The Reconstruction of $f(\phi)R$ and Mimetic Gravity from Viable
  Slow-roll Inflation
  In this work, we extend the bottom-up reconstruction framework of $F(R)$
gravity to other modified gravities, and in particular for $f(\phi)R$ and
mimetic $F(R)$ gravities. We investigate which are the important conditions in
order for the method to work, and we study several viable cosmological
evolutions, focusing on the inflationary era. Particularly, for the $f(\phi)R$
theory case, we specify the functional form of the Hubble rate and of the
scalar-to-tensor ratio as a function of the $e$-foldings number and
accordingly, the rest of the physical quantities and also the slow-roll and the
corresponding observational indices can be calculated. The same method is
applied in the mimetic $F(R)$ gravity case, and in both cases we thoroughly
analyze the resulting free parameter space, in order to show that the viability
of the models presented is guaranteed and secondly that there is a wide range
of values of the free parameters for which the viability of the models occurs.
In addition, the reconstruction method is also studied in the context of
mimetic $F(R)=R$ gravity. As we demonstrate, the resulting theory is viable,
and also in this case, only the scalar-to-tensor ratio needs to be specified,
since the rest follow from this condition. Finally, we discuss in brief how the
reconstruction method could function for other modified gravities.
