Squeezing $f_{\rm NL}$ out of the matter bispectrum with consistency relations

We show how consistency relations can be used to robustly extract the amplitude of local primordial non-Gaussianity ($f_{\rm NL}$) from the squeezed limit of the matter bispectrum, well into the non-linear regime. First, we derive a non-perturbative relation between primordial non-Gaussianity and the leading term in the squeezed bispectrum, revising some results present in the literature. This relation is then used to successfully measure $f_{\rm NL}$ from $N$-body simulations. We discuss the dependence of our results on different scale cuts and redshifts. Specifically, the analysis is strongly dependent on the choice of the smallest soft momentum, $q_{\rm min}$, which is the most sensitive to primordial bispectrum contributions, but is largely independent of the choice of the largest hard momentum, $k_{\rm max}$, due to the non-Gaussian nature of the covariance. We also show how the constraints on $f_{\rm NL}$ improve at higher redshift, due to a reduced off-diagonal covariance. In particular, for a simulation with $f_{\rm NL} = 100$ and a volume of $(2.4 \text{ Gpc}/h)^3$, we measure $f_{\rm NL} = 98 \pm 12$ at redshift $z=0$ and $f_{\rm NL} = 97 \pm 8$ at $z=0.97$. Finally, we compare our results with a Fisher forecast, showing that the current version of the analysis is satisfactorily close to the Fisher error. We regard this as a first step towards the realistic application of consistency relations to constrain primordial non-Gaussianity using observations.