Extended theories of gravity have been extensively investigated during the last thirty years, aiming at fixing infrared and ultraviolet shortcomings of General Relativity and of the associated $$\varLambda $$ Λ CDM cosmological model. Recently, non-local theories of gravity have drawn increasing attention due to their potential to ameliorate both the ultraviolet and infrared behavior of gravitational interaction. In particular, Integral Kernel theories of Gravity provide a viable mechanism to explain the late time cosmic acceleration so as to avoid the introduction of any form of unknown dark energy. On the other hand, these models represent a natural link towards quantum gravity. Here, we study a scalar-tensor equivalent model of General Relativity corrected with non-local terms, where corrections are selected by the existence of Noether symmetries. After performing the weak field limit and generalizing the results to extended mass distributions, we analyse the non-local model at galaxy cluster scales, by comparing the theoretical predictions with gravitational lensing observations from the CLASH program. We obtain agreement with data at the same level of statistical significance as General Relativity. We also provide constraints for the Navarro–Frenk–White parameters and lower bounds for the non-local length scales. The results are finally compared with those from the literature.

Alternative cosmological models have been under deep scrutiny in recent years, aiming to address the main shortcomings of the ΛCDM model. Moreover, as the accuracy of cosmological surveys improved, new tensions have risen between the model-dependent analysis of the Cosmic Microwave Background and lower redshift probes. Within this framework, we review two quantum-inspired non-locally extended theories of gravity, whose main cosmological feature is a geometrically driven accelerated expansion. The models are especially investigated in light of the Hubble and growth tension, and promising features emerge for the Deser–Woodard one. On the one hand, the cosmological analysis of the phenomenological formulation of the model shows a lowered growth of structures but an equivalent background with respect to ΛCDM. On the other hand, the study of the lensing features at the galaxy cluster scale of a new formulation of non-local cosmology, based on Noether symmetries, makes room for the possibility of alleviating both the H0 and σ8 tension. However, the urgent need for a screening mechanism arises for this non-local theory of gravity.