Inflation in exponential scalar model and finite-time singularity
  induced instability
  We investigate how a Type IV future singularity can be included in the
cosmological evolution of a well-known exponential model of inflation. In order
to achieve this we use a two scalar field model, in the context of which the
incorporation of the Type IV singularity can be consistently done. In the
context of the exponential model we study, when a Type IV singularity is
included in the evolution, an instability occurs in the slow-roll parameters,
and in particular on the second slow-roll parameter. Particularly, if we
abandon the slow-roll condition for both the scalars we shall use, then the
most consistent description of the dynamics of the inflationary era is provided
by the Hubble slow-roll parameters $\epsilon_H$ and $\eta_H$. Then, the second
Hubble slow-roll parameter $\eta_H$, which measures the duration of the
inflationary era, becomes singular at the point where the Type IV singularity
is chosen to occur, while the Hubble slow-roll parameter $\epsilon_H$ is
regular there. Therefore, this infinite singularity indicates that the
occurrence of the finite-time singularity is responsible for the instability in
the scalar field model we study. This sort of instability has it's imprint on
the dynamical system that can be constructed from the cosmological equations,
with the dynamical system being unstable. In addition, the instability due to
the singularity mechanism we propose, is discussed in the context of other
inflationary scalar potentials. Finally, we discuss the implications of such a
singularity in the Hubble slow-roll parameters and we also critically discuss
qualitatively, what implications could this effect have on the graceful exit
problem of the exponential model.
