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 JCAP 01 (2019) 017 [arXiv:1810.01047
[gr-qc]], this is a type-II minimally modified gravity theory. The theory is
characterized by the gravitational constant $G_{\rm N}$ and a function
$V(\phi)$ of a non-dynamical auxiliary field $\phi$ that plays the role of dark
energy. Once one fixes a homogeneous and isotropic cosmological background, the
form of $V(\phi)$ is determined and the theory no longer possesses a free
parameter or a free function, besides $G_{\rm N}$. For $V'(\phi) = 0$ the
theory reduces to general relativity (GR) with $G_N$ being the Newton's
constant and $V=const.$ being the cosmological constant. For $V'(\phi) \ne 0$,
it is shown that gravity behaves differently from GR but that GR with $G_{\rm
N}$ being the Newton's constant is recovered for weak gravity at distance and
time scales sufficiently shorter than the scale associated with $V(\phi)$.
Therefore this theory provides the simplest framework of cosmology in which
deviations from GR can be tested by observational data.
