Most casual observers would assume that the cosmos is a space that expands into infinity, but the answer is not as simple as gazing into a starry sky and hazarding a measurement. Einstein's theory of general relativity, when paired with estimates of the relative amounts of matter and energy in the cosmos, allows for only one possible solution -the universe is infinite.
General relativity requires that the universe remain the same throughout (homogeneity) and appear the same in all directions (isotropy). Therefore, the shape of the universe is the result of the push and pull of gravity and dark energy. This may sound familiar. The same characteristics determine the universe's three possible fates: the Big Crunch, the Big Rip, and the Big Chill.
Just as a universe with an energy density less than its gravitational pull will eventually collapse in on itself (the Big Crunch scenario), the same gravity will overcome dark energy to mold the universe into a sphere. A spherical universe implies that there is a finite amount of space (just as there is a finite amount of surface on a sphere), that two lines appearing parallel will eventually converge (just as lines of longitude on Earth converge as they approach the poles from the equator), and that by traveling far enough we can return to our original position.
Conversely, a universe with an energy density greater than its gravitational pull will exhibit the opposite geometry, better resembling a saddle than a sphere. In such a universe, the overwhelming force of dark energy pulls the universe into an inverted curve where initially parallel lines will gradually diverge. Much like the previous scenario, this universe is still finite.
However, just as cosmologists are fairly confident that the cosmos will not end its life in a Big Rip or Big Crunch, they are equally confident that the geometry of the universe is neither spherical nor saddle- shaped. When both gravity and dark energy reach a balance in their effect on the cosmos, the math implies that the universe will simply stretch out forever as an infinite flat plane. In this universe, two initially parallel lines remain parallel forever, and we will never be able to return to our starting point by traveling any distance in the same direction.
It is worth noting that confidence in this measurement depends on the correctness of Einstein's assumptions about homogeneity and isotropy as well as the accuracy of the current understanding of dark matter. These assumptions underlie the standard models of cosmology, but should they prove even marginally inaccurate, we could be living in a much different universe indeed.
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