What is the fate of the universe? This is clearly one of the fundamental questions of modern cosmology. The answer depends in part on the distribution of mass on very large scales. The gravitational effect of this distributed mass is one of the factors that determine how the universe evolves.
To see how gravity affects the expansion of the universe, it will help to recall gravity’s effects on the motion of projectiles. The fate of a projectile fi red straight up from the surface of the Moon depends on its speed. If the speed is less than the Moon’s escape velocity (2.4 kilometers per second, or km/s), the Moon’s gravity will eventually stop the projectile and pull it back to the Moon’s surface. But if the speed of the projectile is greater than the Moon’s escape velocity, then the projectile will escape from the Moon entirely.
Just as the mass of the Moon gravitationally pulls on a projectile to slow its climb, the mass contained in the universe gravitationally slows the universe’s expansion. If there is enough mass in the universe, then gravity will be strong enough to stop the expansion. The universe will slow, stop, and eventually collapse in on itself. But if there is not enough mass, then the expansion of the universe may slow, but it will never stop. The universe will expand forever.
"Gravity slows the expansion
of the universe."
A planet’s mass and radius determine the escape velocity from its surface. The “escape velocity” of the universe is also determined by its mass and size specifically, its average density. If the universe is denser on average than a particular value, called the critical density, then gravity is strong enough to eventually stop and reverse the expansion. If the universe is less dense than the critical density, gravity is too weak and the universe will expand forever.
The faster the universe is expanding, the more mass is needed to turn that expansion around. For this reason, the critical density depends on the value of the Hubble constant, H0. Assuming that H0 = 72 kilometers per second per megaparsec (km/s/Mpc) and that gravity is the only thing we have to worry about, the universe’s critical density has a value of 8 × 10⁻27 kilograms per cubic meter (kg/m3), or fewer than 5 hydrogen atoms in every cubic meter. Rather than trying to keep track of such awkward numbers, we will instead talk about the ratio of the actual density of the universe to this critical density. We call this ratio Ω mass (pronounced “omega sub mass”). Because it is a ratio of two densities, Ω mass has no units.
The expansion of different possible mass-dominated universes is shown in the picture below, which plots the scale factor R U versus time for different values of Ω mass. If Ω mass is greater than 1, the universe will eventually collapse. Conversely, if Ωmass is less than 1, the universe will be slowed by gravity, but it will still expand forever. The dividing line, where mass equals 1, corresponds to a universe that expands more and more slowly but never quite stops.
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| Three possible scenarios for the fate of the universe, based on the critical density of the universe but ignoring any cosmological constant. |
Until the closing years of the 20th century, most astronomers thought that this straightforward application of gravity to the universe was all there was to the question of expansion and collapse. Researchers carefully measured the mass of galaxies and collections of galaxies in the hope that this would reveal the density and therefore the fate of the universe.
The luminous matter gives a value for mass of about 0.02. Galaxies contain about 10 times as much dark matter as normal matter, so adding in the dark matter in galaxies pushes the value of mass up to about 0.2. When we include the mass of dark matter between galaxies, mass could increase to 0.3 or higher. Still, by this accounting there is only about one-third as much mass in the universe as is needed to stop the universe’s expansion.
The Accelerating Universe Revives Einstein’s “Biggest Blunder”
If the expansion of the universe is in fact slowing down, as these simple models using gravity predict, then when the universe was young it must have been expanding faster than it is today. Objects that are very far away (so that we see them as they were long ago) should therefore have larger velocities than our local Hubble’s law would lead us to expect.
During the 1990s, astronomers tested this prediction. They measured the brightness of Type Ia supernovae in very distant galaxies and compared each of those brightnesses with the expected brightness based on the redshifts of those galaxies. The findings of these studies sent a wave of excitement through the astronomical community. Rather than showing that the expansion of the universe is slowing, the data indicated that it is speeding up. For this to be true, a force must be pushing the entire universe outward in opposition to gravity.
The idea of a repulsive force opposing the attractive force of gravity is not new. When Einstein used general relativity to calculate the structure of spacetime in the universe, he was greatly troubled. The theory clearly indicated that any universe containing mass could not be static, any more than a ball can hang motionless in the air. However, Einstein’s formulation of spacetime came more than a decade before Hubble discovered the expansion of the universe, and the conventional wisdom at the time was that the universe was indeed static that it neither expands nor collapses.
To force his new general theory of relativity to allow for a static universe, Einstein inserted a “fudge factor” called the cosmological constant into his equations. The cosmological constant acts as a repulsive force in the equations, opposing gravity and allowing galaxies to remain stationary despite their mutual gravitational attraction. When Hubble announced his discovery that the universe is expanding, Einstein realized his mistake. General relativity demands that the structure of the universe be dynamic.
Instead of inventing the cosmological constant, Einstein could have predicted that the universe must either be expanding or contracting with time. What a coup it would have been for his new theory to successfully predict such an amazing and previously unsuspected result. He called the introduction of his fudge factor, the cosmological constant, the “biggest blunder” of his career as a scientist. It is ironic that with our measurements of the brightness of Type Ia supernovae, Einstein’s biggest blunder has returned to center stage. The repulsive force represented by the infamous cosmological constant is just what is needed to describe a universe that is expanding at an ever-accelerating rate.
Today, we write this constant as Λ (pronounced “omega sub lambda”). If Λ is not zero, something is effectively pushing outward from within the universe, adding to its expansion. Gravity will have a harder time turning the expansion around. Figure be slow hows plots of the scale factor RU versus time that are similar to those shown in first picture, but now we have included the effects of a nonzero cosmological constant. If mass is sufficiently large, a universe will collapse back on itself, regardless of whether Λ is zero. In contrast, if a universe expands forever, its evolution will depend on whether or not Λ is zero.
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| Plots of scale factor RU versus time for cosmologies with and without a cosmological constant, ΩΛ. If there is enough mass in a universe, gravity could still overcome the cosmological constant and cause that universe to collapse. Any universe without enough mass to eventually collapse will instead end up expanding at an ever-increasing rate.
While a universe is young and compact, gravity dominates the effect of the cosmological constant. As a universe expands, gravity gets weaker because the mass spreads out. Since the cosmological constant remains, well, constant, it becomes increasingly important. Unless gravity is able to turn the expansion around, the cosmological constant wins in the end, causing the expansion to continue accelerating forever. Even if mass is greater than one, a large enough cosmological constant could overwhelm gravity and make the universe expand forever.
When Einstein added the cosmological constant to his equations of general relativity, he considered it a new fundamental constant, similar to Newton’s universal gravitational constant G. Today we realize that the vacuum can have distinct physical properties of its own. For example, the vacuum can have a nonzero energy even in the total absence of matter. We call this energy dark energy.
Dark energy produces exactly the kind of repulsive force that Einstein’s cosmological constant calls for. Figure below shows the range of values for mass and Λ that are allowed by current observations. Each colored region represents the data from a different experiment. Values of mass and Λ outside of these regions are ruled out by these experiments, so the allowed values of mass and Λ must lie within the area on the graph where all of these regions overlap. The allowed values for mass and Λ are about 0.3 and 0.7, respectively. These values are most tightly constrained by the data from WMAP, in bright red. The expansion of our universe is dominated by the effect of dark energy.
Current observations from different sources Type Ia supernovae (yellow), measurements of mass in galaxies and clusters (orange), and detailed observations of the structure of the cosmic microwave background (pink and bright red) suggest that the best current estimate for Ω Λ is about 0.7 and about 0.3 for Ωmass, which means thatthe expansion of the universe is accelerating.
The dark diagonal line in picture above shows where mass + Λ = 1. The experiments tightly constrain the value of mass + Λ to lie along this line. This means that on very large scales, the universe we live in is “fl at”—it is only locally warped by the mass inside, like the rubber sheet we learned about in Chapter 13. This, in turn, means that ordinary Euclidean geometry describes circles, triangles and parallel lines in the universe as a whole.
"The universe appears to be accelerating and will expand forever"
Other geometries exist. For example, the universe could have been positively curved on the largest scales, like the surface of a ball, so that the circumference of a circle is less than 2π times the radius. Parallel lines would converge (think of lines of longitude on Earth), and the sum of a triangle’s angles would be greater than 180°. Or the universe could have been negatively curved, like the middle of a Pringles potato chip. In this case, the circumference of a circle is more than 2π times the radius.
Parallel lines would diverge, and the sum of a triangle’s angles would be less than 180°.
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