I'd like to clarify something real quick. In my last post, I talked about the density "becoming infinite" at ##t=0##. But what's that mean exactly?
If we look at a very simple example, ##y=\frac{1}{x}##, then, like what I said in my previous post, there is no answer when ##x = 0##. It's undefined. When we say that the density of the universe at a point "is infinite", what we mean is that the density goes to infinity in a finite time. What's this mean exactly?
Well, let's look at probably the simplest example ever, ##y = x##. This formula tells us that ##y## increases linearly as we increase ##x##. As ##x## goes to infinity, so does ##y##. In other words, ##y## increases without limit as ##x## does.
This is different than something like ##y=\frac{1}{5-x}##. Here we have a problem. As ##x## approaches 5, ##y## increases without limit. The closer ##x## is to 5, the larger ##y## becomes. We can say that ##y## goes to infinity in a finite time. In this example, ##x## does not need to increase without limit for ##y## to do so. But remember that when ##x=5##, ##y## is undefined, not infinity. "Going to infinity" is a process. It's a description of the behavior of the function. It's not a number.
This last example is what happens as we look backwards in time. The density of the universe goes to infinity even though ##t## does not. In other words, the density of the universe increases without limit as ##t## approaches a single value.
That's my understanding at least. As always, someone please correct me if I'm wrong.
