String-Inspired Running Vacuum, the "Vacuumon", and the Swampland Criteria

Abstract: We elaborate further on the compatibility of the "vacuumon potential" that characterises the inflationary phase of the Running Vacuum Model (RVM) with the Swampland criteria. The work is motivated by the fact that, as demonstrated recently by the authors [1-3], the RVM framework can be derived as an effective gravitational field theory stemming from underlying microscopic (critical) string theory models with gravitational anomalies, involving condensation of primordial gravitational waves. Although believed to be a classical scalar field description, not representing a fully fledged quantum field, nonetheless we show here that the vacuumon potential satisfies certain Swampland criteria for the relevant regime of parameters and field range. We link the criteria to the Gibbons-Hawking entropy that has been argued to characterise the RVM during the de Sitter phase. These results imply that the vacuumon may, after all, admit under certain conditions, a r^ole as a quantum field during the inflationary (almost de Sitter) phase of the running vacuum. The conventional slow-roll interpretation of this field, however, fails just because it satisfies the Swampland criteria. The RVM effective theory derived from the low-energy effective action of string theory does, however, successfully describe inflation thanks to the $\sim H^4$ terms induced by the gravitational anomalous condensates. In addition, the stringy version of the RVM involves the Kalb-Ramond (KR) axion field, which, in contrast to the vacuumon, does perfectly satisfy the slow-roll condition. We conclude that the vacuumon description is not fully equivalent to the stringy formulation of the RVM.