The Hubble parameter experimental puzzle in the context of Modfied Gravity models

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During the last year, we have been investigating how modified gravity models could face the tensions present in cosmology.In particular, cosmology is facing two tensions: 1) the determination of H0, thetoday's rate of expansion of the universe; 2) the rate of formation of structures in our universe. The predictions of general relativity in the presence of a cold dark matter fluid and of a cosmological constant seem to be not consistent with the most recent observations of these two observables. In particular, the first tension is about 5 sigmas, and at the moment, people seem to have checked all the possible systematics which can occur in the experimental determination of H0.In the models, I have proposed with my collaborators, mostly Prof. Mukohyama (YITP) and Prof. Tsujikawa (Waseda U.), we have been pushing forward the idea that modified gravity models could account for both these tensions. We have used both powerful calculations and theoretical methods in order to explore the strengths and weaknesses of some of the models we have recently introduced in the literature.I do hope that in the near future experimental data will be further tested to show the real presence of these tensions. If so, general relativity will require some modifications. Evidently, I want to further explore methods and ways to falsify (or prove) the theories I have been working on. The future might look very bright, hopefully. Since April 2022, I have published 9 papers in famous peer-review journals (JCAP, PRD, and PRL).