Astronomy 162: Professor Barbara Ryden

Monday, March 10


``This is the way the world ends,
Not with a bang but with a whimper.''
- T. S. Eliot

Key Concepts

(1) Most of the density of the universe is contributed by dark energy.

As I was saying last week, space is curved on small scales by small massive objects such as people, planets, stars, galaxies, clusters of galaxies, and superclusters of clusters. On very large scales, however, space seems to be flat. That is, it is neither positively curved, like the 3-d equivalent of a sphere, nor negatively curved, like the 3-d equivalent of a hyperboloid. Rather, it is the 3-d equivalent of a plane, and obeys all of Euclid's laws of geometry. The laws of general relativity tell us that for an expanding universe to be flat, its density must be equal to a critical density which is proportional to the square of the Hubble constant. For our universe, the critical density is about 9 x 10-27 kilograms per cubic meter. (This extraordinarily low density is equivalent to taking the matter in a small drop of water and spreading it over the volume of the Earth.)

Small though the critical density is, the amount of matter in the universe seems to be still smaller than the critical density. The amount of matter clumped up into clusters and superclusters only contributes about 30 percent of the critical density. What provides the rest of the mass (or energy, since E = mc2)? Photons provide energy, but all the photons in the universe, from the Cosmic Microwave Background, from starlight, from your computer screen, and from all other sources, provide far less than one percent of the critical density. The rest of the critical density is provided by some form of energy (or mass, since m = E/c2) which is smoothly distributed throughout the universe, instead of being clumped up into clusters and superclusters. This mysterious form of energy is called dark energy.

The name ``dark energy'' is something of a confession of ignorance. Cosmologists could just as well have called it ``whatchamacallit''. What do we know about dark energy?

(2) The expansion of the universe currently seems to be speeding up.

The standard way of determining the properties of dark matter is to look for its influence on luminous matter. A similar analysis can be done of dark energy. In particular, the dark energy strongly affects the expansion of the universe. Remember, the universal expansion of the universe causes galaxies to move away from each other. However, galaxies have mass, and hence the mutual gravitational attraction of the galaxies will tend to make the expansion slow down. Now consider the effect of adding dark energy to the universe. The energy (or mass) of the dark energy will add more gravitational attraction to the universe, and hence will tend to amek the expansion slow down even more. HOWEVER, if the dark energy has pressure as well, the pressure could act to speed up the expansion.

So, how do astronomers go about determining whether the expansion of the universe is speeding up or slowing down? The current rate of expansion is given by the Hubble constant (H0). To find the Hubble constant, you can plot the distance (d) of standard candles as a function of their recession speed (v). If the resulting plot has a shallow slope (that is, if galaxies with large recession speed have a relatively small distance), then the universe is expanding rapidly and H0 is large. By contrast, if the plot of distance as a function of recession speed has a steep slope, then the universe is expanding slowly and H0 is small.

But H0 tells us how rapidly the universe is expanding now; how can we tell how rapidly the universe was expanding in the past? Remember, ``a telescope is a time machine''. The speed of an extremely distant galaxy tells us how fast the universe was expanding at the time the light we observe was emitted; that may have been billions of years ago, when the universe was much younger (and may have been expanding much faster or slower). If a plot of distance as a function of recession speed has a shallower slope at large recession speeds, then the universe was expanding more rapidly in the past than it is now. If a plot of distance versus recession speed is steeper at large recession speeds, then the universe was expanding more slowly in the past. (This verbal description, I realize, may be a bit confusing; a look at Figure 28-17 of the textbook should make things clearer.)

During the past decade, research groups have carefully observed Type Ia supernovae with large redshifts (and hence at large distances). After observing dozens of distant supernovae, they plotted the distances (deduced from their apparent brightness) as a function of their recession velocity (deduced from their redshift).

(3) The acceleration of the universe might be caused by a cosmological constant.

What could be the source of dark energy, the weird stuff filling the universe and causing its expansion to speed up? Actually, Albert Einstein had an answer to that question as long ago as 1917, two years after he published his Theory of General Relativity. In 1917, Edwin Hubble had not yet discovered the Hubble Law; hence, Einstein assumed that the universe was neither expanding nor contracting. This was a problem, since Einstein knew that gravitational attraction would make the universe collapse. Thus, he was searching for a repulsive force that would exactly balance the attractive force of gravity, and thus leave the universe in equilibrium. What Einstein did to ensure the equilibrium of the universe was insert a ``fudge factor'' into the equations describing gravity. This fudge factor was symbolized by the Greek letter Lambda (I don't know why - maybe Einstein picked that letter out of a hat), and was given the name of the cosmological constant. Einstein's cosmological constant, when inserted into the equations of general relativity, provides: Thus, the cosmological constant acts just the way observations tell us that the ``dark energy'' behaves.

When Einstein first introduced the cosmological constant in 1917, he regarded it as an ugly mathematical contrivance. (He thought it marred the elegant simplicity of his equations.) When Hubble pointed out, a decade or so later, that the universe was not static, Einstein happily jettisoned the cosmological constant.

Now, however, the acceleration of the universe has caused scientists to bring Einstein's cosmological constant out of the attic and dust it off. Moreover, where Einstein thought of the cosmological constant as a purely mathematical ``fudge factor'', modern physicists have found a physical reality behind it. The cosmological constant, according to the laws of quantum mechanics, could represent the energy and pressure of the vacuum. In classical Newtonian physics, a vacuum is totally empty, and thus can't have an energy or pressure. In the world of quantum mechanics, however, vacuums are not totally empty. In apparently empty space, pairs of particles and antiparticles are constantly being created, only to annihilate with each other a short time later. Protons and anti-protons, neutrons and anti-neutrons, electrons and anti-electrons (also called positrons) - they are constantly popping out of nowhere, only to be destroyed a tiny fraction of a second later. These virtual pairs of particles, as they are called, provide energy and pressure during the short time of their existence.

Cosmologists have put together, within the broad frame of the Big Bang Theory, a more specfic model of the universe. The observations on which the model is based:

The deduced properties of the model: Although the Big Bang Theory has been around for the best part of a century, this specific model (flat yet accelerating) is a relatively recent development. The details (70 percent dark energy? 60 percent? 80 percent?) are being vigorously debated by cosmologists.
Prof. Barbara Ryden (

Updated: 2003 Mar 9

Copyright 2003, Barbara Ryden