$F(R)$ Gravity with an Axion-like Particle: Dynamics, Gravity Waves,
  Late and Early-time Phenomenology
  In this work we investigate several theoretical and phenomenological
implications of a scalar -$F(R)$ gravity containing a non-minimal coupling to
the scalar curvature. This kind of model is a generalization of axion-$F(R)$
gravity models, so we shall examine several implications of the latter theory.
Firstly we study in detail the Einstein frame picture of the model, and also we
discuss the dynamics of the cosmological system. By appropriately using the
equations of motion, we demonstrate that an arbitrary cosmological evolution
can be realized. Also we study the gravitational waves of the theory, and we
demonstrate that the speed of their propagation is the same as in $F(R)$
gravity, but there is the possibility of enhancement or dissipation of the
gravitational waves, an effect quite similar to the propagation of gravity
waves in a viscous fluid. Finally, we examine the energy momentum tensor and we
investigate which quantities related to it are conserved. We also present the
constraints imposed by the radiation domination era on the non-minimal coupling
of the axion scalar field to the scalar curvature.
