Kinematic consistency relations of large-scale structures

We describe how the kinematic consistency relations satisfied by density correlations of the large-scale structures of the Universe can be derived within the usual Newtonian framework. These relations express a kinematic effect and show how the $(\mathrm{\ell }+n)$-density correlation factors in terms of the $n$-point correlation and $\mathrm{\ell }$ linear power spectrum factors, in the limit where the $\mathrm{\ell }$ soft wave numbers become linear and much smaller than the $n$ other wave numbers. We describe how these relations extend to multifluid cases. In the standard cosmology, these consistency relations derive from the equivalence principle. A detection of their violation would indicate non-Gaussian initial conditions, non-negligible decaying modes, or a modification of gravity that does not converge to General Relativity on large scales.