Do we owe our existence to a Gravitational Anomaly?

I review a mechanism for a potential explanation of the dominance of matter over antimatter in the Universe (``our existence''), based on a (3+1)-dimensional string-inspired cosmological model, characterised by (uncancelled) gravitational Chern-Simons (gCS) anomalies in the premordial epochs. The model is consistent with general covariance but entails spontaneous Lorentz (LV) and CPT Violation (CPTV), due to appropriate backgrounds of axion fields, which couple to the anomaly terms. Upon condensation of the gCS terms, due to primordial gravitational waves, these axions acquire a linear potential, which drives inflation of a Running-Vacuum-Model (RVM) type, while the effective action is also characterised by imaginary parts as well, which imply metastability of this RVM inflation. The axion backgrounds remain undiluted until the exit from inflation, and are responsible for yielding unconventional LV and CPTV Leptogenesis, which is then communicated to the baryon sector (Baryogenesis) via sphaleron processes. The model provides a phenomenologically consistent cosmological evolution from inflation till the modern era.