Despite the vacuum energy and in general the dark energy (DE) is postulated as being the physical cause that produces the measured accelerated expansion of the universe, the usual prediction in quantum field theory (QFT) is much too large as compared to the critical density of the Universe. The mismatch is ultimately responsible for the "Cosmological Constant Problem", which is the biggest conundrum of fundamental physics ever. The main problem stems from the usually proposed renormalization procedures of the energy-momentum tensor in QFT, which lead to contributions proportional to the quartic powers of the mass of the quantum fields (i.e. proportional to $m^4$). However, if one considers an appropriate variant of the adiabatic regularization and renormalization of quantum fields in a FLRW background it is possible to provide a well defined result which can elude the $m^4$ terms and ultimately provides a dynamical vacuum energy density $\rho_{vac}(H)$ evolving as powers of the Hubble rate, H, and its derivatives, therefore sufficiently small.
In this work, the student will have the opportunity to perform such a QFT calculation in an appropriate context, in which matter is represented by a scalar field non-minimally coupled to curvature. He/she will verify that $\rho_{vac}(H)$ evolves as a constant term (representing the cosmological constant part of the final result) plus some dynamical components of order $H^2$ and $H^4$. The higher powers are relevant for the early universe only, where they can trigger fast inflation through effective string-theory effects generating those terms. These are induced by the presence of anomalous Chern-Simon couplings involving the Kalb-Ramond field of the gravitational multiplet. At present, $\rho_{vac}(H)$ is dominated by the additive constant term accompanied by a tiny dynamical component of order $H^2$. This result tells us that the vacuum energy density is not just constant but evolves with time. Its current variation is small enough, but still potentially measurable. At the phenomenological level, it looks as a kind of effective quintessence behavior. In point of fact, however, there is not a real quintessence field but an underlying QFT vacuum changing ('running') very mildly with the cosmic expansion.