Cosmological Constant and Dark Energy

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Shortly after the development of his theory of general relativity, Albert Einstein recognized a potential problem with his new theory. General relativity was the first theory to describe not only the dynamics of objects within space-time but also the dynamics of space-time itself. As such, the theory offered the possibility of providing a first-principles understanding of the evolution of the universe itself. However, the fact that it reproduced Newtonian gravity in weak gravitational fields meant that no stable static cosmological solution of Einstein's equations existed involving merely matter and radiation. Since the gravitational attraction of all such sources of energy is universally attractive, any initially static system of mass points, such as galaxies, will inevitably collapse inward. In 1917, however, it appeared that the universe on large scales was indeed static.

In an effort to resolve this problem, Einstein recognized that he could preserve the symmetries that led him to develop the theory of general relativity by modifying his equation with the addition of an extra term, which he dubbed the "cosmological term." Such a term could produce, on large scales, a small repulsive force throughout the universe that might serve to counterbalance the standard gravitational attraction of distant masses, while leading to no observable effects on terrestrial scales that might disagree with existing observations.

Unfortunately, however, almost immediately problems arose with Einstein's idea. The physicist Willem de Sitter demonstrated that a consistent cosmological solution existed in the presence of a cosmological constant in which an otherwise empty universe could continue to expand forever. Einstein found such a possibility distasteful. But, a far more serious concern arose as it was recognized over the next decade, largely due to the work of the astronomer Edwin Hubble, that the universe is not static but is in fact expanding. In such a case, no additional repulsive force would be needed to counterbalance conventional gravity. In an expanding universe gravity could merely work to slow the expansion. Whether the observed expansion would stop was thus a simple question of initial conditions. The attempt to determine the expansion rate of the universe and its mass density—the two factors needed to ascertain the ultimate fate of the Universe—then became the central focus of cosmology for much of the rest of the twentieth century, while the cosmological constant faded from interest.

The question of the possible existence of a cosmological constant emerged again, however, following World War II as theoretical physicists began to grapple with the quantum mechanical properties of elementary particles. It was soon recognized that empty space need not be precisely empty. Virtual particle-antiparticle pairs could spontaneously appear and disappear again, as long as they did so in a time interval short enough so that no direct observations of the violation of the conservation of energy and momentum could be observed. With this recognition came the recognition that quantum mechanically, at least, one would in general expect the vacuum state of nature to possess energy. An examination of the form that this energy would take, dictated by the fundamental symmetry called Lorentz invariance, demonstrated that such vacuum energy produced an additional contribution to Einstein's equations identical in form to Einstein's cosmological term.

A new challenge then arose, which has since been termed the cosmological constant problem. The lack of any observed repulsive force in nature governing the expansion of the universe placed very strong constraints on the size of any possible vacuum energy density today. When compared with naïve theoretical estimates based on extrapolating our current knowledge of elementary particle physics to scales as small as the Planck length, the observational upper limit on the cosmological constant is about 125 orders of magnitude smaller. This is perhaps the worst prediction in all of physics!

It was generally assumed by particle physicists that an ultimate resolution of the cosmological constant problem would involve some mechanism, perhaps based on an unknown symmetry argument that required the ultimate vacuum energy density of the universe to be precisely zero. Only in this case could one hope to gain accord with observations while not requiring some unprecedented fine-tuning mechanism that might reduce the vacuum energy density by 125 orders of magnitude.

Here the situation remained until the last decade of the twentieth century. A host of new observational data began to expose some inconsistencies in the standard model of cosmology, which involved a flat universe dominated by some new form of nonbaryonic matter, conventionally called cold-dark matter. The problems arose from three separate fronts: (1) The age of a flat-matter-dominated universe was predicted to be less than about 10 billion years old, given the measured expansion rate of the universe. However, a determination of the age of the oldest stars in our own galaxy suggested that these stars were at least 12 to 20 billion years old. (2) Observations of the clustering properties of matter on the largest observable scales gave some estimate of the overall density of matter, and these estimates repeatedly began to suggest that there was not sufficient mass density in the universe to result in its being spatially flat.(3) Direct probes of the matter density in clusters of galaxies, based on X-ray measurements of the hot gas contained there, and on their evolution as a function of time, both put upper limits on the total matter density that fell far short of the amount needed to result in a flat universe today.

In order to resolve the latter two problems, some additional source of unclustered, spatially uniform energy would have to exist in the universe in order to provide the extra energy necessary for a flat universe. If such energy was unclustered, then both X-ray and gravitational probes on the scale of galaxy clusters would not be sensitive to it. One possible source might be vacuum energy density. Such energy would have the additional impact of causing a net acceleration to the expansion of the universe that could resolve the apparent discrepancy with the expansion age of the universe compared to stellar ages. If the universe had expanded at a rate slower than that by which it is now measured to be expanding, then it would have taken longer for galaxies observed a certain distance away from Earth to have achieved that separation. Thus, the expansion age could be longer than it would otherwise be if vacuum energy plays a dynamical role today.

Between 1998 and 2000 these indirect arguments were bolstered by two significant developments from cosmology. First, observations of distant supernovae allowed a comparison of the relationship between physical distance and redshift that could probe the temporal evolution of the expansion. Much to the surprise of a large segment of the scientific community, evidence was obtained that the expansion of the universe is indeed accelerating, precisely at the amount required if vacuum energy is the dominant energy density of the universe today and leads to a flat universe. Concurrently, in 1999, observations of small fluctuations in the temperature of the cosmic microwave background (CMB) radiation that permeates space as a remnant of the Big Bang allowed the direct measurement of the large-scale geometry of space-time. As the geometry of the universe changes from open to closed, the angular scale corresponding to a fixed physical location observed from a distance changes. By comparing the angular scale of anisotropies in the CMB radiation, which has traveled unimpeded since matter became neutral about 300,000 years into the Big Bang, with model predictions, three different experiments confirmed with good accuracy that the universe is flat.

The combination of these two observations definitely suggests that some energy appears to be associated with empty space, indeed, this is the dominant energy density in the universe today! All cosmological observations are now consistent with the notion of a flat universe, in which about 30 percent of the total energy density results from matter and about 70 percent results from some form of dark energy.

The term dark energy is used to describe the energy associated with empty space because existing observations cannot distinguish between a true cosmological constant and some other unclustered form of energy permeating all of space. Much of the inquiry in present-day cosmology focuses on trying to find techniques that might distinguish a cosmological constant from something else. This observational activity will be very challenging, however.

The existence of dark energy presents one of the greatest puzzles in all of physics. As of yet scientists have no good theoretical grasp of what causes this energy, nor do they understand why it has its apparent value. Moreover, the presence of such dark energy has completely altered perceptions of the possible future evolution of the universe. Such energy violates a condition in general relativity called the strong energy condition that allows a one-to-one connection between geometry and the ultimate fate of an expanding universe. As a result, although one of the central goals of cosmology in the twentieth century, to determine the geometry of the universe, has now been achieved, the future is unfortunately far less certain than expected. Allowing for the presence of dark energy, a closed universe can expand forever, and an open or flat universe can recollapse. If one is to ever unambiguously be able to predict the future evolution of the universe, an understanding of the nature of dark energy arising from the fundamental theories of particle physics will be required.

See also:Big Bang; Cosmology; Inflation


Krauss, L. M. "Cosmological Antigravity." Scientific American280 (1), 34–41 (1999).

Krauss, L. M. Quintessence (Basic Books, New York, 2000).

Weinberg, S. "The Cosmological Constant Problem." Reviews of Modern Physics61 , 1–24 (1989).

Lawrence M. Krauss