This is my second post in the Just Science Week Challenge.

This 2003 paper in Physical Review Letters puts a lower limit on the size of the universe at no smaller than 46.5 billion light years in radius. If the universe is geometrically flat, that is.

In this video I made on the Hubble Deep Field, I mentioned this number and was immediately inundated with questions and comments from people screaming that that number could not possibly be correct. How can the universe be that big if the fastest anything can travel is the speed of light? The universe simply CAN’T be larger than the distance light travels during age of the universe, right?

Wrong.

It is true that the universe is 13.5 billion years old, and it is also true that nothing can travel faster than the speed of light. But it does NOT follow that the size of the universe is simply the distance light traveled in 13.5 billion years. You can’t stop there. Why?

**Because the universe is expanding**, and has been for 13.5 billion years.

Remember yesterday’s post? Everything in the entire universe is flying away from each other at a rate linearly proportional to its distance. That’s Hubble’s Law. The distance that light has to travel over time is continuously getting bigger and you MUST take that into account.

Technorati Tags: cosmology

Remember in my last post, we’ve established that the universe is expanding at roughly the Hubble Constant, and that number is a function of time. It matters WHEN you take your measurement of the redshifts of far away galaxies. Right now, the universe is expanding at about 71 km/sec/Mpc and is accelerating.

A somewhat simpler way to think of the expansion rate of the universe is that it expands at roughly the age of the universe to the 2/3 power: AgeOfUniverse^(2/3). Unfortunately, it’s not simply a plug and chug formula, since the expansion is occurring continuously, you need to apply some calculus. Here’s the formula, but I’ll go through a simple example a little later:

Illustration Credit: Ned Wright

The above integral just takes the ratio of elapsed expansion time to the age of the universe raises it to the 2/3 power and does this over the entire time the expansion is occurring.

What all of this means is that whenever you discuss the size of the universe, you need to apply a scale factor that is relevant TO THE TIME you are interested in. The issue of when is very important because the size of the universe, and the rate at which it was expanding has changed since the universe began.

So, for RIGHT NOW, the size of the universe has expanded to roughly 46.5 billion light years since the Big Bang.

Let’s break down the above integral into some smaller intervals and watch what happens. Let’s use 13 billion years as the age of the universe and let the universe expand for an average of five billion years at three different points in time: 2, 7, and 12 billion years after the Big Bang:

- At age of universe = 2 billion light years: the universe has expanded by a factor of (13/2)^2/3 = 3.48
- At age of universe = 7 billion light years: the universe has expanded by a factor of (13/7)^2/3 = 1.51
- At age of universe = 12 billion light years: the universe has expanded by a factor of (13/12)^2/3 = 1.08

So combining these scale factors over the two intervals above, the universe has expanded to a size of:

(average distance light travels over interval of interest) * (sum of all scale factors).

Plugging in the numbers (we used an elapsed time interval of 5 billion years):

**(5 billion light years) * (3.48+1.51+1.08) = 30.37 billion light years**.

The 5 billion light year number above is the average distance light traveled in 5 billion years so the units are in light years.

Now, this is a discreet example, taking only three points in time, but already we have a number bigger than 13 billion light years. Since the universe is expanding continuously, we actually need to do the integral above and when you do that, the answer approaches 47 or so billion light years.

Actually, this is a little misleading, the number cited in the above paper does a different analysis and I’m doing something a little different that what the authors of that paper did, so I’m trying to make a number fit that was derived using different techniques in my example above and it won’t work out that way. Still, the end results are similar and nothing is really lost by doing that.

But, ignoring the details for a minute, what I’m really trying to do here is show that **the size of the universe isn’t simply the light travel time over the age of the universe**. The expansion of the universe requires that you apply a scale factor as outlined above.

If the math is confusing you, just remember this: that scale factor is important. It accounts for the distance the universe has expanded over the time period you’re interested in. It doesn’t go far enough to say that the size of the universe is simply the distance light travels over the course of the age of the universe. Since the creation of time, everything has gotten farther apart, remember Hubble’s Law: everything is speeding away from everything else, all the time.

So, when the Hubble telescope took the deep field images, it provided us with the deepest views we’ve ever seen into the universe. Those galaxies were approaching the farthest edges of our cosmic home, and they weren’t 13.5 billion light years away, they were much, much farther.

UPDATE: Here’s a good graph I found on Universe-Review.ca, they have kindly given permission for me to post this.

Illustration Credit: Universe-Review.ca

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