Resolving the Cosmological Constant Problem

Imagine you're a quantum physicist and you just got through calculating the energy density of the vacuum. To your horror you find that your calculation is around a hundred orders of magnitude greater than vacuum energy density measurements! This, in a nutshell, is the cosmological constant problem. According to your numbers, the cosmological constant should be huge. According to WMAP measuremments, the cosmological constant is too small to tweet about. In this post we shed some light on this problem. First, here are the variables we will be using:

Below we derive the cosmological constant value based on the WMAP vacuum energy density measurement of approximately 10^-10 Joules per cubic meter (see equations 3 and 4):

The next order of business is to consider the limitations caused by the Compton wavelength formula and the Heisenberg uncertainty principle--so we derive equation 8:

There are two ways we can interpret equation 8: 1. If delta-x grows smaller, energy (delta-E) grows bigger. This makes perfect sense when you consider that smaller spaces contain shorter wavelengths, and shorter wavelengths correspond to higher energies. 2. If delta-x grows smaller, delta-E becomes more uncertain. In any case, we use equation 8 as our means to make sense of the cosmological constant problem.

Equation 9 below is equation 8 modified. On the left side we have the vacuum energy measurement (E) and one meter (x). Of course the product of these values is far greater than Planck's constant, so the right side has the scale factor n:

OK, here's what we need to do: we need to reduce the vacuum's volume while maintaining a constant energy density. This can be done, but only up to a point. The question becomes, up to what point? Or, if you prefer, down to what point? How small can we make the energy and space before things get weird? Here's the answer:

Equation 10 above is the formula for finding the cutoff, or, the Heisenberg uncertainty limit. The product of the energy and space can't go below h-bar/2 without violating the uncertainty principle.

At 11 and 12 below, we plug-n-chug:

At the cutoff, the distance is around 10^-4 meters. The volume is 10^-12 cubic meters. The energy is approximately 10^-22 Joules. If the space is reduced further, the energy value will explode! With an upper limit of infinity! Obviously, if the energy skyrockets, or becomes more uncertain, the energy density won't be the same--and any calculation done at that level will yield ridiculous results.

The diagram below shows vacuum energy density matches observations at or above the cutoff. Any meaningful measurement is taken above that point. But if you prefer infinities and uncertainties, feel free to go smaller than the Heisenberg limit.

Thus, the cosmological constant problem appears to be a natural consequence of doing calculations and measurements below the cutoff limit.