11. ACCELERATION The cosmological constant returns:

Einstein called the cosmological constant his greatest blunder. Why bring it up again? Mainly, because sometimes we encounter observations that appear utterly inconsistent with the big bang model of the universe. But if the cosmological constant is included in Einstein's equations, the solutions allow a much greater variety of possible models for the cosmic expansion than the Friedmann solutions. Just a few years ago, it seemed that there was a fundamental contradiction between different measures of the age of the universe. At that time, the Hubble constant seemed to be in the range 75 - 100 km/s/Mpc, implying a universe with age less than 10 billion years today. Moreover, the best estimate of the ages of the globular clusters was about 15 billion years. But you can't have a universe younger than the stars in it! That would be a scientific crisis. But with a cosmological constant, the universe could be older than 1/H₀, and the crisis would be resolved. Cosmologists are loath to include a cosmological constant in their equations. It replaces a very simple set of solutions (the Friedmann models) by a somewhat more complicated set. They would like the universe to be simple. But if forced to choose between including a cosmological constant or giving up the theory of the expanding universe, they'll include it.

Now, two recent observations have forced us to bring the cosmological constant back into the picture. First, recent observations of distant supernovae by two different groups indicate that the universe was expanding more slowly when it was about half its present age than it is today -- i.e., the expansion of the universe is accelerating, not decelerating. (See Breakthrough of the Year: Cosmic Motion Revealed and The High-Z Supernova Search Team) The only way to have an accelerating universe today is to have some kind of long-range repulsion in the universe that overcomes gravity. That's exactly what a cosmological constant does.

In April 2000, new measurements of the fluctuations of the Cosmic Background Radiation (CBR) have provided evidence indicating that there is a cosmological constant. The idea is this: the curvature of the universe is determined by the total energy density in the universe. The energy density is partly the energy density due to the matter (most of which is dark matter) according to Einstein's formula E = Mc², but also the energy density due to the cosmological constant, which astronomers have begun to call "dark energy". The curvature of the universe can magnify or de-magnify the fluctuations of the CBR at the epoch of recombination, as illustrated below.

The following diagram illustrates what the combined results of the observations of distant supernova observations and the CBR tell us:

In the diagram above, W _m is the closure parameter of the universe due to the dark matter, which astronomers have determined has a value W _m = 0.3 - 0.4. W _L is a comparable measure of dark energy due to the cosmological constant. It appears that the most likely values are W _m = 0.35 and W _L = 0.65, implying that the dark energy is about twice that of the dark matter and that the universe is accelerating now.

The matter/energy content of the universe. The green wedge represents ordinary matter, including stars and gas between the stars. The dark blue wedge represents dark matter which, like ordinary matter, attracts. The red wedge represents dark energy.

The age of the universe, revisited: As I mentioned at the top of this section, introducing the cosmological constant changes the relationship between the age of the universe and the Hubble constant. If the universe is accelerating now, it can be older than 2/(3H₀). In fact, the best current estimate of the age of the universe is about 14 billion years (see Nick Gnedin's Cosmological Calculator). This value is comfortably greater than the age of the oldest stars.

What would Einstein think? Einstein introduced the cosmological constant into his equations in order to find a mathematical description of a universe that was static -- never changing, no motion. To do that, he had to set the cosmological constant to have a value exactly right to counteract gravity. But now we know that the universe is actually expanding. Therefore, the cosmological constant that we need to describe the actual universe today does not have the value that Einstein chose, and it does not balance gravity. In fact, the acceleration due to the cosmological constant exceeds the deceleration due to gravity. Evidently, Einstein introduced the cosmological constant into his equations for the wrong reason, and so he chose the wrong value. But the idea of the cosmological constant seems right today.

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Last modified April 15, 2002
Copyright by Richard McCray