Testing gravity with the cosmic microwave background: constraints on modified gravity with two tensorial degrees of freedom

We provide a cosmological test of modified gravity with two tensorial degrees of freedom and no extra propagating scalar mode. The theory of gravity we consider admits a cosmological model that is indistinguishable from the $\Lambda$CDM model at the level of the background evolution. The model has a single modified-gravity parameter $\beta$, the effect of which can be seen in linear perturbations, though no extra scalar mode is propagating. Using the Boltzmann code modified to incorporate the present model, we derive the constraints $-0.047 < \beta < -0.028$ at 68$\%$ confidence from Planck CMB data. Since our modified gravity model can hardly be constrained by the Solar System tests and gravitational-wave propagation, our result offers the first observational test on the model.