Inflationary physics and transplanckian conjecture in the Stringy Running-Vacuum-Model: from the phantom vacuum to the true vacuum

In previous works we have embedded the Running Vacuum Model (RVM) of Cosmology in the framework of string theory. Specifically, we considered a string-inspired Cosmology with primordial gravitational waves (GW) and gravitational anomalies, which were argued to lead, via appropriate condensation during the very-early-universe era, to dynamical inflation, of RVM type, without the need for extra inflaton fields. A crucial role for the associated slow-roll nature of the inflationary era was played by the fundamental axion field that exists in the gravitational multiplet of strings, viz. the Kalb-Ramond (KR) axion. In this paper, we study further this model and demonstrate several novel facts, completing our previous studies. We clarify the different roles played by the background KR axions, which constitute a form of stiff matter that dominates a pre-RVM-inflationary epoch of the Universe. We show that the KR axion when combined with the gravitational Chern-Simons contribution obey together a peculiar equation of state $p=-\rho$, with negative energy density $\rho < 0$, which we call `phantom vacuum'. Eventually, this state transmutes into the standard vacuum state thanks to the contribution from the gravitational Chern-Simons condensate, which makes the total $\rho_{\rm total} =-p_{\rm total} >0$. At this point the RVM vacuum picture emerges naturally, with an overall equation of state of true vacuum type. We find that our scenario is consistent with the trans-Planckian censorship hypothesis, given that the rate of change of the KR axion takes only sub-Planckian values. We also discuss some explicit scenarios on the creation of GWs within the context of supergravity models that could be embedded in our string theory framework. {Finally, we argue how the RVM in our context can help to alleviate the current tensions of the $\Lambda$CDM.}