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Full calibration of the tomographic redshift distribution from the HSC PDR3 Shape Catalog with DESI

This paper presents a complete calibration of all four tomographic redshift bins in the HSC PDR3 weak lensing catalog using DESI DR1 and DR2 data via clustering redshifts, revealing small redshift shifts compared to previous analyses and proposing refined modeling techniques to account for galaxy bias and magnification effects.

Original authors: J. Choppin de Janvry, S. Gontcho A Gontcho, U. Seljak, A. Baleato Lizancos, E. Chaussidon, W. d'Assignies, J. DeRose, S. Heydenreich, E. Paillas, D. Valcin, T. Zhang, J. Aguilar, S. Ahlen, D. Bianchi

Published 2026-03-31

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CC BY 4.0

Original authors: J. Choppin de Janvry, S. Gontcho A Gontcho, U. Seljak, A. Baleato Lizancos, E. Chaussidon, W. d'Assignies, J. DeRose, S. Heydenreich, E. Paillas, D. Valcin, T. Zhang, J. Aguilar, S. Ahlen, D. Bianchi, D. Brooks, F. J. Castander, T. Claybaugh, A. Cuceu, A. de la Macorra, P. Doel, S. Ferraro, A. Font-Ribera, J. E. Forero-Romero, E. Gaztañaga, G. Gutierrez, H. K. Herrera-Alcantar, K. Honscheid, M. Ishak, R. Joyce, S. Juneau, R. Kehoe, D. Kirkby, T. Kisner, A. Kremin, O. Lahav, C. Lamman, M. Landriau, L. Le Guillou, M. Manera, A. Meisner, R. Miquel, S. Nadathur, N. Palanque-Delabrouille, W. J. Percival, C. Poppett, F. Prada, I. Pérez-Ràfols, G. Rossi, E. Sanchez, D. Schlegel, M. Schubnell, J. Silber, D. Sprayberry, G. Tarlé, B. A. Weaver, R. Zhou

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 you are an astronomer trying to map the history of the universe. You have a giant, high-resolution camera (the HSC) that takes pictures of millions of distant galaxies. But there's a catch: your camera can only tell you roughly how far away a galaxy is based on its color, not its exact distance. It's like looking at a crowd of people from a mile away and guessing their ages based on whether they look "young," "middle-aged," or "old."

To do serious science, you need to know the exact distribution of ages in that crowd. If you get the ages wrong, your calculations about how the universe is expanding or how dark matter is distributed will be off.

This paper is about fixing those age guesses using a new, super-accurate tool called DESI.

The Problem: The "Blurry" Photo

The HSC camera groups galaxies into four "bins" (like sorting people into age groups: 30s, 60s, 90s, and 120s). In the past, astronomers tried to figure out the exact average age of each group using only the blurry color data. They suspected their guesses were wrong, but they didn't have a way to prove it, especially for the oldest, most distant groups (Bins 3 and 4).

The Solution: The "ID Card" Check

Enter DESI (Dark Energy Spectroscopic Instrument). Think of DESI as a team of detectives who can walk up to a few thousand of those galaxies and scan their "ID cards" (spectra) to get their exact distance.

The authors of this paper used a clever trick called "Clustering Redshifts."

The Analogy: Imagine you have a huge crowd of people (HSC galaxies) whose ages you only guess. You also have a small group of people (DESI galaxies) whose ages you know exactly.
The Trick: You don't just look at the people individually. Instead, you look at how they cluster or group together. If the people you know are 50 years old tend to hang out in the same neighborhood as the people you guess are 50, then your guess was probably right. If the "50-year-old" experts are actually hanging out with the "60-year-old" crowd, you know your guess was off.

By measuring how the "known" galaxies cluster with the "guessed" galaxies, the team could mathematically reconstruct the true age distribution of the whole crowd.

What They Did Differently

Previous attempts had two big problems:

They couldn't see the oldest galaxies: The old tools didn't have "ID cards" for galaxies older than a certain point, so the oldest bins (3 and 4) were a mystery. This paper used DESI's new data, which includes very old galaxies (Quasars and Emission Line Galaxies), finally solving the mystery for the oldest bins.
They ignored the "lens" effect: Gravity acts like a magnifying glass. It can make distant galaxies look brighter or dimmer, which messes up the counting. The authors built a complex correction system to account for this "cosmic magnification," ensuring their counts were accurate.

The Results: "We Were Off, But Not by Much"

After doing all this math and correction, here is what they found:

Bin 1 (Youngest): The galaxies are actually slightly closer (younger) than the original guess.
Bin 2 (Middle): The original guess was spot on.
Bins 3 & 4 (Oldest): The galaxies are slightly farther (older) than the original guess.

Why does this matter?
In previous studies, the errors in the oldest bins were huge, leading to big uncertainties in our understanding of the universe. This paper shows that while there are small shifts, they are much smaller than feared. It's like realizing your guess of the crowd's age was off by a few months, not a few years.

The Big Picture

This work is like calibrating a ruler. Before you can measure the size of the universe (cosmology), you have to make sure your ruler (the redshift distribution) is accurate.

The authors have now provided a fully calibrated ruler for the HSC camera. This means that when future telescopes (like the ones on the Roman Space Telescope or Euclid) take their data, they can use these new, precise measurements to unlock the secrets of Dark Energy and the fate of the universe with much greater confidence.

In short: They took a blurry photo, used a high-tech scanner to check a few samples, corrected for optical illusions, and realized their map of the universe's history was slightly off—but now they have the exact coordinates to fix it.

1. Problem Statement

Weak lensing cosmological analyses rely heavily on accurate knowledge of the tomographic redshift distributions ( $n(z)$ ) of source galaxies. Imaging surveys like the Hyper Suprime-Cam (HSC) Subaru Strategic Program (SSP) use photometric redshifts ( $z_p$ ) to bin galaxies, but these estimates suffer from systematic biases and scatter due to limited spectral information.

The Challenge: Previous HSC analyses (e.g., Year 3) relied on "self-calibration" (fitting shift parameters to cosmic shear data) or incomplete spectroscopic calibration. Specifically, the HSC Year 3 analysis lacked spectroscopic calibration for the highest redshift bins (Bin 3: $0.9 \le z_p < 1.2$ and Bin 4: $1.2 \le z_p < 1.5$ ) because the available spectroscopic samples (like LRGs) did not extend to high enough redshifts.
Consequence: This led to significant uncertainties in the mean redshift ( $\langle z \rangle$ ) of these bins, resulting in large shifts in cosmological parameters (specifically $S_8$ and $\Omega_m$ ) when compared to other analyses.

2. Methodology

The authors employ the clustering redshifts technique to calibrate the HSC PDR3 Shape Catalog using the DESI (Dark Energy Spectroscopic Instrument) Data Releases 1 and 2 (DR1, DR2).

Data Sources

Photometric Sample: HSC-SSP PDR3 Shape Catalog (S19A), split into four tomographic bins ( $0.3 \le z_p < 1.5$ ). A specific calibration cut was applied to the first two bins to remove outliers with multi-modal photo-z PDFs.
Spectroscopic Calibration Samples: DESI DR1 and DR2, utilizing multiple tracer classes to cover the full redshift range:
- BGS (Bright Galaxy Survey): $z \lesssim 0.6$
- LRG (Luminous Red Galaxies): $0.3 \lesssim z \lesssim 1.2$
- ELG (Emission Line Galaxies): $0.7 \lesssim z \lesssim 1.6$ (New in DR2, crucial for Bin 3/4).
- QSO (Quasars): $0.7 \lesssim z \lesssim 2.8$ (Crucial for high-redshift calibration).

Modeling & Systematics Mitigation

The core estimator relates the angular cross-correlation between spectroscopic slices and photometric bins to the true $n(z)$ . The authors address three major systematic effects:

Galaxy Bias Evolution: Instead of assuming a constant linear bias, the authors relax this assumption. They assume the cross-correlation coefficient between the spectroscopic and photometric samples is constant over redshift within a bin, but they explicitly model the evolution of the auto-correlation (bias) for both samples.
- Spectroscopic Bias: Measured via auto-correlations on large scales ( $30-65 h^{-1}$ Mpc).
- Photometric Bias: Modeled using a power-law fit to the corrected auto-correlation of the photometric sample.
Magnification Bias: The authors model the magnification effects ( $\mu \times g$ and $g \times \mu$ ) which arise from the correlation between the selection function and the matter distribution. They construct a "magnification matrix" to solve for the corrected $n(z)$ , incorporating the log-number count slope ( $s_\mu$ ) for each tracer.
Scale Cuts: To mitigate non-linear bias and fiber assignment incompleteness (especially for ELGs), the analysis uses two scale cuts:
- Fiducial: $0.3 - 3 h^{-1}$ Mpc (maximizing signal-to-noise).
- Conservative: $1 - 5 h^{-1}$ Mpc (minimizing systematics).
- Result: Consistency between these cuts validates the bias modeling.

Parametrization

The measured $n(z)$ points are modeled using B-splines within a Bayesian framework (PyMC).

The spline coefficients are constrained by a Dirichlet prior, refined using Non-Negative Least Squares (NNLS) to encourage sparsity where data is consistent with zero.
This approach regularizes the distribution, preventing unphysical negative values and smoothing statistical fluctuations.

3. Key Contributions

Full Redshift Coverage: This is the first clustering redshift calibration of HSC that includes $z > 1.2$ sources (ELGs and QSOs from DESI DR2), allowing for a complete calibration of all four tomographic bins, including the previously uncalibrated Bin 4.
Relaxed Bias Assumptions: The work moves beyond the standard linear bias assumption, demonstrating that assuming a constant cross-correlation coefficient while modeling bias evolution yields robust results even at smaller scales ( $0.5 h^{-1}$ Mpc).
Comprehensive Systematics Treatment: A rigorous inclusion of magnification bias corrections and galaxy bias evolution for both spectroscopic and photometric samples.
Validation: The consistency of results across different scale cuts ($0.3-3$ vs $1-5 h^{-1}$ Mpc) and the successful use of QSOs for high-redshift calibration provide strong validation for the method.

4. Key Results

The authors report the shift ( $\Delta z$ ) of the mean redshift of the calibrated distributions relative to the HSC Year 3 "Total" (PhotZ + WX) reference. A negative $\Delta z$ implies the new measurement prefers a higher mean redshift than the reference.

Bin 1 ( $0.3 \le z_p < 0.6$ ): Exhibits a small shift towards lower redshifts ( $\Delta z_1 \approx +0.029$ ). This is attributed to strong bias evolution in the BGS sample used for calibration.
Bin 2 ( $0.6 \le z_p < 0.9$ ): Shows excellent agreement with previous photometric calibrations ( $\Delta z_2 \approx -0.003$ ).
Bin 3 ( $0.9 \le z_p < 1.2$ ): Exhibits a shift towards higher redshifts ( $\Delta z_3 = -0.039^{+0.020}_{-0.021}$ $Δ z_{3} = - 0.03 9_{- 0.021}^{+ 0.020}$ ).
- Comparison: This shift is significantly smaller than the shifts found in HSC Year 3 cosmic shear analyses (which found $\Delta z_3 \approx -0.075$ to $-0.115$).
- Cause: The new calibration captures a "tail" of high-redshift galaxies ( $z \sim 1.3-1.8$ ) that was missed by the previous CAMIRA LRG sample (limited to $z \lesssim 1.2$ ).
Bin 4 ( $1.2 \le z_p < 1.5$ ): Exhibits a shift towards higher redshifts ( $\Delta z_4 = -0.048^{+0.012}_{-0.012}$ $Δ z_{4} = - 0.04 8_{- 0.012}^{+ 0.012}$ ).
- Comparison: Much smaller than the HSC Year 3 cosmic shear shift ( $\Delta z_4 \approx -0.15$ to $-0.19$).
- Significance: The inclusion of QSOs and high-redshift ELGs allows for a robust calibration of this bin, which was previously unconstrained by clustering redshifts.

Summary of Shifts (This Work vs. HSC Y3 Cosmic Shear):

Bin	This Work ( $\Delta z$ )	HSC Y3 Cosmic Shear ( $\Delta z$ )
Bin 3	$-0.039 \pm 0.020$	$-0.075$ to $-0.115$
Bin 4	$-0.048 \pm 0.012$	$-0.157$ to $-0.192$

5. Significance

Reduced Systematic Uncertainty: The calibrated distributions show significantly smaller shifts than those inferred from cosmic shear self-calibration. This suggests that the large shifts previously seen in HSC Y3 analyses may have been driven by modeling degeneracies in the shear analysis rather than large photo-z errors.
Validation of Clustering Redshifts: The study confirms that clustering redshifts, when combined with modern spectroscopic data (DESI) and careful bias/magnification modeling, can provide a robust, independent calibration for weak lensing surveys.
Future Surveys: The methodology and results serve as a critical benchmark for upcoming Stage IV surveys (LSST, Euclid, Roman), demonstrating the necessity of high-redshift spectroscopic tracers (QSOs, ELGs) to break degeneracies in the highest redshift bins.
Cosmological Impact: The companion paper (referenced in the conclusion) uses these calibrated distributions to perform a new cosmic shear analysis, likely yielding tighter and more reliable constraints on cosmological parameters.