A theory of type-II minimally modified gravity

We propose a modified gravity theory that propagates only two local gravitational degrees of freedom and that does not have an Einstein frame. According to the classification in [2], this is a type-II minimally modified gravity theory. The theory is characterized by the gravitational constant $G_N$ and a function $V\left(\varphi \right)$ of a non-dynamical auxiliary field $\varphi$ that plays the role of dark energy. Once one fixes a homogeneous and isotropic cosmological background, the form of $V\left(\varphi \right)$ is determined and the theory no longer possesses a free parameter or a free function, besides $G_N$. For $V^{\prime }\left(\varphi \right)=0$ the theory reduces to general relativity (GR) with $G_N$ being the Newton's constant and $V=const.$ being the cosmological constant. For $V^{\prime }\left(\varphi \right)\ne 0$, it is shown that gravity behaves differently from GR but that GR with $G_N$ being the Newton's constant is recovered for weak gravity at distance and time scales sufficiently shorter than the scale associated with $V\left(\varphi \right)$. Therefore this theory provides the simplest framework of cosmology in which deviations from GR can be tested by observational data.