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First full-shape joint analysis of the two- and three-point correlation functions on real data: $\Lambda$ CDM cosmological constraints from BOSS DR12

This paper presents the first full-shape joint cosmological analysis of the two- and three-point correlation functions using BOSS DR12 data, demonstrating that combining these statistics yields significant improvements in constraining key $\Lambda$ CDM parameters compared to using the two-point function alone.

Original authors: Massimo Guidi, Michele Moresco, Alfonso Veropalumbo, Lorenzo Cavazzini, Antonio Farina, Andrea Labate

Published 2026-06-26

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CC BY 4.0

Original authors: Massimo Guidi, Michele Moresco, Alfonso Veropalumbo, Lorenzo Cavazzini, Antonio Farina, Andrea Labate

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, three-dimensional city made of stars and galaxies. For decades, astronomers have been trying to understand how this city was built and how it's expanding. To do this, they usually look at how galaxies are spaced out from one another, measuring the distance between pairs of neighbors. This is like looking at a map and measuring the distance between every two houses to figure out the layout of the neighborhood. This method, called the "two-point correlation," has been very successful.

However, this paper argues that looking at pairs isn't enough. Just like knowing the distance between two houses doesn't tell you if they are part of a cul-de-sac, a triangle, or a straight line, looking only at pairs misses a lot of the story. The authors decided to look at triangles of galaxies instead. They measured the "three-point correlation," which looks at how three galaxies relate to each other simultaneously.

Here is a breakdown of what they did and found, using simple analogies:

1. The Challenge: The "Fuzzy" Map

When we look at galaxies, their positions aren't perfectly clear. Galaxies are moving, and this motion distorts how we see them, kind of like how a car looks stretched out when it speeds past you in a photo. This is called "redshift-space distortion."

Previous attempts to use the "triangle" method (3-point) were stuck because:

Too much math: Calculating triangles for millions of galaxies is incredibly hard, like trying to count every possible group of three people in a stadium of 100,000 people.
No good models: Scientists didn't have a reliable "rulebook" to predict what these triangles should look like in a distorted, moving universe.

2. The Solution: A New "Rulebook" and a Super-Computer

The authors created a new, sophisticated rulebook (a theoretical model) that accounts for the motion of galaxies and the way they cluster. They used a clever mathematical trick called the "Velocity Difference Generating Function" (VDG). Think of this as a special lens that corrects the "fuzziness" caused by the galaxies' movement, allowing them to see the true shape of the cosmic triangles.

To make the math fast enough to run on a computer, they built a neural network emulator. Imagine a student who has studied millions of practice exams (simulations) and can now instantly guess the answer to a new question without doing the heavy math from scratch. This "emulator" allowed them to test their theories against real data quickly.

3. The Data: The "BOSS" Survey

They tested their new method on data from the BOSS DR12 survey, which mapped out over a million galaxies. They split these galaxies into two groups: "low-z" (closer to us, like the suburbs) and "high-z" (farther away, like the distant countryside).

4. The Results: Triangles Reveal More Secrets

When they combined the old method (pairs/2-point) with their new method (triangles/3-point), the results were significantly better.

The "Hubble Constant" (How fast the universe is expanding): This is the most important number they measured. By adding the triangle data, they improved the precision of this measurement by 29%. It's like going from guessing the speed of a car within 10 mph to guessing it within 7 mph.
Dark Matter Density: They improved the precision on how much invisible "dark matter" exists by 10%.
Clumping of Matter: They improved the precision on how much matter clumps together by 24%.

Why did triangles help?
The authors explain that the "triangle" configurations contain a hidden map of the Baryon Acoustic Oscillations (BAO). You can think of BAO as a "standard ruler" left over from the Big Bang. While the "pair" method sees this ruler, the "triangle" method sees it from different angles and in different shapes, providing extra clues that help break up confusing overlaps (degeneracies) in the data. It's like trying to guess the shape of a hidden object: looking at it from the side (pairs) is good, but looking at it from the corner (triangles) gives you a much clearer 3D picture.

5. The Conclusion

This paper marks a major milestone: it is the first time scientists have successfully used this full "triangle" method on real, messy astronomical data to measure the fundamental properties of the universe.

They found that their new model works well down to a certain scale (about 60 million light-years). Below that, the "fuzziness" of the galaxy movements gets too complicated for the current rules. But within that safe zone, the new method proves that looking at cosmic triangles is a powerful way to tighten our understanding of the universe's expansion and composition, offering a clearer view than looking at pairs alone.

Technical Summary: First Full-Shape Joint Analysis of 2PCF and 3PCF on Real Data

Problem Statement
While the three-point correlation function (3PCF) encodes cosmological information beyond the two-point correlation function (2PCF), a full-shape joint analysis using real observational data in redshift space has been historically lacking. Previous efforts have largely focused on the bispectrum in Fourier space due to the availability of efficient estimators and perturbation theory (PT) frameworks. Configuration-space analyses face significant hurdles, including the $O(N^3)$ computational cost of triplet counting and the absence of accurate, efficient theoretical models for the redshift-space 3PCF suitable for full-shape likelihood analyses. Although perturbative models for the real-space 3PCF have been validated, extending this to redshift space—where redshift-space distortions (RSD) and Alcock–Paczyński (AP) effects must be modeled—remained a critical gap.

Methodology
This work presents the first full-shape joint cosmological analysis of 2PCF and 3PCF multipoles in redshift space using BOSS DR12 data. The analysis splits the sample into two effective redshift bins: low- $z$ ( $\bar{z} \simeq 0.38$ ) and high- $z$ ( $\bar{z} \simeq 0.61$ ).

Theoretical Framework: The authors employ the Velocity Difference Generating Function (VDG) framework. This approach provides a non-perturbative treatment of line-of-sight (LOS) velocity statistics and their damping effects (Fingers-of-God), combined with a one-loop Standard Perturbation Theory (SPT) description of the density field. The model includes a complete Eulerian galaxy bias expansion (incorporating non-local and higher-derivative terms) and implements Infrared (IR) resummation to accurately capture Baryon Acoustic Oscillations (BAO).
Data and Covariance: The analysis utilizes BOSS DR12 data and estimates the covariance matrix from 2048 MultiDark-Patchy mock catalogues. To handle the high dimensionality of the joint data vector, an optimal data-vector compression scheme based on Singular Value Decomposition (SVD) is applied. This compresses the data into a low-dimensional subspace that retains the vast majority of cosmological information while ensuring a stable covariance inversion.
Inference and Emulation: A Bayesian inference framework is used to constrain cosmological parameters ( $10^9 A_s$ , $h$ , $\omega_{cdm}$ ) and nuisance parameters (bias, counter-terms, velocity dispersion). Given the computational cost of evaluating the 3PCF, a neural-network emulator is trained on a Latin Hypercube Sampling of the parameter space to predict the full joint data vector. The emulator's systematic error is verified to be negligible compared to statistical uncertainties.
Scale Selection: The validity of the perturbative model is assessed via goodness-of-fit tests ( $\chi^2_{red}$ ) and posterior stability diagnostics. The analysis adopts a conservative minimum scale cut of $r^{2PCF}_{min} = 20 \, h^{-1}\text{Mpc}$ for the 2PCF and $r^{3PCF}_{min} = 60 \, h^{-1}\text{Mpc}$ for the 3PCF (with an elongation threshold $\eta_{min}=1$ ), where the model provides a statistically acceptable description of the data.

Key Contributions

First Real-Data Joint Analysis: This is the first full-shape cosmological inference that jointly combines 2PCF and 3PCF on real observational data, extending previous real-space validations to the full redshift-space case including RSD and AP effects.
Redshift-Space 3PCF Modeling: The redshift-space 3PCF is modeled for the first time within the VDG framework, extending the bispectrum approach to configuration space.
Optimal Compression: The application of optimal data-vector compression to a joint 2PCF+3PCF likelihood analysis enables the stable inversion of the joint covariance matrix without discarding cosmologically informative 3PCF configurations.
Emulator Development: A dedicated emulator is constructed to facilitate the computationally intensive likelihood sampling required for full-shape configuration-space analysis.

Results
The joint analysis yields significant improvements in cosmological constraints compared to the 2PCF-only baseline:

Precision Gains: The joint analysis reduces the uncertainty on the Hubble parameter $h$ by approximately 29% ( $\sigma(h)$ ), on the scalar amplitude $A_s$ by 24%, and on the cold dark matter density $\omega_{cdm}$ by 10%.
Parameter Constraints: At the reference configuration ( $r^{3PCF}_{min} = 60 \, h^{-1}\text{Mpc}$ ), the joint constraints are $10^9 A_s = 1.687^{+0.141}_{-0.154}$ , $h = 0.687 \pm 0.0035$ , and $\omega_{cdm} = 0.1288^{+0.0056}_{-0.0052}$ .
Information Source: The improvements are primarily driven by the BAO cosmological information encoded in the 3PCF triangle configurations, which provides additional geometric constraints on $h$ and shape information on $\omega_{cdm}$ that are not fully accessible via the 2PCF alone.
Model Validity: The perturbative model provides a good description of the joint data vector down to $r^{3PCF}_{min} \sim 60 \, h^{-1}\text{Mpc}$ , with posterior stability confirmed across different scale cuts and elongation thresholds.

Significance and Claims
The paper claims this work constitutes a milestone in large-scale structure analysis. It demonstrates that full-shape joint analyses of 2PCF and 3PCF are feasible on real data, offering a robust alternative to Fourier-space methods by naturally handling complex survey geometries and selection effects without window function convolutions. The authors state that the results are consistent with Fourier-space full-shape analyses of the same dataset but achieve tighter constraints on $h$ due to the specific choice of fixed parameters (e.g., $\omega_b$ and $n_s$ ) which removes certain degeneracies. The work establishes a framework directly applicable to upcoming Stage IV surveys (DESI, Euclid), where the increased volume and galaxy density will further amplify the constraining power of joint higher-order statistics. The authors modestly note that while the current analysis fixes $\omega_b$ and $n_s$ , the framework is designed to be extended to broader parameter spaces, including primordial non-Gaussianity and massive neutrinos.

1. The Challenge: The "Fuzzy" Map

2. The Solution: A New "Rulebook" and a Super-Computer

3. The Data: The "BOSS" Survey

4. The Results: Triangles Reveal More Secrets

5. The Conclusion

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