Universe can be treated as a closed system for
which $dS \geq0$. The maximum entropy of a closed system, in this case (Figure 2)
with $L=2 \pi l_{\Lambda}$, the circumference of a circle with radius $l_{\Lambda}$, is obtained when all the energy $E_{H}$
within the system is worked, i.e. degraded into the smallest bits
possible. All energy is converted into minimal energy dark photons with wavelengths as large as the system $L$. From the Compton wavelength relation, the minimum quanta of energy $E_{0}$ of this system is then:
\begin{equation}
\notag
E_{0}=\frac{hc}{\lambda_{max}}=\frac{\hslash c}{l_{\Lambda}}=m_{0}c^2
\end{equation}
A long time ago, Wesson conjectured that in a Universe with a positive cosmological constant, there must be a quantum-scale rest mass $m_p$. In fact, it is fairly simple to show that $m_p=m_0$ (Wesson used the Planck constant, as he was only utilizing dimensional analysis).
Now, quantum field theory can be formulated in terms of harmonic oscillators. So, if we consider our Figure 1
system as a quantum harmonic oscillator, with $E_0 = m_0 c^2$ as the ground state (zero-point)
energy and $\omega_{0}$ being the ground state angular frequency.
\begin{equation}
\notag
E_{0}=\hslash \frac{\omega_{0}}{2}
\end{equation} Now, the de Broglie relation is then:
\begin{equation}
\notag
E_{0}=\hslash \omega_{B}
\end{equation} As $\omega_{0}=2\omega_{B}$ we might consider $\omega_{0}$ as the zitter frequency of vacuum. Things get even more interesting when we realise that $\omega_{B} = c/l_{\Lambda} =H$ implying that the Hubble Constant can be considered as a `de Broglie' frequency of vacuum; or equivalently, that time is a geometric property of space.
Wait, I hear some of you saying, $H_0$ is usually presented as (km/s/Mpa) rather than ($s^{-1}$), and you are right, but that simply due to cosmologists confusing things, using more convenient units.
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