The cosmological constant problem and running vacuum in the expanding universe

It is well known that quantum field theory (QFT) induces a huge value of the cosmological constant (CC), Lambda, which is outrageously inconsistent with cosmological observations. We review here some aspects of this fundamental theoretical conundrum (the cosmological constant problem, CCP) and strongly argue in favour of the possibility that the cosmic vacuum density rho(vac) may be mildly evolving with the expansion rate H. Such a 'running vacuum model' (RVM) proposal predicts an effective dynamical dark energy without postulating new ad hoc fields (quintessence and the like). Using the method of adiabatic renormalization within QFT in curved space-time, we find that rho(vac)(H) acquires a dynamical component O(H-2) caused by the quantum matter effects. There are also O(H-n) (n=4,6, ...) contributions, some of which may trigger inflation in the early universe. Remarkably, the evolution of the adiabatically renormalized rho(vac)(H) is not affected by dangerous terms proportional to the quartic power of the masses (similar to m(4)) of the quantized matter fields. Traditionally, these terms have been the main source of trouble as they are responsible for the extreme fine-tuning feature of the CCP. In the context under study, however, the late time rho(vac)(H) around H-0 is given by a dominant term (rho(0)(vac)) plus the aforementioned mild dynamical component alpha nu(H-2-H-0(2)) (with |nu| << 1), which makes the RVM to mimic quintessence. Finally, on the phenomenological side, we show that the RVM may be instrumental in alleviating some of the most challenging problems (so-called 'tensions') afflicting nowadays the observational consistency of the 'concordance' Lambda CDM model, such as the H-0 and sigma(8) tensions. This article is part of the theme issue 'The future of mathematical cosmology, Volume 2'.