The Big Rip Ripoff?

In the 2009 February issue of Astronomy magazine, by Liz Kruesi, Liz explains that it is possible the universe could rip itself to infinity instead of just simply expanding into the cold wilderness. The usual theory now is that the universe will continue expanding until galaxies, stars, planets, rocks, and even atoms are incredible distances apart, and they will continue to expand forever. This is called the Big Chill. What is proposed in this article is that instead, the universe will continue to expand for a while, and then the expansion will increase, slowly at first, then more rapidly. At some time, maybe a few hundred billion years from now, it will expand so rapidly that it will tear everything apart and send everything to infinity in a finite amount of time, a time called the Big Rip.

This reminds me of another situation. Suppose you have a population that is growing at a rate proportional to the population itself; for example, if it is humans, then the women have a constant rate of childbearing. To see what kind of growth rate results, differential equations can help. Suppose you have an initial population, and that the growth rate is some constant r times the population. If we let t denote the time, and x the population, the equation is:

x' = rx

or dx/dt = rx

The way to solve this is to take this last equation and invert it:

dt/dx = 1/rx

You then integrate both sides with respect to x to get:

t = log(rx) - B

where B is some constant to be determined by initial conditions. Solve for x and you get:

x = Ce^rt

where C = e^B. This is an exponential function. An exponential function grows rapidly, and the more you go out, the more rapidly it grows. Note that x = x¹, as anything to the 1 power is itself again. The number 1 here is a growth factor that in some ways tells how fast the function goes.

But suppose the increase is more than exponential. Suppose instead that we use some number greater than 1 as the power to which we take x. Suppose we take 2 instead. Then we get:

x' = rx²

or dx/dt = rx²

The way to solve this is to take this last equation and invert it:

dt/dx = 1/rx²

This is a power function, and the integral of 1/rx² is -1/rx + D. That is,

t = D - 1/rx

or upon solving for x,

x = 1/r(t-D)

I chose D because it stands for "Doomsday". Now we have t in the denominator, so something really off the wall happens when t rises to become equal to D. That means the denominator becomes 0, with the numerator not 0. Such a division can't be done, but it can be thought of in a way as representing infinity, and indeed the value of x increases without bound as t approaches the dreaded Doomsday D.

Doesn't this sound like the Big Rip scenario? If we let a represent the exponent (which is 1 for the first example and 2 for the second), then it turns out that when a is 1, then the function stays finite forever, but when it is greater than 1, then the function reaches infinity in a finite amount of time. Is there such an a around in cosmology? Yes there is, according to the article. It is a number called the "equation of state" w. If this number is -1, then the universe will always exist. If it is less than -1, the universe will cease to exist eventually. The article associates -1 with the cosmological constant, and less than -1 with something called "phantom energy" - that maybe some dark energy out there somewhere makes w less than -1 and hence the exponent is greater than one, so an infinite blowup in a finite amount of time occurs. It is also interesting that w is the negative of the a I use in these two examples.

Can such a thing as a Big Rip happen? I don't think so, because I believe in the Infinity Principle: you cannot have a far infinity in the real world. This means a mass can't have infinite density, a distance can't be infinite and so forth. You have to allow for "deep" infinity, however, else we can't go anywhere, as Zeno's paradox shows, but an infinity that is way far out there, so to speak, cannot exist in our realm of existence. So I believe that before the Rip can occur, it will stop when quantum effects stop the expansion, leaving the Universe an extremely sparse sea of elementary particles.