Everything You Always Wanted To Know About The Cosmological Constant Problem (But Were Afraid To Ask)

TL;DR

The paper analyzes the cosmological constant problem by separating classical and quantum contributions to vacuum energy and examining their gravitational effects. It shows that while classical tuning is insufficient due to phase transitions, quantum zero-point fluctuations introduce divergent, regulator-dependent vacuum energy that can be correctly treated with Lorentz-invariant schemes like dimensional regularization. The Gaussian effective potential and detailed field-type analyses (scalar, fermion, vector) reveal that, after renormalization, vacuum energy scales with particle masses and logs rather than with a naïve cut-off, but the resulting magnitude remains vastly larger than the observed cosmological constant unless new physics (e.g., exact or softly broken supersymmetry, or other mechanisms) acts to suppress it. The curved-space treatment confirms these flat-space conclusions and connects them to cosmological measurements, which indicate a small but nonzero $\Lambda$, driving the accelerating expansion observed via the Hubble diagram. Overall, the work clarifies the structure of vacuum energy, the role of bubble diagrams, and the rigorous regularization required to relate quantum vacuum effects to cosmology, while underscoring the need for novel solutions to the cosmological constant problem.

Abstract

This article aims at discussing the cosmological constant problem at a pedagogical but fully technical level. We review how the vacuum energy can be regularized in flat and curved space-time and how it can be understood in terms of Feynman bubble diagrams. In particular, we show that the properly renormalized value of the zero-point energy density today (for a free theory) is in fact far from being 122 orders of magnitude larger than the critical energy density, as often quoted in the literature. We mainly consider the case of scalar fields but also treat the cases of fermions and gauge bosons which allows us to discuss the question of vacuum energy in super-symmetry. Then, we discuss how the cosmological constant can be measured in cosmology and constrained with experiments such as measurements of planet orbits in our solar system or atomic spectra. We also review why the Lamb shift and the Casimir effect seem to indicate that the quantum zero-point fluctuations are not an artifact of the quantum field theory formalism. We investigate how experiments on the universality of free fall can constrain the gravitational properties of vacuum energy and we discuss the status of the weak equivalence principle in quantum mechanics, in particular the Collela, Overhausser and Werner experiment and the quantum Galileo experiment performed with a Salecker-Wigner-Peres clock. Finally, we briefly conclude with a discussion on the solutions to the cosmological constant problem that have been proposed so far.

Figures (14)

• Figure 1: The effective potential given by Eq. (\ref{['eq:effectivepotT']}). Before the transition, for $T>T_{\rm cri}$, the minimum of the potential is located at the origin and the vacuum energy is given by $V_0+\lambda v^4/4$. After the transition, for $T<T_{\rm cri}$, the minimum is located at $\Phi=v$ and the corresponding vacuum energy has changed and now equals $V_0$. It is clear that $V_0$ can always be chosen such that the vacuum energy vanishes either before or after the transition. It is equally clear that one cannot choose the parameters of the potential such that $\rho_{_{\rm vac}}$ is zero before and after the phase transition.
• Figure 2: Effective potential of the Higgs boson before and after the electroweak phase transition. The left panel corresponds to a situation where the vacuum energy vanishes at high temperature. As a consequence $\rho_{_{\rm vac}}$ is negative at temperature smaller than the critical temperature. This is the situation treated in the text where the quantity $-m^4/(4\lambda)$ is explicitly calculated. On the right panel, the off-set parameter $V_0$ is chosen such that the vacuum energy is zero after the transition. As a consequence, it does not vanish at high temperatures.
• Figure 3: Contours in the complex time plane used to calculate the Feynman propagators of the one-dimensional harmonic oscillator. As usual, there are two poles on the real axis that can be avoided by going along the two small red circles. The final contour is closed by a large blue circle of radius $R$.
• Figure 4: Sketch of the Riemann coordinates used in order to derive an approximate expression for the Green function, see Eq. (\ref{['eq:propergreen']}). The plot aims at illustrating the fact that the Riemann coordinates are a local concept. Since it is sufficient to establish the Green function locally (and not in the entire curved manifold), one can always endow the neighborhood of a point $x'$ (denoted here by the region surrounded by the closed dotted line) with special coordinates $x^{\mu}$ such that the calculation is simplified. It is important to notice that this does not restrict the generality of the obtained results.
• Figure 5: Evolution of the vacuum energy density [more precisely the first term in Eq. (\ref{['eq:rhovacstandard']}) versus the renormalization scale $\mu$. In the range considered here, the vacuum energy density is negative. The "divergence" observed around $\log_{10}\mu \simeq 5$ does not correspond to a new physical effect but just signal that $\rho_{_{\rm vac}}$ becomes positive.