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We address the impact of consistent modifications of gravity on the largestobservable scales, focusing on relativistic effects in galaxy number counts andthe cross-correlation between the matter large scale structure (LSS)distribution and the cosmic microwave background (CMB). Our analysis applies toa very broad class of general scalar-tensor theories encoded in the HorndeskiLagrangian and is fully consistent on linear scales, retaining the fulldynamics of the scalar field and not assuming quasi-static evolution. Asparticular examples we consider self-accelerating Covariant Galileons,Brans-Dicke theory and parameterizations based on the effective field theory ofdark energy, using the \hiclass\, code to address the impact of these models onrelativistic corrections to LSS observables. We find that especially effectswhich involve integrals along the line of sight (lensing convergence, timedelay and the integrated Sachs-Wolfe effect -- ISW) can be considerablymodified, and even lead to $\mathcal{O}(1000\%)$ deviations from GeneralRelativity in the case of the ISW effect for Galileon models, for whichstandard probes such as the growth function only vary by $\mathcal{O}(10\%)$.These effects become dominant when correlating galaxy number counts atdifferent redshifts and can lead to $\sim 50\%$ deviations in the total signalthat might be observable by future LSS surveys. Because of their integratednature, these deep-redshift cross-correlations are sensitive to modificationsof gravity even when probing eras much before dark energy domination. Wefurther isolate the ISW effect using the cross-correlation between LSS and CMBtemperature anisotropies and use current data to further constrain Horndeskimodels (abridged).

Gravity at the horizon: on relativistic effects, CMB-LSS correlations and ultra-large scales in Horndeski's theory