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 todays 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.
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