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 - Start with three horizon quantities:

• Horizon entropy: $S=\dfrac{\pi k_Bc^3R^2}{G\hbar}$
• de Sitter temperature: $T_{\rm dS}=\dfrac{\hbar c}{2\pi k_BR}$
• Maximum entropy throughput, meaning one horizon's information per radial light-crossing time: $\dot S_{\max}\equiv\dfrac{S}{R/c} =\dfrac{\pi k_Bc^4R}{G\hbar}$

Therefore,

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

For a same-radius black hole, $T_{\rm BH}=\frac12T_{\rm dS}$, because its surface gravity is $\kappa_{BH}=\frac{c^2}{2R}$, while for de Sitter, $\kappa_{dS}=\frac{c^2}{R}$.  The Schwarzschild Smarr relation supplies the missing factor of two: $E_{\rm BH}=2T_{\rm BH}S_{\rm BH}$. Hence,

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

The black hole is half as hot, but both horizons have the same maximum radial-throughput power.  

The underlying relation $P=T\dot S$ is the rate form of the Clausius identity $\delta Q=T dS$ used in Jacobson's thermodynamic derivation of Einstein's equation. Saturating it with the horizon entropy and radial causal-time bounds gives $c^5/(2G)$. Schiller's related maximum-force formulation instead uses the full diameter and obtains $c^5/(4G)$.

Finally, each radial thermal channel obeys

$$P_{\rm ch}\propto T^2, \qquad \dot S_{\rm ch}\propto T \quad\Longrightarrow\quad \dot S_{\rm ch}\propto\sqrt{P_{\rm ch}}.$$

Thus the CEH's radiative entropy flow is one-dimensional in the same channel-theoretic sense identified by Bekenstein and Mayo, 2001  Black holes are one-dimensional.