I've written quite a bit on big numbers because I find them fascinating. Two of the areas where you find interesting big numbers are astronomy and cosmology. For example the observable universe is thought to be 93 billion light years across (That's \(8.8x10^{26}\) meters.)

Time is also an important measure and it too can be represented by big numbers. For example, we think that the Universe is 13.7 billion years old.

Physicist Don Page gets the award for using the longest time ever used in a proof for his paper on Information Loss in Black Holes and/or Conscious Beings. In the paper he calculates the Poincaré recurrence time for the quantum state of a hypothetical box containing a black hole with an estimated mass of the entire Universe. In laymen's terms, he calculated the amount of time that needs to pass before our Universe arrives back to the exact point it is in right now. Let me explain.

#### Poincaré Recurrence Time

In mathematics, Poincaré recurrence theorem states that certain systems will, after a sufficiently long but finite time, return to the initial state. The theorem assumes that the system is finite.

Poincaré recurrence time is the amount of time it takes to get back into the recurring state.

Since the Universe is believed to be a finite volume, after some very long period of time it would eventually come back around to repeat itself. That means that after the Poincaré recurrence time you would be born again, live the exact same life with every single atom in the exact state that it is now.

Don Page calculated the amount of time before this happens and put it in his paper. The Poincaré recurrence time for our Universe according to Don Page is:

$$ 10^{10^{10^{10^{10^{1.1}}}}} \text{ years} $$

Let that sink in. After that amount of time the Universe's history will repeat itself arbitrarily many times due to properties of statistical mechanics.

There is a great numberphile video that discusses Don's paper:

Poincaré Recurrence Time is a great example of big numbers in action. The size of this number is so great that the Universe would sooner likely suffer from heat death long before it is reborn and then reaches recurrence time.

#### Heat Death of the Universe

One possible ultimate fate of the Universe is called the Heat Death. This is a diminished state where thermodynamic free energy is all used up and therefore nothing can sustain processes that increase entropy. The Universe would reach thermodynamic equilibrium.

The amount of time before the heat death occurs in roughly \( 10^{100} \) years. Don't worry though, long after the so called dark ages it is possible that another Universe could be created by random quantum fluctuations or quantum tunneling. Want to know how long you would have to wait before the new Universe is formed?

$$ 10^{10^{10^{56}}} \text{ years} $$

That is a long time but not nearly as long as the time it would take for the Universe to repeat itself again.

Another interesting feature on Poincaré Recurrence Time is that there is no way to observe it happening. Since the Universe would need to die and then be reborn long before recurrence time, it would be impossible for an observer to capture it happening.

Big numbers are so cool.