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Remarks on the cosmological constant and minimal acceleration

As Lineweaver explained, because our Universe is expanding at an accelerating rate, our Universe has a cosmic event horizon (CEH).  

Its an event horizon Jim, but not as we know it...Image credit: DALL.E2 by SR Anderson

Events beyond the CEH will never be observed. The CEH is also the source of de Sitter radiation, which has a specific temperature $T_{dS}$. It is the minimum possible temperature of the Universe, and, it is not absolute zero (zero Kelvins). Numerically $T_{dS} \approx 2.4 \times 10^{-30}K$, the universal minimum (black body) `absolute cold' local temperature of the future dS state. As a comparison, in 2020 the NASA Cold Atom lab was able to cool an atom to a record low $\sim 2 \times 10^{-7}K$. 

Now, in any theory one may think of temperature as an energy, and from the semi-classical Unruh relationship, temperature $\sim$ acceleration. Therefore:  

\begin{equation}
\notag
E_{dS} = T_{ds} \ k_B= \frac {\hslash a}{2\pi c}
\end{equation}

What we get from Unruh is $a=cH$, the universal background (local) minimum acceleration. $H$ is the future Hubble constant (as you probably know, this constant is not actually...constant). You could also express this as saying that our present Universe has a de Sitter attractor in our infinite far future.

Gibbons taught us that a theory with a minimum length should have a maximal acceleration and a maximal temperature. Now, we all know that in classical General Relativity (GR) there is no such thing as a minimum length. However, in most approaches to quantum gravity, which includes semi-classical approximations such as Hawking's black hole temperature and the Unruh relationship, there is such a beast. 

Here of course, we are talking about the local minimum universal acceleration, which implies the existence of a maximum length scale, being the de Sitter characteristic length $l_{\Lambda}$ the future cosmic event horizon radius. Numerically $l_{\Lambda} \sim 16 \ Gly$. You can also write $l_{\Lambda} =\sqrt{ {3}/{\Lambda}}$, showing that the cosmological constant is the only parameter in dS space. 

 

(Lambda, the symbol used for the cosmological constant)

 
 

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