The universe records itself with a quiet, stubborn accuracy. For decades, cosmologists have listened most intently to the two-point correlation function, a statistical heartbeat that tells us how galaxies cluster with their nearest neighbors. It’s like tuning a radio to catch the main signal—the big, loud chorus of structure in the cosmos. But the real plot twists hide in the gaps and overlaps, where galaxies tell stories that a single number can’t capture. Enter the three-point correlation function, a higher-gear measure that listens for the way triplets of galaxies align, revealing the fingerprints of nonlinear gravity, galaxy bias, and the shape of the primordial fluctuations that seeded all cosmic structure. The paper from the Euclid Collaboration asks: can we mature from a comfortable, two-channel reading to a richer, simultaneous readout of both the 2PCF and the 3PCF in real space, and what would that buy us for mapping the Universe?
Funded by the European Space Agency and a global consortium of institutions—from the University of Bologna to the Max Planck Institute and beyond—the study is led by Massimo Guidi and a large team of collaborators. Their work is framed as a preparation step for Euclid, a Stage IV mission designed to chart the distribution of galaxies and the subtle distortions of light to unveil the nature of dark energy and gravity. The central achievement isn’t a Nobel-like discovery so much as a technical and methodological advance: a validated pipeline that can predict, fast and reliably, how both 2PCF and 3PCF should look in a universe shaped by known physics as Euclid peers into 0.9 ≤ z ≤ 1.8, millions of galaxies at a time. And crucially, they do this in real space, before redshift distortions shuffle the signals into complication.
What follows is a guided tour of that achievement—what the paper tested, how it did so, and why it matters for the near future of cosmology. The authors don’t pretend to have solved every wrinkle of galaxy bias or every possible systematic. But they do demonstrate that a joint, full-shape analysis of the 2PCF and the 3PCF can tighten our grip on the fabric of the cosmos and sharpen our estimates of three core cosmological parameters. The work also introduces a practical engine—the emulator—that makes such a joint analysis computationally feasible in a data-rich era where Euclid will be a literal factory of galaxy maps.
A new lens on the big map
The 2-point function has long been the workhorse of large-scale structure. It quantifies how much more likely you are to find a pair of galaxies a certain distance apart than you would by random chance. The 3-point function, by contrast, looks at the likelihood of triangles formed by three galaxies. It’s a step into non-Gaussianity, the realm where gravity’s nonlinearity, galaxy bias (how galaxies trace the underlying matter), and redshift-space distortions leave their mark. In the clean, idealized world of real space, the 3PCF carries information that complements the 2PCF, encoding how structure evolves and how galaxies bias their surroundings in a way that a pair statistic alone cannot disentangle.
Guidi and collaborators push this idea beyond a cautious, single-statistic analysis. They test the full perturbative model for both the 2PCF and the 3PCF against synthetic galaxy catalogs that emulate Euclid’s spectroscopic sample. The catalogs come from the Flagship I N-body simulation, a vast, high-fidelity mock that mimics the density field and clustering amplitude expected in the ideal absence of observational hiccups like purity and completeness. The study probes four redshift snapshots—z = 0.9, 1.2, 1.5, and 1.8—spanning the window Euclid will observe. For the 3PCF, they consider all triangular configurations above a minimum separation, a challenging task because the estimator’s cost scales steeply with the number of galaxies.
Two threads run through their approach. The first is physical: they adopt a real-space perturbative model grounded in Eulerian perturbation theory, with a bias expansion that captures linear and nonlinear galaxy bias plus nonlocal operators tied to gravitational evolution. The second is computational: to make a joint analysis tractable, they build an emulator—a small, fast neural-network–driven surrogate that predicts the 2PCF and the 3PCF for a three-parameter cosmology (the scalar amplitude As, the cold dark matter density ωcdm, and the reduced Hubble parameter h). The emulator-augmented pipeline is then used to probe how the 2PCF and 3PCF respond together to changes in those parameters, within a Gaussian-risk framework that already assumes the covariance structure under a realistic, large-volume estimate.
Crucially, the researchers also explore how to separate their data vector into scale ranges that stay within the theory’s safe zone. They test different minimum separations, rmin, and, for the 3PCF, a parameter ηmin that effectively excludes configurations that are nearly isosceles. The reason is practical as well as physical: the tree-level 3PCF prediction is most fragile for certain triangle shapes and small scales, where the perturbative expansion stretches its validity. The result is a nuanced prescription for what scales should be trusted when combining two- and three-point statistics.
From model to data: the emulator bridges speed and accuracy
The theoretical backbone is a perturbative model that describes the galaxy power spectrum to one loop and the 3PCF at tree level, all embedded in the framework of the effective field theory of large-scale structure (EFTofLSS). In plain terms, the model tries to separate what gravity does on large scales from the messy business of how galaxies form and cluster, by including bias terms and a counterterm that accounts for small-scale physics folded into larger scales. The authors’ treatment includes several nuisance parameters tied to bias (b1, b2, bG2, bΓ3) and a counterterm c0 that encodes higher-derivative corrections. They also examine bias-relations that reduce the parameter space without sacrificing fit quality, enabling more robust cosmological inference.
To tame the computational challenge, they build an emulator in PyTorch that can predict the joint 2PCF and 3PCF quickly across a physically motivated region of parameter space. The emulator learns how each term contributes to the signal, with the cosmology dependence factored in analytically where possible and only the remaining piece learned by the network. This separation is not just an elegance trick; it keeps the physics at the forefront while riding the speed of modern machine learning. The payoff is dramatic: what used to take minutes per cosmology in a full calculation now takes milliseconds, a difference that matters when you’re exploring thousands of cosmological models in a likelihood analysis.
Validation sits at the heart of the paper. They test the emulator’s predictions against the exact (non-emulated) calculations across a grid of cosmological and nuisance parameters, verifying that the emulated residuals stay well below the data’s statistical uncertainties. The upshot is a robust demonstration that the emulator can stand in for the full model across the scales Euclid will probe, at least within the tested region. That validation matters not just for this particular study, but as a blueprint for future joint-analyses that push configuration-space statistics into mainstream cosmology.
The paper also spells out three performance metrics to quantify success: goodness of fit (GoF), a figure of bias (FoB) that tracks how far the recovered parameters drift from fiducial values due to modeling systematics, and a figure of merit (FoM) that quantifies how tightly the parameter space is constrained. These metrics are computed in several scenarios: with all bias parameters free, with bias-relations imposed, and with the joint 2PCF+3PCF data vectors across different rmin and ηmin choices. The pattern is clear: the 3PCF, when blended with the 2PCF, shrinks uncertainties and helps break degeneracies that plague 2PCF-only analyses, especially for As and the linear bias b1.
What the joint 2PCF and 3PCF buys cosmology
Perhaps the most exciting result is not a single number but a methodological gain: a real-space, joint, full-shape analysis that leverages the complementary information in two- and three-point statistics to tighten constraints on three cosmological parameters—As, ωcdm, and h. The study confirms that including the 3PCF can meaningfully reduce the degeneracy between As and b1 that often weakens 2PCF-only constraints. In practical terms, this means a clearer read of the amplitude of primordial fluctuations and the matter content of the universe, with the Hubble parameter constrained in tandem as the distance scales mapped by Euclid come into play.
As a rule of thumb, the authors propose conservative scale cuts to preserve the model’s reliability: a 3PCF minimum scale around rmin ≈ 40 h−1 Mpc with ηmin ≈ 3, or, in a more aggressive but still acceptable setup, rmin ≈ 30 h−1 Mpc with ηmin ≈ 2. These choices balance the desire to exploit small-scale information against the risk of bias from a perturbation theory that isn’t perfectly tuned to those regimes. The results show that when the data vector aggressively reaches into smaller scales or into nearly isosceles configurations, the inferred As can become biased or the GoF deteriorates, especially at the lowest redshift. This is a candid reminder that more data isn’t always better if the theory isn’t ready to interpret it.
Beyond scale choices, the paper tests the practical value of imposing known bias-relations to reduce the parameter space. Two relations—linking bG2 to b1 and bΓ3 to b1—helped shrink uncertainties in the nuisance sector without compromising the fit. In the cosmological-inference context, those priors translate into crisper constraints on As and the combination of cosmological parameters that govern the growth of structure. The upshot is a cleaner path to extracting physics from the same data: you borrow structure from theory to sharpen the empirical lens.
One standout technical triumph is the emulator’s performance. Compared with the original calculation, the emulator reduces the time to predict the joint 2PCF+3PCF from minutes to milliseconds per model, enabling a practical exploration of a high-dimensional likelihood space. This is a crucial capability because the Euclid data challenge will be immense, and the science payoff depends not only on clever statistcs but on the ability to iterate through cosmologies rapidly enough to test robustness and to quantify uncertainties comprehensively.
While the work is grounded in real-space modeling, the authors are quick to acknowledge the leap needed to translate everything to redshift space—the domain where Euclid will observe real galaxies, with their peculiar velocities imprinting anisotropies on the clustering signal. The current results set a strong baseline and demonstrate the viability of a joint, full-shape approach. The path forward includes extending the emulator to redshift-space interpretations and merging with other probes like weak lensing to harness Euclid’s full multi-tracer potential. The paper’s stance is pragmatic: prove the method in a clean setting first, then build the full, messy, real-world engine.
The broader implication is that cosmology may move toward a more integrative usage of information embedded in the cosmic web. The 2PCF and 3PCF aren’t competing channels; they are complementary voices in a single confession about how the universe grew from tiny fluctuations to the rich structure we see today. The joint analysis does not just improve error bars. It deepens the narrative: the way galaxies bias their environment, how nonlinear gravity shapes triangles as well as pairs, and how the initial conditions of the universe still echo in modern structure. The work provides a blueprint for maximizing the science return from Euclid’s galaxy maps without being blindsided by modeling systematics at small scales.
In a sentence: the study proves that a carefully calibrated, emulator-fueled joint analysis of real-space 2PCF and 3PCF can extract more cosmological wisdom from the same data, and it does so with a level of methodological rigor that fits the ambition of the Euclid mission. It’s not a jaw-dropping discovery so much as a demonstration of what’s possible when theory, computation, and data come into alignment. The cosmic map is vast and intricate, but Guidi and colleagues have sharpened a lens that lets us hear not just the chorus of galaxies, but the harmonies that underlie the chorus.
Behind the work is a robust institutional backbone. The Euclid Collaboration spans dozens of universities and research centers, with leadership and authorship anchored in Europe’s major physics and astronomy communities. The lead author, Massimo Guidi, and a broad team including collaborators from Università di Bologna, INAF, and other European institutes, carry forward the mission’s promise to translate the statistical poetry of galaxies into a clearer picture of dark energy, gravity, and the history of our expanding universe.
