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The cosmological constant and the minimum length-scale

In a previous post, we showed how, if the ultimate fate of our Universe is space empty of matter...but not quite....of energy (a de Sitter space), then this future cosmic event horizon (CEH) radius of our current, quantum, Universe set a natural maximum length-scale. Amazingly, the future CEH radius also defines the cosmological constant $\Lambda$.

  

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 $E_H$

What about a Universal minimum length-scale? Might that also be related to a de Sitter space?

For a long time, it has been observed that the evolution of our Universe can be considered as two asymptotic de Sitter epochs connected by a transition phase parameterized by the brief moment of matter-radiation equality.  We are presently living in the second epoch - our accelerating Universe is quasi-de Sitter. 

Perhaps then, the most natural state of our Universe is dS space, as Sean Carroll posited. The past dS state is during and up to the end of inflation. Although then, by logical extension (i.e. we exist!) it also implies that dS space is highly unstable, giving common ground with Swampland, i.e. in string theory it is basically impossible to construct a meta-stable de Sitter theory. 

This implies that the Universe before inflation would be the actual initial past boundary condition, with the dS-like start of inflation being a later phase change. A key takeaway here is that during and up during and up to the end of inflation, the proper distance $l_{\Lambda}$ to the CEH stays constant (i.e. a dS-like state), while the proper distance between two points increases exponentially.  I know, this is a little difficult to get your head around. What it means is that inflation is not superluminal expansion.

As Lineweaver explains, during inflation (a period of around 60 e-folds) all the energy is in the inflaton which has very few degrees of freedom and low entropy. Inflation ends with a period of reheating (between 2-3 e-folds), during which the inflation's energy is transferred into a relativistic fluid. This is also known as the Hot Big Bang. After reheating, the CEH entropy and the CEH proper radius increases as $l_{\Lambda} \propto a$ , where $a$ the cosmic scale factor (refer Figure below, also from Lineweaver). As this is a power law function, vacuum displays scale invariance. 

Alright then, but what about the minimal length-scale? Well, there is an argument (derived from the holographic principle), that the entire universe must be contained within the past horizon of a so-called eternal observer. The postulate then; the entropy of the CEH just before the start of inflation is $4\pi$. We can write the rest energy of the CEH $E_H$ at the start of inflation as:

 \begin{equation}
\notag
E_{H}=m_{CEH} \ c^2= \frac {c^4}{2G} l_{\Lambda}
\end{equation}

With $S_{dS}=4\pi$ we get $E_H=E_{Planck}$ the Planck energy, and we also get   $l_{\Lambda}=2l_{Planck}$. OK, so we now have a length-scale $l_{\Lambda}=2l_{Planck}$  and it is appropriately tiny. However, this is not quite all. 

Boehmer and Harko proved that in classical GR, in the presence of a positive cosmological constant, there is a minimum (local) mass $m_{GR}$. If we use our minimum length-scale in their equation, we find:
\begin{equation}
\notag
m_{GR} \geqslant \frac{\Lambda_{I} c^2}{12G} {\left(2L_{Planck}\right)}^3 \geqslant \frac{m_{Planck}}{2}
\end{equation}

We can also see that in dS space, $2m_{GR}=m_{CEH}$. If we also think back to a connection with a previous post, here, we know the energy of reheating must have escaped its initial self-gravity, so a black hole horizon cannot have been present. Writing the Schwarzschild radius equation:
\begin{equation}
\notag
R \geqslant \frac{2GM}{c^2}
\end{equation} With $M=m_{Planck}/2$, we get $R=L_{Planck}$. Which implies the semi-classical minimum length, in the presence of a positive cosmological constant, is the Planck length, which we kinda expected. 

Now that we have confirmed our semi-classical minimum length scale as the Planck Length , we can also consider what the maximum local Universal acceleration $a_I$  is. As we now know, in dS space, $H=c/l_{\Lambda}$ so:
\begin{equation}
\notag
a_{I}=c \ H_I=\frac{m_{GR} \ c^3}{\hslash}=\frac{a_{Planck}}{2}
\end{equation}

This maximum local acceleration $a_I$ is the same form as the critical acceleration of an electron subject to the Schwinger field from QED.  

 


 


 

 

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