Euclid data release 1 (DR1) - SIR spectroscopic calibration

Introduction

One of the foremost scientific goals of the Euclid space mission is to constrain the expansion history of the Universe through precise measurements of baryon acoustic oscillations (BAO) and redshift-space distortions (RSD). Achieving these objectives requires a spectroscopic data set that is simultaneously accurate, complete, and statistically homogeneous over an exceptionally large volume.

This translates into three key requirements: (i) a wide-area coverage of $\sim14\,000\ \mathrm{deg}^{2}$ to minimise sample variance; (ii) high-precision redshifts, with $\sigma(z)/(1+z) \le0.001$, necessitating a minimum resolving power of $R \ge500$ for a 0.5-diameter reference source to sufficiently resolve the $H\alpha$ line and preserve both the sharpness of the BAO feature and the small-scale anisotropies probed by redshift-space distortions; and (iii) an accurate, spatially uniform flux calibration, ensuring a position-independent $H\alpha$ line-flux limit of $\sim2 \times10^{-16}\ \mathrm{erg\ s}^{-1}\mathrm{cm}^{-2}$, which is essential for maintaining the required completeness of the galaxy tracer population across the entire survey footprint.

To achieve these survey-level requirements, Euclid relies on the near-infrared slitless spectroscopic channel of the Near-Infrared Spectrometer and Photometer (NISP, [9]), hereafter NISP-S. This specific design choice allows for the massive, simultaneous collection of spectra for all sources within the field of view without the need for a physical slit, thereby maximising the survey’s efficiency and cosmological volume coverage. A comprehensive overview of the strategies and frameworks developed for the calibration of these spectroscopic data can be found in [18].

However, slitless spectroscopy also introduces intrinsic complexities. In the absence of slits, spectral traces are affected by overlaps between neighbouring spectra, self-contamination arising from the entanglement of spatial and spectral information along the dispersion direction, and a strong dependence of the effective resolving power on the intrinsic morphology of the source ([7], hereafter SIR-Q1). The $4 \times4$ detector mosaic, with physical gaps of approximately 6 mm and 3 mm along and across the dispersion direction respectively (see e.g. Figure 1), further complicates the prediction of the exact position and shape of each dispersed spectrum, particularly when part of the trace falls in inter-detector regions.

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Delivering the accuracy required for BAO and RSD therefore requires spectra calibration and spectra reduction pipelines capable of modelling and correcting all instrumental signatures in a globally consistent manner. Such pipelines must be able to reconstruct the geometric distortions, the dispersion law, the point-spread function (PSF) over the entire focal plane (FP), and an accurate spectro-photometric response, which is strictly dependent on the characterisation of the detectors ([10]). This effort builds upon the initial optical models and results obtained during the ground-based calibration campaigns ([8]), ensuring that the measured wavelengths, fluxes, and uncertainties remain stable at the sub-per cent level across the whole Euclid survey.

Within the Euclid science ground segment ([16]), the SIR science pipelines (SIR-Q1) Q1, Euclid Collaboration: Scodeggio et al. [19], hereafter SIR-DR1) are responsible for transforming the raw NISP-S exposures from spacecraft into wavelength- and flux-calibrated spectra. Their performance relies on a comprehensive set of calibration products that describe the full spectroscopic behaviour of the NISP-S instrument. This work presents the SIR calibration pipelines developed to generate all the spectroscopic calibration products required by the SIR science pipelines for Euclid Data Release 1 (2026). These include (i) the geometric models (Sect. 3) most directly related to redshift accuracy requirements, and (ii) the spectro-photometric models (Sect. 4), which ensure the fulfilment of sample completeness requirements.

We present the calibration observing strategy, with a focus on the dedicated calibration fields (Sect. 2), the mathematical models and the calibration algorithms (Sects. 3 and 4), and the validation procedures (Sect. 5) employed to derive these products and assess their performance against Euclid’s mission-level requirements within the Data Release 1 (DR1) framework (Euclid Collaboration: Schirmer et al. [18]).

The SIR science pipelines also incorporate a dedicated preprocessing stage responsible for the removal of instrumental signatures from the raw data. Except for the quantum efficiency (QE) correction (see Sect. 4.1), the preprocessing algorithms and their associated calibrations have been developed by the NIR processing function (Euclid Collaboration: Polenta et al. [14]) and are therefore excluded from the scope of this paper.

NISP spectroscopic observations

NISP-S operates in the near-infrared regime, covering the 950–2000 nm range using a grism-wheel assembly (GWA) hosting four grisms, three red (RGS000/180/270) and one blue (BGS000). Each grism consists of a fused-silica grating-prism combination with a slightly curved base designed to guarantee optimal focus across the entire FP.

The Euclid Wide Survey (EWS, Euclid Collaboration: Scaramella et al. [17]; Euclid Collaboration: Tereno et al. [20]) employs two out of three¹ RGSs, RGS000 and RGS180, which cover the same nominal spectral domain, $RGE = 1206$–1892 nm, with an effective dispersion of 1.372 nm px$^{-1}$ and a resolving power $R > 480$ for a typical 0.′′5-diameter source. To support the spectral decontamination process, the EWS uses these two contra-dispersing RGSs as designed, as well as two configurations where an additional $\pm4^\circ$ tilt is introduced in the GWA orientation to further diversify the dispersion angles. Accordingly, the EWS acquires spectroscopic frames using four distinct red grism-tilt configurations: RGS000+0, RGS180+4, RGS000−4, and RGS180+0. Hereafter, this specific sequence of four acquired frames is referred to as a reference observing sequence (ROS).

In addition to the EWS, Euclid carries out the Euclid Deep Survey (EDS), a distinct observational programme targeting a limited number of specific fields with significantly longer exposure times. Unlike the EWS, which exclusively uses the RGSs, the EDS incorporates observations from both the red and blue grisms. The BGS covers the nominal pass-band, $BGE = 926$–1366 nm, with a dispersion of 1.239 nm px$^{-1}$ and a resolving power $R > 400$.

To process the data from both surveys, the SIR calibration pipelines independently characterise five distinct and independent grism-tilt configurations: the four RGS configurations used in the EWS and EDS, and the BGS setup specific to the EDS.

NISP-S calibration observing strategy

To ensure the high precision required by the NISP-S scientific objectives (Euclid Collaboration: Mellier et al. [12]), a dedicated calibration campaign was designed, targeting specific astronomical sources with well-known properties (Euclid Collaboration: Schirmer et al. [18]). Unlike the standard scientific observations characterised by a single ROS, the calibration strategy often involves multiple ROSs or specialised dither sequences. These are executed at different positions within the same field to ensure an accurate and uniform coverage of the FP, allowing the same calibration source to be sampled across multiple detector positions. The geometric and spectro-photometric calibration framework primarily relies on three dedicated fields obtained during the performance-verification (PV) phase:

CALBLOCK-PV-001: This self-calibration field is fundamental for monitoring the stability of the instrument’s spectro-photometric response against space weathering or out-gassing. The spectroscopic pipelines responsible for the optical, curvature, and PSF models (see Sect. 3), as well as the relative flux model (Sect. 4.2), leverage the high density of bright, non-saturated point sources within this field. These sources are utilised to map the spatial variations of the modelled quantities across the entire FP. The calibration pipelines rely on this field, which is acquired in four visits and each visit produces 15 frames for each RGS configuration and 16 frames for the BGS. To maximise the source density across the FP and ensure the derivation of a more robust and reliable map, the total number of available frames is processed simultaneously for each grism-tilt configuration.

CALBLOCK-PV-009: This field is used to derive the wavelength solution (Sect. 3.5). The strategy involves observing a compact planetary nebula (PN) characterised by strong emission lines. For each of the 16 detectors, the PN is stepped across five different positions, forming a quincunx (i.e., dice-five) pattern. This results in a total of 80 dithered positions for each grism-tilt configuration (see Figure 1). To avoid uncertainties in the GWA repositioning, which would compromise the self-consistency of the dispersion solution, all 80 frames are completed before moving the GWA.

The PN SMC-SMP-20 was selected for this purpose due to its compact spatial extent, optimal visibility, and lack of significant contamination from neighbouring sources (Euclid Collaboration: Paterson et al. [13]).

CALBLOCK-PV-003: Following a strategy similar to the PV-009 field, this block is dedicated to absolute flux calibration (see Sect. 4.3), using white dwarf (WD) GRW+70 5824 as flux reference, to derive the absolute sensitivity curve of the instrumental setup.

We emphasise that DR1 calibrations rely exclusively on observations of the PV fields, under the implicit assumption of instrument stability, as discussed in Sect. 6; repeated visits of the CALBLOCK-F-001 and CALBLOCK-F-010 are not used in this context.

Geometric calibrations

Geometric models refer to the set of semi-analytical descriptions of the optical distortions of NISP-S instrument. These models are essential for the SIR pipelines to properly identify spectrograms on the dispersed science image (specifically first- and zeroth-orders for DR1), and to characterise their precise location and extent. Consequently, given the astronomical coordinates $(\alpha, \delta)$ of a source, as provided by the input MER catalogue (Euclid Collaboration: Kümmel et al. [11]), the geometric calibration allow the accurate location within the frame of the position at any wavelength of its associated spectrogram.

The geometric description comprises five distinct models: the FP array metrology (FPA, Sect. 3.1), optical (OPT, Sect. 3.2), curvature (CRV, Sect. 3.3), PSF (Sect. 3.4), and inverse dispersion solution (IDS, Sect. 3.5) models, which are used in conjunction as follows:

• FPA relates physical positions in the FP (in mm) to pixel-coordinates of individual detectors;

• OPT maps sky coordinates $(\alpha, \delta)$ to reference positions $(x_{\mathrm{r}}, y_{\mathrm{r}})$ in the FP for zeroth- and first-order dispersions;

• CRV describes the spectral trace curvature $\delta y$ as a function of the position $\delta x$ along the first-order spectrogram for a given reference position in the FP; similarly, PSF describes the point-source width $\sigma$ of this spectrogram;

• IDS relates, for first-order spectrograms, the wavelength $\lambda$ to position $\delta x$ along the spectral trace.

Ultimately, these models jointly define, for each of the five grism-tilt instrumental configurations, a mapping between the position of a source on the sky, the wavelength, and the dispersion order, and the pixel coordinates in the FP.

The NISP FP consists of a $4 \times4$-detector matrix separated by substantial physical gaps (see Figure 1). These regions, while lacking active sensors, are nonetheless illuminated by the incident beam. As a result, the grism also disperses light from sources located within these gaps, producing split spectrogram captured by adjacent detectors. The exact geometry of these fragmented spectra depends on the object’s position within the gap. However, calibration pipelines cannot directly measure those positions due to the absence of detectors in those regions.

To address this situation, geometric models cannot be defined solely on the sensitive area of the FP; they must also map regions in between and marginally outside the detector extents. Consequently, all mentioned models, except the FPA model, need to be implemented as global models (as opposed to detector-specific local models): they provide a continuous mapping of the full FP. This approach enables the models to predict the behaviour and position of sources even when located within the gaps or slightly outside the detector footprint, utilising the FPA model to subsequently convert these physical coordinates in the FP into pixel positions on the relevant detector.

As mentioned in SIR-Q1, the SIR pipelines work in a dedicated coordinate system where the spectrograms are approximately horizontal: $(X,Y)_{\mathrm{SIR}} \equiv(+Z,-Y)_{\mathrm{MOSAIC}}$ (Euclid Collaboration: Jahnke et al. [9]). Unless otherwise specified, all subsequent geometric formulations follow this convention.

Focal plane assembly (FPA) model

The FPA model (also referred to as detector metrology) describes the physical arrangement of the detectors in the FP. It serves as the fundamental link to convert physical FP coordinates (in mm) into detector-specific coordinates (in pixels), accounting for the imperfect distribution and alignment of the detectors in the FP. When not properly accounted for, such misalignments introduce spatial discontinuities between detectors that continuous analytical models cannot accurately represent.

In practice, the FPA model consists of a collection of 16 linear transformations $\mathcal{D}_l$, mapping, for each detector $l$, from the FP coordinates (in mm) to pixel coordinates, and stored as per-detector world coordinate systems (WCSs, Calabretta & Greisen [6]); for DR1 data, the $\mathcal{D}_l$ transformations only involve rigid translations of the detectors (potential rotations and scalings are postponed to DR2). This model quantifies the positional deviations of each detector relative to the detector grid, arising from three primary factors: translational shifts in $X_{\mathrm{SIR}}$ and $Y_{\mathrm{SIR}}$ within the FP, rotational misalignments relative to the detector grid, and a lack of perfect coplanarity among the detectors. The latter could result in slight pixel-scale discrepancies between adjacent sensors.

The first spectroscopic frames acquired during the PV phase revealed that ground-based metrology, acquired at room temperature, does not fully reflect the actual in-flight configuration of the FP. Consequently, an effective FPA model was developed from in-flight data. The calibration procedure uses

• a photometric frame where each detector has been astrometrically calibrated (Euclid Collaboration: Polenta et al. [15]),

• a catalogue of bright, non-saturated point-like sources at sky coordinates $(\alpha_i,\delta_i)$,

• and the corresponding measured positions $\mathbf{p}_i^{(l)}$ of these objects for each detector $l$, in pixels.

Starting from ground-based estimates, the procedure establishes an initial global WCS $\mathcal{G}$, which incorporates distortions via a fifth-order polynomial (Shupe et al. [26]) to map the sky object coordinates $(\alpha_i,\delta_i)$ into FP coordinates. By composing $\mathcal{G}$ with $\mathcal{D}_l$, the predicted positions of the sources onto detector $l$ are derived; a minimisation routine is then applied to adjust the individual detector translational offsets (specifically updating CRVAL1 and CRVAL2 WCS keywords only for DR1) to minimise the residuals between the measured pixel positions and those predicted by the FPA models,

As this specific procedure was not originally scheduled for DR1, an independent recipe does not yet exist, and its output was directly integrated into the OPT DpdSirOptModel2² (see below); consequently, no stand-alone FPA-specific data product currently exists. A dedicated metrology pipeline and data product, including detector rotation and piston, will be implemented for the forthcoming data releases.

Optical (OPT) model

The primary objective of the OPT model is to derive the reference position $(x_r, y_r)$ within a spectroscopic frame of the spectrogram of a source in the field of view for a reference wavelength $\lambda_r$. For DR1, the OPT model accounts for both the first- and zeroth-dispersion orders; other dispersion orders – notably $-1$ and $+2$, with peak flux only a per cent-level of the main dispersion orders – are not yet considered in this model, and are therefore ignored in all subsequent steps (including decontamination).

The OPT model provides a continuous mapping from celestial coordinates $(\alpha, \delta)$ to physical FP SIR coordinates (in mm). In the context of dispersed images, ‘localisation’ of a spectrogram specifically refers to the identification of a reference wavelength associated with a detectable spectral feature. Throughout the following sections, any mention to the object’s reference position within the FP implicitly refers to the coordinates of its reference wavelength.

The first-order reference position is the one of the stellar feature at $\lambda_r^{\mathrm{RGS}} = 1504$ nm (blended Mg I) for the RGSs (see Figure 2), and of the Paschen Pa $\beta$ absorption feature at $\lambda_r^{\mathrm{BGS}} = 1282$ nm for the BGS. The zeroth-order spectra are concentrated on a few pixels and exhibit two distinct peaks (e.g., [8]); the OPT model adopts the bluest peak as a reference for the RGSs, and the reddest peak for the BGS.

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For each grism-tilt combination and spectral order ($+1$ and $0$), the model first employs a linear transformation $O$ to establish an approximate sky-to-FP mapping³

This is subsequently refined by two-dimensional ordinary polynomials represented by the matrices $\mathbf{P}^{(X)}$ and $\mathbf{P}^{(Y)}$ (of order $L = 5$ in DR1), and accounting for the non-linear optical distortions,

Each grism-tilt and spectral order has its own set of $\mathbf{P}^{(X,Y)}$ matrices.

The calibration is performed using data from the self-calibration field CALBLOCK-PV-001; the calibration pipeline works in parallel on each grism-tilt producing a single DpdSirOptModel for dispersion orders $+1$ and $0$. It selects bright yet non-saturated point-like sources from the Two Micron All Sky Survey catalogue (2MASS, [27] with $J$-band magnitude between 12 and 16. The procedure simultaneously utilises multiple frames (15 per grism-tilt configuration), to ensure as extensive and uniform a coverage of the FP as possible. The calibration process is organised in three stages.

Pointing registration. Using an initial OPT model derived and validated during the PV phase, the pipeline locates zeroth-order spots (on the four central detectors with a better optical quality) to determine the effective pointing for each individual frame. These features are then measured using the DAOStarFinder algorithm ([28]), as implemented in the photutils package ([5]), with a configuration optimised for the morphology of zeroth orders.

Zeroth-order calibration. Once the effective pointing of a frame is adjusted, the pipeline measures the positions of all zeroth orders on the FP to fit the two global polynomials $\mathbf{P}^{(X,Y)}$, by minimising the $\chi^2$ between the measured and predicted positions.

Since the localisation of the zeroth-orders during the pointing-adjustment stage is dependent on the OPT model, the newly fitted model is utilised to derive a more accurate pointing correction; this, in turn, allows for the recalibration of an improved OPT model. This iterative process is interrupted when convergence is reached (typically within three iterations).

First-order calibration. With a stable pointing solution established, the pipeline calibrates the first-order model, measuring the positions of stellar absorption features in the first-order spectrograms. The absorption feature positions are measured independently along the dispersion ($X_{\mathrm{SIR}}$) and cross-dispersion ($Y_{\mathrm{SIR}}$) directions (Figure 2). Similarly to the procedure adopted for the zeroth orders, the pipeline utilises a pre-calibrated model derived from the PV phase to define the regions of interest where the absorption features are expected to be located.

Within a predefined window, the spectrogram is summed along the cross-dispersion $Y_{\mathrm{SIR}}$-axis to produce a raw spectrum, whose local continuum is removed by fitting a linear baseline. A Gaussian function is then adjusted to the continuum-subtracted spectrum to localise the reference absorption feature. The resulting Gaussian centroid $\mu$, converted to SIR coordinates through the FPA model, is a measurement of $x_{\mathrm{r}}$ for this source (Figure 2, lower-right panel).

Similarly, to determine the position along the cross-dispersion $Y_{\mathrm{SIR}}$-axis, the spectral region is first untilted according to the grism tilt angle and then collapsed along the dispersion $X_{\mathrm{SIR}}$-axis to produce a cross-dispersion profile. The centroid of a Gaussian fit on this profile defines the $y_{\mathrm{r}}$ coordinate of the spectrogram (Figure 2, lower-left panel).

Fits that do not meet specific quality criteria on amplitude, width, and minimal number of data points are discarded from further analysis (typically 30% of the fits). Finally, the polynomials $P^{(X,Y)}$ are derived from $\chi^2$-minimisation between observed and expected positions $(x_{\mathrm{r}}, y_{\mathrm{r}})$ for all selected sources in the FP. The number of sources used for calibration depends on the specific pointing sequences and consequently varies across grism configurations, between 4300 and 5400 sources for the DR1 calibration. The model fitting procedures were performed using $6 \times6$ ($L = 5$) P matrices.

Curvature (CRV) model

The CRV pipeline aims at describing their spatial morphology, characterised by the deviation of the first-order spectrogram traces from an ideal straight line aligned with the horizontal $X_{\mathrm{SIR}}$-axis (see Figure 2). The CRV model describes these deviations and is fundamental for accurately tracing the signal during the 2D recti-linear resampling of the spectrograms and the subsequent spectral extraction process (see SIR-Q1). Like the OPT model calibration, this procedure relies on bright, non-saturated point source spectrograms identified within the self-calibration field CALBLOCK-PV-001.

Starting from the first-order reference position of the spectrogram as delivered by the OPT model, the pipeline then scans along the dispersion direction $X_{\mathrm{SIR}}$, extracting and collapsing spectrogram slices at predefined intervals of 10 px along the spectrogram length – a choice that captures the spectrogram curvature while increasing profile signal-to-noise ratio (S/N) and providing a sufficient number of points along the trace. The overall spectrogram length in the dispersed images – defined as the extended and measurable signal – depends on the grism and on the position in the FP (approximately 530 px for the RGSS and 350 px for the BGS). The cross-dispersion integrated profiles are then fitted with a Gaussian error-function (to account for the marginally sampled profile), to generate

• a set of $\{\delta x_i\}$, the centre of each slice, corresponding to the distances along $X_{\mathrm{SIR}}$-axis with respect to $x_{\mathrm{r}}$ reference position,

• a set of $\{\delta y_i\}$ which represent the offsets between the resulting Gaussian centroids $\{\mu_i\}$ and the $y_{\mathrm{r}}$ reference position (Figure 2).

The associated Gaussian widths $\{\sigma_i\}$ are also preserved for the subsequent PSF model (see Sect. 3.4).

For each individual source at reference position $(x_{\mathrm{r}}, y_{\mathrm{r}})$, the set of discrete data points $\{(\delta x_i,\delta y_i)\}$ is used to model the spectral trace. Specifically, a $N^{\mathrm{th}}$-order Chebyshev polynomial of the first kind is fitted to these points to describe the spatial offset $\delta y$ along the $Y_{\mathrm{SIR}}$-axis as a function of the $X_{\mathrm{SIR}}$-distance $\delta x$ from the reference position,

where $T_k$ is the Chebyshev polynomial of order $k$. This is the so-called ‘local’ model, computed individually for selected calibration sources all over the FP.

Due to the optical distortions of the NISP instrument, these curvature coefficients $\alpha_{x_{\mathrm{r}},y_{\mathrm{r}}}^{(k)}$ vary across the FP. To establish a ‘global’ solution, which maps the entire FP, each local coefficient is itself modelled using a 2D $M^{\mathrm{th}}$-order Chebyshev polynomial as a function of the object’s reference coordinates $(x_{\mathrm{r}}, y_{\mathrm{r}})$,

The $A^{(k)} = [a_{i,j}^{(k)}]$ matrices for each grism-tilt configuration are stored in the DpdSirCrvModel data product, with $N = 3$ and $M = 2$ for DR1.

As mentioned earlier, this global model, based on typically more than 5500 objects over the full FP per grism configuration, allows the prediction of the local curvature coefficients $\alpha_{x,y}^{(k)}$ for any source within the FP – including sources partially masked by detector gaps or slightly outside of the FP extent, – ensuring a continuous and accurate description of the spectroscopic traces; see Sect. 5.3 for illustrations of the CRV chromatic and spatial performance.

During the DR1 production, it appeared that the CRV model, as initially computed from CALBLOCK-PV-001 observations, was sometimes not reaching the sub-pixel precision required by the optimal extraction procedure; this was traced back to metrology model inaccuracies and GWA sub-degree positioning jitter, resulting in a small uncertainty on the mean dispersion direction. This prompted the need for an additional ad hoc residual adjustment per exposure and per detector, based on an analysis of bright star spectrograms (see SIR-DR1); a more accurate CRV procedure is under consideration for Data Release 2 (DR2).

Point-spread function (PSF) model

The intrinsic resolution $\sigma$ of NISP-S is estimated from the cross-dispersion width of point-source spectrograms (SIR-Q1) at the same time as the CRV calibration (Sect. 3.3). The resulting measurements, covering selected stars and wavelengths, are then robustly combined into five spectral bins spanning the full spectral range. At this stage, the wavelength solution is not yet finalised (see Sect. 3.5); consequently, the conversion between detector position along the dispersion direction ($\delta x$, in pixels) and wavelength $\lambda$ remains approximate, though without any practical impact.

Similarly to the CRV expansions (4) and (5), the variations of the intrinsic resolution $\sigma(x,y,\lambda)$ as function of the wavelength and of the position in the FP are modelled using local (chromatic) and global (spatial) Chebyshev polynomial expansions,

The $C^{(k)} = [c_{i,j}^{(k)}]$ matrices are stored in the DpdSirSpectral-PsfModel product following the same structural convention as the CRV model. As anticipated from the instrument design and its excellent optical quality, no meaningful differences in spectral resolution are observed between the various red grism-tilt configurations (SIR-Q1).

Inverse dispersion solution (IDS) model

The final component of the geometric calibrations is the IDS, which provides the spectroscopic wavelength solution, i.e., the mapping, for a given source, between a given wavelength $\lambda$ and its corresponding physical displacement $\delta x$ from the reference position $x_{\mathrm{r}}$ in the FP.

The IDS calibration pipeline follows the fundamental principles of standard spectroscopic wavelength calibration: by observing sources with known emission lines, a relationship is established between the observed pixel positions and the corresponding reference wavelengths. In the SIR framework, the transformation is defined from $\lambda$ to FP coordinates at a given reference position, in accordance with the geometric requirements detailed in Sect. 3.1. Since dispersed imaging cannot use traditional arc lamps as reference, the procedure relies on astronomical targets with well-characterised spectral properties, specifically PN SMC-SMP-20, acquired through dedicated observations of the CALBLOCK-PV-009 field (see Sect. 2.1).

The calibration is performed in parallel for each grism-tilt. It starts with the identification within the dispersed image of the reference PN target through a cross-match against the input MER catalogue. Euclid’s mission database (MDB) provides sky coordinates and reference vacuum wavelengths (Euclid Collaboration: Paterson et al. [13]) for selected emission lines of the PN both in BGE and RGE ranges (see Figure 3).

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Using the previously calibrated OPT and CRV models, along an initial guess of the IDS derived during the PV phase, 80 wavelength-calibrated spectra are extracted for the PN at different positions in the FP (see Figure 1). For each spectrum associated with a PN reference position $(x_{\mathrm{r}}, y_{\mathrm{r}})$ in the FP, a Gaussian fit is performed within search windows centred on the reference wavelengths tabulated in the MDB; the resulting Gaussian centroids $\mu$ are assumed to be the true position of the emission lines. A local IDS is then derived by fitting a Chebyshev polynomial of the first kind: the independent variable corresponds to the tabulated wavelengths $\{\lambda^{R}\}$ (7 for BGS, 11 for RGSs, see Figure 3), while the dependent variable represents the distance $\{\delta^{R}x\}$ of measured line positions $\{\mu_i\}$ with respect to reference position $x_{\mathrm{r}}$,

Once the 80 local models are established, a global spatial model is computed using the same expansion described previously,

The resulting global matrices $B^k = [b_{i,j}^{(k)}]$, which represent the final IDS, mapping the wavelength $\lambda$ to the offset $\delta x$ along dispersion direction at any reference position within the entire FP, are stored in the DpdSirIdsModel product. The DR1 calibration assumes $M = 4$ and $N = 3$ for RGSs, and $N = 2$ for BGS. The smaller $N$ for BGS accounts for the smaller number of reference emission lines over the smaller wavelength domain.

An illustration of the accuracy – mean and root mean square (RMS) – of the procedure as function of position in the FP is presented in Figure 1 for RGS000+0, where the accuracy remains within 0.24px; other grism-tilt configurations present similar behaviours (see Sect. 5.4).

Spectro-photometric calibrations

The second suite of calibration pipelines comprises those dedicated to the flux calibration process, responsible for converting instrumental signal units into astrophysical flux units. By characterising the overall instrumental response and the variations in throughput, these workflows provide the necessary sensitivity models and zero-points required to obtain spectra that are accurately calibrated in flux.

Detector-scaling model

The detector-scaling procedure aims to compensate for signal variations both within and across individual detectors, primarily arising from pixel-specific differences in QE (Euclid Collaboration: Kubik et al. [10]). For the DR1, a dedicated in-flight calibration procedure is not implemented yet; instead, the correction is still performed by combining ground-based calibration data (Euclid Collaboration: Gillard et al. [8]), stored in DpdSirDetectorScaling with an estimate of the zodiacal light spectrum derived from each individual exposure (SIR-Q1).

Relative flux calibration (RFX) model

Large-scale transmission variations across the NISP FP arise from a combination of optical throughput, detector response, and vignetting. These variations can locally reach the $\sim10\%$ level for the tilted-grism configurations (Euclid Collaboration: Mellier et al. [12]) and must be corrected to ensure spatially uniform spectrophotometry.

Euclid measures these variations using repeated observations of bright stars (typically $16 < H_{\mathrm{E}} < 18$) obtained at many dither positions for all grism-tilt configurations in the self-calibration field. Because each star is observed multiple times at different FP positions during these self-calibration runs, the ensemble of observations provides strong statistical constraints on the large-scale relative flux (RFX) solution that must be applied.

At a high level, the calibration proceeds in three stages:

1. gather per-star and per-visit photometric measurements from the extracted spectra, in narrow wavelength bins;

2. fit a self-calibration model that simultaneously solves for the intrinsic magnitude of each calibration star, a smooth 2D large-scale flat pattern per detector, and per-detector QE/gain offsets, using overlapping regions between detectors;

3. interpolate the fitted solution onto a regular FP grid for each wavelength bin and grism-tilt configuration, producing a 3D relative flux solution with associated uncertainties and quality information.

For DR1, the relative response is found to be largely achromatic (but see discussion in Sect. 6); to maximise the S/N on the solution, a wavelength-averaged solution is therefore derived that depends only on FP coordinates for each grism-tilt configuration (Figure 4). This solution (stored in DpdSirRelativeFluxScaling) successfully flattens repeated observations of bright sources and shows clear signatures of the expected vignetting pattern from optical modelling. The module derives and applies large-scale relative flux corrections to NISP slitless spectra so that repeated observations of the same source yield consistent instrumental fluxes, independent of detector position, grism-tilt configuration, or epoch.

Potential time variability of the relative flux solution has not been measured for DR1, and is being investigated for DR2. However, tests on flux stability across the survey (see Sect. 5.5) indicate small variation in the solution.

Absolute flux calibration (AFX) model

The absolute flux (AFX) calibration pipeline is designed to create a sensitivity function that is used to convert instrumental signal units (electrons per sampling element) into astrophysical flux units ($\mathrm{erg\,s^{-1}\,cm^{-2}\,Å^{-1}}$). The sensitivity function is created by first averaging repeated observations of the flux calibration star GRW+70 5824, a DA2.4 WD (Gianninas et al. [23]) acquired during the PV phase in a quincunx pattern on each detector (Euclid Collaboration: Jahnke et al. [9]), independently for each of the five grism-tilt combinations (RGS000+0/-4, RGS180+0/+4 and BGS000). For DR1, a separate sensitivity function is created for each grism-tilt combination (four red and one blue) from the pull-clipped weighted average of $16 \times5 = 80$ single-dither spectra (after relative flux scaling) of the standard star.

The single-dither spectra of the reference star are first extracted (using the same optimal extraction as for science pipeline, see SIR-DR1) and averaged, and the instrumental flux units are converted into units of $\mathrm{e^{-}\,s^{-1}\,Å^{-1}}$. These spectra are then divided by a suitably matched reference spectrum, namely model spectrum grw_70d5824_mod_001.fits from CALSPEC⁴ (Bohlin et al. [4]), first resampled to $0.1$ nm, convolved to an effective spectral resolution of $3.3$ nm (close to the spectral resolution of the RGS for a point source), and finally rebinned onto the SIR wavelengths. The DR1 sensitivity curves for all five grism-tilt configurations are stored in the DpdSirAbsoluteFluxScaling and shown in Figure 5 (top panel).

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This procedure assumes the sensitivity curve is uniform across the FP; this is confirmed across most of the spectrum at the $< 1\%$-level through observations of the WD. The sensitivity function, however, shows FP-dependent variations at the edges of the bands for both the RG$_{\mathrm{E}}$ and BG$_{\mathrm{E}}$ domains, but this affects only a few spectral pixels. The net effect would be increased noise in the extracted (pre-combined) spectrum for a few pixels near the edge of the band for a minority of FP source positions. This effect is being handled by special masking in future releases of the SIR pipeline.

It is noted that the spectral extraction, based on a 2D-resampled spectrogram of a bright (16th magnitude) point source, led to some extra low frequency pattern being added to the spectra used to calculate the sensitivity function (see Figure 6). Although this structure was seen on some individual spectra (especially the tilted configurations), much of it is suppressed when averaging (for a given grism) the sensitivity functions over all available observations. The pattern is the result of Moiré fringing caused by the under-sampling of the PSF of the star and the slight tilt of the spectrograms (SIR-DR1); we are investigating ways to ameliorate this effect in future analyses. However, a small (but statistically insignificant) ripple is introduced into the final sensitivity functions for the two tilted RGSs.

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Here we discuss only the formal uncertainties in the sensitivity functions. Figure 5 (bottom panel) shows the relative error (RMS scatter in the observations divided by the final sensitivity value, expressed as a percentage) as a function of wavelength for each of the grism-tilt configuration. This result is obtained using the optimal extraction method on the WD spectra. As can be seen from the figure, the relative error is mostly below 1%, except for BGS for which it reaches 1.5% for the bluest wavelengths. The results are still above mission requirement (0.5%), but we think that this does not undermine the validity of the final sensitivity function for two reasons: (i) the final sensitivity functions are fitted with a noiseless function to avoid imprinting any noise onto the calibrated spectra, and (ii) we have shown through further analysis, that the relative error is inflated because a minority of these very bright WD spectra are slightly displaced from the expected FPA positions when optimal extraction is made. Optimal extraction is very sensitive to small displacements in the assumed cross-dispersion profile, leading to a systematic underestimate of the extracted fluxes. This effect causes a small percentage ($\sim5\%$) of the spectra to show enhanced scatter compared with the vast majority of WD spectra. However, since we generate the mean sensitivity function using a pull-clipping algorithm (see SIR-Q1), the final result is almost unaffected by this problem, even if it does affect the formal uncertainty (see Figure 6). SIR is investigating improvements to the optimal extraction method for very bright sources, which should bring the final relative error below the formal requirement.

Quality of calibrated data

Upon completion of the calibration processes, a set of data quality control (DQC) parameters is computed to identify potential anomalies, in the generated products and to verify compliance with the mission requirements. A dedicated DQC pipeline, using DQC values embedded in the calibration products, generates detailed diagnostic reports, essential for recognizing peculiarities, and monitors the quality parameters associated with each product.

The overall quality of the calibration products is intrinsically linked to the final performance and accuracy of the extracted spectra. Any residual systematic error in the calibration models directly propagates into the scientific data, potentially affecting the precision of wavelength and flux measurements.

Focal plane assembly (FPA) model

Although DR1 still lacks a dedicated calibration pipeline for the FPA model, evaluating its current performance remains crucial. To this end, we used a DpDnirCalibratedFrameCatalog ([14]) containing paired ceestial $(\alpha,\delta)$ and detector pixel $(x_{\mathrm{px}},y_{\mathrm{px}})$ coordinates for a given frame. We applied the FPA model to convert pixel coordinates into FP positions $(x_{\mathrm{mm}},y_{\mathrm{mm}})$ and fit a global WCS, $\mathcal{G}$, mapping the sky and the FP. Concatenating $\mathcal{G}$ with the FPA model yields predicted pixel positions

For each detector, we computed the distances $\delta r$ between measured $(x_{\mathrm{px}},y_{\mathrm{px}})$ and expected $(\hat{x}_{\mathrm{px}},\hat{y}_{\mathrm{px}})$ positions (in pixels). A deeper inspection reveals spatial structures in the residuals of certain detectors that a simple translation in the current model fails to capture (see Figure 7); nevertheless, these remaining uncertainties are subsequently partially absorbed by the OPT, CRV and IDS models. Clearly, these global models cannot fully account for local variations that differ from one detector to another. Consequently, certain structures in the $\delta r$ distribution will persist, even if mitigated.

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Optical (OPT) model

The OPT calibration pipeline uses the geometric models (2) and (3) to compare the predicted and observed positions of key spectral reference points, specifically a peak of the zeroth-order spots and a stellar absorption feature within the first-order spectra (see Sect. 3.2).

The model’s accuracy is evaluated using the distribution of residuals $(\Delta X,\Delta Y)$ to the reference positions, which are pre-processed to remove outliers via the inter-quartile range (IQR) method: data points are considered as outliers and rejected if they fall outside the interval $[Q_{25\%} \pm1.5\ \mathrm{IQR}]$. This clipping procedure ensures that the final statistical estimators are representative of the core model performance, effectively mitigating the impact of spurious spectral measurement or zeroth-order spot misidentifications. Table 1 summarises the median and the normally-scaled median absolute deviation (sMAD) of these distributions, both in $X_{\mathrm{SIR}}$ (along dispersion) and $Y_{\mathrm{SIR}}$ (across dispersion) directions, for the zeroth- and first-order reference positions.

Configuration	Zeroth-order $\Delta X$ [px]	Zeroth-order $\Delta Y$ [px]	First-order $\Delta X$ [px]	First-order $\Delta Y$ [px]
BGS000	$-0.004 \pm0.121$	$0.003 \pm0.130$	$-0.029 \pm0.305$	$0.000 \pm0.152$
RGS000+0	$0.004 \pm0.128$	$-0.003 \pm0.131$	$0.013 \pm0.230$	$-0.003 \pm0.135$
RGS000-4	$0.008 \pm0.127$	$0.026 \pm0.143$	$0.013 \pm0.281$	$-0.005 \pm0.147$
RGS180+0	$0.006 \pm0.126$	$0.000 \pm0.136$	$-0.012 \pm0.231$	$-0.015 \pm0.138$
RGS180+4	$0.006 \pm0.123$	$-0.002 \pm0.130$	$-0.011 \pm0.283$	$0.015 \pm0.161$

Figure 8 shows that first-order residuals $(\Delta X,\Delta Y)$ exhibit no significant trends across the FP. The larger deviations observed at the edges of few individual detectors are attributed to current limitations in the FPA model, resulting in fictitious discontinuities in the global model of the FP (see Sect. 6).

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The residual statistics for the first-order reference positions demonstrate that the first stage for the geometrical calibrations, which define the detailed spectrogram geometry and wavelength solution, is well within the required tolerances. It should be further noted that the quoted precision of the OPT model in the $X_{\mathrm{SIR}}$ direction does not represent a hard limit for the accuracy of the wavelength calibration, since the zero-point term of the IDS model – $\beta_{x,y}^{(0)}$ in Eq. (8) – can still compensate for residuals in the reference position along the dispersion axis; specifically, $\delta^{R}_{X}(\lambda_{r})$ is not constrained to zero.

Curvature (CRV) model

The CRV model calibration pipeline uses the current model to predict the expected trace positions for bright, point-like sources. For each of these reference objects, the pipeline performs a direct comparison between the CRV-predicted and measured positions at regular intervals along the dispersion direction. Table 2 summarises the distribution of these positional residuals over the full FP – both before ($\Delta Y$ column) and after ($\Delta Y_{\star}$ column) the ad hoc CRV correction discussed below –, categorised by grism, highlighting the statistical offset and the dispersion of the trace modelling accuracy.

Configuration	$\Delta Y_{\bullet}$ [px]	$\Delta Y_{\star}$ [px]
BGS000	$0.000 \pm0.147$	$0.000 \pm0.104$
RGS000+0	$-0.002 \pm0.136$	$0.000 \pm0.104$
RGS000-4	$0.009 \pm0.144$	$0.000 \pm0.100$
RGS180+0	$0.023 \pm0.132$	$0.000 \pm0.101$
RGS180+4	$-0.018 \pm0.164$	$0.000 \pm0.107$

Figure 9, analogously to Figure 8, illustrates the spatial distribution of the robust RMS residuals of $\Delta Y$ across the FP; no significant systematic trends are observed. The larger residuals noted at the detector edges are likely attributable to the limitations of the current FPA model, which does not yet fully account for all detector misalignments (improvements of this model are discussed in the Sect. 6).

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An analysis of the rRMS for each grism as a function of the $\delta x$-distance from the reference position – equivalent to evaluating evolutions across the wavelength range – reveals that the residuals are consistently distributed below the mission requirement of 0.3 px at all wavelengths (see Figure 10). This analysis also reveals a slight colour dependent deflection limited to a few hundredths of a pixel.

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As mentioned in Sect. 3.3 and detailed in Euclid Collaboration: Scodeggio et al. (2026), it should be noted that an ad hoc CRV correction is applied to the DR1 data on a per-pointing and per-detector basis to account for astrometry log inaccuracies and GWA jitter. Consequently, the initial CRV residuals, although already within the mission requirements, are further improved by this refinement procedure, as reported in Table 2 and in Figure 10.

Inverse dispersion solution (IDS) model

Similarly to the other geometric models, the performance of the IDS is evaluated by assessing its accuracy in predicting the positions of the emission lines for the observed PN. For each PN and every emission line utilised for calibration, the positions on the frame are measured. These observed positions are then compared with the values predicted by the model, followed by a detailed analysis of the residuals. The distribution of these residuals is presented in Figure 11 for five grism-tilt configurations, and a statistical summary averaged over all the reference lines is presented in Table 3. We note that all residuals are below the mission requirement of 0.3 px, that is 0.37 nm for blue grism and 0.40 nm for red grisms.

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Configuration	rRMS [nm]
BGS000	0.205 $\pm$ 0.021
RGS000+0	0.152 $\pm$ 0.046
RGS000−4	0.189 $\pm$ 0.018
RGS180+0	0.146 $\pm$ 0.024
RGS180+4	0.137 $\pm$ 0.034

In the RGSs, the reference line at 1282.596 nm appears to be problematic, with sMAD sometimes larger than 0.3 px; however, this is attributed to line measurement challenges rather than IDS model inaccuracy, as this line is blended with a line at 1279.281 nm, while the DR1 procedure does not yet account for blended lines. Overall, the polynomial local model shows excellent agreement with the experimental data.

Flux calibrations

The stability of the flux solution was examined by checking the scaling of measured SIR fluxes for stars with Gaia [22] as a function of sky position and time. Euclid galaxy clustering science puts a tight constraint on allowable spatial or temporal variations in the flux solution, with allowable fluctuations below 0.7%.

The flux stability analysis looked at the relation between Gaia and SIR flux as a function of stellar type, as encoded by position on the stellar locus in $G-G_{\mathrm{RP}}$ versus $G_{\mathrm{BP}}-G$. The tight relation between star position on the Gaia stellar locus and ratio of SIR flux to Gaia flux is illustrated in Figure 12. We used $\sim6 \times10^{5}$ stars spread over DR1 to check for deviations in the relation. To remove galactic extinction as a confounding factor, we divided the sky into HEALPix tiles of $\sim3.36\ \mathrm{deg}^{2}$ ($N_{\mathrm{side}}=32$), computed the average galactic $E(B-V)$ within each tile, and restricted the analysis to pixels with average $E(B-V)<0.02$. The measured stability was quantified by examining the spread in measured scaling for regions across the DR1 footprint, which will be sensitive to positional and temporal variations. Figure 13 indicates that SIR flux variations are of the order of 0.7%, consistent with meeting the mission requirement.

Conclusions and planned future work

In this paper, the SIR geometric and flux calibration pipelines have been detailed, describing their overall architecture, the nature of the generated calibration products (part of the DR1) and their functional interdependencies. The consistency of this calibration chain is the primary factor determining the final accuracy in both wavelength and flux for the spectra produced by the SIR science pipelines (SIR-DR1). Sections 5.2 to 5.4 demonstrated that the geometric calibrations, which directly influence the wavelength precision, remain within the mission requirements; correspondingly, Sect. 5.5 showed a comparable level of quality for the spectro-photometric calibrations, which govern the accuracy of the flux calibration.

While data products delivered for DR1 meet the baseline requirements defined in the top-level Euclid Mission Requirements Document [18], our analysis identifies specific areas where further optimisation would significantly enhance the spectral quality for DR2. Our current development efforts for the upcoming release are focused on the following key areas.

To improve spectral localisation and ensure a more precise wavelength zero-point calibration, the interplay between the FPA and OPT/CRV models is currently being refined. Key developments are underway:

FPA model optimisation: a dedicated calibration pipeline for the FPA model, based on NIR photometric data, is being implemented. This pipeline could go beyond rigid translations and also derives rotation and scale factors per detector. This approach aims to enhance the overall detector metrology in the FP and geometric FP-based model accuracy.

GWA jitter analysis: the impact of GWA positioning uncertainty, estimated to be $\lesssim0.1$-RMS, on the OPT and CRV models is being quantified.

As mentioned in Sect. 3.3, it was observed in the DR1 processing that the CRV model still required a frame-by-frame sub-pixel adjustment at the detector level (SIR-DR1); this correction appeared essential to ensure that 2D spectrograms are accurately located, enabling the rigorous application of optimal extraction, very sensitive to the miscentering of the assumed cross-dispersion profile, and ultimately preventing flux inaccuracies. Research is therefore underway on the proper way to model and include this GWA jitter correction into the OPT and CRV geometric models.

Wavelength calibration (IDS) accuracy is being improved through two complementary strategies:

Model stabilisation: due to the sparse distribution of calibration sources – approximately 80 PN positions in the FP per grism-tilt configuration (see Figure 1) –, global polynomial expansions (such as Eq. (8)) are susceptible to Runge’s phenomenon, leading to unphysical oscillations at the edges of the interpolation domain. To mitigate this, stabilisation techniques and regularisation methods are being introduced to ensure a smooth and reliable solution across the entire FP.

Emission-line centroiding: to improve the measurement of the PN emission lines, cosmic-ray rejection techniques and optimal extraction of the spectra are being deployed. These refinements are expected to yield more precise reference line positions, leading to a better wavelength solution.

Although no specific updates are planned for the relative flux calibration (RFX), this pipeline component will directly benefit from scheduled improvements to the science pipeline (SIR-DR1), namely a better background subtraction and a more precise masking of zeroth-order spectrograms.

For the absolute flux calibration (AFX), several improvements are planned for DR2, including:

Variation of the grism passband: based on an analysis of the way in which the actual wavelengths of the sensitivity curve are dependent on the position in the FP, we have developed a grism-dependent strategy to mask (as ‘un-usable’ and ‘suspicious’) domains near the edge of the band where this effect is most prominent.

‘Suspicious’ AFX-scaling masking: the spectral domain over which the AFX scaling is deemed ‘suspicious’ will change from 25–85% of the band-edge throughput for DR1, to 25–50% in DR2. When combined with the extra masking discussed in the previous bullet, the net effect is to free-up a few spectroscopic samples across all grisms at the edges of the band. It was considered that the DR1 limits were more conservative than they needed to be.

AFX calibration cross-checks: check the absolute calibration, based on the primary calibrator GRW+70 5824, with other spectrophotometric standard stars, including several faint WDs which lie in the self-cal/EDF-N and EDF-S fields

(Deustua et al. in prep.). This is similar to the strategy used by OU-NIR.

It is important to highlight that the DR1 calibrations are exclusively computed from PV observations, and therefore do not account for any potential time evolution in these models since then. For the geometric part, no major source of variability in the intrinsic optical properties – directly related to the physical characteristics of lenses and grisms, e.g. distortion pattern, dispersive power – has been identified. This is supported by the fact that the overall wavelength accuracy – a direct consequence of the quality of the geometric models – is well controlled and does not degrade over time (see SIR-DR1). Regarding flux calibration, the situation is less straightforward, as some temporal evolution of the detectors has been identified, mostly because of ice contamination which primarily affects BGS, but has some impact on RGSs as well [14, 18]. However, the results shown in Figure 13 indicate that the effect on the RGSs does not induce a coherent and significant flux calibration offset over the DR1 data footprint. Nevertheless, temporally evolving zero points due to ice contamination are being monitored and will be accounted for in future data releases. We estimate that the impact on RGS spectra is $\lesssim1\%$ over DR1 (consistent with the results in Figure 13), with variations potentially as large as 5 to 15% for BGS, particularly at the shortest wavelengths.

A temporal analysis of the calibration products will be carried out as part of DR2, based on the monthly self-calibration observations (for all but IDS and AFX calibrations). Regarding wavelength solution (IDS), we will conduct a comparison between initial PN calibration sequence (CALBLOCK-PV-009) and more recent one (CALBLOCK-F-010); as mentioned above, absolute flux calibration (AFX) will be cross-checked with regularly observed secondary calibrators.

The refinements currently being implemented for the DR2 calibration products, coupled with upcoming enhancements to the SIR science pipeline, are expected to yield a significantly more stable and accurate spectroscopic data set. These improvements will minimise systematic uncertainties, ensuring that the DR2 products reach the full scientific potential of the Euclid mission.

Acknowledgements. Funded by the European Union – Next Generation EU, Mission 4 Component 1 Large Scale Lab (LaScaLa), CUP C53D23001390006. This work has made use of the Euclid Data Release 1 (DR1) from the European Space Agency (ESA) mission Euclid, available at https://www.cosmos.esa.int/web/euclid/data-release-dr1. The Euclid Consortium acknowledges the European Space Agency and a number of agencies and institutes that have supported the development of Euclid, in particular the Agenzia Spaziale Italiana, the Austrian Forschungsförderungsgesellschaft funded through BMIMI, the Belgian Science Policy, the Canadian Euclid Consortium, the Deutsches Zentrum für Luft- und Raumfahrt, the DTU Space and the Niels Bohr Institute in Denmark, the French Centre National d’Etudes Spatiales, the Fundaçāo para a Ciência e a Tecnologia, the Hungarian Academy of Sciences, the Ministero di Scienza, Innovacion y Universidades, the National Aeronautics and Space Administration, the National Astronomical Observatory of Japan, the Netherlands Onderzoekschool Voor Astronomie, the Norwegian Space Agency, the Research Council of Finland, the Romanian Space Agency, the Swiss Space Office (SSO) at the State Secretariat for Education, Research, and Innovation (SERI), and the United Kingdom Space Agency. A complete and detailed list is available on the Euclid web site (www.euclid-ec.org/consortium/community/). This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. In the development of our pipeline, we acknowledge use of the Python libraries Numpy [24], Scipy [30], Matplotlib [25], Astropy [3, 2, 1] and Pandas [29].

Authors and affiliations

Euclid Collaboration: M. Fumana{sup}⋆1, Y. Copin{sup}2, P. N. Appleton{sup}3, J. Jacobson{sup}3, D. C. Masters{sup}4, S. Hemmati{sup}3, D. Tavagnacco{sup}5, A. Gargiulo{sup}1, C. Mancini{sup}1, A. Carle{sup}2, G. Herzog{sup}1, A. Alavi{sup}3, R. Chary{sup}4, 6, S. Conseil{sup}7, 2, I. Das{sup}3, A. L. Faisst{sup}3, W. Gillard{sup}8, V. Le Brun{sup}9, C. Scarlata{sup}10, N. R. Stickley{sup}11, H. I. Teplitz{sup}4, Q. Xie{sup}3, M. Scodeggio{sup}1, B. Altieri{sup}12, S. Andreon{sup}13, N. Auricchio{sup}14, H. Aussel{sup}15, C. Baccigalupi{sup}16, 5, 17, 18, M. Baldi{sup}19, 14, 20, A. Balestra{sup}21, S. Bardelli{sup}14, P. Battaglia{sup}14, A. N. Belikov{sup}22, 23, A. Biviano{sup}5, 16, E. Branchini{sup}24, 25, 13, M. Brescia{sup}26, 27, J. Brinchmann{sup}28, 29, 30, S. Camera{sup}31, 32, 33, G. Cañas-Herrera{sup}34, V. Capobianco{sup}33, C. Carbone{sup}1, J. Carretero{sup}35, 36, F. J. Castander{sup}37, 38, M. Castellano{sup}39, G. Castignani{sup}14, S. Cavuoti{sup}27, 40, K. C. Chambers{sup}41, A. Cimatti{sup}42, C. Colodro-Conde{sup}43, G. Congedo{sup}44, C. J. Conselice{sup}45, L. Conversi{sup}46, 12, A. Costille{sup}9, F. Courbin{sup}47, 48, 49, H. M. Courtois{sup}50, J.-G. Cuby{sup}51, 9, A. Da Silva{sup}52, 53, R. da Silva{sup}39, 54, H. Degaudenzi{sup}55, S. de la Torre{sup}9, G. De Lucia{sup}5, A. M. Di Giorgio{sup}56, H. Dole{sup}57, F. Dubath{sup}55, X. Dupac{sup}12, S. Dusini{sup}58, S. Escoffier{sup}8, M. Farina{sup}56, R. Farinelli{sup}14, S. Ferriol{sup}2, S. Fotopoulou{sup}59, N. Fourmanoit{sup}8, M. Frailis{sup}5, E. Franceschi{sup}14, P. Franzetti{sup}1, L. Gabarra{sup}60, S. Galeotta{sup}5, K. George{sup}61, B. Gillis{sup}44, C. Giocoli{sup}14, 20, J. Gracia-Carpio{sup}62, B. R. Granett{sup}13, A. Grazian{sup}21, F. Grupp{sup}62, 63, L. Guzzo{sup}64, 13, 65, W. G. Hartley{sup}55, S. V. H. Haugan{sup}66, G. Helou{sup}11, 3, H. Hoekstra{sup}34, W. Holmes{sup}67, F. Hormuth{sup}68, A. Hornstrup{sup}69, 70, M. Huertas-Company{sup}43, 71, 72, K. 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Zucca{sup}14, V. Allevato{sup}27, 1, I. T. Andika{sup}63, M. Ballardini{sup}114, 115, 14, P. Bergamini{sup}14, S. Borgani{sup}89, 16, 5, 17, 116, E. Bozzo{sup}55, C. Burigana{sup}117, 118, R. Cabanac{sup}102, M. Calabrese{sup}119, 1, A. Cappi{sup}105, 14, F. Caro{sup}39, T. Castro{sup}120, 5, 16, G. Daste{sup}9, D. Di Ferdinando{sup}20, J. A. Escartin Vigo{sup}62, G. Fabbian{sup}57, J. García-Bellido{sup}101, T. Gasparetto{sup}39, A. Gregorio{sup}89, 5, 17, C. M. Gutierrez{sup}43, 121, A. Hall{sup}44, Y. Kang{sup}106, A. Loureiro{sup}122, 123, J. Macias-Perez{sup}124, R. Maoli{sup}125, 39, R. B. Metcalf{sup}84, 14, M. Pöntinen{sup}76, E. Sarpa{sup}5, V. Scottez{sup}126, 127, M. Sereno{sup}14, 20, K. Tanidis{sup}128, M. Tenti{sup}20, M. Tucci{sup}55, M. Viel{sup}16, 5, 18, 117, M. Wiesmann{sup}66, J. A. Acevedo Ponce{sup}67, Y. Akrami{sup}101, 29, G. Angora{sup}27, 114, N. Anselmi{sup}58, 100, 130, M. Archidiacono{sup}64, 65, F. Atrio-Barandela{sup}131, M. Aubert{sup}2, M. Baes{sup}132, L. Bazzanini{sup}114, 14, J. R. Bermejo-Climent{sup}133, 134, D. Bertacca{sup}100, 21, 58, M. Bethermin{sup}135, F. Beutler{sup}44, A. Blanchard{sup}102, F. Bouché{sup}136, 40, M. L. Brown{sup}45, S. Bruton{sup}11, A. Calabro{sup}39, B. Camacho Quevedo{sup}16, 18, 5, P. Carrilho{sup}137, C. S. Carvalho{sup}138, Y. Charles{sup}9, F. Cogato{sup}14, T. E. Collett{sup}139, A. R. Cooray{sup}140, P. Corcho-Caballero{sup}22, M. Correntin{sup}39, 54, B. Csizi{sup}106, O. Cucciat{i}^{14}, S. Davini{sup}60, T. de Boer{sup}41, F. De Paolis{sup}141, 142, G. Desprez{sup}22, A. Díaz-Sánchez{sup}144, S. Di Domizio{sup}24, 25, 100, J. M. Diego{sup}145, F. Dufresne{sup}9, W. Duret{sup}146, 14, A. Enia{sup}63, Y. Fang{sup}13, 25, A. F. Farina{sup}13, 25, A. M. N. Ferguson{sup}44, A. Finoguenov{sup}76, A. Frano{sup}141, 142, 143, A. Fumagalli{sup}5, Y. Fu{sup}34, 22, K. Ganga{sup}86, E. Gaztanaga{sup}37, 38, 139, Z. Ghaffari{sup}5, 16, F. Giacomini{sup}20, F. Gianotti{sup}14, G. Gozaliaslid{sup}147, 76, A. Gruppuso{sup}14, 20, M. Guidi{sup}19, 14, C. Hernández-Monteagudo{sup}121, 43, T. Heuberger{sup}81, H. Hildebrandt{sup}148, J. Hjorth{sup}149, S. Hosseini{sup}102, M. Jauzac{sup}150, 85, 151, 152, 102, S. Joudaki{sup}35, 139, J. J. E. Kajava{sup}153, 154, 155, V. Kansal{sup}156, 157, D. Karagiannis{sup}114, 158, 159, J. Kim{sup}60, C. C. Kirkpatrick{sup}3, A. Kovács{sup}132, I. Kovačić{sup}139, K. Koyama{sup}12, S. Kruk{sup}44, M. L. L.amp{sup}44, M. Lattanzio{sup}160, S. Lee{sup}106, L. Legrand{sup}161, 57, M. Lembo{sup}162, M. D. Lepinzina{sup}5, J. Lesgourgues{sup}163, T. J. Liaudat{sup}164, L. Linke{sup}106, S. J. Liu{sup}56, G. Maggio{sup}5, F. Mannucci{sup}165, F. R. Marleau{sup}106, C. J. A. P. Martins{sup}166, 29, M. Miluzio{sup}12, 167, S. Nadathur{sup}139, K. Naidoo{sup}139, 73, P. Natoli{sup}114, 115, A. Navarro-Alsina{sup}82, S. Nesseris{sup}101, L. Nicastro{sup}14, F. Oppizzi{sup}25, 58, 100, F. Pace{sup}31, 32, 33, 53, E. Palazzi{sup}14, D. Paoletti{sup}14, 118, F. Passalacqua{sup}58, K. Paterson{sup}73, L. Patrizii{sup}20, C. Pattison{sup}139, A. Pisani{sup}8, L. Pizzuti{sup}168` D. Potter$^{169}$, M. Poulain$^{170}$, G. W. Pratt$^{15}$, S. Quai$^{84, 14}$, M. Radovich$^{21}$, F. Ricci$^{171, 39}$, G. Rodighiero$^{100, 21}$, W. Roster$^{62}$, M. Sahlén$^{172}$, D. B. Sanders$^{41}$, A. Schneider$^{169}$, M. Schultheis$^{105}$, D. Sciotti$^{39, 83}$, E. Sellentin$^{173, 34}$, S. Serjeant$^{174}$, L. C. Smith$^{175}$, J. G. Sorce$^{176, 57}$, I. Szapudi$^{41}$, M. Talia$^{84, 14}$, G. Testera$^{25}$, R. Teyssier$^{177}$, S. Tosi$^{24, 13, 25}$, A. Troja$^{5}$, C. Uhlemann$^{178, 179}$, C. Valieri$^{20}$, A. Venhola$^{170}$, D. Vergani$^{14}$, E. Vilenius$^{108}$, S. Vinciguerra$^{9}$, M. von Wietersheim-Kramsta$^{85, 150}$, N. A. Walton$^{175}$, A. H. Wright$^{148}$, L. Zalesky$^{41}$, and D. Scott$^{180}$

$^{1}$ INAF-IASF Milano, Via Alfonso Corti 12, 20133 Milano, Italy

$^{2}$ Université Lyon 1, CNRS/IN2P3, IP2I Lyon, Villeurbanne, France

$^{3}$ Caltech/IPAC, 1200 E. California Blvd., Pasadena, CA 91125, USA

$^{4}$ Infrared Processing and Analysis Center, California Institute of Technology, Pasadena, CA 91125, USA

$^{5}$ INAF-Osservatorio Astronomico di Trieste, Via G. B. Tiepolo 11, 34143 Trieste, Italy

$^{6}$ University of California, Los Angeles, CA 90095-1562, USA

$^{7}$ Centre de Recherche Astrophysique de Lyon, UMR5574, CNRS, Université Claude Bernard Lyon 1, ENS de Lyon, 69230, Saint-Genis-Laval, France

$^{8}$ Aix-Marseille Université, CNRS/IN2P3, CPPM, Marseille, France

$^{9}$ Aix-Marseille Université, CNRS, CNES, LAM, Marseille, France

$^{10}$ Minnesota Institute for Astrophysics, University of Minnesota, 116 Church St SE, Minneapolis, MN 55455, USA

$^{11}$ California Institute of Technology, 1200 E California Blvd, Pasadena, CA 91125, USA

$^{12}$ ESAC/ESA, Camino Bajo del Castillo, s/n., Urb. Villafranca del Castillo, 28692 Villanueva de la Cañada, Madrid, Spain

$^{13}$ INAF-Osservatorio Astronomico di Brera, Via Brera 28, 20122 Milano, Italy

$^{14}$ INAF-Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via Piero Gobetti 93/3, 40129 Bologna, Italy

$^{15}$ Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France

$^{16}$ IFPU, Institute for Fundamental Physics of the Universe, via Beirut 2, 34151 Trieste, Italy

$^{17}$ INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste TS, Italy

$^{18}$ SISSA, International School for Advanced Studies, Via Bonomea 265, 34136 Trieste TS, Italy

$^{19}$ Dipartimento di Fisica e Astronomia, Università di Bologna, Via Gobetti 93/2, 40129 Bologna, Italy

$^{20}$ INFN-Sezione di Bologna, Viale Berti Pichat 6/2, 40127 Bologna, Italy

$^{21}$ INAF-Osservatorio Astronomico di Padova, Via dell’Osservatorio 5, 35122 Padova, Italy

$^{22}$ Kapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands

$^{23}$ ATG Europe BV, Huygensstraat 34, 2201 DK Noordwijk, The Netherlands

$^{24}$ Dipartimento di Fisica, Università di Genova, Via Dodecaneso 33, 16146, Genova, Italy

$^{25}$ INFN-Sezione di Genova, Via Dodecaneso 33, 16146, Genova, Italy

$^{26}$ Department of Physics "E. Pancini", University Federico II, Via Cinthia 6, 80126, Napoli, Italy

$^{27}$ INAF-Osservatorio Astronomico di Capodimonte, Via Moiariello 16, 80131 Napoli, Italy

$^{28}$ European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching, Germany

$^{29}$ Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, PT4150-762 Porto, Portugal

$^{30}$ Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre, 4150-007 Porto, Portugal

$^{31}$ Dipartimento di Fisica, Università degli Studi di Torino, Via P. Giuria 1, 10125 Torino, Italy

$^{32}$ INFN-Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy

$^{33}$ INAF-Osservatorio Astrofisico di Torino, Via Osservatorio 20, 10025 Pino Torinese (TO), Italy

$^{34}$ Leiden Observatory, Leiden University, Einsteinweg 55, 2333 CC Leiden, The Netherlands

$^{35}$ Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Avenida Complutense 40, 28040 Madrid, Spain

$^{36}$ Port d’Informació Científica, Campus UAB, C. Albareda s/n, 08193 Bellaterra (Barcelona), Spain

$^{37}$ Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, s/n, 08193 Barcelona, Spain

$^{38}$ Institut d’Estudis Espacials de Catalunya (IEEC), Edifici RDIT, Campus UPC, 08860 Castelldefels, Barcelona, Spain

$^{39}$ INAF-Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monteporzio Catone, Italy

$^{40}$ INFN – Sezione di Napoli, Via Cinthia 6, 80126, Napoli, Italy

$^{41}$ Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USA

$^{42}$ Dipartimento di Fisica e Astronomia "Augusto Righi" - Alma Mater Studiorum Università di Bologna, Viale Berti Pichat 6/2, 40127 Bologna, Italy

$^{43}$ Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain

$^{44}$ Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK

$^{45}$ Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, UK

$^{46}$ European Space Agency/ESRIN, Largo Galileo Galilei 1, 00044 Frascati, Roma, Italy

$^{47}$ Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona (IEEC-UB), Martí i Franquès 1, 08028 Barcelona, Spain

$^{48}$ Institució Catalana de Recerca i Estudis Avançats (ICREA), Passeig de Lluís Companys 23, 08010 Barcelona, Spain

$^{49}$ Institut de Ciències de l’Espai (IEEC-CSIC), Campus UAB, Carrer de Can Magrans, s/n Cerdanyola del Vallès, 08193 Barcelona, Spain

$^{50}$ Université Lyon 1, CNRS/IN2P3, IUF, IP2I Lyon, Villeurbanne, France

$^{51}$ Canada-France-Hawaii Telescope, 65-1238 Mamalahoa Hwy, Kamuela, HI 96743, USA

$^{52}$ Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Edifício C8, Campo Grande, PT1749-016 Lisboa, Portugal

$^{53}$ Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal

$^{54}$ Space Science Data Center, Italian Space Agency, via del Politecnico snc, 00133 Roma, Italy

$^{55}$ Department of Astronomy, University of Geneva, ch. d’Ecogia 16, 1290 Versoix, Switzerland

$^{56}$ INAF-Istituto di Astrofisica e Planetologia Spaziali, via del Fosso del Cavaliere, 100, 00100 Roma, Italy

$^{57}$ Université Paris-Saclay, CNRS, Institut d’astrophysique spatiale, 91405, Orsay, France

$^{58}$ INFN-Padova, Via Marzolo 8, 35131 Padova, Italy 59 School of Physics, HH Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, UK

60 Department of Physics, University of Oxford, Keble Road, Oxford OX1 3RH, UK

61 University Observatory, LMU Faculty of Physics, Scheinerstr. 1, 81679 Munich, Germany

62 Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1, 85748 Garching, Germany

63 Universitäts-Sternwarte München, Fakultät für Physik, Ludwig-Maximilians-Universität München, Scheinerstr. 1, 81679 München, Germany

64 Dipartimento di Fisica "Aldo Pontremoli", Università degli Studi di Milano, Via Celoria 16, 20133 Milano, Italy

65 INFN-Sezione di Milano, Via Celoria 16, 20133 Milano, Italy

66 Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, 0315 Oslo, Norway

67 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA, 91109, USA

68 Felix Hormuth Engineering, Goethestr. 17, 69181 Leimen, Germany

69 Technical University of Denmark, Elektrovej 327, 2800 Kgs. Lyngby, Denmark

70 Cosmic Dawn Center (DAWN), Denmark

71 Université PSL, Observatoire de Paris, Sorbonne Université, CNRS, LERMA, 75014, Paris, France

72 Université Paris-Cité, 5 Rue Thomas Mann, 75013, Paris, France

73 Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany

74 NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA

75 Université de Genève, Département de Physique Théorique and Centre for Astroparticle Physics, 24 quai Ernest-Ansermet, CH-1211 Genève 4, Switzerland

76 Department of Physics, P.O. Box 64, University of Helsinki, 00014 Helsinki, Finland

77 Helsinki Institute of Physics, Gustaf Hällströminkatu 2, University of Helsinki, 00014 Helsinki, Finland

78 Laboratoire d’etude de l’Univers et des phenomenes eXtremes, Observatoire de Paris, Université PSL, Sorbonne Université, CNRS, 92190 Meudon, France

79 SKAO, Jodrell Bank, Lower Withington, Macclesfield SK11 9FT, UK

80 Centre de Calcul de l’IN2P3/CNRS, 21 avenue Pierre de Coubertin 69627 Villeurbanne Cedex, France

81 University of Applied Sciences and Arts of Northwestern Switzerland, School of Computer Science, 5210 Windisch, Switzerland

82 Universität Bonn, Argelander-Institut für Astronomie, Auf dem Hügel 71, 53121 Bonn, Germany

83 INFN-Sezione di Roma, Piazzale Aldo Moro, 2 - c/o Dipartimento di Fisica, Edificio G. Marconi, 00185 Roma, Italy

84 Dipartimento di Fisica e Astronomia "Augusto Righi" - Alma Mater Studiorum Università di Bologna, via Pietro Gobetti 93/2, 40129 Bologna, Italy

85 Department of Physics, Institute for Computational Cosmology, Durham University, South Road, Durham, DH1 3LE, UK

86 Université Paris Cité, CNRS, Astroparticule et Cosmologie, 75013 Paris, France

87 CNRS-UCB International Research Laboratory, Centre Pierre Binétruy, IRL2007, CPB-IN2P3, Berkeley, USA

88 Institute of Physics, Laboratory of Astrophysics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland

89 Dipartimento di Fisica - Sezione di Astronomia, Università di Trieste, Via Tiepolo 11, 34131 Trieste, Italy

90 Telespazio UK S.L. for European Space Agency (ESA), Camino bajo del Castillo, s/n, Urbanizacion Villafranca del Castillo, Villanueva de la Cañada, 28692 Madrid, Spain

91 Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona), Spain

92 School of Mathematics and Physics, University of Surrey, Guildford, Surrey, GU2 7XH, UK

93 European Space Agency/ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands

94 Waterloo Centre for Astrophysics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

95 Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

96 Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada

97 Centre National d’Etudes Spatiales – Centre spatial de Toulouse, 18 avenue Edouard Belin, 31401 Toulouse Cedex 9, France

98 Institute of Space Science, Str. Atomistilor, nr. 409 Măgurele, Ilfov, 077125, Romania

99 Institute for Particle Physics and Astrophysics, Dept. of Physics, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093 Zurich, Switzerland

100 Dipartimento di Fisica e Astronomia "G. Galilei", Università di Padova, Via Marzolo 8, 35131 Padova, Italy

101 Instituto de Física Teórica UAM-CSIC, Campus de Cantoblanco, 28049 Madrid, Spain

102 Institut de Recherche en Astrophysique et Planétologie (IRAP), Université de Toulouse, CNRS, UPS, CNES, 14 Av. Edouard Belin, 31400 Toulouse, France

103 Université St Joseph, Faculty of Sciences, Beirut, Lebanon

104 Departamento de Física, FCFM, Universidad de Chile, Blanco Encalada 2008, Santiago, Chile

105 Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, Bd de l’Observatoire, CS 34229, 06304 Nice cedex 4, France

106 Universität Innsbruck, Institut für Astro- und Teilchenphysik, Technikerstr. 25/8, 6020 Innsbruck, Austria

107 Department of Physics and Helsinki Institute of Physics, Gustaf Hällströminkatu 2, University of Helsinki, 00014 Helsinki, Finland

108 Mullard Space Science Laboratory, University College London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK

109 Cosmic Dawn Center (DAWN)

110 Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen, Denmark

111 Universidad Politécnica de Cartagena, Departamento de Electrónica y Tecnología de Computadoras, Plaza del Hospital 1, 30202 Cartagena, Spain

112 European University of Technology EUt+, European Union

113 Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität Heidelberg, Mönchhofstr. 12-14, 69120 Heidelberg, Germany

114 Dipartimento di Fisica e Scienze della Terra, Università degli Studi di Ferrara, Via Giuseppe Saragat 1, 44122 Ferrara, Italy

115 Istituto Nazionale di Fisica Nucleare, Sezione di Ferrara, Via Giuseppe Saragat 1, 44122 Ferrara, Italy

116 ICSC - Centro Nazionale di Ricerca in High Performance Computing, Big Data e Quantum Computing, Via Magnanelli 2, Bologna, Italy

117 INAF, Istituto di Radioastronomia, Via Piero Gobetti 101, 40129 Bologna, Italy

118 INFN-Bologna, Via Irnerio 46, 40126 Bologna, Italy

119 Astronomical Observatory of the Autonomous Region of the Aosta Valley (OAVdA), Loc. Lignan 39, I-11020, Nus (Aosta Valley), Italy 120 Department of Mathematical Physics, Institute of Physics, University of São Paulo, R. do Matão 1371, 05508-090, São Paulo, SP, Brazil

121 Universidad de La Laguna, Dpto. Astrofísica, E-38206 La Laguna, Tenerife, Spain

122 Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, Stockholm, SE-106 91, Sweden

123 Astrophysics Group, Blackett Laboratory, Imperial College London, London SW7 2AZ, UK

124 Univ. Grenoble Alpes, CNRS, Grenoble INP, LPSC-IN2P3, 53, Avenue des Martyrs, 38000 Grenoble, France

125 Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy

126 Institut d’Astrophysique de Paris, 98bis Boulevard Arago, 75014, Paris, France

127 ICL, Junia, Université Catholique de Lille, LITL, 59000 Lille, France

128 Center for Astrophysics and Cosmology, University of Nova Gorica, Nova Gorica, Slovenia

129 CERCA/ISO, Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106, USA

130 Laboratoire Univers et Théorie, Observatoire de Paris, Université PSL, Université Paris Cité, CNRS, 92190 Meudon, France

131 Departamento de Física Fundamental. Universidad de Salamanca. Plaza de la Merced s/n. 37008 Salamanca, Spain

132 Universiteit Gent, Department of Physics and Astronomy, Proeftuinstraat 86 N3, 9000 Ghent, Belgium

133 Konkoly Observatory, HUN-REN CSFK, MTA Centre of Excellence, Budapest, Konkoly Thege Miklós út 15-17. H-1121, Hungary

134 MTA-CSFK Lendület Large-Scale Structure Research Group, Konkoly-Thege Miklós út 15-17, H-1121 Budapest, Hungary

135 Université de Strasbourg, CNRS, Observatoire astronomique de Strasbourg, UMR 7550, 67000 Strasbourg, France

136 Scuola Superiore Meridionale, Via Mezzocannone 4, 80138, Napoli, Italy

137 Department of Physics, Astronomy and Mathematics, University of Hertfordshire, College Lane, Hatfield AL10 9AB, UK

138 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, Tapada da Ajuda, 1349-018 Lisboa, Portugal

139 Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth PO1 3FX, UK

140 Department of Physics & Astronomy, University of California Irvine, Irvine CA 92697, USA

141 Department of Mathematics and Physics E. De Giorgi, University of Salento, Via per Arnesano, CP-I93, 73100, Lecce, Italy

142 INFN, Sezione di Lecce, Via per Arnesano, CP-193, 73100, Lecce, Italy

143 INAF-Sezione di Lecce, c/o Dipartimento Matematica e Fisica, Via per Arnesano, 73100, Lecce, Italy

144 Departamento Física Aplicada, Universidad Politécnica de Cartagena, Campus Muralla del Mar, 30202 Cartagena, Murcia, Spain

145 Instituto de Física de Cantabria, Edificio Juan Jordá, Avenida de los Castros, 39005 Santander, Spain

146 CEA Saclay, DFR/IRFU, Service d’Astrophysique, Bat. 709, 91191 Gif-sur-Yvette, France

147 Department of Computer Science, Aalto University, PO Box 15400, Espoo, FI-00 76, Finland

148 Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), German Centre for Cosmological Lensing (GCCL), 44780 Bochum, Germany

149 DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 155, 2200 Copenhagen, Denmark

150 Department of Physics, Centre for Extragalactic Astronomy, Durham University, South Road, Durham, DH1 3LE, UK

151 Astrophysics Research Centre, University of KwaZulu-Natal, Westville Campus, Durban 4041, South Africa

152 School of Mathematics, Statistics & Computer Science, University of KwaZulu-Natal, Westville Campus, Durban 4041, South Africa

153 Department of Physics and Astronomy, Vesilinnantie 5, University of Turku, 20014 Turku, Finland

154 Finnish Centre for Astronomy with ESO (FINCA), Quantum, Vesilinnantie 5, University of Turku, 20014 Turku, Finland

155 Serco for European Space Agency (ESA), Camino bajo del Castillo, s/n, Urbanizacion Villafranca del Castillo, Villanueva de la Cañada, 28692 Madrid, Spain

156 ARC Centre of Excellence for Dark Matter Particle Physics, Melbourne, Australia

157 Centre for Astrophysics & Supercomputing, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia

158 Department of Physics and Astronomy, University of the Western Cape, Bellville, Cape Town, 7535, South Africa

159 Van Swinderen Institute, University of Groningen, Nijenborgh 3, 9747 AG Groningen, The Netherlands

160 Ohio University, Physics & Astronomy Department,¹ Ohio University, Athens, OH 45701, USA

161 Brazilian Center for Research in Physics (CBPF), Dr. Xavier Sigaud st. 150, zip 22290-180, Rio de Janeiro, RJ, Brazil

162 Institut d’Astrophysique de Paris, UMR 7095, CNRS, and Sorbonne Université, 98 bis boulevard Arago, 75014 Paris, France

163 Institute for Theoretical Particle Physics and Cosmology (TTK), RWTH Aachen University, 52056 Aachen, Germany

164 IRFU, CEA, Université Paris-Saclay 91191 Gif-sur-Yvette Cedex, France

165 INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125, Firenze, Italy

166 Centro de Astrofísica da Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal

167 HE Space for European Space Agency (ESA), Camino bajo del Castillo, s/n, Urbanizacion Villafranca del Castillo, Villanueva de la Cañada, 28692 Madrid, Spain

168 Dipartimento di Fisica “G. Occhialini”, Università degli Studi di Milano Bicocca, Piazza della Scienza 3, 20126 Milano, Italy

169 Department of Astrophysics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland

170 Space physics and astronomy research unit, University of Oulu, Pentti Kaiteran katu 1, FI-90014 Oulu, Finland

171 Department of Mathematics and Physics, Roma Tre University, Via della Vasca Navale 84, 00146 Rome, Italy

172 Theoretical astrophysics, Department of Physics and Astronomy, Uppsala University, Box 524, 751 20 Uppsala, Sweden

173 Mathematical Institute, University of Leiden, Einsteinweg 55, 2333 CA Leiden, The Netherlands

174 School of Physical Sciences, The Open University, Milton Keynes, MK7 6AA, UK

175 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK

176 Univ. Lille, CNRS, Centrale Lille, UMR 9189 CRIStAL, 59000 Lille, France

177 Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA

178 Fakultät für Physik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany

179 School of Mathematics, Statistics and Physics, Newcastle University, Herschel Building, Newcastle-upon-Tyne, NE1 7RU, UK

180 Department of Physics and Astronomy, University of British Columbia, Vancouver, BC V6T 1Z1, Canada