Getting to the Observable Universe

Roll up your sleeves. Some junior high school math with lots of powers of ten is coming up.

After all, honestly, it’s not as if I’d have the good sense to leave well enough alone. In the previous post, we discussed the current size of the “observable universe” (10 to the 28th power centimeters), and cosmology models that depend on “inflation” to explain how we got so big so fast. If, that is, you accept that the 14 billion year age of the universe is “so fast”.

But light travels pretty fast. If the universe only expanded at the speed of light, and (almost by definition, you’d think) has done so ever since the Big Bang 14 billion years ago, that would make a pretty big sphere today. Do we really need “inflation” to account for the 10^28 cm size of today’s “observable universe”?

We only need to know the speed of light and number of years this has been going on, and compare this to the known size of the universe, to figure this out – and we need to do it all in centimeters. Why centimeters? Ask the scientists.

If you ever calculated long-wire radio antenna lengths, you still remember that the speed of light is about 3 million meters per second, so we don’t have to convert from 186,284 miles per hour. That’s 3×10^8 cm per second. We need to convert that to centimeters per year – I make that some 9.4608 x 10^13 cm/yr.

And we need a distance for this over 14 billion years, say (9.5 x 10^13 cm/yr) x (14 x 10^9) yr equals around 1.3 x 10^15 centimeters. That would be the radius of a light sphere originating at the time and point of the Big Bang. Double that to 2.6×10^15 cm for the size of the observable universe.

Heck, I don’t mind showing my work. It’s probably wrong. I did it all longhand, errors and all, while watching Cash Cab on TV. If we have the right power of 10, we’re close enough.

So if the observable universe is 10^28 cm in size, and light alone could have sped out into a sphere roughly 10^15 in size, how did the universe outdistance the speed of light by so much? The difference is a respectable 10^13 centimeters.

After thinking about this article and waking up in the morning, that’s not a subtractive difference, but a factor in powers of ten. The universe we could account for by the speed of light alone, as huge as that sphere is, is a tiny mote compared to what cosmologists call the “observable universe”.

Let’s say we want to compare 100,000 (10^5) to 100 (10^2). While the difference is 99,900 (almost 10^5), normally we would just say that 10^5 is 1,000 times as big as 10^2, or 10^3 times as big. We subtract exponents to get the multiplier.

So the observable universe is about 10^13 times as big as what we could account for by expansion at the speed of light alone. That’s inflation!

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