The Cosmological Constant Problem
Leah Jenks, PhD Student
Hello! I’m Leah, a second year PhD student at Brown University, working on a wide variety of problems at the intersections of gravity, cosmology, and high energy theory. Today, in an attempt to think about something other than the pandemic, I’m going to talk to you about one of the biggest problems in modern cosmology—the cosmological constant problem.
“Physics is an experimental science” is the favorite refrain from one of my graduate school professors, who wanted to drill into us the very concrete experimental underpinnings of Jackson E&M. As a theorist, whose forays into experiment have included frustration to the point of tears while attempting to use an oscilloscope and blowing up a superconducting magnet, I always bristled at the insinuation that the experimental aspect of physics was somehow more important or more fundamental. However, I have since realized, although there is certainly fundamental value in theoretical work, at the end of the day, if it doesn’t match experimental evidence, we aren’t doing physics any more. I am certainly not the first person to come to this conclusion, and I will not be the last, but fundamental theories have to match observations and experiments if we want to get closer to explaining the universe. One great example of this was with Albert Einstein and his theory of general relativity, in which he inadvertently stumbled upon one of the biggest puzzles in modern cosmology.
Today we take it as a given that the universe is expanding. In any introductory astronomy or cosmology class, we learn Vr = Ho D - Hubble’s law, where Ho is a constant which accounts for the expansion of the universe. In particular, Ho determines the proportionality between the recessional velocity Vr of a distant object (i.e. the velocity at which the distant object moves away or 'recedes' from the observer) and the distance D between the object and the observer. Thus, the further away an object is, the faster it moves away from us. However, an expanding universe was not always taken for granted. In fact, our standard ‘Big Bang’ cosmology was actually a nickname tossed out with derision in 1949 by Fred Hoyle, an astronomer who was steadfastly convinced that the universe was static, to mock those who were silly enough to think that the universe was expanding. In the early to mid 1900s, there was a great divide between those who thought that the universe was static and those who thought it was expanding. Initially, Albert Einstein was staunchly on the side of those who posited the universe to be an unchanging, eternal entity.
In 1915, Einstein wrote down his famous field equations, describing the curvature of spacetime due to matter. The ten coupled, nonlinear, hyperbolic-elliptic partial differential equations can by neatly summarized in the tensor equation:
This set of equations compactly describes the theory of general relativity, elegantly summed up by John Wheeler as ‘spacetime tells matter how to move; matter tells spacetime how to curve’. The left hand side of the equation encodes information about the curvature of spacetime, while the right-hand side describes the matter that is present in spacetime.
However, there is a missing piece here. When Einstein initially wrote down these equations, he was dismayed to find that they indicated the universe is a dynamic entity, contradicting his (widely accepted) belief that the universe is static. Assuming there was something wrong in the equations, which needed to describe the physical universe as it was perceived to be, Einstein introduced a ‘fudge factor’—a cosmological constant term Λ—which would hypothetically keep the universe in equilibrium. The full equations then became
Everything seemed all well and good until 1929, when observations by Edwin Hubble confirmed that the universe was indeed expanding. Einstein regretted his choice to introduce the cosmological constant into his equations, allegedly referring to it as his ‘biggest blunder’. In hindsight, Einstein shouldn’t have been so hard on himself. We now know that the cosmological constant in the field equations represents the vacuum energy of the universe, which is in fact causing the universe to expand! So, in the end, although Einstein’s attempt to match his theory to the physical universe was initially misguided, he did end up with a theory that describes observations, just not the ones he thought.
Einstein’s ‘biggest blunder’ actually ended up being correct, but all is not well with the cosmological constant yet. There is another issue, what is generally referred to as ‘the cosmological constant problem,’ which is a ~120 order of magnitude discrepancy between the observed and theoretical values of Λ. Yes, you read that correctly. This conundrum unfortunately cannot be resolved trivially. Given data from the Planck satellite, Λ has an experimental value of 4.33 x 10^-66 eV^2. On the theoretical side, effective field theories predict that Λ should be on the order of the Planck mass squared, or ~10^54 eV^2. Again, to emphasize, these two values differ by 120 orders of magnitude. Although it may be tempting to throw up our hands, add in a ‘~’ instead of an ‘=’, and shrug off the discrepancy, because after all, what are a few orders of magnitude between friends, this is a more difficult problem. There is clearly either something going very wrong with the measurements, or there is a problem with the theory. This may seem hopeless, but fortunately for us there are many smart people working on this problem. There are many possibilities to explain this issue, though none have done so definitively. Three categories of paths to a solution include (but are certainly not limited to):
The anthropic principle: Using anthropic arguments, one could consider that there are many different values of the cosmological constant in many different parts of the universe (or multiverse), and we just happen to observe the small value because we live in a patch that happens to have a small Λ value. These arguments tend to be unpopular, but if you are willing to accept a multiverse and that physical quantities can be randomly assigned, then this may be the solution for you.
Modifications to general relativity: General relativity is a complete theory, which has been verified to great observational precision. However, there is potential for some modification or extension of the theory that resolves the cosmological constant problem. For example (to include my own research for a moment), one could imagine a scenario in which Λ is allowed to vary. If Λ itself is dynamical, then it is certainly feasible for it to have multiple values and the discrepancy is not as much as an issue.
QFT solutions: One can also look at the underlying quantum field theory processes that govern Λ and resolve it this way. For example, some sort of unknown scalar field dynamics in the early universe could 'relax' the value of the Λ in the same way that has been suggested for the very similar electroweak hierarchy problem.
This is of course not an exhaustive list of possibilities. There are a myriad of papers on approaching this problem from all possible angles, and yet we still do not have a definitive solution. The cosmological constant problem is still a huge puzzle and an active area of research in cosmology. If you have any thoughts or ideas about the cosmological constant problem, I would love to discuss them with you! Even if you think your idea might be blatantly wrong, just remember, what Einstein thought was one of his greatest failures ended up leading to one of the most important components of modern cosmology. You never know!
Einstein Image from Wikipedia - by Ferdinand Schmutzer - http://www.bhm.ch/de/news_04a.cfm?bid=4&jahr=2006 [dead link], archived copy (image), Public Domain, https://commons.wikimedia.org/w/index.php?curid=34239518