The angle-averaged squeezed limit of nonlinear matter N-point functions

We show that in a certain, angle-averaged squeezed limit, the $N$-point function of matter is related to the response of the matter power spectrum to a long-wavelength density perturbation, $P^{-1}d^nP(k|\delta_L)/d\delta_L^n|_{\delta_L=0}$, with $n=N-2$. By performing N-body simulations with a homogeneous overdensity superimposed on a flat Friedmann-Robertson-Lema\^itre-Walker (FRLW) universe using the \emph{separate universe} approach, we obtain measurements of the nonlinear matter power spectrum response up to $n=3$, which is equivalent to measuring the fully nonlinear matter $3-$ to $5-$point function in this squeezed limit. The sub-percent to few percent accuracy of those measurements is unprecedented. We then test the hypothesis that nonlinear $N$-point functions at a given time are a function of the linear power spectrum at that time, which is predicted by standard perturbation theory (SPT) and its variants that are based on the ideal pressureless fluid equations. Specifically, we compare the responses computed from the separate universe simulations and simulations with a rescaled initial (linear) power spectrum amplitude. We find discrepancies of 10\% at $k\simeq 0.2 - 0.5 \,h\,{\rm Mpc}^{-1}$ for $5-$ to $3-$point functions at $z=0$. The discrepancy occurs at higher wavenumbers at $z=2$. Thus, SPT and its variants, carried out to arbitrarily high order, are guaranteed to fail to describe matter $N$-point functions ($N>2$) around that scale.