  In the theories described by a volume normalized action functional, an
arbitrary cosmological constant is eliminated from the physical picture of our
Universe, and dynamical alternatives must be responsible for today's
accelerated expansion as well as for the conjectured early-time inflation in
the Universe. A few well-known such scenarios realized by a single homogeneous
scalar field are examined in this new context, and their diverse fates are
elucidated. Typical inflationary models are not affected at the level of the
background evolution, and also give rise to the scalar perturbations equivalent
to those obtained in the standard General Relativity; however, the primordial
quantum tensor fluctuations are absent in the new framework, irrespective of
the inflationary model. As a consequence, our proposal would be ruled out
should the primordial tensor modes, or their indirect consequences, be
discovered in observations.
