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Section8.4Final Remarks

The universe appears to be homogeneous and isotropic. The presence of cosmic microwave background radiation is evidence of this: it is coming to us from every direction with more or less constant temperature. This uniformity can be explained by the inflationary universe theory. The theory, pioneered in the 1980s by Alan Guth and others, states that during the first \(10^{-30}\) seconds (or so) after the big bang, the universe expanded at a stupendous rate, causing the universe to appear homogeneous, isotropic, and also flat.

The assumptions of isotropy and homogeneity are remarkably fruitful when one approaches the geometry and topology of the universe from a mathematical point of view. Under these assumptions, three possibilities exist for the geometry of the universe - the three geometries that have been the focus of this text. Each geometry type has possible universe shapes attached to it, and Section 8.1 showcases some of the leading compact candidates.

The mathematical point of view gives us our candidate geometries, but attempts at detecting the geometry from the mathematical theory have proved unsuccessful. For instance, no enormous, cosmic triangle involving parallax has produced an angle sum sufficiently different from \(\pi\) radians to rule out Euclidean geometry.

Adopting a physical point of view, we have another way to approach the geometry of the universe. Einstein's theory of general relativity ties the geometry of the universe to its mass-energy content. If the universe has a high mass-energy content, then the universe will have elliptic geometry. If the universe has a low mass-energy content, then it will be hyperbolic. If it has just a precise amount, called the critical mass density, the universe will be Euclidean. From a naive point of view, this makes it seem highly unlikely that our universe is Euclidean. If our mass-energy content deviates by the mass of just one hydrogen atom from this critical amount, our universe fails to be Euclidean.

A little notation might be helpful. It turns out that from Einstein's field equations, the mass-energy density of the universe, \(\rho\text{,}\) is related to its curvature \(k\) by the following equation, called the Friedmann Equation: \begin{equation*} H^2 = \frac{8\pi G}{3}\rho - \frac{k}{a^2}. \end{equation*}

Here \(G\) is Newton's gravitational constant; \(H\) is the Hubble constant measuring the expansion rate of the universe; \(k = -1,0\text{,}\) or \(1\) is the curvature constant; and \(a\) is a scale factor. In fact, \(a\) and \(H\) are both changing in time, but may be viewed as constant during the present period. Current estimates of \(H\) (see  [17] or  [22]) are in the range of 68 to 70 kilometers per second per megaparsec, where 1 megaparsec is 3,260,000 light-years.

In a Euclidean universe, \(k = 0\) and solving the Friedmann equation for \(\rho\) gives us the critical density \begin{equation*} \rho_{c} = \frac{3H^2}{8\pi G}. \end{equation*}

This critical density is about \(1.7 \times 10^{-29}\) grams per cubic centimeter, and is the precise density required in a Euclidean universe.

We let \(\Omega\) equal the ratio of the actual mass-energy density \(\rho\) of the universe to the critical one \(\rho_c\text{.}\) That is, \begin{equation*} \Omega = \frac{\rho}{\rho_c}. \end{equation*}

Then, if \(\Omega \lt 1\) the universe is hyperbolic; if \(\Omega > 1\) the universe is elliptic; and if \(\Omega = 1\) on the nose then it is Euclidean.

Until the late 1990s all estimates of the mass-energy content of the universe put the value of \(\Omega\) much less than 1, suggesting a hyperbolic universe. In fact, different observational techniques for estimating the total mass-energy content of the universe put the value of \(\Omega\) at about 1/3, contrary to the value of 1 predicted by the inflationary universe model.

But at the dawn of the 21st century, this all changed with detailed measurements of the cosmic microwave background radiation, and the remarkable discovery that the universe is expanding at an accelerated rate.

Careful analysis of the WMAP data on the cosmic background radiation 6  suggests that the universe is flat, or nearly so, in agreement with the theory of inflation. (This analysis is different than the circles in the sky method, which searches for shape.) The five-year estimate from the WMAP data (see [21]) put the value of \(\Omega\) at \begin{equation*} \Omega = 1.0045\pm .013. \end{equation*}

So if the mass-energy density is about 1/3 of what is required to get us to a Euclidean universe, but it appears from the CMB that the universe is Euclidean, or nearly so, some other form of energy must exist. Evidence for this was presented in 1999 (see, for instance,  [18]) in the form of observations of distant exploding stars called Type Ia supernovae. These distant supernovea are fainter than expected for a universe whose expansion rate is slowing down, suggesting that the universe is accelerating its expansion. In 2011, Saul Perlmutter, Brian Schmidt, and Adam Riess won the Nobel Prize in Physics for their work on this discovery.

When Einstein first proposed the 3-sphere as the shape of the universe, his theory predicted that the 3-sphere should be expanding or collapsing. The idea of a static universe appealed to him, and he added a constant into his field equations, called the cosmological constant, whose role was to counteract gravity and prevent the universe from collapsing in on itself. But at roughly the same time, Edwin Hubble, Vesto Slipher and others discovered that galaxies in every direction were receding from us. Moreover, galaxies farther away were receding at a faster rate, implying that the universe is expanding. Einstein withdrew his constant.

But now the constant has new life, as it can represent the repulsive dark energy that seems to be counteracting gravity and driving the accelerated expansion of the universe. So the density parameter in the Friedmann equation can have two components: \(\rho_M\text{,}\) which is the mass-energy density associated with ordinary and dark matter (the mass-energy density that cosmologists have been estimating by observation); and \(\rho_{\Lambda}\text{,}\) which is the dark energy, due to the cosmological constant. In this case, Friedmann's equation becomes \begin{equation*} H^2 = \frac{8\pi G}{3}(\rho_M + \rho_{\Lambda}) - \frac{k}{a^2} \end{equation*} and dividing by \(H^2\) we have \begin{equation*} 1 = \frac{8\pi G}{3H^2}\rho_M + \frac{8\pi G}{3H^2}\rho_{\Lambda} - \frac{k}{(aH)^2}. \end{equation*}

We let \begin{equation*} \Omega_M = \frac{\rho_M}{\rho_c},~~~~~~~ \Omega_{\Lambda} = \frac{\rho_{\Lambda}}{\rho_c},~~~~~~\Omega_k = \frac{-k}{(aH)^2}. \end{equation*}

So the simple equation \begin{equation*} 1 = \Omega_M + \Omega_\Lambda + \Omega_k \end{equation*} fundamentally describes the state of the universe. The inflationary universe model suggests that \(\Omega_k \approx 0\text{,}\) which is supported by recent reports. Nine-year analysis of the WMAP data combined with measurements of the Type Ia Supernaovae (see [22]) suggest \begin{equation*} -0.0066 \lt \Omega_k \lt 0.0011, \end{equation*} with \(\Omega_\Lambda \approx .72\) and \(\Omega_M \approx 0.28\text{.}\) As the universe evolves, the values of the density parameters may change, though the sum will always equal one.

Now is probably as good a time as any to tell you that the fate of the universe is tied to its mass-energy content and the nature of the dark energy, which is tied to the the geometry of the universe, which is tied to the topology of the universe.

If the cosmological constant is zero then the relationship is simple: if \(\Omega > 1\) so that we live in an elliptic 3-manifolds, the universe will eventually begin to fall back on itself, ultimately experiencing a “big crunch”. If \(\Omega = 1\text{,}\) a finite Euclidean universe will have one of 10 possible shapes (6 if we insist on an orientable universe) and its expansion rate will asymptotically approach 0, but it will never begin collapsing. If \(\Omega \lt 1\text{,}\) the universe will continue to expand until everything is so spread out we will experience a “big chill.”

With a nontrivial vacuum energy the situation changes. While the gravitational force due to the usual mass-energy content of the universe tends to slow the expansion of the universe, it seems the dark energy causes it to accelerate. Whether the universe is hyperbolic, elliptic, or Euclidean, if the dark energy wins the tug of war with gravity, the curvature of the universe would approach 0 as it continued to accelerate its expansion, and the density of matter in the universe would approach 0.

The European Space Agency launched the Planck satellite in 2009. In its orbit at a distance of 1.5 million kilometers from the Earth, the Planck satellite has given us improved measurements of the CMB temperature, enabling sharper estimates of cosmological parameters, as well as more refined data on which to run circles-in-the-sky tests. Alas, no circles have been detected. Regarding the curvature of the universe, they conclude in [17], in agreement with the WMAP team, that our universe appears to be flat to a one standard deviation accuracy of \(0.25\%\text{.}\)

In short, the universe appears to be homogeneous, isotropic, nearly flat, and dominated by dark energy. The current estimates for \(\Omega_k\) leave the question of the geometry of the universe open, though just barely. It is still possible that our universe is a hyperbolic or an elliptic manifold, but the curvature would have to be very close to 0. If the universe is a compact, orientable Euclidean manifold, we have six different possibilities for its shape. Since Euclidean manifold volumes aren't fixed by curvature, there is no reason to expect the dimensions of a Euclidean manifold to be close to the radius of the observable universe. But if the size is right, the circles-in-the-sky method would reveal the shape of the universe through matching circles.

Perhaps we will be treated to matching circles some day. Perhaps not. Perhaps the universe is just too big. In any event, pursuing the question of the shape of the universe is a remarkable feat of the human intellect. It is inspiring to think, especially looking up at a clear, star filled night sky, that we might be able to determine the shape of our universe, all without leaving our tiny planet.