Euclid: Improving redshift distribution reconstruction using a deep-to-wide transfer function

Technical Summary: Euclid: Improving redshift distribution reconstruction using a deep-to-wide transfer function

Problem Statement
The Euclid mission aims to constrain the nature of dark energy and the expansion history of the Universe through cosmic shear measurements. Achieving the required scientific precision necessitates an extremely accurate reconstruction of the true redshift distribution, $n(z)$, for galaxies in tomographic bins. The systematic error on the mean redshift, $\sigma_{\langle z \rangle}$, must satisfy $\sigma_{\langle z \rangle} < 0.002(1+z)$ per bin.

Current calibration methods rely on reference samples with spectroscopic redshifts to correct photometric redshifts. However, a significant source of systematic uncertainty arises from the mismatch in photometric properties between the deep reference fields (Euclid Auxiliary Fields, EAFs) and the shallow wide survey (Euclid Wide Survey, EWS). The EWS is approximately two magnitudes shallower than the EAFs, leading to different signal-to-noise ratios (S/N) and distinct scatter in color space due to photometric uncertainties. Consequently, identical sources observed in deep and wide configurations occupy different regions of the Self-Organising Map (SOM) used for calibration, leading to biased $n(z)$ reconstructions. While image-based simulation methods like Balrog (used by the Dark Energy Survey) can address this, they are computationally expensive and require substantial modifications to complex pipelines.

Methodology
The authors propose a catalogue-level Multi-Passband Transfer (MPT) method to degrade the photometry of deep-sample objects so that they statistically mimic wide-survey observations.

1. Transfer Function Construction:

   • Unlike simple single-passband transfer (SPT) which adds independent Gaussian noise, MPT preserves the correlations between fluxes, flux errors, and across different passbands.
   • For a deep-sample object with fluxes $d_{X,i}$ and errors $\sigma_{X,i}$, the method identifies nearest neighbors in the wide-sample catalogue (using a k-d tree) based on flux space.
   • It computes a likelihood $L$ that a wide-sample object is identical to the deep-sample one, considering the combined variance of both measurements.
   • A wide-sample object is drawn based on these likelihoods, and its flux errors ($\tau_{X,i}$) are applied to the deep object. The new "wide-like" fluxes are drawn from a normal distribution $N(d_{X,i}, \tau_{X,i}^2)$.
   • This process ensures that the multi-dimensional probability distribution of the degraded sample matches the wide sample, including error correlations.

2. Selection Function:

   • An empirical selection function is applied to the transformed sample to account for the fact that faint deep sources would not be detected in the wide survey. This is derived from the flux distribution of the wide field.

3. Calibration Pipeline Integration:

   • The method is integrated into the SOM-based calibration framework (Masters et al. 2015).
   • The authors test several scenarios (A–G) to determine the optimal implementation:
     • Scenario A: Original Masters et al. method (Deep sample used directly).
     • Scenario C: The Deep calibration subset is transformed 50 times using MPT to create an "MPT-Mock" sample, which is then projected onto a Deep-trained SOM.
     • Scenario F: The entire pipeline (SOM training, bin definition, and projection) uses the MPT-Mock sample.
     • Scenario G: Objects are assigned to tomographic bins based on individual photo-zs (following Roster et al. 2025) using the MPT-Mock sample.

Key Contributions

• Development of MPT: A computationally efficient, catalogue-level algorithm that degrades deep photometry to wide-like conditions while preserving multi-passband flux-error correlations.
• Validation against Balrog: The method is tested against the DES Y3 Balrog catalogue (image-based injection). The authors demonstrate that MPT recovers the statistical properties of the wide sample (flux and error distributions) with a Wasserstein distance comparable to or better than Balrog, despite being significantly faster (processing 1 million sources in 8 bands in ~5 minutes).
• Systematic Evaluation of Calibration Steps: By isolating specific steps in the calibration pipeline (SOM training, bin definition, and reference projection), the paper identifies that the most critical factor for accuracy is ensuring the reference objects (spectroscopic sample) are distributed in the color space in the same way as the wide-survey objects.

Results
The methods were tested using the Euclid Flagship Mock Galaxy Simulation, which provides ground-truth redshifts.

• Performance Metrics: The mean redshift bias ($\Delta z / (1+z)$) was measured across ten tomographic bins.
• Scenario Comparison:
  • Scenario A (Original): Failed to meet Euclid requirements in most bins, with biases as high as $0.124/(1+z)$.
  • Scenario C (MPT on reference projection): Showed significant improvement, with a non-negligible probability of meeting requirements in many bins.
  • Scenario G (MPT + Individual Photo-z binning): Achieved the best performance, with a probability of meeting the $\sigma_{\langle z \rangle} < 0.002(1+z)$ requirement in all bins (ranging from 31% to 77% depending on the bin).
• Distribution Shape: Beyond the mean, the method successfully reproduces the full shape of the $n(z)$ distribution. For the bin $1.0 < z \le 1.25$, the variance of the reconstructed $n(z)$ matched the true variance within 1% (Scenario C and F), whereas the uncorrected method (Scenario B) showed a 15% error in variance.
• SOM Coverage: Using MPT to generate multiple realisations of the reference sample reduced the fraction of unpopulated (uncalibratable) SOM cells from 46.5% (Deep sample only) to 6.7% (Scenario C).

Significance and Claims
The paper claims that the MPT method provides a robust and feasible solution for the Euclid Data Release 1 (DR1) and beyond.

• Feasibility: The approach demonstrates that accurate $n(z)$ reconstruction is achievable without computationally prohibitive image simulations, provided the photometric properties of the reference sample are correctly degraded to match the survey depth.
• Critical Insight: The primary driver for calibration success is not necessarily retraining the SOM on wide-like data (which performed poorly in Scenario E), but rather ensuring the projection of the reference sample onto the SOM reflects the wide-survey color distribution.
• Broader Applicability: The method preserves the full statistical structure of the redshift distribution (including higher moments), making it suitable for applications beyond weak lensing, such as angular clustering.
• Modest Outlook: The authors note that while the method meets requirements for DR1, further improvements (e.g., better reference samples) will be necessary for subsequent data releases. The work establishes a solid foundation for future calibration pipelines.