The Most General Four-Derivative Unitary String Effective Action with Torsion and Stringy Running Vacuum Model Inflation: Old Ideas from a Modern Perspective

The string-inspired running vacuum model (StRVM) of inflation is based on a Chern–Simons (CS) gravity effective action in which the only four-spacetime-derivative-order term is a gravitational anomalous CS–Pontryagin density coupled to an axion. In this work, we revisit curvature-squared string-inspired effective actions from the point of view of appropriate local field redefinitions, leaving the perturbative string scattering matrices invariant. We require simultaneously unitarity and torsion interpretation of the field strength of the Kalb–Ramond antisymmetric tensor, features characterizing the (3+1)-dimensional StRVM cosmology. Unlike the higher-dimensional case, the above features are possible in the context of (3+1)-dimensional spacetimes, obtained after string compactification. We demonstrate that the unitarity and torsion interpretation requirements lead to a single type of extra four-derivative terms in the effective gravitational action, not discussed in the previous literature on StRVM, which is, however, shown to be subleading by many orders of magnitude compared to the terms of the StRVM framework. Hence, its presence has no practical implications for the relevant inflationary (and, hence, postinflationary) physics of the StRVM. This demonstrates the phenomenological completeness of the StRVM cosmological scenario, which is thus fully embeddable in the UV-complete (quantum gravity-compatible) string theory framework.