  In this work, we elaborate further on a cosmological model of inflation that
characterises Chern-Simons (CS) gravity models inspired from string theory.
Such models are known to belong to the class of the so-called String-Inspired
Running Vacuum Cosmologies. In particular, by applying methods of dynamical
systems, commonly used in scalar-field cosmology, we examine in detail, for the
first time, the passage from a pre-inflationary era dominated by a
stiff-axion-matter equation of state, characteristic of the model, to inflation
of Running Vacuum Model (RVM) type. By a careful discussion of the formation of
the condensate of the CS gravitational anomaly term, induced by populations of
primordial gravitational waves at the end of the stiff-axion-matter era, we
show that an effectively linear axion-monodromy potential arises. This
eventually causes the transition from the matter to the RVM inflation. By
taking into account terms that have previously been ignored in the relevant
literature of weak-graviton quantisation, we show that the effect of such terms
is to diminish the value of the condensate by half, remaining however in the
same order of magnitude. This, in turn, implies that the qualitative
conclusions of previous works on the subject remain valid. Moreover, on
assuming the approximate cosmic-time independence of the gravitational-CS
condensate, we also provide an estimate of the number of sources of the
primordial gravitational waves, upon the requirement of respecting the
transplanckian conjecture.
