Quantum-Ordering Ambiguities in Weak Chern-Simons 4D Gravity and Metastability of the Condensate-Induced Inflation

In this work, we elaborate further on a 4D cosmological Running-Vacuum-type Model (RVM) of inflation that characterises string-inspired Chern-Simons (CS) gravity. It has been shown that inflation in such models is caused by a condensation of the gravitational CS (gCS) terms induced by primordial gravitational waves (GW), which leads to a linear-axion potential, thus breaking the shift symmetry, and lifting its periodicity (monodromy). We demonstrate here that this inflationary phase may be metastable, due to the existence of non-trivial imaginary parts of the gCS condensate. These imaginary parts are quantum effects, proportional to appropriate commutators of GW perturbations. As we stress, their existence is quantum-ordering-scheme dependent. We argue here in favor of a physical importance of such imaginary parts, which we compute to second order in the GW (tensor) perturbations in the framework of a specific gauge-fixed effective Lagrangian, within a (mean field) weak-quantum-gravity path integral approach. We pay specific attention to the various space-time boundary terms. We thus provide an estimate of the life time of inflation. On matching our results with the relevant inflationary phenomenology, we fix the quantum-ordering ambiguities, and obtain an order-of-magnitude constraint on the ratio of the string energy scale $M_s$ in this model to the (four-spacetime-dimensional) reduced Planck mass $M_{\rm Pl}$, specifically, $M_s/M_{\rm Pl} = \mathcal{O}(10^{-1})$. This is consistent with the corresponding estimate obtained in previous analyses by the authors in this framework, based on a dynamical-system approach to linear-axion RVM inflation. Finally, we examine the role of periodic modulations in the axion potential induced by non-perturbative stringy effects on the slow-roll inflationary parameters, and find compatibility with the cosmological data.