Flatness Problem

The Flatness Problem in Cosmology

The universe is very likely flat because it was created flat. A flat universe starts with minimal information, which is fine since our cosmic creation process can create all the information that we have in the universe today.

When I was a first-year graduate student in astrophysics at Harvard University in 1958, I encountered two problems that have remained with me all these years. One was the fundamental problem of information philosophy - “What creates the information structures in the universe?” The other was the flat universe.

At that time, the universe was thought to be positively curved and "closed." A particle traveling in a fixed direction would eventually return to its starting position. Edwin Hubble’s red shifts of distant galaxies showed that they did not have enough kinetic energy to overcome the gravitational potential energy, so they should all eventually collapse back together in a "Big Crunch." Textbooks likened the universe to the surface of an expanding balloon decorated with galaxies (themselves not expanding) moving away from one another.

George Gamow's illustration from his book One Two Three... Infinity

Gamow's balloon popped for me when Walter Baade came to Harvard to describe his work at Mount Wilson during the war years when brownouts made the nighttime California sky dark and his German citizenship prevented him being drafted into war work, like all his America colleagues. So Baade was alone at the telescope, with more observing time and better observing conditions than one individual had ever had.

Baade took many images with long exposures of nearby galaxies and discovered there are two distinct populations of stars. And in each population there was a different kind of Cepheid variable star. The period of the Cepheid’s curve of light variation indicated its absolute brightness, so they could be used as “standard candles” to find the distances to star clusters in the Milky Way and to our spiral galaxy neighbor Andromeda.

Baade realized that the Cepheids being used to calculate the distance to Andromeda were 1.6 magnitudes brighter than the ones used in our galaxy. Baade said that must mean Andromeda is twice as far away as Hubble had thought.

As I listened to Baade, for me the universe went from being positively curved to negatively curved. It jumped right over the flat universe! I was struck that we seemed to be within observational error of being flat. Some day a physicist will find the reason for perfect flatness, I thought.

I used to draw a line with tick marks for powers of ten in density around the critical density ρ_c to show how close we are. Given so many orders of magnitude of possible densities, it seemed improbable that we were just close by accident.

We could increase the density of the universe by thirty powers of ten before it would have the same density as the earth (way too dense!). But on the lighter side, there are an infinite number of powers of ten. We can’t exclude a universe with average density zero, which still allows us to exist, but little else in the distance.

In the long run we are approaching a universe with average density zero. Some say all the non-gravitationally bound systems will slip over our light horizon as the expansion takes more and more of them faster than the velocity of light. At this time, galaxies with a redshift greater than z = 1.8 are already over our light horizon. We will always still see them, but can never exchange signals with them.

But note that we may always be able to see back to the cosmic microwave background, all the same contents of the universe that we see today will always be visible, just extremely red-shifted!

When Alan Guth presented his theory of inflation at Harvard as an explanation of how the universe appears to be flat, I asked him the simple question, "What if the universe has always been flat?" "That's too easy," he replied.

When I wrote to Steven Weinberg about the possibility of a flat universe he replied that flatness might be the simplest solution. Years later in his 2008 book Cosmology he wrote...

[W]e will see later that the total energy density of the present universe is still a fair fraction of the critical density. How is it that after billions of years, ρ is still not very different from ρ_crit? This is sometimes called the flatness problem.
The simplest solution to the flatness problem is just that we are in a spatially flat universe , in which K = 0 and ρ is always precisely equal to ρ_crit. A more popular solution to the flatness problem is provide by the inflationary theories... In these theories K may not vanish. and ρ may not start out close to ρ_crit, but there is an early period of rapid growth in which ρ/ρ_crit rapidly approaches unity.

Cosmology, p.39

The simplest solution to the flatness problem is therefore correct within observational error. There is nothing in general relativity theory that can explain that observational fact, but there is another simple observation that might throw light on the overall or average curvature of the universe, as opposed to the curvature of space near large gravitating objects.

The Universe is Spherically Symmetric

We can start from the simple fact that the observable universe looks more or less the same in all directions.

As far back as 380,000 years after the Big Bang, when free electrons were combining with protons to form hydrogen atoms, light that was being scattered by the previously ionized gas (the last remnant of the original plasma) was now free to travel across the now transparent universe.

Visible light at that time had temperature 5000K (about the same as today's solar surface). It now appears to us as cooled down microwave radiation at 2.7K, as roughly predicted by George Gamow in the 1940's. This cosmic microwave background radiation (CMB) was first discovered by Arno Penzias and Robert Wilson in 1965 as faint noise coming in from all directions.

Now the latest sky surveys (see especially the Sloan Digital Sky Survey) show areas of the sky a bit more populated than others, and the CMB is marked with tiny regions that are slightly cooler than others (which may correspond to closer regions with fewer galaxy clusters), but overall the universe is remarkably uniform.

We can ask what the gravitational force would be on a particle of light or matter traveling through space today. If it's traveling near a large gravitational object, its path will be curved or bent toward that object as Einstein explained. But what if it's traveling in some part of space far from most masses?

If we averaged out the matter in all directions, the universe density would be uniformly ρ_crit in all directions, and the net gravitational force on our test particle would be zero!

We can now see that a spherically symmetric universe would exert no net force on a test particle. It would travel in a straight line. It would experience no spatial curvature!

We conclude that the average curvature of the universe is simply zero, except in the neighborhood of large gravitational masses. When Walter Baade's observations in the 1940's suggested that the universe was not closed and finite as previously thought, the universe could first be seen as open and infinite in all directions.

The Problem of Missing Mass

Given our assumption that the universe is exactly flat, the missing mass problem is that there is not enough observable material so that in Newtonian cosmology the gravitational binding energy can exactly balance the kinetic energy. The visible (luminous mass) accounts for only about 4-5 percent of the needed mass. Studying the rotation curves of galaxies and galaxy clusters reveals an invisible mass (called dark matter) contained inside the galaxies and clusters that amounts to perhaps 6 times the visible matter, which then accounts for about 30 percent of the critical mass density needed to make the universe exactly flat. Current theory accounts for the balance by “dark energy,” an interpretation of the cosmological constant Einstein considered adding to his equations as a pressure to keep it from collapsing. But the missing mass could just be more invisible dark matter between the galaxies and clusters. About three times their dark matter halos around galaxies would do.

This much material can close the universe and explain its flatness. But it would not explain the apparent acceleration of the expansion seen in distant Type 1a supernovae. The acceleration might be an artifact of the assumption the supernovae are perfect “standard candles.” Recent evidence suggests that distant Type 1a supernovae are in a different population than those nearby, something like Baade’s two populations of stars.

It seems a bit extravagant to assume the need for an exotic form of vacuum energy on the basis of observations that could have unknown but significant sources of error. And I am delighted that observations of mean density today are within a factor of three of the critical density ρ_c.

When Baade found the universe was open in the 1950’s, we needed thirty times more matter to get back to a flat universe. Now we need only three times more.

The discovery of more invisible dark matter or the hypothetical dark energy will only move us closer to a universe with zero curvature, with no need for the cosmological constant that Einstein told us he never should have added to his general theory of relativity.

More than ever, the universe is obviously flat! The large-scale average curvature of the universe is zero.

Something from Nothing?

A possible answer to Gottgried Leibniz’s great question, “Why is there something rather than nothing?” might be “the universe is made out of something and the opposite of that something.”

In classical Newtonian mechanics, the question of whether the universe was closed or open was usually framed in terms of motion (kinetic) energy and binding (gravitational potential) energy.

Is the kinetic energy of the receding galaxies less than the gravitational potential energy that is slowing the expansion down? If so, the expansion will slow down, stop, and turn around to a contracting universe. In this case, the universe is closed and bounded (it has a finite volume with edges).

Is the kinetic energy of the receding galaxies greater than the binding gravitational potential energy? Then the universe is open, infinite, and unbounded. It will expand forever.

So is the classical answer to Leibniz' question that the negative gravitational binding energy Is that exact opposite to the positive kinetic motion energy? So that the total universe energy is zero. "Nothing and come from nothing," they say. Is the current universe one way of looking at "nothing?" Sadly, no.

The relativistic Einstein universe is more complicated, but the boundary between open and closed suggests the same flatness problem. Einstein added the cosmological constant to his field equations as a pressure term, a repulsion, to resist the gravitational attraction that he feared would set the universe matter in motion, when he thought observational evidence favored a static universe (the "fixed stars").

The best observational evidence today for mean density puts us within a factor of three of a flat universe, well within the famous "astronomical accuracy" of an order of magnitude (a factor of ten).

What's lacking is a theorist who can explain this without adding ad hoc assumptions like Einstein's cosmological constant or Alan Guth's assumption of early universe inflation.

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