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Since the discovery of the accelerated expansion of the present Universe,significant theoretical developments have been made in the area of modifiedgravity. In the meantime, cosmological observations have been providing morehigh-quality data, allowing us to explore gravity on cosmological scales. Tobridge the recent theoretical developments and observations, we present anoverview of a variety of modified theories of gravity and the cosmologicalobservables in the cosmic microwave background and large-scale structure,supplemented with a summary of predictions for cosmological observables derivedfrom cosmological perturbations and sophisticated numerical studies. Wespecifically consider scalar-tensor theories in the Horndeski and DHOST family,massive gravity/bigravity, vector-tensor theories, metric-affine gravity, andcuscuton/minimally-modified gravity, and discuss the current status of thosetheories with emphasis on their physical motivations, validity, appealingfeatures, the level of maturity, and calculability. We conclude that theHorndeski theory is one of the most well-developed theories of modifiedgravity, although several remaining issues are left for future observations.The paper aims to help to develop strategies for testing gravity with ongoingand forthcoming cosmological observations.

Cosmological gravity probes: connecting recent theoretical developments to forthcoming observations