🔭 astrophysics

Growth rate measurements from a joint analysis of the large-scale galaxy clustering in Fourier and configuration space

This paper presents a robust framework for simultaneously analyzing redshift-space distortions in both Fourier and configuration spaces using effective field theory, demonstrating unbiased constraints on simulated data and yielding a growth rate measurement of $f\sigma_8 = 0.463 \pm 0.052$ from BOSS+eBOSS luminous red galaxies that aligns with official 2020 results.

Original authors: Vincenzo Aronica, Julian E. Bautista, Arnaud de Mattia, Hector Gil-Marín

Published 2026-02-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Vincenzo Aronica, Julian E. Bautista, Arnaud de Mattia, Hector Gil-Marín

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, expanding loaf of raisin bread. As the dough rises, the raisins (galaxies) move apart. But they don't just drift away randomly; they also have their own little "jiggles" caused by the gravity of nearby raisins pulling on them.

For decades, astronomers have been trying to measure these jiggles to understand how fast the universe is growing and how gravity works. To do this, they look at the "fingerprint" left by billions of galaxies.

This paper is about a new, smarter way to read that fingerprint.

The Two Ways of Looking at the Map

Imagine you are trying to describe a complex piece of music to a friend.

Method A (Configuration Space): You describe the music by saying, "There is a drum beat every 2 seconds, and a violin note 5 seconds later." You are looking at the distance between events.
Method B (Fourier Space): You describe the music by saying, "There is a strong bass frequency of 60Hz and a high-pitched whistle at 400Hz." You are looking at the patterns and rhythms (frequencies) of the sound.

In astronomy, scientists have traditionally used one method or the other to study galaxy clusters.

Configuration Space looks at how far apart galaxies are from each other.
Fourier Space looks at the waves and patterns of their distribution.

Both methods use the exact same data (the same galaxies), so in a perfect world, they should give the exact same answer. But in the messy real world, they are like two different translators: they hear the same song, but they might emphasize different parts of the melody or get confused by background noise in slightly different ways.

The Problem: Two Answers, One Truth

Usually, when scientists get two slightly different answers from these two methods, they have to pick one or take a simple average. It's like asking two chefs to taste a soup and then just averaging their salt recommendations. It works, but you might miss the nuance of why they tasted it differently.

Sometimes, the "noise" in the data makes the answer from one chef look weird (non-Gaussian), and simple averaging fails to capture the full picture.

The Solution: The "Joint Space" Chef

The authors of this paper developed a new framework called Joint Space Analysis. Instead of asking the two chefs to taste the soup separately and then averaging their notes, they put the two chefs in the same kitchen and asked them to taste the soup together at the same time.

They built a single mathematical model that listens to both the "distances" (Configuration Space) and the "rhythms" (Fourier Space) simultaneously.

The Analogy: Imagine trying to solve a puzzle. Method A sees the edge pieces clearly but struggles with the middle. Method B sees the middle clearly but gets confused by the edges. The "Joint Space" method puts both puzzle boxes on the table and solves the whole picture at once, using the strengths of both views to fill in the gaps of the other.

The Test Drive

Before applying this to real data, the authors tested their new "Joint Kitchen" on a massive computer simulation (the AbacusSummit suite).

They created a fake universe with known rules.
They ran their new method on it.
Result: The method worked perfectly. It gave the correct answer, just like the old methods, but it was more robust and didn't get confused by the "noise" that sometimes trips up the single-method approaches.

The Real Deal: The BOSS/eBOSS Survey

Finally, they applied this new method to real data from the BOSS and eBOSS surveys, which mapped hundreds of thousands of red galaxies.

The Result: They measured the growth rate of the universe (how fast structures are forming) to be 0.463.
The Verdict: This number matches the official results from previous years very closely. This is a huge win. It proves that their new "Joint Space" method is reliable.

Why Does This Matter?

Think of the universe as a mystery novel.

Dark Energy is the villain slowing things down.
Dark Matter is the invisible glue holding things together.
Gravity is the plot.

By measuring how fast the "raisins" (galaxies) are clustering together, we can figure out if our understanding of gravity (General Relativity) is correct, or if there's a new force at play.

The authors' new method is like upgrading from a magnifying glass to a high-definition microscope. It allows us to look at the universe's growth rate with more confidence, knowing that we aren't just relying on one perspective. It confirms that our current understanding of the universe is solid, while giving us a better tool for the next generation of telescopes (like DESI) to find even deeper secrets.

In short: They built a better way to combine two different ways of looking at the universe, proved it works on fake data, and confirmed it gives the right answer on real data. It's a more robust, "all-in-one" approach to understanding how our cosmic home is growing.

1. Problem Statement

Redshift-space distortions (RSD) in galaxy surveys provide a powerful probe of the growth rate of cosmic structure ( $f\sigma_8$ ), testing theories of gravity. Traditionally, RSD analyses are conducted separately in Configuration Space (CS) using the two-point correlation function $\xi(r)$ or in Fourier Space (FS) using the power spectrum $P(k)$ .

The Issue: While both methods use the same underlying data, they differ in noise properties, sensitivity to systematic effects, and the specific scales used for inference.
Limitations of Current Consensus Methods: Previous attempts to combine CS and FS results (e.g., Sánchez et al. 2017) relied on a Gaussian Approximation (GA), assuming that the parameter posteriors from individual analyses are Gaussian. This assumption often breaks down with noisy data or complex models, leading to potentially biased consensus results.
Goal: The authors aim to develop and validate a framework for a joint analysis that simultaneously fits CS and FS multipoles, accounting for their full covariance without assuming Gaussianity, to obtain more robust and precise cosmological constraints.

2. Methodology

The authors extend a joint fitting framework previously tested for Baryon Acoustic Oscillations (BAO) to full RSD analysis.

A. Data and Simulations

Real Data: The combined BOSS and eBOSS Luminous Red Galaxy (LRG) sample ( $0.6 < z < 1.0$ ), containing ~377,000 galaxies.
Validation Simulations:
- AbacusSummit: 25 high-fidelity N-body simulations (cubic boxes) to test theoretical model biases and scale ranges.
- EZmocks: 1,000 approximate mocks reproducing the survey geometry and clustering statistics of the BOSS/eBOSS sample to estimate covariance matrices and test statistical properties.

B. Theoretical Model

Framework: Effective Field Theory of Large-Scale Structure (EFTofLSS) in a Lagrangian frame (LPT).
Model Components:
- Includes one-loop corrections, counterterms ( $\alpha_i$ ), and stochastic terms ( $s_{n_i}$ ) to model non-linearities and galaxy bias.
- ShapeFit: Incorporates a parameter $m$ to model the slope of the linear power spectrum, capturing early-time physics (variations in $\omega_b, \omega_m, n_s$ ) without varying the full cosmology.
- Alcock-Paczynski (AP) Distortions: Modeled via dilation parameters $q_\parallel$ and $q_\perp$ .
Implementation: Uses the velocileptors code within the desilike framework, with a Taylor expansion emulator to speed up loop integrals.

C. Inference Strategies

The paper compares four approaches:

CS Only: Fitting correlation function multipoles.
FS Only: Fitting power spectrum multipoles.
Gaussian Approximation (GA): Combining CS and FS posteriors assuming they are Gaussian and estimating cross-covariances from mocks.
Joint Space (JS): A single likelihood fit combining CS and FS data vectors.
- JSsep (Baseline): A variation where nuisance parameters (bias terms $b_2, b_s$ and counterterms $\alpha_i$ ) are allowed to vary independently between the CS and FS models to resolve tensions observed in the standard joint fit.

D. Statistical Corrections

Covariance: Estimated from 1,000 mocks.
Hartlap Correction: Applied to the precision matrix to correct for bias due to finite mock numbers.
Scaling Factors ( $m_1, m_2$ ): Applied to parameter uncertainties to account for the uncertainty in the covariance matrix estimation (Percival et al. 2014).

3. Key Contributions

Joint RSD Framework: Successfully extends the joint CS+FS fitting methodology from BAO to full RSD analysis, including the complex EFTofLSS model.
Resolution of Nuisance Tensions: Identified that forcing nuisance parameters to be identical across CS and FS in a joint fit leads to biases (due to different sensitivities to small-scale physics). The introduction of JSsep (separate nuisance parameters) resolves these tensions, yielding unbiased results.
Validation of Non-Gaussianity: Demonstrated that the GA method can fail when posteriors are non-Gaussian (e.g., for the shape parameter $m$ ), whereas the joint fit naturally handles non-Gaussian likelihood shapes.
Robust Covariance Testing: Highlighted the sensitivity of joint analyses to the accuracy of cross-covariance terms derived from mocks, suggesting that a larger number of mocks may be required for high-precision joint analyses.

4. Key Results

A. Validation on Simulations (AbacusSummit & EZmocks)

Unbiasedness: The JSsep method yields unbiased constraints on $f\sigma_8$ , $q_\parallel$ , and $q_\perp$ on simulated data, consistent with individual CS/FS analyses.
Fit Quality: Standard joint fits (JS) showed poor $\chi^2$ due to tensions in nuisance parameters. JSsep reduced the reduced $\chi^2$ to $\sim 1.5$ , comparable to single-space fits.
Uncertainty Gains: Joint analyses (JSsep) provided constraints on $f\sigma_8$ that were slightly tighter (smaller uncertainties) than CS or FS alone, though the gain was modest.
Covariance Sensitivity: Tests showed that using a limited number of mocks (1,000) to estimate the cross-covariance for joint fits can introduce noise, potentially shifting contours. Increasing the mock count improved stability.

B. Application to Real Data (BOSS+eBOSS LRG)

The authors applied the framework to the DR16 LRG sample:

Consistency: All methods (CS, FS, GA, JSsep) produced results consistent within $1\sigma$ .
Final Measurements (JSsep):
- $q_\parallel = 1.031 \pm 0.044$
- $q_\perp = 0.995 \pm 0.021$
- $f\sigma_8 = 0.463 \pm 0.052$
Comparison to Official Results: The result is in good agreement with the official 2020 eBOSS results (FS: $0.454 \pm 0.046$ , CS: $0.460 \pm 0.050$ , GA: $0.449 \pm 0.044$ ).
Model Differences: The slight shift in $f\sigma_8$ and larger uncertainty in $q_\parallel$ compared to official results are attributed to the use of the EFTofLSS model (with additional counterterms and shot-noise terms) and the inclusion of cross-covariance terms.
Fit Quality: The joint fit (JSsep) achieved a reduced $\chi^2$ of 1.34, indicating a good fit to the data.

5. Significance and Conclusion

Robustness: The joint space analysis (specifically the JSsep variant) is demonstrated to be a robust method that avoids the Gaussian assumptions of the GA method and the potential loss of information from analyzing spaces separately.
Future Surveys: The authors argue that this framework should be adopted for future high-precision surveys (like DESI and Euclid) to maximize the extraction of cosmological information from two-point statistics.
Caveat: The method is highly sensitive to the precision of the covariance matrix, particularly the cross-covariance terms between CS and FS. Future work must ensure the use of sufficiently large mock catalogs to minimize noise in these estimates.

In summary, this paper provides a rigorous validation of a joint configuration/Fourier space analysis for RSD, demonstrating that it yields unbiased, consistent, and slightly more precise constraints on the growth rate of structure compared to traditional methods, provided that nuisance parameters are treated flexibly and covariance matrices are accurately estimated.