Capturing primordial non-Gaussian signatures in the late Universe by multi-scale extrema of the cosmic log-density field

We construct two new summary statistics, the scale-dependent peak height function (scale-PKHF) and the scale-dependent valley depth function (scale-VLYDF) of matter density, and forecast their constraining power on primordial non-Gaussianity and cosmological parameters based on \textsc{Quijote} and \textsc{Quijote-PNG} simulations at $z=0$. With the Fisher analysis, we demonstrate that these statistics outperform the power spectrum and bispectrum. Key findings include: (1) the constraint on the scalar spectral index $n_s$ obtained from the scale-VLYDF/scale-PKHF is 1.59/1.10 times tighter than that from the joint analysis of power spectrum and bispectrum; (2) the combination of the two statistics yields a slight improvement in constraining $\{f_\mathrm{NL}^\mathrm{local}, f_\mathrm{NL}^\mathrm{equil}\}$ over the power spectrum-bispectrum combination, and provides a 1.39-fold improvement in the constraint on $f_\mathrm{NL}^\mathrm{ortho}$; (3) after incorporating the power spectrum with our new statistics, parameter constraints surpass those from power spectrum-bispectrum combination by factors up to 2.93. This work offers an effective scheme for extracting primordial signals from the late Universe, paving the way for further breakthroughs in precision cosmology.