I have investigated many types of modified gravity theories and considered the various applications based on different approaches. I briefly introduce my research accomplishments for each modified gravity theory. (Last update: 2021/2/6)
The scalar-tensor theory is the gravitational theory coupling with scalar field. The simplest case is expressed as the GR with the minimally coupling canonical scalar field. The most general scalar-tensor theory with the single scalar field is known as the Horndeski theory, which produces the second-order equation of motion.
The action of F(R) gravity is replaced by the function of the Ricci scalar instead of the Einstein-Hilbert action of GR. The functional degrees of freedom in F(R) gravity is written in terms of the scalar field, and then, it is known that the F(R) gravity is equivalent to the scalar-tensor theory at the classical level.
The Born-Infeld gravity is a non-linear extension of GR motivated by the QG. This theory is different from GR in the high energy scale although it restores GR in the low energy scale. If the Born-Infeld gravity is treated as an effective theory of QG, we can evaluate the quantum effect in this theory.
The de Rham-Gabadaze-Tolley (dRGT) massive gravity describes non-linear interacting massive spin-2 field, which is referred to the massive graviton. The interaction terms are constructed to avoid the ghost mode and realize the Vainshtein mechanism.
The bigravity theory includes two independent metric tensors. This theory contains a massive spin-2 field called massive graviton in addition to the ordinary massless spin-2 gravity, and it has been successfully constructed as the generalization of dRGT massive gravity.