Geometry, Omega, and the Universe

FAS Astronomers Blog, Volume 30, Number 12.

Way back when we were in high school, many of us studied geometry. We learned about triangles, rectangles, and parallel lines. We found that the sum of the angles in a triangle equals 180^o. What we didn’t know then is that this geometry was first described by Euclid in The Elements around 300 BC. Euclid’s geometry, referred to as Euclidian geometry, is applicable to a flat surface or space.

Euclidian geometry is built from many “propositions” (mathematical theorems). They were derived from five postulates and five common notions, which were statements about the world (geometry) that are not proven but assumed to be true. Euclid wasn’t quite sure about one of them, the fifth postulate, which essentially says that only one line can be drawn through a point parallel to a given line. Euclid tried to prove this from his other nine but couldn’t. Many years later, mathematicians discovered that Euclid’s 5^th postulate wasn’t always true. In fact, if you change it, you can create different geometries other that the one described by Euclid.

Eventually, Carl Frederic Gauss, Wolfgang & Johan Bolyai, and Nikolai Lobatchevsky developed Hyperbolic Geometry in the early 19^th century. An interesting consequence of this geometry is that the sum of the angles of a triangle are less than 180^o and an infinite number of lines can be drawn through a point parallel to a given line. Hyperbolic Geometry is often represented by a Saddle with negative curvature.

Gauss’ student Georg Bernhard Riemann was next. He described Elliptic Geometry in 1854 with the publication On the Hypotheses Which Lie at the Foundation of Geometry. Riemann introduced a geometry where the sum of the angles of a triangle are greater than 180^o and there are no parallel lines. Elliptic Geometry is represented by a sphere with positive curvature.

The surface of the Earth exhibits an Elliptic Geometry. The shortest distance between any two locations is described by a great circle (geodesic) that seems to curve when drawn on a flat surface. This is why airplanes appear to fly across an arc that bends to the north when flying between North American and Europe (see Mapping the World).

Today, most astronomers believe that the universe began with something called the Big Bang and it has been expanding ever since. For a timeline of the universe, see The History of the Universe. The rate of expansion is captured in Hubble’s constant, H₀. For more on Hubble’s constant, see Cosmic Distances, Stellar Brightness, and The Hubble Constant.

The large-scale structure of the universe is described by Einstein’s theory of General Relativity. In fact, three astronomers (de Sitter, Friedmann, and Lemaire) used Einstein’s theory to build the first models of an expanding universe. More about them in another article.

Einstein proposed something called the cosmological principle, which he thought applied when viewing the universe as a whole. This principle is an extension of the Copernican principle, which states that we’re not in a special place. The cosmological principle says that the universe, on a large scale, is both homogeneous and isotropic. Homogeneous says that it is the same everywhere and isotropic says that it looks the same in all directions. Astronomers soon realized that a homogeneous and isotropic universe could be described by the three geometries discussed above.

The universe has stuff in it (mass). It also has a volume. Putting the two together, the universe has a density (r), which is the stuff (mass) per unit volume. It turns out that the density of the universe becomes an important determination of its geometric structure. Alexander Friedmann took Einstein’s equations of General Relativity and develop a relationship between the universe’s density and curvature (see below).

The Big Bang explains the beginning of the universe. But, what about the end of the universe? Astronomers have thought about and debated this for years. In general, there are three possibilities, each associated with a different geometry and the universe’s density relative to “the critical density” required for a flat universe.

To make things easier, astronomers focus on a number called omega (Ω), which is the ratio of the universe’s density to the critical density.

• If Ω = 1, we have a flat universe, and it will slowly continue to expand.
• If Ω < 1, the universe is hyperbolic, and it will continue to expand resulting in a “Deep Freeze”.
• If Ω > 1, the universe is elliptic, and it will eventually collapse in on itself in a “Big Crunch”.

For many years, the standard model of the universe was described by an Einstein-de Sitter Universe, which contains mass and is spatially flat. Unfortunately, when astronomers measure the density of ordinary matter, they came up short. There simply wasn’t enough visible stuff to achieve a flat universe. Then astronomers discovered that ordinary visible matter isn’t all there is. First, Dark Matter, which seems to hold galaxies together, was discovered. This was followed by Dark Energy, which is pushing the acceleration of the universe’s expansion.

Today, the standard model of a universe that began with a Big Bang and has both Dark Matter and Dark Energy is described by the Lambda Cold Dark Matter (ΛCDM) model. This model is based on something called the Friedmann–Lemaitre–Robertson–Walker metric and describes a universe that is homogeneous (the same everywhere) and isotropic (looks same in all directions) but could have a curvature of zero, positive, or negative.

Astronomers now consider Ω to be composed of four components, Ω_B, Ω_D, Ω_R and Ω_L, one each for ordinary matter, dark matter, radiation, and dark energy. Interestingly, different components dominated the universe at different points in time. Just after the Big Bang, radiation dominated the universe. However, as the universe cooled, matter (both ordinary and dark) came to dominate. Then as the universe continued to expand, the density of matter became less and less. After 8 or 9 billion years, dark energy took over and now drives the acceleration of the universe’s expansion.

Although the ΛCMD model allows for all three geometries, when you measure and add the individual components up, Ω comes out remarkably close to 1. Evidence for this comes from the Cosmic Microwave Background (CMB). This is the afterglow from the time, 380,000 years after the Big Bang, when the universe first became transparent, and light could escape and travel through the cosmos. Three spacecraft (COBE, WMAP, and Planck) have measured the CMB with increasing precision. Their results have been consistent with the LCDM model and have supported the view that the universe is flat (with Ω = 1).

Despite all this, astronomers were puzzled by the measurement of Ω close to 1. The standard Big Bang model couldn’t explain why the universe should be flat. Then along came Alan Guth and Andrei Linde. They introduced the concept of cosmic inflation, where shortly after the Big Bang, the universe expanded exponentially (some 10²⁶ times) in a fraction of a second (some 10^-32 seconds). This inflated everything from subatomic size to a more macro size and smoothed out (flattened) the structure of the universe (at least as far as we can see). Therefore, we see a flat cosmos the same way we see a flat horizon although the Earth on a larger scale is curved.

One remaining question is whether the entire universe (beyond what is visible), is flat or curved. The answer is, we just don’t know. We can “see” approximately 46 billion light years in either direction. This gives us a visible universe with a diameter of 92 billion light years. What is beyond this? Again, we don’t know.

As far as the fate of the universe, if dark energy is constant (i.e., the cosmological constant), it will keep expanding (and accelerating) until we reach a deep freeze. However, recently in 2024, astronomers have suggested that dark energy might be weakening, and this might lead to a different conclusion.