# What conceivable connection is there between the cosmological constant and radius of the Universe?

You probably noticed that the radius of the (observable) Universe in Planck lengths $R$ (about $10^{60}$) and the cosmological constant $\Lambda$ (about $10^{-120}$) in natural units obey approximately:

$$\Lambda R^2 \approx1$$

Suggesting that the amount of dark energy is proportional to the surface area of the cosmological horizon.

In Hawking's black hole theory the entropy and temperature of a black hole is proportional to its surface area.

This suggests that dark energy should be inversely proportional to the information content of the Universe or proportional to the temperature of the cosmological horizon by the Unruh effect (or seen from the "outside"?)

(Also, it is known that a black hole at a certain size will increase exponentially, as if it has it's own "dark energy" because it is colder than the thermal equilibrium of outer space. Could there be a connection here between how black holes expand and the Universe expands?)

And yet.... the cosmological constant is supposed to be constant.

So is the relationship merely a coincidence? Are there any current theories that an account for this relationship?

• Your value for $R$ is the radius of the observable universe, at the current time. The universe may be much bigger than the observable universe; in fact, current curvature measurements point to it probably being infinite. The radius of the observable universe does not necessarily correspond to the cosmological horizon, especially not anything like an event horizon of a black hole. – probably_someone Jun 21 '18 at 16:08
• Well depends what you mean by "current time". If I take the temporal slice equivalent to the past light cone it has a finite size. If I take the slice at current "cosmological" time it is infinite but mostly unobservable by ourselves until the future when the light reaches us. – zooby Jun 21 '18 at 16:44
• You are right the observable horizon, essentially looking back towards the big bang is not like a black hole. More like a black hole turned inside out! – zooby Jun 21 '18 at 16:46
• It is not clear what reason except pure numerology you have to believe that the cosmological constant and the radius of the observable universe are connected, i.e. as written, the reasoning in this question about the cosmological constant being not constant is rather reminiscient of England drifting out to sea. – ACuriousMind Jun 21 '18 at 16:47
• One could compare the hawking radiation of a black hole with the cosmic background radiation. Both decrease as the surface area (of black hole or cosmic horizon) gets bigger. – zooby Jun 21 '18 at 16:48