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), [Formula: see text], 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 [Formula: see text] may be mildly evolving with the expansion rate [Formula: see text]. 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 [Formula: see text] acquires a dynamical component [Formula: see text] caused by the quantum matter effects. There are also [Formula: see text] ([Formula: see text]) contributions, some of which may trigger inflation in the early universe. Remarkably, the evolution of the adiabatically renormalized [Formula: see text] is not affected by dangerous terms proportional to the quartic power of the masses ([Formula: see text]) 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 [Formula: see text] around [Formula: see text] is given by a dominant term ([Formula: see text]) plus the aforementioned mild dynamical component [Formula: see text] (with [Formula: see text]), 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' [Formula: see text]CDM model, such as the [Formula: see text] and [Formula: see text] tensions. This article is part of the theme issue 'The future of mathematical cosmology, Volume 2'.