Modified Gravity and Cosmology: An Update by the Cantata Network

Introduction.- General Relativity.- Foundations of gravity - modifications and extensions.- Part I: Theories of Gravity.- Introduction.- A flavour on f(R) theories: theory and observations.- Horndeski/Galileon theories.- Massive Gravity and Bimetric Gravity.- Gravity in extra dimensions.- Non-local models.- Metric-Affine gravity.- Geometric Foundations of Gravity.- Palatini theories of gravity.- Hybrid metric-Palatini gravity and cosmology.- Teleparallel and f(T) Gravity: Foundations and Cosmology.- Finsler gravity .- Gravity's Rainbow.- Quantum Cosmology in modified theories of gravity.- Part II: Testing Relativistic Effects.- Introduction.- Laboratory Constraints .- Screening mechanisms.- Microscopic effects of modified gravity.- Compact stars as tests of modified gravity.- Compact objects in General Relativity and beyond.- Parametrized post-Newtonian formalism.- Gravitational Waves .- Gravitational lensing.- Classicalizing gravity.- Part III: Cosmology and Observational Discriminators.- Introduction.- Phenomenological tests of gravity on cosmological scales.- Relativistic effects.- Cosmological constraints from the effective field theory of dark energy .- The H0 tensions to discriminate among concurring models.- σ8 tension. Is gravity getting weaker at low z? Observational evidence and theoretical implications.- Testing gravity with standard sirens: challenges and opportunities.- Testing the dark universe with cosmic shear.- Galaxy clusters and modified gravity.- Probing screening modified gravity with non-linear structure formation .- Conclusions.- Outlook.- The end of the beginning.