Gravity at the horizon: on relativistic effects, CMB-LSS correlations
  and ultra-large scales in Horndeski's theory
  We address the impact of consistent modifications of gravity on the largest
observable scales, focusing on relativistic effects in galaxy number counts and
the cross-correlation between the matter large scale structure (LSS)
distribution and the cosmic microwave background (CMB). Our analysis applies to
a very broad class of general scalar-tensor theories encoded in the Horndeski
Lagrangian and is fully consistent on linear scales, retaining the full
dynamics of the scalar field and not assuming quasi-static evolution. As
particular examples we consider self-accelerating Covariant Galileons,
Brans-Dicke theory and parameterizations based on the effective field theory of
dark energy, using the \hiclass\, code to address the impact of these models on
relativistic corrections to LSS observables. We find that especially effects
which involve integrals along the line of sight (lensing convergence, time
delay and the integrated Sachs-Wolfe effect -- ISW) can be considerably
modified, and even lead to $\mathcal{O}(1000\%)$ deviations from General
Relativity in the case of the ISW effect for Galileon models, for which
standard probes such as the growth function only vary by $\mathcal{O}(10\%)$.
These effects become dominant when correlating galaxy number counts at
different redshifts and can lead to $\sim 50\%$ deviations in the total signal
that might be observable by future LSS surveys. Because of their integrated
nature, these deep-redshift cross-correlations are sensitive to modifications
of gravity even when probing eras much before dark energy domination. We
further isolate the ISW effect using the cross-correlation between LSS and CMB
temperature anisotropies and use current data to further constrain Horndeski
models (abridged).
