Consistency Conditions and Constraints on Generalized f(R) Gravity with Arbitrary Geometry-Matter Coupling

We study the consistency conditions of the generalized f(R) gravity by extending f(R) gravity with non-minimal coupling to the generalized f(R) with arbitrary geometry-matter coupling. Specifically, we discuss the two particular models of generalized f(R) by means of consistency conditions. It is found that the second model is not physically viable so as to be ruled out. Moreover, we further constrain the first model using the Dolgov–Kawasaki stability criterion, and give the value ranges of the parameters in the first model. It is worth stressing that our results include the ones in f(R) gravity with non-minimal coupling as the special case of Q(L_\rm m)=L_\rm m.