How many bits of information could you fit in the whole universe?

Let's figure it out!  For starters, let's say we mean the observable universe.  The universe may be infinite in size.  But when we look back all the way to when the hot gas of the early universe first cooled down and became transparent, everything we see fits in a finite-sized ball centered at us.  That ball has by now expanded to be much larger.  This larger present-day ball is called the observable universe

Maybe it should be called the 'once-observed universe', since we can't see what distant galaxies are doing now.  But regardless of what it's called, this ball is about 9 × 10^26 meters in diameter.  How much information could we fit in it?

When you keep trying to stuff more information into some region, eventually you get a black hole.  The information becomes inaccessible, so then it's unknown information, also known as entropy.  The amount of entropy is proportional to the surface area of the black hole.  And this is just 4π times the black hole's radius squared.

(At least this is true if the black hole isn't rotating or charged... and it's sitting in otherwise flat space.   Luckily, while the universe is expanding, space at any moment seems close to flat.)

So let's work out the surface area of the observable universe, by taking 4π times its radius squared.  We get 9.5 × 10^54 square meters. 

How much information fits onto this area?  It turns out that for a black hole, each nat of information takes an area of 4 times the Planck length squared.  A nat is a bit divided by the natural logarithm of 2.  Computer scientists like base 2, but black holes seem to like base e. 

4 times the Planck length squared is 10^-69 square meters.   So, calculating a bit, we see the number of nats of information we can pack into the observable universe is roughly 10^124.   I hope you check my work!

But I know you prefer bits, so let's divide by the natural logarithm of 2.  We get: the most information we can fit into the observable universe is 1.4 × 10^124 bits.

Of course, doing this would require turning the observable universe into a black hole!  But if we did so, the black hole would have lots of quantum states.  If the entropy of a system is N bits, its number of quantum states is 2^N.  So, this black hole would have 2^(10^124) quantum states.

That's the biggest number I know that has any good physical meaning.  It's  big... but still tiny compared to plenty of numbers I can easily name, like Graham's number.   And next I'll tell you about some some numbers that make Graham's number look tiny.

By the way, don't take this picture very seriously!  It's just cute.  But it comes from a good page on black hole entropy:

#bigness #astronomy  
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