arXiv subjects

Axel Guinot

Publications and source records attributed to Axel Guinot.

At least 19 recordsLinked to original sources

arXiv2026

Measuring satellite galaxy subhalo masses in redMaPPer clusters with UNIONS weak lensing data

In this work, we present the observed galaxy-galaxy lensing signal of satellite galaxies in rich clusters obtained from the Ultraviolet Near-Infrared Optical Northern Survey (UNIONS). Such observations are crucial for understanding dark matter halo physics and interaction dynamics in the cluster environment. Our goal is to investigate the tidal stripping of cluster subhalos. Theoretical predictions, supported by recent studies, suggest that as satellite galaxies fall into clusters, their dark matter halos are stripped and dispersed into the cluster's main halo, while their stellar mass remains relatively intact. A robust method to probe this phenomenon is through statistical measurements of the excess surface mass density of satellite galaxies at different cluster-centric distances. Our results reveal a significant lensing signal, with strong statistical power from the large number of lens-source pairs. A simple model used in the fitting process to constrain subhalo masses effectively reproduces the observed lensing signal at relevant satellite-centric distances. However, it lacks the complexity to accurately fit the excess surface density at larger distances, where the host cluster halo dominates. We measure the subhalo-to-stellar mass ratio as a function of cluster-centric radius and find a strong positive trend. These results confirm tidal stripping or other environmental processes that impact the relative masses of a satellite galaxy's stellar component and its dark matter halo.

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arXiv2026

Robust Photometry for Roman High-Latitude Imaging Survey Cosmology Using Roman and Rubin Imaging

Accurate and precise photometry is essential for Roman cosmology because it affects photometric-redshift performance, galaxy and cluster selection, tomographic binning, and redshift-distribution characterization. Achieving robust photometry is challenging because Roman's depth leads to substantial source blending, particularly in joint Roman--Rubin analyses, where the two surveys have different spatial resolutions. In this work, we develop and validate slimfarmer, a model-fitting photometry pipeline designed for Roman High-Latitude Imaging Survey (HLIS) cosmological analyses. Building on The Farmer, slimfarmer introduces treatments of the correlated noise present in Roman images and of astronomical shot noise, together with model-fitting configurations tuned for Roman imaging. We validate the pipeline using Roman coadded simulated images and perform forced photometry on matched Rubin image simulations. We find that slimfarmer recovers galaxy colors with the mode of the color residuals within 20 millimag for Roman-based colors and within 70 millimag for Rubin-based colors. Properly accounting for correlated noise is essential for uncertainty quantification, as photometric uncertainties are otherwise underestimated by up to a factor of ~3. We also find that applying multi-object fitting consistently to both Roman and Rubin imaging substantially mitigates blending-induced color systematics. In crowded regions, single-object fitting produces environment-dependent color biases of up to ~0.5 mag, leading to environment-dependent degradation of photometric-redshift performance. Together, these results establish slimfarmer as a prototype photometric pipeline for Roman HLIS cosmology applications and identify the correlated-noise corrections, astronomical shot-noise treatment, and joint Roman--Rubin multi-object fitting required for robust photometric-redshift characterization.

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arXiv2026

Differentiable Forward Modeling for Efficient and Accurate Shear Inference

Forthcoming Stage-IV dark energy optical surveys, such as LSST, have the ambitious goal of measuring cosmological parameters at sub-percent precision. Realizing their full scientific potential requires very precise measurement of the cosmic shear signal and control of corresponding systematics. In this work, we present a modern implementation of the Bayesian shear inference framework in Schneider et al. (2015), in the case that the PSF and sky background are known. This framework automatically propagates the pixel-noise measurement error from each galaxy into the final shear estimate, and thus requires no external calibration to handle noise bias. As a first application of this new implementation, we infer the cosmic shear posterior from simulated images consisting of isolated exponential galaxies with semi-realistic noise levels. In this simplified scenario, we estimate the absolute multiplicative bias $|m|$ of our approach to be below $0.9 \times 10^{-3}\,[3\sigma]$ when the intrinsic distribution of galaxy properties is known, and below $1.3 \times 10^{-3}\,[3\sigma]$ when these distributions are inferred alongside shear. Additionally, we make progress towards the algorithm's computational feasibility in the context of modern wide-field surveys, where billions of galaxies must be processed, by leveraging differentiable forward models of galaxies, gradient-based samplers, and GPUs. Our final galaxy-fitting MCMC produces $300$ effective samples of galaxy properties in $0.45$ seconds per galaxy using a single A100 GPU. In the future, we seek to generalize our algorithm to handle selection, detection, and model shear biases so it can be applied to real survey data.

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arXiv2026

cosmo-numba: B-modes and COSEBIs computations accelerated by Numba

Weak gravitational lensing is a widely used probe in cosmological analysis. It allows astrophysists to understand the content and evolution of the Universe. We are entering an era where we are not limited by the data volume but by systematic uncertainties. It is in this context that we present here a simple python-based software package to help in the computation of E-/B-mode decomposition, which can be use for systematic checks or science analysis. As we demonstrate, our implementation has both the high precision and speed required to perform this kind of analysis while avoiding a scenario wherein either numerical precision or computational time is a significant limiting factor.

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arXiv2026

Simulation-Based Inference for Probabilistic Galaxy Detection and Deblending

Stage-IV dark energy wide-field surveys, such as the Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST), will observe an unprecedented number density of galaxies. As a result, the majority of imaged galaxies will visually overlap, a phenomenon known as blending. Blending is expected to be a leading source of systematic error in astronomical measurements. To mitigate this systematic, we propose a new probabilistic method for detecting, deblending, and measuring the properties of galaxies, called the Bayesian Light Source Separator (BLISS). Given an astronomical survey image, BLISS uses convolutional neural networks to produce a probabilistic astronomical catalog by approximating the posterior distribution over the number of light sources, their centroids' locations, and their types (galaxy vs. star). BLISS additionally includes a denoising autoencoder to reconstruct unblended galaxy profiles. As a first step towards demonstrating the feasibility of BLISS for cosmological applications, we apply our method to simulated single-band images whose properties are representative of year-10 LSST coadds. First, we study each BLISS component independently and examine its probabilistic output as a function of SNR and degree of blending. Then, by propagating the probabilistic detections from BLISS to its deblender, we produce per-object flux posteriors. Using these posteriors yields a substantial improvement in aperture flux residuals relative to deterministic detections alone, particularly for highly blended and faint objects. These results highlight the potential of BLISS as a scalable, uncertainty-aware tool for mitigating blending-induced systematics in next-generation cosmological surveys.

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arXiv2025

Cluster properties as a function of dynamical state in the DESI Legacy x UNIONS surveys

We investigate how the dynamical state of galaxy clusters influences their galaxy populations and mass distributions. Using photometrically selected clusters from the DESI Legacy Imaging Survey cross-matched with the UNIONS galaxy shear catalogue, we classify clusters as evolved or evolving based on their rest-frame r-band magnitude gaps and stellar mass ratios between the brightest cluster galaxies (BCGs) and bright satellites. We measure the stellar mass functions, weak-lensing profiles, and radial number density and red-fraction profiles of stacked clusters in both subsamples. Evolved clusters exhibit more concentrated lensing profiles, bimodal stellar mass functions dominated by massive BCGs, and a deficit of intermediate-mass satellites, while evolving clusters show flatter central lensing signals and an excess of massive satellites. Applying the same selection to IllustrisTNG clusters reproduces these trends and links the observed differences to distinct mass accretion histories. These results demonstrate the close link between cluster galaxy populations and the overall dynamical state of their underlying dark matter halo.

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arXiv2025

Lensing the darkness: The matter density profile in cosmic voids from UNIONS

We measure the distribution of matter contained within the emptiest regions of the Universe: cosmic voids. We use the large overlap between the Ultraviolet Near-Infrared Optical Northern Survey (UNIONS) and voids identified in the LOWZ and CMASS catalogues of the Baryon Oscillation Spectroscopic Survey (BOSS) to constrain the excess surface mass density of voids using weak lensing. We present and validate a novel method for computing the Gaussian component of the conventional weak lensing covariance, adapted for use with void studies. We detect the stacked weak lensing void density profile at the $6.2\sigma$ level, the most significant detection of void lensing from spectroscopically-identified voids to date. We find that large and small voids have different matter density profiles, as expected from numerical studies of void profiles. This difference is significant at the $2.3\sigma$ level. Comparing the void profile to a measurement of the void-galaxy cross-correlation to test the linearity of the relationship between mass and light, we find good visual agreement between the two, and a galaxy bias factor of $2.45\pm0.36$, consistent with other works. This work represents a promising detection of the lensing effect from underdensities, with the goal of promoting its development into a competitive cosmological probe.

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arXiv2025

UNIONS: The Ultraviolet Near-Infrared Optical Northern Survey

The Ultraviolet Near-Infrared Optical Northern Survey (UNIONS) is a "collaboration of collaborations" that is using the Canada-France-Hawai'i Telescope, the Pan-STARRS telescopes, and the Subaru Observatory to obtain $ugriz$ images of a core survey region of 6250 deg$^2$ of the northern sky. The $10\sigma$ point source depth of the data, as measured within a 2-arcsecond diameter aperture, are $[u,g,r,i,z] = [23.7, 24.5, 24.2, 23.8, 23.3]$\ in AB magnitudes. UNIONS is addressing some of the most fundamental questions in astronomy, including the properties of dark matter, the growth of structure in the Universe from the very smallest galaxies to large-scale structure, and the assembly of the Milky Way. It is set to become the major ground-based legacy survey for the northern hemisphere for the next decade and provides an essential northern complement to the static-sky science of the Vera C. Rubin Observatory's Legacy Survey of Space and Time. UNIONS supports the core science mission of the {\it Euclid} space mission by providing the data necessary in the northern hemisphere for the calibration of the wavelength dependence of the {\it Euclid} point-spread function and derivation of photometric redshifts in the North Galactic Cap. This region contains the highest quality sky for {\it Euclid}, with low backgrounds from the zodiacal light, stellar density, extinction, and emission from Galactic cirrus. Here, we describe the UNIONS survey components, science goals, data products, and the current status of the overall program.

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arXiv2025

Cosmology from UNIONS weak lensing profiles of galaxy clusters

Cosmological information is encoded in the structure of galaxy clusters. In Universes with less matter and larger initial density perturbations, clusters form earlier and have more time to accrete material, leading to a more extended infall region. Thus, measuring the mean mass distribution in the infall region provides a novel cosmological test. The infall region is largely insensitive to baryonic physics, and provides a cleaner structural test than other measures of cluster assembly time such as concentration. We consider cluster samples from three publicly available galaxy cluster catalogues: the Spectroscopic Identification of eROSITA Sources (SPIDERS) catalogue, the X-ray and Sunyaev-Zeldovich effect selected clusters in the meta-catalogue M2C, and clusters identified in the Dark Energy Spectroscopic Instrument (DESI) Legacy Imaging Survey. Using a preliminary shape catalogue from the Ultraviolet Near Infrared Optical Northern Survey (UNIONS), we derive excess surface mass density profiles for each sample. We then compare the mean profile for the DESI Legacy sample, which is the most complete, to predictions from a suite of simulations covering a range of $\Omega_{\rm m}$ and $\sigma_8$, obtaining constraints of $\Omega_{\rm m}=0.34\pm 0.06$ and $\sigma_8=0.77 \pm 0.04$. We also measure mean (comoving) splashback radii for SPIDERS, M2C and DESI Legacy Imaging Survey clusters of $1.39^{+0.21}_{-0.18} {\rm cMpc}$, $1.77^{+0.20}_{-0.18} {\rm cMpc}/h$ and $1.42^{+0.11}_{-0.12} {\rm cMpc}/h$ respectively. Performing this analysis with the final UNIONS shape catalogue and the full sample of spectroscopically observed clusters in DESI, we can expect to improve on the best current constraints from cluster abundance studies by a factor of 2 or more.

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arXiv2024

Galaxy-Point Spread Function correlations as a probe of weak-lensing systematics with UNIONS data

Weak gravitational lensing requires precise measurements of galaxy shapes and therefore an accurate knowledge of the PSF model. The latter can be a source of systematics that affect the shear two-point correlation function. A key stake of weak lensing analysis is to forecast the systematics due to the PSF. Correlation functions of galaxies and the PSF, the so-called $\rho$- and $\tau$-statistics, are used to evaluate the level of systematics coming from the PSF model and PSF corrections, and contributing to the two-point correlation function used to perform cosmological inference. Our goal is to introduce a fast and simple method to estimate this level of systematics and assess its agreement with state-of-the-art approaches. We introduce a new way to estimate the covariance matrix of the $\tau$-statistics using analytical expressions. The covariance allows us to estimate parameters directly related to the level of systematics associated with the PSF and provides us with a tool to validate the PSF model used in a weak-lensing analysis. We apply those methods to data from the Ultraviolet Near-Infrared Optical Northern Survey (UNIONS). We show that the semi-analytical covariance yields comparable results than using covariances obtained from simulations or jackknife resampling. It requires less computation time and is therefore well suited for rapid comparison of the systematic level obtained from different catalogs. We also show how one can break degeneracies between parameters with a redefinition of the $\tau$-statistics. The methods developed in this work will be useful tools in the analysis of current weak-lensing data but also of Stage IV surveys such as Euclid, LSST or Roman. They provide fast and accurate diagnostics on PSF systematics that are crucial to understand in the context of cosmic shear studies.

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arXiv2024

The Blending ToolKit: A simulation framework for evaluation of galaxy detection and deblending

We present an open source Python library for simulating overlapping (i.e., blended) images of galaxies and performing self-consistent comparisons of detection and deblending algorithms based on a suite of metrics. The package, named Blending Toolkit (BTK), serves as a modular, flexible, easy-to-install, and simple-to-use interface for exploring and analyzing systematic effects related to blended galaxies in cosmological surveys such as the Vera Rubin Observatory Legacy Survey of Space and Time (LSST). BTK has three main components: (1) a set of modules that perform fast image simulations of blended galaxies, using the open source image simulation package GalSim; (2) a module that standardizes the inputs and outputs of existing deblending algorithms; (3) a library of deblending metrics commonly defined in the galaxy deblending literature. In combination, these modules allow researchers to explore the impacts of galaxy blending in cosmological surveys. Additionally, BTK provides researchers who are developing a new deblending algorithm a framework to evaluate algorithm performance and make principled comparisons with existing deblenders. BTK includes a suite of tutorials and comprehensive documentation. The source code is publicly available on GitHub at https://github.com/LSSTDESC/BlendingToolKit.

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arXiv2024

MADNESS Deblender: Maximum A posteriori with Deep NEural networks for Source Separation

Due to the unprecedented depth of the upcoming ground-based Legacy Survey of Space and Time (LSST) at the Vera C. Rubin Observatory, approximately two-thirds of the galaxies are likely to be affected by blending - the overlap of physically separated galaxies in images. Thus, extracting reliable shapes and photometry from individual objects will be limited by our ability to correct blending and control any residual systematic effect. Deblending algorithms tackle this issue by reconstructing the isolated components from a blended scene, but the most commonly used algorithms often fail to model complex realistic galaxy morphologies. As part of an effort to address this major challenge, we present MADNESS, which takes a data-driven approach and combines pixel-level multi-band information to learn complex priors for obtaining the maximum a posteriori solution of deblending. MADNESS is based on deep neural network architectures such as variational auto-encoders and normalizing flows. The variational auto-encoder reduces the high-dimensional pixel space into a lower-dimensional space, while the normalizing flow models a data-driven prior in this latent space. Using a simulated test dataset with galaxy models for a 10-year LSST survey and a galaxy density ranging from 48 to 80 galaxies per arcmin2 we characterize the aperture-photometry g-r color, structural similarity index, and pixel cosine similarity of the galaxies reconstructed by MADNESS. We compare our results against state-of-the-art deblenders including scarlet. With the r-band of LSST as an example, we show that MADNESS performs better than in all the metrics. For instance, the average absolute value of relative flux residual in the r-band for MADNESS is approximately 29% lower than that of scarlet. The code is publicly available on GitHub.

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arXiv2024

Black-Hole-to-Halo Mass Relation From UNIONS Weak Lensing

This letter presents, for the first time, direct constraints on the black-hole-to-halo-mass relation using weak gravitational lensing measurements. We construct type I and type II Active Galactic Nuclei (AGNs) samples from the Sloan Digital Sky Survey (SDSS), with a mean redshift of 0.4 0.1 for type I (type II) AGNs. This sample is cross-correlated with weak lensing shear from the Ultraviolet Near Infrared Northern Survey (UNIONS). We compute the excess surface mass density of the halos associated with $36,181$ AGNs from $94,308,561$ lensed galaxies and fit the halo mass in bins of black-hole mass. We find that more massive AGNs reside in more massive halos. We see no evidence of dependence on AGN type or redshift in the black-hole-to-halo-mass relationship when systematic errors in the measured black-hole masses are included. Our results are consistent with previous measurements for non-AGN galaxies. At a fixed black-hole mass, our weak-lensing halo masses are consistent with galaxy rotation curves, but significantly lower than galaxy clustering measurements. Finally, our results are broadly consistent with state-of-the-art hydro-dynamical cosmological simulations, providing a new constraint for black-hole masses in simulations.

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arXiv2023

Bayesian multi-band fitting of alerts for kilonovae detection

In the era of multi-messenger astronomy, early classification of photometric alerts from wide-field and high-cadence surveys is a necessity to trigger spectroscopic follow-ups. These classifications are expected to play a key role in identifying potential candidates that might have a corresponding gravitational wave (GW) signature. Machine learning classifiers using features from parametric fitting of light curves are widely deployed by broker software to analyze millions of alerts, but most of these algorithms require as many points in the filter as the number of parameters to produce the fit, which increases the chances of missing a short transient. Moreover, the classifiers are not able to account for the uncertainty in the fits when producing the final score. In this context, we present a novel classification strategy that incorporates data-driven priors for extracting a joint posterior distribution of fit parameters and hence obtaining a distribution of classification scores. We train and test a classifier to identify kilonovae events which originate from binary neutron star mergers or neutron star black hole mergers, among simulations for the Zwicky Transient Facility observations with 19 other non-kilonovae-type events. We demonstrate that our method can estimate the uncertainty of misclassification, and the mean of the distribution of classification scores as point estimate obtains an AUC score of 0.96 on simulated data. We further show that using this method we can process the entire alert steam in real-time and bring down the sample of probable events to a scale where they can be analyzed by domain experts.

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arXiv2022

The shape of dark matter haloes: results from weak lensing in the Ultraviolet Near-Infrared Optical Northern Survey (UNIONS)

Cold dark matter haloes are expected to be triaxial, and so appear elliptical in projection. We use weak gravitational lensing from the Canada-France Imaging Survey (CFIS) component of the Ultraviolet-Near Infrared Optical Northern Survey (UNIONS) to measure the ellipticity of the dark matter haloes around Luminous Red Galaxies (LRGs) from the Sloan Digital Sky Survey Data Release 7 (DR7) and from the CMASS and LOWZ samples of the Baryon Oscillation Spectroscopic Survey (BOSS), assuming their major axes are aligned with the stellar light. We find that DR7 LRGs with masses $M \sim 2.7\times10^{13} \mathrm{M}_{\odot}/h$ have halo ellipticities $e=0.46\pm0.10$. Expressed as a fraction of the galaxy ellipticity, we find $f_h = 2.2\pm0.6$. For BOSS LRGs, the detection is of marginal significance: $e = 0.20\pm0.10$ and $f_h=0.7\pm0.7$. These results are in agreement with other measurements of halo ellipticity from weak lensing and, taken together with previous results, suggest an increase of halo ellipticity of $0.10\pm0.06$ per decade in halo mass. This trend agrees with the predictions from hydrodynamical simulations, which find that at higher halo masses, not only do dark matter haloes become more elliptical, but that the misalignment between major axis of the stellar light in the central galaxy and that of the dark matter decreases.

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arXiv2022

Impact of Point Spread Function Higher Moments Error on Weak Gravitational Lensing II: A Comprehensive Study

Weak gravitational lensing, or weak lensing, is one of the most powerful probes for dark matter and dark energy science, although it faces increasing challenges in controlling systematic uncertainties as \edit{the statistical errors become smaller}. The Point Spread Function (PSF) needs to be precisely modeled to avoid systematic error on the weak lensing measurements. The weak lensing biases induced by errors in the PSF model second moments, i.e., its size and shape, are well-studied. However, Zhang et al. (2021) showed that errors in the higher moments of the PSF may also be a significant source of systematics for upcoming weak lensing surveys. Therefore, the goal of this work is to comprehensively investigate the modeling quality of PSF moments from the $3^{\text{rd}}$ to $6^{\text{th}}$ order, and estimate their impact on cosmological parameter inference. We propagate the \textsc{PSFEx} higher moments modeling error in the HSC survey dataset to the weak lensing \edit{shear-shear correlation functions} and their cosmological analyses. We find that the overall multiplicative shear bias associated with errors in PSF higher moments can cause a $\sim 0.1 \sigma$ shift on the cosmological parameters for LSST Y10. PSF higher moment errors also cause additive biases in the weak lensing shear, which, if not accounted for in the cosmological parameter analysis, can induce cosmological parameter biases comparable to their $1\sigma$ uncertainties for LSST Y10. We compare the \textsc{PSFEx} model with PSF in Full FOV (\textsc{Piff}), and find similar performance in modeling the PSF higher moments. We conclude that PSF higher moment errors of the future PSF models should be reduced from those in current methods to avoid a need to explicitly model these effects in the weak lensing analysis.

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arXiv2022

UNIONS: The impact of systematic errors on weak-lensing peak counts

UNIONS is an ongoing deep photometric multi-band survey of the Northern sky. As part of UNIONS, CFIS provides r-band data which we use to study weak-lensing peak counts for cosmological inference. We assess systematic effects for weak-lensing peak counts and their impact on cosmological parameters for the UNIONS survey. In particular, we present results on local calibration, metacalibration shear bias, baryonic feedback, the source galaxy redshift estimate, intrinsic alignment, and the cluster member dilution. For each uncertainty and systematic effect, we describe our mitigation scheme and the impact on cosmological parameter constraints. We obtain constraints on cosmological parameters from MCMC using CFIS data and MassiveNuS N-body simulations as a model for peak counts statistics. Depending on the calibration (local versus global, and the inclusion of the residual multiplicative shear bias), the mean matter density parameter $\Omega_m$ can shift up to $-0.024$ ($-0.5\sigma$). We also see that including baryonic corrections can shift $\Omega_m$ by $+0.027$ ($+0.5 \sigma$) with respect to the DM-only simulations. Reducing the impact of the intrinsic alignment and cluster member dilution through signal-to-noise cuts can lead to a shift in $\Omega_m$ of $+0.027$ ($+0.5 \sigma$). Finally, with a mean redshift uncertainty of $\Delta \bar{z} = 0.03$, we see that the shift of $\Omega_m$ ($+0.001$ which corresponds to $+0.02 \sigma$) is not significant. This paper investigates for the first time with UNIONS weak-lensing data and peak counts the impact of systematic effects. The value of $\Omega_m$ is the most impacted and can shift up to $\sim 0.03$ which corresponds to $0.5\sigma$ depending on the choices for each systematics. We expect constraints to become more reliable with future (larger) data catalogues, for which the current pipeline will provide a starting point.

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arXiv2022

ShapePipe: a new shape measurement pipeline and weak-lensing application to UNIONS/CFIS data

UNIONS is an ongoing collaboration that will provide the largest deep photometric survey of the Northern sky in four optical bands to date. As part of this collaboration, CFIS is taking $r$-band data with an average seeing of 0.65 arcsec, which is complete to magnitude 24.5 and thus ideal for weak-lensing studies. We perform the first weak-lensing analysis of CFIS $r$-band data over an area spanning 1700 deg$^2$ of the sky. We create a catalogue with measured shapes for 40 million galaxies, corresponding to an effective density of 6.8 galaxies per square arcminute, and demonstrate a low level of systematic biases. This work serves as the basis for further cosmological studies using the full UNIONS survey of 4800 deg$^2$ when completed. Here we present ShapePipe, a newly developed weak-lensing pipeline. This pipeline makes use of state-of-the-art methods such as Ngmix for accurate galaxy shape measurement. Shear calibration is performed with metacalibration. We carry out extensive validation tests on the Point Spread Function (PSF), and on the galaxy shapes. In addition, we create realistic image simulations to validate the estimated shear. We quantify the PSF model accuracy and show that the level of systematics is low as measured by the PSF residuals. Their effect on the shear two-point correlation function is sub-dominant compared to the cosmological contribution on angular scales <100 arcmin. The additive shear bias is below 5x$10^{-4}$, and the residual multiplicative shear bias is at most $10^{-3}$ as measured on image simulations. Using COSEBIs we show that there are no significant B-modes present in second-order shear statistics. We present convergence maps and see clear correlations of the E-mode with known cluster positions. We measure the stacked tangential shear profile around Planck clusters at a significance higher than $4\sigma$.

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