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Frequentist Cosmological Constraints from Full-Shape Clustering Measurements in DESI DR1

This paper presents a frequentist analysis of DESI DR1 full-shape clustering data, revealing that frequentist constraints on cosmological parameters, particularly in extended $w_0w_a$ CDM models, differ significantly from Bayesian results due to the latter's sensitivity to prior choices, though these discrepancies diminish when supernova data are included.

Original authors: James Morawetz, Hanyu Zhang, Marco Bonici, Will Percival, Andrea Crespi, Jessica Nicole Aguilar, Steven Ahlen, Davide Bianchi, David Brooks, Francisco Javier Castander, Todd Claybaugh, Shaun Cole, And

Published 2026-08-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: James Morawetz, Hanyu Zhang, Marco Bonici, Will Percival, Andrea Crespi, Jessica Nicole Aguilar, Steven Ahlen, Davide Bianchi, David Brooks, Francisco Javier Castander, Todd Claybaugh, Shaun Cole, Andrei Cuceu, Axel de la Macorra, Arnaud de Mattia, Biprateep Dey, Peter Doel, Simone Ferraro, Andreu Font-Ribera, Jaime E. Forero-Romero, Enrique Gaztañaga, Satya Gontcho A Gontcho, Gaston Gutierrez, ChangHoon Hahn, Klaus Honscheid, Dragan Huterer, Mustapha Ishak, Dick Joyce, Robert Kehoe, David Kirkby, Theodore Kisner, Ofer Lahav, Andrew Lambert, Martin Landriau, Laurent Le Guillou, Marc Manera, Ramon Miquel, Eva-Maria Mueller, Seshadri Nadathur, Jeffrey A. Newman, Gustavo Niz, Nathalie Palanque-Delabrouille, Francisco Prada, Ignasi Pérez-Ràfols, Graziano Rossi, Lado Samushia, Eusebio Sanchez, David Schlegel, Michael Schubnell, Joseph Harry Silber, David Sprayberry, Gregory Tarlé, Benjamin Alan Weaver, Pauline Zarrouk, Rongpu Zhou, Hu Zou

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 balloon covered in a complex pattern of dots. These dots are galaxies, and the way they cluster together holds the secret code to how our universe began, how it grows, and what mysterious force is pushing it apart. Scientists call this "Large-Scale Structure." To crack the code, they use a cosmic ruler called "Baryon Acoustic Oscillations" (BAO)—a fossilized ripple from the Big Bang that helps measure distances. They also study the "full shape" of the galaxy patterns, which tells them how fast structures are growing and how much "dark energy" (the invisible force accelerating the expansion) is at work. The big question everyone is asking is: Is the universe behaving exactly as our standard textbook models predict, or is there something new and strange hiding in the data?

This is where the Dark Energy Spectroscopic Instrument (DESI) comes in. It's a massive telescope project that has just released its first batch of data (DR1), measuring the positions of millions of galaxies. But there's a catch: when scientists analyze this data, they have to make choices about how to do the math. One popular method, called "Bayesian," relies heavily on "priors"—which are like starting guesses or assumptions about what the answer might be before looking at the data. The other method, "Frequentist," ignores those guesses and lets the data speak for itself, asking, "If we repeated this experiment a thousand times, how often would we get this result?"

The paper you are about to read is a detective story about these two mathematical detectives. The authors, led by J. Morawetz, took the same fresh DESI data and ran it through both the Bayesian and Frequentist lenses to see if the starting guesses were secretly twisting the results. They found that for the simplest model of the universe (called $\Lambda$ CDM), both detectives agree perfectly. However, for a more complex model that allows dark energy to change over time (called $w_0w_a$ CDM), the Bayesian detective's results were significantly shifted by its starting guesses, while the Frequentist detective stuck to the facts.

Here is what they found:

The Simple Universe vs. The Complicated One
When the scientists looked at the standard, simple model of the universe ( $\Lambda$ CDM), both methods gave very similar answers. For example, they found that the universe is expanding at a rate of about $68.96$ km/s/Mpc (a measure of how fast galaxies are moving away from us) and that matter makes up about $30.34\%$ of the universe. The "starting guesses" didn't mess things up here; the data was strong enough to override any bias, with the two methods differing by less than $1\sigma$ (a standard statistical measure of deviation).

The "Prior" Trap
The story gets wilder when they tested the complex model where dark energy changes over time ( $w_0w_a$ CDM). In this scenario, the Bayesian method (which uses starting guesses) produced results that were statistically distinct from the Frequentist method (which doesn't).

The Bayesian Result: Without supernovae data, it suggested the universe is expanding much faster ( $H_0 \approx 77.0$ km/s/Mpc) and that dark energy is behaving very strangely.
The Frequentist Result: It suggested a much slower expansion ( $H_0 \approx 63.7$ km/s/Mpc) and different values for dark energy.

The authors calculated the statistical distance between these results and found massive shifts: the Bayesian mean was $4.1\sigma$ away from the Frequentist estimate for the expansion rate ( $H_0$ ), and even larger shifts for other parameters like dark energy ( $6.3\sigma$ and $6.6\sigma$ ). The authors realized the Bayesian result was being "pulled" by its own starting assumptions. Because the data for this complex model is a bit fuzzy, the "prior" (the guess) acted like a heavy magnet, dragging the final answer toward where the guess was, rather than where the data pointed. The Frequentist method, refusing to look at the magnet, found a different, more data-driven path.

The Magic of Supernovae
There was a twist, though. When the scientists added data from Type Ia supernovae (exploding stars that act as another cosmic ruler) to the mix, the two methods finally agreed. The extra data from the supernovae was so strong that it broke the "degeneracy" (the confusion) between the parameters. It forced the Bayesian method to drop its biased guesses and align with the Frequentist method, reducing the statistical shifts to less than $1\sigma$ . Without the supernovae, the Bayesian results were heavily dependent on the choice of starting assumptions, leading to conclusions that differed significantly from the data-driven Frequentist analysis.

The Bottom Line
The paper concludes that for complex cosmological models, relying solely on Bayesian methods with standard starting guesses can lead to misleading conclusions. The Frequentist approach provided a crucial reality check, showing that the "weird" results seen in previous Bayesian analyses might just be mathematical artifacts of the prior choice, not new physics.

Specifically, the Frequentist analysis of the DESI data combined with other measurements gave these confidence intervals:

For the simple model ( $\Lambda$ CDM): $\sigma_8 = 0.863^{+0.048}_{-0.040}$ , $H_0 = 68.96^{+0.81}_{-0.80}$ km s $^{-1}$ Mpc $^{-1}$ , and $\Omega_m = 0.3034 \pm 0.0110$ .
For the complex model ( $w_0w_a$ CDM) without supernovae: $\sigma_8 = 0.782^{+0.060}_{-0.036}$ , $H_0 = 63.7^{+4.2}_{-2.0}$ km s $^{-1}$ Mpc $^{-1}$ , $\Omega_m = 0.378^{+0.024}_{-0.047}$ , $w_0 = -0.16^{+0.10}_{-0.50}$ , and $w_a = -3.0^{+1.7}$ .

The authors emphasize that while the Bayesian method is powerful, it must be used with extreme caution when the data isn't perfectly clear, as the "prior" can easily hijack the conclusion. By using the Frequentist method as a cross-check, they ensured that the map of our universe is drawn based on the terrain, not just on where we thought it might be.

Technical Summary: Frequentist Cosmological Constraints from Full-Shape Clustering Measurements in DESI DR1

Problem Statement
Modern cosmological inference, particularly regarding Dark Energy Spectroscopic Instrument (DESI) Data Release 1 (DR1), has largely relied on Bayesian statistical frameworks. While powerful, Bayesian inferences for extended cosmological models (such as $w_0w_a$ CDM) can be highly sensitive to the choice of priors, especially when dealing with effective field theory (EFT) models of galaxy clustering that involve a large number of weakly constrained nuisance parameters. Previous studies have identified "prior effects"—comprising prior weight effects (PWE) and prior volume projection effects (PVE)—which can shift the maximum a posteriori (MAP) estimates away from the maximum likelihood estimate (MLE) and distort marginal posterior distributions. These effects are parameterization-dependent and can lead to misleading conclusions if the prior is not physically well-motivated. This paper addresses the need to quantify these prior effects by performing a parallel frequentist analysis that is inherently free of prior assumptions.

Methodology
The authors perform a frequentist analysis of DESI DR1 full-shape (FS) clustering measurements, comparing the results directly with the baseline Bayesian analysis from the official DESI collaboration.

Data and Modeling: The analysis utilizes the same datasets as the baseline Bayesian work: DESI DR1 full-shape power spectrum multipoles, post-reconstruction baryon acoustic oscillation (BAO) measurements, Planck "lite" CMB likelihood, and Type Ia supernovae compilations (Union3, DES Year 5, PantheonPlus). The modeling employs the same EFT-based perturbation theory models and nuisance parameters (galaxy bias, counterterms, stochastic parameters) used in the Bayesian analysis.
Computational Framework: To handle the computational cost of frequentist profiling, the authors employ surrogate models (emulators) Effort.jl for full-shape power spectra and Capse.jl for CMB anisotropies. These differentiable codes enable the use of gradient-based optimization methods.
Statistical Approach:
- Bayesian: Standard Markov Chain Monte Carlo (MCMC) using the No-U-Turn Sampling (NUTS) algorithm to derive marginalized posteriors and credible intervals.
- Frequentist: The profile likelihood (PL) method is used to construct confidence intervals. This involves maximizing the likelihood over all nuisance parameters for fixed values of the parameters of interest. The authors calculate the MLE and construct 1D and 2D confidence intervals based on $\Delta\chi^2$ thresholds (1 and 2.30 for 68.3% confidence, respectively), relying on Wilks' theorem.
- Comparison: The study compares the MLE (frequentist), MAP (Bayesian), and marginalized posterior means across various dataset combinations for the $\Lambda$ CDM and $w_0w_a$ CDM models.

Key Contributions

Direct Comparison of Paradigms: The paper provides a rigorous side-by-side comparison of frequentist confidence intervals and Bayesian credible intervals using identical data and physical models, isolating the impact of prior choices.
Quantification of Prior Effects: It quantifies the magnitude of prior effects in DESI DR1 constraints, demonstrating that while $\Lambda$ CDM constraints are robust, extended models ( $w_0w_a$ CDM) exhibit significant shifts between MLE and MAP when external data (CMB, Supernovae) are not sufficiently constraining.
Validation of Coverage: A coverage test using 100 mock realizations confirms that the profile likelihood-based confidence intervals maintain correct frequentist coverage (approx. 68.3%) even in scenarios with strong prior effects, whereas Bayesian credible intervals interpreted as frequentist intervals show significant under-coverage for the $w_0w_a$ CDM model.
Computational Efficiency: The study demonstrates that high-precision frequentist constraints can be obtained with relatively low computational cost (~1000 CPU hours) using modern emulators and gradient-based optimizers, making such analyses feasible for future large-scale surveys.

Results

$\Lambda$ CDM Model: For the standard $\Lambda$ CDM model, the frequentist and Bayesian results show close agreement. The MLE and MAP fall within the 1 $\sigma$ Bayesian credible intervals for all parameters ( $\sigma_8, H_0, \Omega_m$ ) and dataset combinations. Shifts between MLE and Bayesian means are minimal (e.g., $<0.8\sigma$ for DESI+BBN+ns10).
$w_0w_a$ CDM Model: Significant discrepancies arise in the extended model.
- Without Supernovae: When fitting DESI full-shape and BAO data alone (combined with BBN constraints), the frequentist MLE is shifted significantly from the Bayesian mean. For example, for $H_0$ , the shift is $4.1\sigma$ ; for $w_0$ , it is $6.3\sigma$ . The Bayesian credible intervals are narrower and centered differently, heavily influenced by the prior volume.
- With Supernovae: The inclusion of Type Ia supernovae data breaks degeneracies among background expansion parameters. This tightens constraints and reduces prior effects, bringing the MLE and MAP into closer agreement (shifts reduced to $<1\sigma$ for most parameters).
Parameter Preferences: The frequentist analysis reveals that the full-shape and BAO probes share similar parameter preferences in the $w_0w_a$ plane. In contrast, the Bayesian full-shape results (without supernovae) are pulled away from the BAO-favored region due to prior effects, a discrepancy that vanishes in the frequentist analysis.

Significance
The paper concludes that while Bayesian methods are standard, their results for extended cosmological models in current DESI data are susceptible to prior-induced biases. The frequentist profile likelihood approach provides a prior-independent benchmark, revealing that the "tension" or specific parameter values derived from Bayesian analyses in the $w_0w_a$ CDM model are partly artifacts of prior choices rather than pure data-driven inferences. The authors argue that complementing Bayesian analyses with frequentist interpretations offers a more nuanced and robust understanding of cosmological constraints, particularly for models with large nuisance parameter spaces. The study does not claim that one method is objectively superior but emphasizes that employing both paradigms helps disentangle data-driven signals from prior-induced artifacts. Future work will apply these methods to DESI DR2 to assess how increased data volume further suppresses prior effects.

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