Model of the Universe: How Average and Critical Density Help


By Joshua N. WinnPrinceton University

To figure out what’s going to happen to our universe, we need to measure the density, and compare it to the critical density. The current value of the critical density is 3 times H-naught-squared over 8pi G, where H-naught is the Hubble constant, 70 kilometers per second per megaparsec. Plugging that in gives a critical density of 9 times 10 to the minus-30 grams per cubic centimeter.

A panoramic view of the entire sky
Instead of comparing the average and critical density, we can use general relativity so we can model the universe more correctly. (Image: T. H. Jarrett, J. Carpenter, & R. Hurt/Public domain)

The Measurement of the Average Density

A more interesting way to express that is 5 and a half proton masses per cubic meter. That seems like a pretty low bar for the universe to jump over, to achieve the critical density. Surely, the actual universe is denser than that! The air in this room has a density of 10 to the 27 proton masses per cubic meter. And what about all that dark matter?

But we need to be careful. We need to remember that the universe is gigantic, and most of it is empty space. Think of those vast expanses between the stars within a galaxy, and the voids of nothingness in that “cosmic web” of galaxies. To measure the average density, we need to assess a representative volume of the universe, large enough for entire galaxies to be like specks of dust.

When astronomers did that, throughout the 1980s and 90s, they found the universe does have an average density on the same order of a few proton masses per cubic meter. Even the dark matter, it turns out, is very dilute. This remarkable result suggested the universe is in that perfectly balanced state, with zero total energy. Most cosmologists found that possibility to be compelling. Zero seems like the most natural possible value for the total energy. 

Average Density Is Not Equal to Critical Density 

But the funny thing is, as the measurements got better, a density equal to the critical density was ruled out. The actual average density of matter is only 30% of the critical density. This rubbed a lot of people the wrong way. Why should the density be of the same order of magnitude as the critical density, but not quite equal to it?

Many theorists suspected the density really is equal to the critical density, but the measurements were off. Maybe, for some reason, astronomers were still missing a lot of the dark matter. Let’s see where that logic leads. We’ll solve the Friedmann equation and find a of t, for the special case of k equals zero.

In that case, the equation says one over a da/dt the quantity squared equals 8pi G over 3 times rho. Both a and rho are functions of time. But, they’re linked by the fact that rho is mass over volume, and since the total mass isn’t changing, rho must be proportional to one over a-cubed. 

That implies one over a da/dt-squared is proportional to one over a-cubed, or equivalently, a times da/dt -squared is a constant. From there, we take the square root, and then integrate, to find that a to the 3/2 power is proportional to time, or, a is proportional to t to the 2/3.

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

The Scale Factor Grows as Time Goes By

We’ve just learned that the cosmological scale factor grows with time, but not at a constant rate. Expansion at a constant rate would imply a is proportional to t. Gravity decelerates the expansion, making it go like t to the 2/3 instead.

Since we’ve agreed to use units such that a equals one at the current time t-naught, we can write a as t over t-naught to the 2/3 power. And we can calculate the value of t-naught, the current age of the universe, based on the measured value of the Hubble constant. In general, H equals one over a da/dt, which in this case is one over a times the derivative of t over t-naught to the 2/3 power.

When we take that derivative, and evaluate it at time t-naught, the left side is the Hubble constant, H-naught, and the right side is 2 over 3t-naught. That means t-naught equals 2 over 3H-naught. Plugging in 70 kilometers per second per megaparsecs for H-naught, the age of the universe comes out to be 9.3 billion years.

A chart of a versus t shows that a starts at time zero, rises, crosses through unity at a time of 9.3 billion years, and keeps growing as t to the 2/3, forever. So, we did it! We figured out the history of the universe.

The Sun
Our Sun is just 5 billion years old. (Image: NASA/Public domain)

…and the Real Problem Appears

Actually, not quite. There’s a problem. The Sun may be only 5 billion years old, but some other stars in our galaxy appear to be 13 billion years old. In particular, there are a few globular clusters in which stars of the same mass as the Sun have already started evolving into giants. That takes more than 10 billion years. How could the stars have had time to do that? How could they be older than the entire universe?

You might think that the issues we’ve just discussed—the density not quite equaling the critical density, and getting the wrong age for the universe—are both artifacts of an oversimplified model. And it’s true, they are. To model the universe correctly, we need to use general relativity, and the relativistic version of the Friedmann equation solves these problems. But it solves them in a very surprising way. More than that, in a shocking and disturbing way.

Common Questions about Comparing Average and Critical Density

Q: How much is the average density of the universe?

The universe has an average density in the same order of a few proton masses per cubic meter. This indicates that our universe is in a perfectly balanced state.

Q: How is the universe’s average density and critical density related?

Once many theorists suspected that these two are equal to each other, but as the measurements improved, the theory was ruled out. The fact is, the actual average density of matter is only 30% of the critical density, and of the same order of magnitude as the critical density to be more accurate.

Q: What do we need to do to model the universe more correctly?

We need to put aside the density and the critical density as they are both artefacts of an oversimplified model. Instead, we should use general relativity, as well as the relativistic version of the Friedmann equation, although the latter solves these problems in a shocking way.

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