  Consistency relations of large-scale structures provide exact nonperturbative
results for cross-correlations of cosmic fields in the squeezed limit. They
only depend on the equivalence principle and the assumption of Gaussian initial
conditions, and remain nonzero at equal times for cross-correlations of density
fields with velocity or momentum fields, or with the time derivative of density
fields. We show how to apply these relations to observational probes that
involve the integrated Sachs-Wolfe effect or the kinematic Sunyaev-Zeldovich
effect. In the squeezed limit, this allows us to express the three-point
cross-correlations, or bispectra, of two galaxy or matter density fields, or
weak lensing convergence fields, with the secondary Cosmic Microwave Background
(CMB) distortion in terms of products of a linear and a nonlinear power
spectrum. In particular, we find that cross-correlations with the integrated
Sachs-Wolfe effect show a specific angular dependence. These results could be
used to test the equivalence principle and the primordial Gaussianity, or to
check the modeling of large-scale structures.
