Cosmological Constant

BACKGROUND

Einstein postulated its existence in 1917 (or so) and then renounced it (on aesthetic grounds) in the 1920's when Hubble announced that the Universe was expanding. Today, we believe that a nonzero Lambda could explain the discrepancy between the ages of the oldest stars in the Milky Way galaxy and the age of the Universe as characterized by 1/H(now).
The notion that the cosmological constant is nonzero is based on the idea that the vacuum can contain energy. In the vacuum, for example, we know that quantum fluctuations lead to the appearance and disappearance of virtual pairs of particles which continuously pop into and then go out of existence. These cannot be measured directly, but they affect the curvature of space. These sum effect of the fluctuations could be positive, negative, or zero depending upon the details of our current theories. They are a major contribution to the background energy of the Universe. We parametrize their strength as:
Lambda = (8 pi G / c4) (vacuum energy) = 8 pi G rho(vacuum) / c2
Comment -- Note that Lambda differs from CC by a factor of 1 / c2. What are the units of Lambda? A simple examination of the above shows that Lambda has units of 1 / length2. What this means is that

If Lambda = 1/(1 km)**2, then one would expect to see effects of the vacuum energy on the space-time for objects as close as 1 kilometer ===> severe distortions could be discerned if you looked at the bookstore from here.

The vacuum can store a lot of energy and not have been noticed before because, in physics, one usually measures changes in energy with respect to the some agreed upon standard. People usually compare things to the vacuum. This is analogous to people defining the heights of mountains with respect to sea level rather than to the center of the Earth. In principle, one could say that Mount Everest was 4,000 miles + 6 miles high, that Mount McKinely was 4,000 miles + ~ 3 miles high, etc. However, because the radius of the Earth is roughly constant, people tend to subtract off the Earth's contribution.

MEASURED VALUE FOR LAMBDA

There is no measured value for Lambda today. There are only upper limits as to how large could be. The simplest argument is that since we do not significant distortions of the space-time on the scale of the Universe, then
Lambda < 1 / (size of the Universe)**2 ~ 1 / (8 - 12 billion light years)
===> Lambda is very small (today).

THEORETICAL PREDICTION FOR LAMBDA

If one sums up the various contributions to the vacuum (that is, one considers quantum fluctuations of particles, unknown things which could be happening on the Planck scale), one arrives at
vacuum energy ~ 1.6 x 10**111 ergs per cubic centimeter.
Oooh, this is big. (The energy density in the center of the Sun is only 2 x 10**17 ergs per cubic centimeter!!)
===>Lambda(predicted) ~ 8 pi G vacuum energy / c**4 ~ 0.6 / {10**(-31} cm}**2 ~ 1 / {10**(-48) light years}**2
A nice scale to characterize this number is to note that the Planck scale is Planck length = sqrt[h G / (2 pi c**3)] ~ 4 x 10**(-33) cm. That is, the simple estimates suggest that Lambda is on the order of 1 / Planck**2.

COMPARISON OF LAMBDAS

Lambda(predicted)/Lambda(observed) ~ (10**10 ly)**2/(10**{-48} ly)**2 ~ 10**116 !! (I used 10**10 ly for convenience -- it doesn't really matter what value I choose to use)

This is not good. In order to bring the observed Lambda and the theoretical estimate for Lambda into line requires that the fluctuations in the vacuum be arranged so that there there is some fierce cancellation going on to make the vacuum energy density very nearly 0 (but not quite 0 according to some people). This is another fine-tuning problem.
The tuning must be very precise. Even if we consider only protons, we would argue that Lambda ~ 1/(1 km)**2. This Lambda is huge and would lead to rather amazing effects in our everyday lives. Lambda clearly cannot be anywhere near thir size.
Some people want the cancelation to be very close to complete, but yet, not complete so that there is a tiny, but nonzero Lambda. A tiny Lambda could explain the age discrepancy between old stars and the Universe. The cancelation needed would be amazing because, it would need to be something like
1.00000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000001 - 1
that is, we want the difference between the two numbers to fall in the 116-th digit. Well, to most rational people, if numbers are this close to canceling it probably means that they cancel exactly. It is hard to make numbers so nearly identical and not really mean that they are identical. But who knows?