3-point correlation functions

This dataset was prepared for the analysis presented in Labate et al. (prep. for submission), which explores the imprint of massive neutrinos on the halo 3-point correlation function (3PCF). If you use this dataset, please cite the above publication.

3PCF multipoles

The measurements were performed with the code MeasCorr (Farina et al. 2024), which implements the algorithm for estimating the isotropic 3PCF presented in Slepian and Eisenstein (2015). The algorithm estimates the Legendre multipoles \(\zeta_\ell\) of the 3PCF \(\zeta\) rather than the 3PCF itself, which substantially reduces the computational cost.

In general, a triangle with vertices 1, 2, and 3 can be parameterized by the lengths of two sides, e.g. \(r_{12}\) and \(r_{13}\) (respectively joining vertices 1–2 and 1–3), and the angle \(\theta\) between them. The multipoles are functions of the two sides, i.e. \(\zeta_\ell = \zeta_\ell(r_{12}, r_{13})\).

If the multipoles are measured in the range \(0 \le \ell \le \ell_{\max}\), then the 3PCF at any angle \(\theta \in [0, \pi]\) can be estimated by computing the sum

\[\zeta(r_{12}, r_{13}, \theta) = \sum_{\ell = 0}^{\ell_{\max}} \zeta_\ell(r_{12}, r_{13}) \, \mathcal{L}_\ell(\cos\theta),\]

where \(\mathcal{L}_\ell\) is the Legendre polynomial of degree \(\ell\).

Dataset

The dataset includes measurements of the 3PCF multipoles for the simulations fiducial, fiducial_ZA, Mnu_ppp, Mnu_pp, Mnu_p, s8_m, and s8_p at \(z = 0, 1, 2\) in real and redshift space. In the latter case, redshift-space distortions are applied along the \(\hat{z}\) axis of the simulation box.

The details of the simulations and realizations for which the measurements are available are summarized in the table below (see Structure and types for the total number of realizations and the values of the cosmological parameters of each simulation set).

Name	simulations	ICs	realization numbers
fiducial	standard	2LPT	0-2000
fiducial	paired fixed	2LPT	NCV_0_0-NCV_0_249 and NCV_1_0-NCV_1_249
fiducial_ZA	standard	Zeldovich	0-499
Mnu_ppp	standard	Zeldovich	0-499
Mnu_pp	standard	Zeldovich	0-499
Mnu_p	standard	Zeldovich	0-499
s8_p	paired fixed	2LPT	NCV_0_0-NCV_0_249 and NCV_1_0-NCV_1_249
s8_m	paired fixed	2LPT	NCV_0_0-NCV_0_249 and NCV_1_0-NCV_1_249

All the measurements are characterized by the following parameters:

\(2.5 \le r_{12}, r_{13} \, [\mathrm{Mpc}/h] \le 147.5\), using bins of width \(\Delta r = 5 \, \mathrm{Mpc}/h\). The centers of the bins are therefore \(5 + i \times \Delta r\), with \(i = 0,1,\ldots,28\).
\(\ell_{\max} = 10\).

Format

The format of the individual 3PCF files is:

r12 [Mpc/h] | r13 [Mpc/h] | ell | zeta

where r12 and r13 are the values of the centers of the \(r_{12}\) and \(r_{13}\) bins, ell is the multipole index \(\ell\), and zeta is the corresponding multipole \(\zeta_\ell(r_{12}, r_{13})\).

The files for a given simulation simname and realization number num are stored in the folder 3PCF/simname/num/. Each file in this folder follows the naming convention Slepian15_zx.x_2.5_147.5_5_11_simname_num_space.csv, where x.x specifies the redshift (0.0, 1.0, 2.0) and space can be either real or redshift.

Example to read the files:

import numpy as np
import pandas as pd

# choose the simulation name, realization number, redshift and space
simname = 'fiducial'
num = '0'
redshift = '0.0'
space = 'real'

# read the file containing the measurements and store them in a dataframe
meas_folder = f'3PCF/{simname}/{num}'
meas_file = f'{meas_folder}/Slepian15_z{redshift}_2.5_147.5_5_11_{simname}_{num}_{space}.csv'
meas_df = pd.read_csv(meas_file, delimiter='\t')

The value of \(\zeta(r_{12}, r_{13}, \theta)\) can then be obtained by running:

from scipy.special import eval_legendre

# estimate the 3PCF for a given triangle (r12, r13, theta)

# choose r12 and r13 (among the available values of bin centers) and the angle theta (here we choose some example values)
r12 = 50  # in Mpc/h
r13 = 80  # in Mpc/h
theta = np.pi/3

# extract from the dataframe the multipoles corresponding to r12 and r13 and compute the Legendre polynomials at theta
slice_condition = (meas_df["r12 [Mpc/h]"] == r12) & (meas_df["r13 [Mpc/h]"] == r13)
ells = meas_df.loc[slice_condition, "ell"].to_numpy()
zeta_vals = meas_df.loc[slice_condition, "zeta"].to_numpy()
zeta_ells = (2 * ells + 1) * zeta_vals  # the prefactor must be included to ensure the correct normalization
legendre_vals = np.array([eval_legendre(l, np.cos(theta)) for l in ells])

# linearly combine the polynomials with coefficients given by the multipoles
zeta = np.dot(zeta_ells, legendre_vals)

Acknowledgments

The measurements were carried out at the Open Physics Hub (OPH) cluster of the Department of Physics and Astronomy (DiFA) “Augusto Righi”, Alma Mater Studiorum – University of Bologna.

Team

Andrea Labate (University of Bologna, INAF OAS – Italian National Institute for Astrophysics - Astrophysics and Space Science Observatory of Bologna)
Michele Moresco (University of Bologna, INAF OAS)
Massimo Guidi (University of Bologna, INAF OAS)
Alfonso Veropalumbo (University of Genoa, INAF, INFN - Italian National Institute for Nuclear Physics)