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A multitracer analysis for the eBOSS galaxy sample based on the effective field theory of large-scale structure

This paper presents a multitracer full-shape analysis of the eBOSS LRG and ELG samples using the Effective Field Theory of Large-Scale Structure, demonstrating that optimizing priors and leveraging cross-correlations improves cosmological constraints by 10–35% and yields results consistent with Planck 2018, including a significant improvement in the measurement of $\sigma_8$ .

Original authors: Ruiyang Zhao, Xiaoyong Mu, Rafaela Gsponer, Jamie Donald-McCann, Yonghao Feng, Weibing Zhang, Yuting Wang, Gong-Bo Zhao, Kazuya Koyama, David Bacon, Robert G. Crittenden

Published 2026-06-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ruiyang Zhao, Xiaoyong Mu, Rafaela Gsponer, Jamie Donald-McCann, Yonghao Feng, Weibing Zhang, Yuting Wang, Gong-Bo Zhao, Kazuya Koyama, David Bacon, Robert G. Crittenden

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 ocean filled with invisible waves. These waves are made of matter—mostly dark matter, with some normal matter like stars and galaxies riding on top. For decades, astronomers have tried to map these waves to understand how the universe expands and how it got its shape.

This paper is like a team of expert cartographers who just finished drawing a new, incredibly detailed map of a specific region of this cosmic ocean. They used data from the eBOSS survey, which is a massive telescope project that took pictures of two different types of "islands" in the ocean:

LRGs (Luminous Red Galaxies): These are the big, old, bright islands.
ELGs (Emission Line Galaxies): These are the smaller, younger, glowing islands.

Here is a simple breakdown of what they did, the problems they solved, and what they found, using everyday analogies.

1. The Problem: The "Fog" of Noise

When you try to measure the waves in a stormy ocean, it's hard to tell if a ripple is a real wave or just random splashing (noise). In cosmology, this "splashing" is called shot noise and cosmic variance.

The Old Way: Traditionally, scientists would look at the big islands (LRGs) and the small islands (ELGs) separately. It's like trying to understand the ocean's currents by only looking at the big waves on the surface, ignoring the smaller ripples. This gives you a blurry picture.
The New Way (Multitracer): This team decided to look at both types of islands at the same time. Imagine you are trying to hear a specific song in a noisy room. If you have two microphones (one for the big islands, one for the small ones), and you compare them, the random noise cancels out, but the real signal (the song) gets louder. This is the Multitracer method.

2. The Tool: A Better Telescope Lens (EFTofLSS)

To interpret the data, they used a sophisticated mathematical tool called the Effective Field Theory of Large-Scale Structure (EFTofLSS).

The Analogy: Think of the universe's structure as a complex machine. Previous methods were like trying to guess how the machine works by only looking at its biggest gears. EFTofLSS is like having a manual that explains how every gear, no matter how small, interacts with the others. It allows them to use much more of the data (up to smaller scales) without getting confused by the noise.

3. The Big Hurdle: The "Volume Effect"

The authors discovered a tricky problem they call the "Volume Effect" (or projection effect).

The Analogy: Imagine you are trying to guess the average height of people in a room, but you only have a small window to look through. Because your view is limited, your brain might trick you into thinking the room is taller or shorter than it really is, even if your math is perfect. In cosmology, because the survey area is finite, the math can get "tricked" into finding fake answers that look good but are actually wrong.
The Solution: They found a mathematical trick using something called a Jeffreys prior. Think of this as a "reality check" filter. It tells the computer, "Don't trust answers that are too extreme or physically impossible." By combining this with a standard "Gaussian prior" (a gentle nudge toward reasonable values), they were able to clear the fog and stop the math from getting tricked.

4. The Results: A Sharper Picture

After cleaning up the data and applying their new methods, they measured the fundamental properties of our universe with much higher precision than before.

The "Hubble Constant" ( $H_0$ ): This measures how fast the universe is expanding. Their result is 70.0 km/s/Mpc.
- Why it matters: This number sits right in the middle of two other famous measurements (one from the early universe and one from nearby stars). It doesn't prove one side is wrong, but it shows the universe is consistent with our current best theories.
The "Clumpiness" ( $\sigma_8$ and $S_8$ ): This measures how much matter is clumped together in galaxies versus spread out.
- The Win: By using the "Multitracer" method (looking at both galaxy types together), they improved the precision of this measurement by about 27%. It's like upgrading from a standard definition TV to 4K; the picture of how the universe is structured is suddenly much clearer.
Consistency: Their results match the famous Planck 2018 satellite data very well. This is a huge relief for physicists because it means our understanding of the universe's history is holding up.

5. Why This Matters

The authors didn't just find new numbers; they proved that looking at different types of galaxies together is a superpower.

The "DESI" Connection: They tested their method on fake data (simulations) that look like what the upcoming DESI survey will see. They found that this method could improve future measurements by 30%.
The Takeaway: By treating the universe's "islands" (galaxies) as a team rather than individuals, and by using a smarter mathematical lens (EFT) with a reality check (Jeffreys prior), we can map the cosmic ocean with unprecedented clarity.

In short: This paper is a success story of using better math and a "team approach" to different types of galaxies to get a sharper, more accurate map of our universe, confirming that our current theories about how the universe works are still on solid ground.

Technical Summary: A Multitracer Analysis for the eBOSS Galaxy Sample Based on the Effective Field Theory of Large-Scale Structure

Problem Statement
The paper addresses the challenge of extracting precise cosmological constraints from large-scale structure (LSS) surveys, specifically the extended Baryon Oscillation Spectroscopic Survey (eBOSS) Data Release 16 (DR16). While traditional template-fitting methods compress data into Alcock–Paczynski (AP) and redshift-space distortion (RSD) parameters, potentially leading to information loss, full-shape analyses using the Effective Field Theory of Large-Scale Structure (EFTofLSS) allow for the utilization of more Fourier modes in the quasi-non-linear regime. However, EFTofLSS analyses are susceptible to a "volume effect" (or projection effect), where non-Gaussian posteriors lead to biased cosmological constraints, particularly when data constraints are weak. Furthermore, joint analyses of multiple tracers (e.g., Luminous Red Galaxies [LRGs] and Emission Line Galaxies [ELGs]) require careful handling of stochastic terms in cross-power spectra and the mitigation of survey systematics, such as the radial integral constraint (RIC) and angular incompleteness.

Methodology
The authors perform a Fourier-space multitracer full-shape analysis on the complete eBOSS DR16 LRG and ELG samples in both the North and South Galactic Caps.

Data and Power Spectrum Measurement: The analysis utilizes the Yamamoto estimator to measure anisotropic power spectrum multipoles ( $\ell=0, 2, 4$ ) up to $k_{\max} = 0.30 \, h \text{Mpc}^{-1}$ . The authors employ a specific normalization scheme (Eq. 7) based on the Cloud In Cell (CIC) assignment to avoid amplitude mismatches caused by angular incompleteness. To mitigate angular systematics in the ELG sample, they utilize the "chained power spectrum" multipole.
Theory Model: The galaxy density field is modeled using the EFTofLSS framework up to one-loop order, including counter-terms and stochastic contributions. The model accounts for the Alcock–Paczynski effect and survey window convolution. Crucially, the cross-power spectrum model includes three additional stochastic degrees of freedom to account for potential correlations between LRG and ELG populations, rather than assuming them to be zero a priori.
Volume Effect Mitigation: The authors investigate the impact of prior choices on the volume effect. They demonstrate that combining a Jeffreys prior (which cancels the volume effect for linear parameters) with a Gaussian prior (which enforces perturbative conditions and excludes unphysical regions) effectively mitigates biases.
Validation: The analysis pipeline is validated using 1,000 eBOSS EZmock realizations. These include "complete" mocks (free of systematics) and "contaminated" mocks (injected with observational systematics). The authors test the pipeline by rescaling covariance matrices to suppress volume effects and verify that the recovered cosmological parameters are unbiased.
Parameter Estimation: Bayesian inference is performed using the Cobaya code. The analysis marginalizes over linear nuisance parameters analytically to further reduce the volume effect. The study compares two prior configurations: $\Pi_A$ (uninformative) and $\Pi_B$ (Jeffreys + Gaussian).

Key Contributions

Multitracer EFTofLSS Implementation: The paper presents a rigorous multitracer analysis combining LRG and ELG samples within the EFTofLSS framework, explicitly modeling the cross-power spectrum stochastic terms.
Prior Optimization: The authors identify that the combination of Jeffreys and Gaussian priors is crucial for multitracer analyses to mitigate the volume effect while avoiding unphysical parameter regions. They show that relying solely on uninformative priors can lead to biased constraints, particularly for $\sigma_8$ and $\ln(10^{10}A_s)$ .
Systematics Handling: The work details the treatment of the radial integral constraint (RIC) using EZmocks and the use of chained power spectra to handle ELG angular systematics, validating these approaches against contaminated mock catalogs.
Stochastic Term Analysis: The study tests the hypothesis of zero stochastic terms in the cross-power spectrum. While the data is consistent with zero, the authors show that fixing these terms to zero (Model MB) breaks degeneracies and significantly tightens constraints compared to the full model (Model MA).

Results
Using the eBOSS DR16 data with the $\Pi_B$ prior and assuming zero stochastic terms in the cross-power spectrum (the baseline result), the authors report the following constraints (consistent with Planck 2018):

$H_0 = 70.0 \pm 2.3 \, \text{km s}^{-1}\text{Mpc}^{-1}$
$\Omega_m = 0.317^{+0.017}_{-0.021}$
$\sigma_8 = 0.787^{+0.055}_{-0.062}$
$S_8 = 0.809^{+0.064}_{-0.078}$

Key Findings:

Improvement over Single Tracer: The multitracer analysis improves constraints on $\sigma_8$ by approximately 27% compared to a single-tracer LRG analysis. Constraints on $\Omega_m$ improve by roughly 10–25%.
Volume Effect: The study confirms that without the Jeffreys prior, the multitracer analysis (especially when combined with the weakly constraining ELG data) suffers from a severe volume effect, shifting the best-fit values.
Systematics: The inclusion of ELG and cross-power spectra shifts $H_0$ by $\sim 1\sigma$ compared to LRG-only analysis, which the authors attribute to residual RIC effects in the cross-spectrum, though the shift remains within statistical uncertainty.
Consistency: The results are consistent with Planck 2018 at the $\lesssim 1.3\sigma$ level and with previous eBOSS template-fitting results at the $\lesssim 0.9\sigma$ level. The Figure of Merit (FoM) for cosmological parameters improves by 62.7% relative to the single LRG analysis.

Significance
The paper claims that its multitracer full-shape analysis demonstrates the efficiency of the EFTofLSS framework in extracting cosmological information from current and future surveys. By successfully mitigating the volume effect through specific prior choices and validating the pipeline against realistic systematics, the authors show that multitracer analysis can significantly enhance constraints on structure growth ( $\sigma_8$ ) even with the limited overlapping area of current eBOSS data. The methodology is presented as directly applicable to future surveys like DESI, which will observe multiple tracers with higher number densities, potentially offering even greater improvements in cosmological constraints by breaking degeneracies between stochastic and cosmological parameters.

1. The Problem: The "Fog" of Noise

2. The Tool: A Better Telescope Lens (EFTofLSS)

3. The Big Hurdle: The "Volume Effect"

4. The Results: A Sharper Picture

5. Why This Matters

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