Climbing the $N$-point Ladder Part I: Information in the Higher-Order Configuration-Space Clustering of Dark Matter Halos

Abstract

We quantify the information content of the configuration-space two-, three-, and connected four-point correlation functions of Quijote dark-matter haloes at $z=0$ and fixed number density. We build Fisher forecasts for $\{{\Omega }_m,{\Omega }_b,h,n_s,{\sigma }_8,M_{\nu }\}$ in real and redshift space from $\sim 38,000$ GPU-accelerated $N$-point measurements. Treating the statistics as a ladder, $2PCF\to +3PCF\to +{\zeta }_{\mathrm{conn}}^{(4)}$, we report the information gained at each rung. The 3PCF supplies most of the accessible higher-order information: it tightens every parameter, most strongly ${\sigma }_8$ and $M_{\nu }$, whose degeneracy it partially breaks, by factors of $1.6$--$6.3$ over the redshift-space $\{{\xi }_0,{\xi }_2\}$ baseline with per-parameter gains consistent with those of the Fourier-space halo bispectrum on the same simulations. The 3PCF gains persist under conservative, modelling-motivated scale cuts: restricted to the tree-level-validated regime with minimum triangle side $\ge 40h^{-1}\mathrm{Mpc}$, it still improves $M_{\nu }$ by $3.9\times$. The connected 4PCF adds a further $\sim 1.2$--$1.5\times$, an increment insensitive to whether the quadrupole is included in the baseline but traceable to small-scale configurations with sides $\lesssim 30h^{-1}\mathrm{Mpc}$. This rung-to-rung increment is stable against derivative-sample noise and compression regularization, whereas the absolute constraints remain limited by the finite simulation ensembles and are reported as preliminary. The configuration-space ladder thus offers an independent and complementary route to the higher-order information probed by the Fourier-space poly-spectra.

37 pages, 8 figures. Accepted in JCAP