The Friedmann Equation and the Possible Destiny of the Universe


By Joshua N. WinnPrinceton University

We’ve seen that a good way to describe the expanding universe is with the cosmological scale factor, a(t). The distance between any pair of galaxies grows in proportion to a. But what equation determines how a varies with time? It’s called the Friedmann equation. Let’s start by assuming that the entire universe is an sphere enormous of uniform density.

Sketched evolution of the scale parameter of the universe for different parameters of the Friedmann equations
The Friedmann equation determines how the distance between galaxies grows over time. (Image: Geek3/Public domain)

What if the Universe Is a Sphere?

It’s only if we look closely that we see the specks of dust spread throughout its volume—those are galaxies. We’ll focus our attention on an arbitrary galaxy, at a distance r from the center. What determines how it moves with time? The only force is gravity, and from Newton’s theorem, the gravitational acceleration is directed inward, with a magnitude G times M-sub-r over r-squared, where M-sub-r is the total mass interior to r.

Does that equation ring a bell? It’s the same one we solved already during our study of planetary motion, and black holes. This case is simpler, though, because there’s just one variable, r, instead of r and theta. The trajectory is purely radial. So, in this model, the motion of the galaxy is the same as that of a spaceship near a black hole, with no more fuel, and no angular momentum. 

Even without solving the equation, we can guess what’s going to happen. If the sphere starts from rest, the galaxy will fall inward. All the interior galaxies will fall, too, so M-sub-r will remain constant as the sphere contracts.

Ultimately Collapsing or Ever-expanding? 

If the initial condition is a Hubble expansion—an expanding sphere, with initial velocity proportional to distance—then gravity will slow it down. Whether a galaxy eventually gets pulled back, or escapes to infinity, depends on how the initial speed compares to the escape velocity.

The Big Crunch
If E is negative, the universe expands for a while, then starts shrinking and ends up collapsing into a black hole. (Image: Ævar Arnfjörð/Public domain)

We can visualize the possible outcomes with the graphical method, as we did for planets and black holes. We start by writing the total energy of the galaxy, E equals 1/2 mv-squared minus GMm over r. We rearrange that to write the kinetic energy as the difference between the total energy and the potential energy. Before, we had another term, L-squared over 2mr-squared, but in this case, L equals zero. 

Then, we sketch the potential energy as a function of r, and make a horizontal line at the level E. The square of the speed at any location is proportional to the difference between the 2 lines: E minus the potential energy.

Suppose E is positive, and a galaxy starts with an initially outward speed. As the galaxy advances with time, the difference between E and the potential energy shrinks, so the galaxy slows down. As r goes to infinity, the potential energy becomes irrelevant, and the speed approaches the square root of 2E over m.

That describes a universe that expands forever, coasting at a constant speed. If the energy is negative, then the lines cross at a certain point. In this case, the galaxy advances to that point, where it stops, turns around, and falls back toward the origin. This would be a universe that expands for a while, then ends up collapsing into a black hole.

This article comes directly from content in the video series Introduction to AstrophysicsWatch it now, on Wondrium.

What If the Total Energy Is Zero?

In between these 2 cases is a special case when the total energy is exactly zero. That’s like a spaceship with an initial speed exactly equal to the escape velocity. It corresponds to a universe that keeps expanding, but at an ever-decreasing rate.

In this model, the fate of the universe depends on its total energy. If we could measure the total energy of the universe, we’d be able to determine its fate. There is a problem, though. On scales of gigaparsecs, we need to describe the universe with general relativity, not classical mechanics. To prepare for general relativity, we’re going to dress up the energy equation in different clothing. First, let’s divide through by little-m, converting everything to units of energy per unit mass. And since E over m is a constant, let’s just call it k.

Now, let’s bring in the scale factor. Instead of r of t, we’ll write a of t times r-naught, where r-naught is an arbitrarily chosen distance scale, say, 100 megaparsecs, the distance from the Milky Way to the Coma cluster. With that, the velocity, dr/dt, becomes da/dt times r-naught. Finally, instead of the enclosed mass, we’ll write the equation in terms of the density of the universe, rho of t. We’ll replace M-sub-r by 4pi over 3 r-cubed times rho.

Universe’s Destiny Depends on Its Density

When we make all those substitutions and then tidy up, by multiplying both sides by 2 and dividing by a r-naught-squared, the end result is the square of one over a da/dt equals 2k over a-squared r-naught-squared plus 8pi G over 3 times rho.

The quantity on the left side, one over a da/dt, might look familiar. It’s the Hubble parameter, H, as we showed a couple of lectures ago. And with that change of notation, voila, we have derived the classical Friedmann equation!

In this new guise, the equation relates the cosmological scale factor, and its time derivative, to the overall density of the universe, at any given time. This allows us to rephrase our statements about the fate of the universe, in terms of its density.

The critical case of zero total energy corresponds to k equals zero. In that case, H-squared equals 8pi G over 3 times rho. Another way to put it is that there’s a critical density of 3H-squared over 8pi G. If the actual density equals the critical density, the universe expands forever at ever-decreasing speed. If the density is higher, the universe collapses. And if it’s lower, the universe ends up coasting at constant speed. So, in this model, destiny depends on density.

Common Questions about the Friedmann Equation and the Possible Destiny of the Universe

Q: What is the Friedmann equation?

The Friedmann equation describes the expanding universe with the cosmological scale factor, a(t), a is the distance between any pair of galaxies that grows in time, and t is the time range.

Q: What are the two possible fates for the universe if E is positive or negative?

If the energy is positive, the universe will expand forever. If the energy is negative, then the cosmos advances to a point, where it stops and turns around, and collapses as a black hole.

Q: On what factor the fate of the universe depends?

The fate of our universe depends on its density. If the actual density of the universe equals its critical density, it expands at an ever-decreasing speed forever. If the density is higher, the universe collapses. And if it’s lower, the universe will ultimately coast at a constant speed.

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