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We consider spatially covariant modified gravity in which the would-be scalardegree of freedom is made non-dynamical and hence there are just two tensorialdegrees of freedom, i.e., the same number of dynamical degrees of freedom as ingeneral relativity. Focusing on a class of such modified gravity theoriescharacterized by three functions of time, we discuss how modified gravity withtwo tensorial degrees of freedom can be distinguished observationally orphenomenologically from general relativity. It is checked that the theory givesthe same predictions as general relativity for weak gravitational fields andthe propagation speed of gravitational waves. We also find that there is nomodification to asymptotically flat black hole solutions. Due to a large degreeof freedom to choose the time-dependent functions in the theory, thehomogeneous and isotropic cosmological dynamics can be made close to or evenidentical to that of the $\Lambda$CDM model. We investigate the behavior ofcosmological perturbations in the long and short wavelength limits and showthat in both limits the effects of modified gravity appear only through themodification of the background evolution. Finally, it is remarked that in thepresence of a galileon field in the matter sector, the scalar degree of freedomis revived, ruining the essential feature of the theory.

Distinguishing modified gravity with just two tensorial degrees of freedom from general relativity: Black holes, cosmology, and matter coupling