  Large-scale surveys allow us to construct cosmological probes such as galaxy
clustering, weak gravitational lensing, the luminosity distance, and cosmic
microwave background anisotropies. The gauge-invariant descriptions of these
cosmological probes reveal the presence of numerous relativistic effects in the
cosmological probes, and they are sensitive (or even divergent) to the long
wave-length fluctuations in the initial conditions. In the standard
$\Lambda$CDM model, this infrared sensitivity is absent due to subtle
cancellations among the relativistic contributions, once the Einstein equation
is used. Here we derive the most general conditions for the absence of infrared
sensitivity in the cosmological probes without committing to general
relativity. We discuss the implications of our results for gravity theories
beyond general relativity.
