Modeling and measuring the anisotropic halo 3-point correlation function: a coordinated study

TL;DR

The paper develops an efficient framework to model and estimate the anisotropic 3-point correlation function (3PCF) using a tripolar spherical harmonic basis within the EFT of Large-Scale Structure, paired with an optimized estimator (MeasCorr) and a fast 2D FFT-Log mapping (Mod3L). By validating against large suites of Minerva and Pinocchio halo catalogs, it demonstrates that including anisotropic 3PCF multipoles helps break the degeneracy between the growth rate $f$ and linear bias $b_1$ and improves the Alcock-Paczyński parameter $\varepsilon$, though the isotropic dilation $\alpha$ remains biased by about $\sim 1\%$ due to small-scale nonlinearities. In joint analyses with the 2PCF, the degeneracy among $f$, $\sigma_8$, and $b_1$ is further broken, yielding tighter constraints, but the anisotropic 3PCF adds limited new information because the tree-level 3PCF cannot fully capture small-scale and squeezed-triangle anisotropies. The results motivate incorporating a 1-loop perturbation theory extension to exploit the full anisotropic information in the 3PCF, which would enhance parameter recovery for upcoming surveys; the study also provides public software (Mod3L and MeasCorr) to enable broader adoption.

Abstract

Ongoing and future spectroscopic galaxy surveys will cover unprecedented volumes with a number of objects large enough to effectively probe clustering anisotropies through higher-order statistics. In this work, we present a novel and efficient implementation of both a model for the multipole moments of the anisotropic 3-point correlation function (3PCF) and of their estimator. To evaluate the performance of our model, we compared its predictions against direct 3PCF measurements obtained with our estimator from a set of 298 dark matter halo catalogs drawn from the $z=1$ snapshots of $N$-body simulations. For the statistical analysis, we employed a covariance matrix estimated from an independent suite of 3000 mock halo catalogs at the same redshift. We then repeated the analysis by combining the 2-point correlation function (2PCF) to the 3PCF, with and without including its anisotropic part. In the 3PCF-only analysis, the addition of the anisotropic component of the 3PCF effectively breaks the degeneracy between the growth rate $f$ and the linear bias $b_1$, significantly reducing their uncertainties. It also significantly improves the precision of the Alcock-Paczynski parameter $\varepsilon$ but does not reduce the $\sim 1$% offset we find in the estimate of the isotropic dilation parameter $α$. The joint 2PCF+3PCF analysis reduces, though does not fully remove, biases in the AP and isotropic dilation parameters and breaks the $f$-$b_1$-$σ_8$ degeneracy, leading to tighter constraints overall. The anisotropic 3PCF adds little to the joint analysis because the tree-level 3PCF model fails to capture the anisotropic information primarily encoded on small scales and in squeezed triangle configurations. A more advanced model will be required to exploit this information fully.

Figures (13)

• Figure 1: Example of a numerical covariance matrix used in one of the joint 2 and 3-point correlation analyses performed in this work. The data vector consists of the 2PCF monopole and quadrupole, measured in the interval $[32.5,\, 147.5]\,\,h^{-1} \mathrm{Mpc}$ in bins of $\Delta r^{\mathrm{2pt}} = 5\,\,h^{-1} \mathrm{Mpc}$, and all the six 3PCF multipole moments measured in Sec.(\ref{['sec:measurements']}), namely $\bm d= \left\{\tilde{\zeta}_{000},\,\hat{\zeta}_{110},\,\tilde{\zeta}_{220},\,\tilde{\zeta}_{202},\,\tilde{\zeta}_{112},\,\tilde{\zeta}_{312} \right\}$. These latter are computed in the separation interval $[35,\, 145]\,\,h^{-1} \mathrm{Mpc}$ in bins of $\Delta r^{3pt} = 10\,\,h^{-1} \mathrm{Mpc}$. Only $\eta \ge 3$ configurations are considered.
• Figure 2: Comparison of variances between the entire Minerva simulation suite (orange squares) and the first 298 Pinocchio realizations (blue circles), which share the same initial conditions as Minerva. Top row: 2PCF monopole and quadrupole. Remaining rows: 3PCF multipoles with $r_{\mathrm{min}}^{3\mathrm{pt}}=45\,h^{-1}\mathrm{Mpc}$ and $\eta = 1$. Each triangle index $i$ denotes entries of the flattened $(r_1,r_2)$ grid and corresponds to a unique triangle configuration. Error-bars are estimated via bootstrap resampling and correspond to 95% confidence intervals.
• Figure 3: Comparison of the correlation indices measured from the full Minerva simulation suite (orange squares) with those extracted from the first 298 Pinocchio realizations (blue circles), which share the same initial conditions as Minerva. The three panels correspond to three different rows of the correlation matrix, as reported in the figure. Error-bars are estimated via bootstrap resampling and correspond to the 95% confidence interval.
• Figure 4: Top six panels: best-fit values of the parameters $f$, $b_1$, $b_2$, $b_{\mathcal{G}_2}$, $\varepsilon$ and $\alpha$ as a function of the minimum triangle size $r_{\mathrm{min}}^{\mathrm{3pt}}$. Different marker styles and colors correspond to distinct triangle configurations defined by the $\eta$ parameter, as indicated in the legend. Error bars represent the 68% uncertainty interval derived from the 1D marginalized posterior distributions. The horizontal black dashed lines, instead, correspond to the reference values of the parameters estimated by Veropalumbo:2022 in real space. Bottom panel: reduced $\chi^2$ value estimated from the 298 Minerva halo catalogs as a function of $r_{\mathrm{min}}^{\mathrm{3pt}}$. The colored bands represent the 95% confidence region associated to a $\chi^2$ distribution characterized by $N_{\mathrm{real}}N_{\mathrm{bin}} - 6$dof. Different colors are used for the different values of $\eta$ considered in the analysis.
• Figure 5: Marginalized 1D and 2D posterior probability contours for the parameters $f$, $b_1$, $b_2$, $b_{\mathcal{G}_2}$, $\varepsilon$ and $\alpha$ estimated from a 3PCF-only analysis. The red contours consider only the isotropic multipole moments ($\tilde{\zeta}_{000}$, $\tilde{\zeta}_{110}$ and $\tilde{\zeta}_{220}$ ), while the blue contours include both isotropic and anisotropic multipoles ($\tilde{\zeta}_{000}$, $\tilde{\zeta}_{110}$, $\tilde{\zeta}_{220}$, $\tilde{\zeta}_{202}$, $\tilde{\zeta}_{112}$, and $\tilde{\zeta}_{312}$). The black dashed lines represent the reference values of the main parameters. The probability contours refer to a survey volume of $1000\,\,h^{-3} \mathrm{Gpc}^{3}$, considering triangles with sizes above the reference value $r_{\min}^{\mathrm{3pt}}=65\,\,h^{-1} \mathrm{Mpc}$ and shape parameter $\eta=3$.