Free Origins: Fourteen Billion Years of Cosmic Evolution by Donald Goldsmith, Neil deGrasse Tyson Page B

resides in this third dimension and appears nowhere on the surface that represents all of space.
    Just as all positively curved surfaces include only a finite amount of area, all positively curved spaces contain only a finite amount of volume. A positively curved cosmos has the property that if you journey outward from Earth for a sufficiently long time, you will eventually return to your point of origin, like Magellan circumnavigating our globe. Unlike positively curved spherical surfaces, negatively curved spaces extend to infinity, even though they are not flat. A negatively curved two-dimensional surface resembles the surface of an infinitely large saddle: it curves “upward” in one direction (front to back) and “downward” in another (side to side).
    If the cosmological constant equals zero, we can describe the overall properties of the universe with just two numbers. One of these, called the Hubble constant, measures the rate at which the universe is expanding now. The other measures the curvature of space. During the second half of the twentieth century, almost all cosmologists believed that the cosmological constant was zero, and saw measuring the cosmic expansion rate and the curvature of space as their primary research agenda.
    Both of these numbers can be found from accurate measurements of the speeds at which objects located at different distances are receding from us. The overall trend between distance and velocity—the rate at which the recession velocities of galaxies increase with increasing distance—yields the Hubble constant, whereas small deviations from this general trend, which appear only when we observe objects at the greatest distances from us, will reveal the curvature of space. Whenever astronomers observe objects many billion light-years from the Milky Way, they look so far back in time that they see the cosmos not as it is now but as it was when significantly less time had elapsed since the big bang. Observations of galaxies located 5 billion or more light-years from the Milky Way allow cosmologists to reconstruct a significant part of the history of the expanding universe. In particular, they can see how the rate of expansion has changed with time—the key to determining the curvature of space. This approach works, at least in principle, because the amount of space’s curvature induces subtle differences in the rate at which the universal expansion had changed through past billions of years.
    In practice, astrophysicists remained unable to fulfill this program, because they could not make sufficiently reliable estimates of the distances to galaxy clusters many billion light-years from Earth. They had another arrow in their quiver, however. If they could measure the average density of all the matter in the universe—that is, the average number of grams of material per cubic centimeter of space—they could compare this number with the “critical density,” a value predicted by Einstein’s equations that describe the expanding universe. The critical density specifies the exact density required for a universe with zero curvature of space. If the actual density lies above this value, the universe has positive curvature. In that case, assuming that the cosmological constant equals zero, the cosmos will eventually cease expanding and start contracting. If, however, the actual density exactly equals the critical density, or falls below it, then the universe will expand forever. Exact equality of the actual and critical values of the density occurs in a cosmos with zero curvature, whereas in a negatively curved universe, the actual density is less than the critical density.
    By the mid-1990s, cosmologists knew that even after including all the dark matter they had detected (from its gravitational influence on visible matter), the total density of matter in the universe only came to about one quarter of the critical density. This result hardly seems astounding, although it does imply that the