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We investigate two Type-IIa Minimally Modified Gravity theories, namely VCDM and Cuscuton theories. We confirm that all acceptable Cuscuton solutions are always solutions for VCDM theory. However, the inverse does not hold. We find that VCDM allows for the existence of exact General Relativity (GR) solutions with or without the presence of matter fields and a cosmological constant. We determine the conditions of existence for such GR-VCDM solutions in terms of the trace of the extrinsic curvature and on the fields which define the VCDM theory. On the other hand, for the Cuscuton theory, we find that the same set of exact GR solutions (such as Schwarzschild and Kerr spacetimes) is not compatible with timelike configurations of the Cuscuton field and therefore cannot be considered as acceptable solutions. Nonetheless, in Cuscuton theory, there could exist solutions which are not the same but close enough to GR solutions. We also show the conditions to determine intrinsic-VCDM solutions, i.e. solutions which differ from GR and do not belong to the Cuscuton model. We finally show that in cosmology a mapping between VCDM and the Cuscuton is possible, for a generic form of the VCDM potential. In particular, we find that for a quadratic potential in VCDM theory, this mapping is well defined giving an effective redefinition of the Planck mass for the cosmological background solutions of both theories., 19 pages+appendices, 1 figure, match with published version

The Minimal theory of Massive Gravity (MTMG) is endowed non-linearly with only two tensor modes in the gravity sector which acquire a non-zero mass. On a homogeneous and isotropic background the theory is known to possess two branches: the self-accelerating branch with a phenomenology in cosmology which, except for the mass of the tensor modes, exactly matches the one of $\Lambda$CDM; and the normal branch which instead shows deviation from General Relativity in terms of both background and linear perturbations dynamics. For the latter branch we study using several early and late times data sets the constraints on today's value of the graviton mass $\mu_{0}$, finding that $(\mu_{0}/H_{0})^{2}=0.119_{-0.098}^{+0.12}$ at $68\%$ CL, which in turn gives an upper bound at $95\%$ CL as $\mu_{0}, Comment: 16 pages, 5 figures, uses revtex

In this letter we propose a reduction of the H0 tension puzzle by means of a theory of minimally modified gravity which is dubbed VCDM. After confronting the theory with the data, a transition in the expansion history of the universe in the low-redshift z≃0.3 is found. From the bestfit values the total fitness parameter is improved by Δχ2=33.41, for the data set considered. We then infer the local Hubble expansion rate today within this theory by means of low redshift Pantheon data. The resulting local Hubble expansion rate today is H0loc=73.69±1.4. Hence the tension is reduced within the VCDM theory.