Inflationary model in minimally modified gravity theories

We have investigated inflationary model constructed from minimally modified gravity (MMG) theories. The MMG theory in the form of $f({\bf H}) \propto {\bf H}^{1+p}$ gravity where, ${\bf H}$ is the Hamiltonian constraint in the Einstein gravity and $p$ is constant, has been studied. An inflation is difficult to be achieved in this theory of gravity unless an additional scalar field playing a role of inflaton is introduced in the model. We have found that the inflaton with exponential potential can drive inflation with graceful exist different from the case of Einstein gravity. The slow-roll parameter for both the exponential and the power-law potentials is inversely proportional to number of e-folding similar to the case of the Einstein gravity. We also have found for the scalar perturbation that the curvature perturbation in the comoving gauge on super Hubble radius scales grows rapidly during inflation unless $p =0$. For the tensor modes, the amplitude of the perturbations is constant on large scales, and sound speed of the perturbations can diviate from unity and can vary with time depending on the form of $f({\bf H})$.