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. 

 

(Lambda, the symbol used for the cosmological constant) 

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.  

 

The de Sitter horizon

A future de Sitter universe has a cosmic event horizon (CEH):

$
l_\Lambda = \sqrt{\frac{3}{\Lambda}} \sim 16 , \text{Gly}.
$

• The associated Gibbons-Hawking temperature is:

$
T_{dS} = \frac{\hbar H}{2 \pi k_B}, \quad H = \sqrt{\frac{\Lambda}{3}}.
$

• The corresponding minimum acceleration:

$
a_{\rm min} = c H.
$ Yes, this is similar to the MOND acceleration, but don't read too much into that.  

---

CEH energy scale

The energy enclosed in the de Sitter horizon (using the Misner-Sharp energy in spherical symmetry) is approximately:

$
E_{\Lambda} = \frac{c^4}{2G} l_\Lambda.
$

• This comes from the analogy with the Schwarzschild radius: $(R_s = 2 G M / c^2 \implies M = c^2 R_s / (2G))$.

• Here $(R_s \to l_\Lambda)$ gives the “mass-energy” associated with the CEH.

So numerically:

$
E_\Lambda = \frac{c^4}{2 G}  l_\Lambda.
$

---

Cosmic maximum power

If we imagine emission at the fastest rate allowed (light crossing time across the CEH):

$
t_\Lambda = \frac{l_\Lambda}{c}.
$

Then the average power is:

$
P_{\rm max}^{\rm CEH} = \frac{E_\Lambda}{t_\Lambda} = \frac{\frac{c^4}{2 G} l_\Lambda}{l_\Lambda / c} = \frac{c^5}{2 G}.
$

It’s exactly the same as the black-hole horizon maximum power! The maximum luminosity allowed by causality in GR is set by horizon formation, whether it’s a black hole or the cosmic event horizon.

Now consider:

• Horizon entropy: $S = \pi k_B c^3 R^2 / (G \hbar)$
• de Sitter temperature: $T_{\rm dS} = \hbar c / (2 \pi k_B R)$
• Maximum entropy production: $\dot S_{\rm max} = S / (R/c) = \pi k_B c^4 R / (G \hbar)$
• Maximum classical power:

$
\boxed{P_{\rm max} = T_{\rm dS} \dot S_{\rm max} = \frac{c^5}{2 G}}
$

 

For a black hole, the expression becomes ${P_{\rm max} = 2T_{\rm BH} \dot S_{\rm max} = \frac{c^5}{2 G}}$,  because $(T_{\rm dS} = 2 T_{\rm BH})$. Or, put another way, the surface gravity of a BH is $\kappa=c^2/2R$ while for dS it is $a=c^2/R$. 

 

A BH horizon processes entropy (and therefore information) twice as fast as a same-size CEH, but at half the temperature. The two horizons are thermodynamically dual in the sense that their $T$ and $\dot{S}$ are exchanged while $P$ is invariant. This could be relevant to complementarity arguments in de Sitter holography.  
 

Actually, this $P_{\rm max}$ expression is the rate form of Jacobson's (1995) identity. Which implies that maximum power is not a consequence of general relativity. It is the thermodynamic constraint from which GR itself emerges. This is what Schiller (via maximum force) has claimed for quite some time.  

 

We can also see that $\dot{S} \propto P^{1/2}$. The CEH is one-dimensional in exactly the same sense as a black hole, as per Bekenstein and Mayo 2001, Black holes are one-dimensional.