One would expect the universe to re-collapse after the Big Bang due to gravity. Astrophysical measurements show that its expansion is accelerating and it is expected to continue to accelerate. This is fixed in the current cosmological model Lambda-CDM with a term called the cosmological constant (also known as dark energy). The cosmological constant can be thought either as an intrinsic curvature of spacetime itself or as an energy density which “pushes” apart due to negative pressure. According to how we observe the expansion of the universe and our cosmological model, this energy of the vacuum should take a value of around 10^-27 kilograms per cubic meter (or 10^-52 negative square meters), which is pretty low (roughly the same as the photon energy density). If one calculates the energy density that we should observe according to quantum mechanics, the result is at least 120 orders of magnitude bigger. This problem is called the vacuum catastrophe and in the following post we will explore it at the fundamental level.
We start with a very small dimensional value which we dont know where does it come from. The first approach to this problem (and what we are going to develop in this post) is dimensional analysis. This is the analysis of relationships between different physical quantities by identifying their base quantities (units of measurement). The physical quantities to examine will be the fundamental constants of nature in order to check wether the cosmological constant is fundamental or not. The best example of how powerful dimensional analysis is the same calculation of the energy of the vacuum acording to quantum mechanics that we mentioned earlier. For quantum electrodynamics (QED), this calculation is very simple as it just involves integrating over the frequencies for a cut off energy contribution: Planck’s energy (if one does not cut the integral with it, the energy of the vacuum would be infinite!). Invoking dimensional analysis and arranging the constants of the speed of light c, the gravitational constant G and Planck’s constant h with the correct powers to give kilograms per cubic meter and we arrive at the same combination as the path of integrating in quantum mechanics (there is only one possible arrangement to give those units). Of course, some integers and mathematical constants will be missing, but one arrives to the orders of magnitude (120 more than observed) which is what we will aim for, but trying out more constants.
We know about 19-22 fundamental constants that one needs to include (for instance, by measurements) in our physics theories. We already mentioned c, G and h, and the rest belong to quantum mechanics such as Yukawa couplings (equivalent to rest masses of quarks and leptons) or two parameters of Higgs field. One can “simplify” these quantum mechanics constants with the Coulomb’s constant / permitivity / permeability of the vacuum, which are introduced in General Relativity (GR). In order to perform dimensional analysis with, for instance, the permeability constant, and obtain kilograms per cubic meter, another constant with units of charge must be also considered (either the charge of the electron e or the quantum charge, 1/3 of the latter).
We could do it defining a set of equations, but why not try all possibilities by computational brute force? Either way, you will find out that there is no way to arrange those constants with reasonable powers (up to 10) to get to the observed value of the cosmological constant. In the list of all combinations that matches the dimensional analysis (around 300 for exponents of integrals from -10 to 10 and half-numbers and their negatives such as 0.5, 1.5 2.5 up to 4.5), one remarkably finds one single combination without Planck’s constant (a pure GR combination, see below). This is interesting since it has been postulated that perhaps the energy of the vacuum has two contributions: one from GR itself as a constant plus the energy of the vacuum from quantum mechanics. Its value yields around 10^70 negative square meters and if set negative arbitrarily, a higher value of positive energy density from quantum mechanics (higher than the QED value) could counteract the first one leading to the observed small value (reducing the vacuum catastrophe to 19 orders of magnitude for QED). This is possible since there are more contributions from quantum mechanics than QED (other fields also contribute), but fine tuning to that precise observed value is still required. Unpromising. Perhaps another constant is missing?
Paul S. Wesson and others did a similar research of dimensional analysis for the cosmological constant including a mass constant, in search of a possible mass quantization. Together with Zel’dovich, they published the three combinations one can create with two of the three main constants, c, G, h and a mass parameter to obtain the cosmological constant, with no success in reaching its value for any reasonable mass of a particle.
Thanks to computation, we can now search by brute force for different masses with the three main constants c, G, h and the cosmological constant. Lets forget about mass quantization since it not our primary objective. The (reasonable) masses we find are: 10^-25, 10^-32, 10^-38 and 10^-48 kilograms. But how can the energy of the vacuum depend on a particular mass which should correspond to a particle in the Standard Model? The first mass of the list above is the interesting one, since its in the orders of magnitude of vacuum expectation value of the Higgs field or the Higgs mass. This is remarkable since both the Higgs parameters and the cosmological constant constitute hierarchy problems: both values should be very sensitive to quantum corrections. Additionally, the Higgs is the only known field with spin zero and a non-zero averge value. So, maybe both constants are related in an ultraviolet theory beyond Quantum Field Theory (a quantum gravity theory) through this equation? Or perhaps one of these mass parameters is the energy cutoff of the correct formula to derive the energy of the vacuum in a more fundamental theory?
Research published in Modern Physics Letters A 36, no 17, 2150114 2021
Available at: Researchgate