Weakening gravity for dark matter in a type-II minimally modified
  gravity
  We propose a new cosmological framework in which the strength of the
gravitational force acted on dark matter at late time can be weaker than that
on the standard matter fields without introducing extra gravitational degrees
of freedom. The framework integrates dark matter into a type-II minimally
modified gravity that was recently proposed as a dark energy mimicker. The idea
that makes such a framework possible consists of coupling a dark matter
Lagrangian and a cosmological constant to the metric in a canonically
transformed frame of general relativity (GR). On imposing a gauge fixing
constraint, which explicitly breaks the temporal diffeomorphism invariance, we
keep the number of gravitational degrees of freedom to be two, as in GR. We
then make the inverse canonical transformation to bring the theory back to the
original frame, where one can add the standard matter fields. This framework
contains two free functions of time which specify the generating functional of
the above mentioned canonical transformation and which are then used in order
to realize desired time evolutions of both the Hubble expansion rate $H(z)$ and
the effective gravitational constant for dark matter $G_{\rm eff}(z)$. The aim
of this paper is therefore to provide a new framework to address the two
puzzles present in today's cosmology, i.e. the $H_0$ tension and the $S_8$
tension, simultaneously. When the dark matter is cold in this framework, we dub
the corresponding cosmological model the V Canonical Cold Dark Matter (VCCDM),
as the cosmological constant $\Lambda$ in the standard $\Lambda$CDM is replaced
by a function $V(\phi)$ of an auxiliary field $\phi$ and the CDM is minimally
coupled to the metric in a canonically transformed frame.
