.. _literature_cosmic_shear: Cosmic Shear ============ Ten published cosmic shear measurements catalogued in ``literature_measurements/lensing/cosmic_shear/``, organised by survey. Each entry has a ``paper.json`` with full provenance (source catalog, shear estimator, tomographic bins, angular scales, key S8 constraint). **Conventions:** S8 ≡ σ₈(Ω\ :sub:`m`\ /0.3)\ :sup:`0.5`. All measurements are from ground-based weak lensing surveys unless noted. Planck 2018 gives S8 = 0.832 ± 0.013 for reference. .. contents:: On this page :local: :depth: 1 ---- Measurement methods ------------------- Two-point shear correlations ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Cosmic shear measures the coherent distortion of background galaxy shapes by large-scale foreground mass structures. The fundamental observables are the **two-point shear correlation functions** in tomographic redshift bins i, j: .. math:: \xi_\pm^{ij}(\theta) = \langle \gamma_t^i \gamma_t^j \rangle(\theta) \pm \langle \gamma_\times^i \gamma_\times^j \rangle(\theta) These are related to the convergence power spectrum via Limber integration: .. math:: C_\kappa^{ij}(\ell) = \int_0^{\chi_H} \frac{q_i(\chi)\,q_j(\chi)}{\chi^2}\, P_m\!\left(\frac{\ell+1/2}{\chi}, z(\chi)\right)\, \mathrm{d}\chi where q\ :sub:`i`\ (χ) is the lensing efficiency kernel: .. math:: q_i(\chi) = \frac{3H_0^2\Omega_m}{2c^2} \frac{\chi}{a(\chi)} \int_\chi^{\chi_H} n_i(\chi')\,\frac{\chi'-\chi}{\chi'}\,\mathrm{d}\chi' The projection means cosmic shear is sensitive to a combination σ₈(Ω\ :sub:`m`\ /0.3)\ :sup:`0.5`\ ≡ S8 rather than σ₈ and Ω\ :sub:`m`\ separately. Constraints from single-probe cosmic shear are thus approximately degenerate along the S8 direction. Alternative statistics ~~~~~~~~~~~~~~~~~~~~~~~ * **COSEBIs** (Complete Orthogonal Sets of E/B Integrals, Schneider+2010): linear combinations of ξ\ :sub:`±`\ optimised to separate E-modes (lensing signal) from B-modes (null test); used in KiDS-1000 (Asgari+2021). * **Bandpower spectra** C\ :sub:`ℓ`\ : Fourier-space estimator; used in HSC (Hikage+2019, Li+2023). More naturally connected to theory predictions; less affected by survey boundary effects than ξ\ :sub:`±`\ . * **Aperture mass statistics** M\ :sub:`ap`\ : third-order statistics (beyond 2pt); sensitive to non-Gaussianity and baryons. Shear measurement ~~~~~~~~~~~~~~~~~ All ground-based surveys require PSF deconvolution to measure the small (< 1%) ellipticity signal. The three main approaches are: * **lensfit** (Miller+2013): forward-model Bayesian fitting of a galaxy model convolved with PSF; outputs per-galaxy posterior on ellipticity. Used by CFHTLenS and KiDS. * **METACALIBRATION** (Sheldon & Huff 2017): applies a small shear to the image and re-measures the shape; self-calibrates the response to shear without image simulations. Used by DES Y3. * **HSMRegauss** (Hirata & Seljak 2003): re-Gaussianization PSF correction; computationally fast; calibrated via image simulations. Used by HSC (same pipeline as BGS+HSC Y3 in this project). Multiplicative bias calibration proceeds through: GREAT3 challenge → per-survey image simulations (MICE, GALSIM) → residual bias \|m\| calibration → uncertainty propagated to C\ :sub:`ℓ`\ . Photometric redshifts and n(z) calibration ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The lensing kernel q(χ) depends critically on the true redshift distribution n(z) of source galaxies. Systematic errors in n(z) shift S8 by up to 0.05. The main calibration approaches are: * **BPZ / LePhare**: template fitting photo-z; validated on spectroscopic overlap (zCOSMOS, VVDS). * **DIR** (direct calibration, Hildebrandt+2017): re-weight a deep spectroscopic training set to match the photometric colour distribution. Used in KiDS. * **SOM** (self-organising map, Wright+2020): group galaxies by colour and calibrate n(z) per cell from deep spectroscopy. Used in KiDS-1000 and HSC. * **DNNz / DEmPz**: deep neural network photo-z; used in HSC Y3. Intrinsic alignments ~~~~~~~~~~~~~~~~~~~~~ The dominant systematic in cosmic shear. The NLA model: .. math:: P_{\rm II}(k,z) = F^2(z)\,P_\delta(k,z), \quad P_{\rm GI}(k,z) = -F(z)\,P_\delta(k,z) where F(z) = -A\ :sub:`IA`\ C₁ ρ\ :sub:`cr`\ Ω\ :sub:`m`\ / D(z). More flexible models (TATT, Blazek+2019) include additional tidal torque and source alignment terms. Baryon feedback ~~~~~~~~~~~~~~~~ Small-scale power suppression from AGN feedback shifts C\ :sub:`ℓ`\ by 10–30% at ℓ > 1000. Parameterised through HMcode2020 (A\ :sub:`bary`\ , η\ :sub:`bary`\ ) or the OwlS/BAHAMAS simulation grid. Most analyses apply angular or k-scale cuts to mitigate this uncertainty, at the cost of information. Code: :mod:`sum_stat.lensing.shear_calib` ---- Status, open questions, and progress -------------------------------------- Current status ~~~~~~~~~~~~~~ As of 2025, cosmic shear from Stage-III surveys (KiDS, DES, HSC) has established: * S8 = 0.76–0.78 from all surveys, consistently below Planck (S8 = 0.832) at the 2–3σ level. The weighted mean across all Stage-III surveys is S8 ~ 0.77 ± 0.02. * The signal is reproducible across independently calibrated shape catalogs (lensfit, METACAL, HSMRegauss), angular scales (configuration and Fourier space), and redshift calibration methods (DIR, SOM, DNNz). * Stage-IV surveys (Euclid, LSST, Roman) will reduce statistical errors by a factor ~5, potentially resolving the S8 tension or establishing it as a definitive cosmological anomaly. Open questions ~~~~~~~~~~~~~~ * **The S8 tension**: Is S8 ~ 0.77 a true cosmological signal (beyond-ΛCDM physics, dynamic dark energy, neutrino mass) or a residual systematic (IA modelling, n(z) tails, PSF leakage at small scales)? The ~3σ level makes it both statistically significant and insufficiently compelling. * **Intrinsic alignments**: NLA model may be inadequate at 1%-level precision. TATT and higher-order models add parameters that weaken S8 constraints. Spectroscopic overlap surveys are needed to break the IA–cosmology degeneracy. * **Baryonic feedback**: Scale cuts discard information. Hydrodynamical simulation predictions (BAHAMAS, IllustrisTNG, FLAMINGO) span a wide range. Joint lensing + tSZ constraints can break the degeneracy. * **Non-limber corrections at low ℓ**: The Limber approximation fails at ℓ < 10 for tomographic bin correlations. Matters for Stage-IV surveys. * **Beyond-2pt**: Third-order statistics (weak lensing bispectrum, aperture mass skewness) carry additional information and constrain non-Gaussianity, but require accurate covariances. Progress over two decades ~~~~~~~~~~~~~~~~~~~~~~~~~ .. list-table:: :header-rows: 1 :widths: 10 90 * - Year - Milestone * - 2000–2002 - First cosmic shear detections (Bacon+2000, Kaiser+2000, Van Waerbeke+2000, Wittman+2000); ~30 deg² each; consistent with ΛCDM but large errors * - 2012 - CFHTLenS (Heymans+2012, Kilbinger+2013): first precision measurement (154 deg², lensfit); first hint of S8 below Planck at 2σ * - 2017 - KiDS-450 (Hildebrandt+2017): 450 deg², 4-bin tomography, DIR n(z); 2.3σ below Planck; S8 tension named and characterised * - 2018 - DES Y1 (Troxel+2018): 1321 deg², METACALIBRATION; independent confirmation at 1.7σ below Planck * - 2019 - HSC Y1 (Hikage+2019): 136 deg² but deepest (i < 24.5, n\ :sub:`eff`\ ~ 22 arcmin⁻²); C\ :sub:`ℓ`\ estimator; S8 = 0.800 * - 2021 - KiDS-1000 (Asgari+2021, Heymans+2021): 1006 deg², 5-bin tomography; three statistics (ξ±, COSEBIs, C\ :sub:`ℓ`\ ) all consistent; 3×2pt S8 = 0.766; 3σ below Planck; most precise ground-based measurement at the time * - 2022 - DES Y3 (Amon+2022, Secco+2022): 4143 deg², 100M sources, METACAL; S8 = 0.772; robustness to baryons and IA tested exhaustively * - 2023 - HSC Y3 (Li+2023, Miyatake+2023): 416 deg², C\ :sub:`ℓ`\ + 3×2pt; S8 = 0.769–0.763; companion analyses internally consistent; ~2σ below Planck * - 2024–25 - Euclid Q1 (Euclid+2025): first space-based cosmic shear with VIS; ~1200 deg²; expected S8 precision ~1%; will distinguish S8 tension from systematics ---- Survey parameter table ----------------------- .. list-table:: :header-rows: 1 :widths: 26 10 8 10 8 9 9 20 * - Survey - Area (deg²) - N\ :sub:`bins` - N\ :sub:`src` - n\ :sub:`eff`\ (arcmin⁻²) - z\ :sub:`src` - Shear - Statistic * - CFHTLenS (Heymans+2012) - 154 - 6 - 4.2 M - 11 - 0.2–1.3 - lensfit - ξ\ :sub:`+`\ /ξ\ :sub:`-` * - CFHTLenS (Kilbinger+2013) - 154 - 1 - 4.2 M - 11 - 0.2–1.3 - lensfit - ξ\ :sub:`+`\ /ξ\ :sub:`-`\ (2D) * - KiDS-450 (Hildebrandt+2017) - 450 - 4 - 15 M - 8.5 - 0.1–0.9 - lensfit - ξ\ :sub:`+`\ /ξ\ :sub:`-` * - DES Y1 (Troxel+2018) - 1321 - 4 - 26 M - 5.9 - 0.2–1.3 - IM3SHAPE+METACAL - ξ\ :sub:`+`\ /ξ\ :sub:`-` * - HSC Y1 (Hikage+2019) - 136 - 4 - 11 M - 22 - 0.3–1.5 - HSMRegauss - C\ :sub:`ℓ` * - KiDS-1000 (Asgari+2021) - 1006 - 5 - 21 M - 6.2 - 0.1–1.2 - lensfit (selfcal) - ξ\ :sub:`±`\ , COSEBIs, C\ :sub:`ℓ` * - KiDS-1000 (Heymans+2021) - 1006 - 5 - 21 M - 6.2 - 0.1–1.2 - lensfit (selfcal) - 3×2pt * - DES Y3 (Amon+2022) - 4143 - 4 - 100 M - 5.6 - 0.0–2.0 - METACALIBRATION - ξ\ :sub:`+`\ /ξ\ :sub:`-` * - DES Y3 (Secco+2022) - 4143 - 4 - 100 M - 5.6 - 0.0–2.0 - METACALIBRATION - ξ\ :sub:`+`\ /ξ\ :sub:`-`\ (scale cuts) * - HSC Y3 (Li+2023) - 416 - 4 - 25 M - 15 - 0.3–1.5 - HSMRegauss - C\ :sub:`ℓ` ---- Summary figure -------------- .. figure:: ../figures/literature/lit_cosmic_shear_s8_timeline.png :width: 80% :alt: S8 timeline from cosmic shear S8 constraints from all Stage-III cosmic shear surveys. The grey band shows Planck 2018 (S8 = 0.832 ± 0.013). The weighted mean of all Stage-III measurements (~0.77) lies ~3σ below Planck. ---- S8 summary ---------- Overview of S8 constraints from all catalogued surveys, ordered by publication year. .. list-table:: :header-rows: 1 :widths: 26 12 9 8 9 26 10 * - Directory - Survey - Area (deg²) - N\ :sub:`src` - z\ :sub:`source` - S8 constraint - Cites * - `Heymans2012_CFHTLenS `_ - CFHTLenS - 154 - 4.2 M - 0.2 – 1.3 - 0.748\ :sup:`+0.058`\ \ :sub:`-0.044` - ~700 * - `Kilbinger2013_CFHTLenS `_ - CFHTLenS - 154 - 4.2 M - 0.2 – 1.3 - 0.79\ :sup:`+0.08`\ \ :sub:`-0.07`\ (2D) - ~600 * - `Hildebrandt2017_KiDS450 `_ - KiDS-450 - 450 - 15 M - 0.1 – 0.9 - 0.745\ :sup:`+0.039`\ \ :sub:`-0.038` - ~700 * - `Troxel2018_DES_Y1 `_ - DES Y1 - 1321 - 26 M - 0.2 – 1.3 - 0.782\ :sup:`+0.027`\ \ :sub:`-0.027` - ~600 * - `Hikage2019_HSC_Y1 `_ - HSC Y1 - 136 - 11 M - 0.3 – 1.5 - 0.800\ :sup:`+0.029`\ \ :sub:`-0.028` - ~400 * - `Asgari2021_KiDS1000 `_ - KiDS-1000 - 1006 - 21 M - 0.1 – 1.2 - 0.763\ :sup:`+0.019`\ \ :sub:`-0.017` - ~600 * - `Heymans2021_KiDS1000 `_ - KiDS-1000 (3×2pt) - 1006 - 21 M - 0.1 – 1.2 - 0.766\ :sup:`+0.020`\ \ :sub:`-0.014` - ~600 * - `Amon2022_DES_Y3 `_ - DES Y3 - 4143 - 100 M - 0.0 – 2.0 - 0.772\ :sup:`+0.018`\ \ :sub:`-0.017` - ~300 * - `Secco2022_DES_Y3 `_ - DES Y3 (scale cuts) - 4143 - 100 M - 0.0 – 2.0 - 0.759\ :sup:`+0.025`\ \ :sub:`-0.023` - ~250 * - `Li2023_HSC_Y3 `_ - HSC Y3 - 416 - 25 M - 0.3 – 1.5 - 0.776\ :sup:`+0.032`\ \ :sub:`-0.033` - ~120 ---- CFHTLenS -------- Canada-France-Hawaii Telescope Lensing Survey: 154 deg² in 5 CFHT MegaCam fields; lensfit shear; PSF modelled from stars using lensfit. .. list-table:: :header-rows: 1 :widths: 28 18 9 45 * - Directory - Statistic - Shear - Reference * - `Heymans2012_CFHTLenS `_ - ξ\ :sub:`+`\ /ξ\ :sub:`-`\ tomographic (6 bins) - lensfit - MNRAS 427, 146 (`arXiv:1210.0032 `_) * - `Kilbinger2013_CFHTLenS `_ - ξ\ :sub:`+`\ /ξ\ :sub:`-`\ non-tomographic (2D) - lensfit - MNRAS 430, 2200 (`arXiv:1212.3338 `_) ---- KiDS ---- Kilo Degree Survey (VST OmegaCAM, r-band shape measurement); lensfit shear; n(z) from DIR cross-calibration with spectroscopic overlaps. .. list-table:: :header-rows: 1 :widths: 28 18 9 45 * - Directory - Statistic - Shear - Reference * - `Hildebrandt2017_KiDS450 `_ - ξ\ :sub:`+`\ /ξ\ :sub:`-`\ tomographic (4 bins) - lensfit - MNRAS 465, 1454 (`arXiv:1606.05338 `_) * - `Asgari2021_KiDS1000 `_ - ξ\ :sub:`+`\ /ξ\ :sub:`-`\ , COSEBIs, bandpowers (5 bins) - lensfit (selfcal) - A&A 645, A104 (`arXiv:2007.15633 `_) * - `Heymans2021_KiDS1000 `_ - 3×2pt: ξ\ :sub:`+`\ /ξ\ :sub:`-`\ + γ\ :sub:`t`\ + w(θ) - lensfit (selfcal) - A&A 646, A140 (`arXiv:2007.15632 `_) ---- DES --- Dark Energy Survey (DECam, riz-band shear); METACALIBRATION shear estimator (Y3); photo-z from BPZ / DNF. .. list-table:: :header-rows: 1 :widths: 28 18 9 45 * - Directory - Statistic - Shear - Reference * - `Troxel2018_DES_Y1 `_ - ξ\ :sub:`+`\ /ξ\ :sub:`-`\ tomographic (4 bins) - IM3SHAPE + METACAL - PhRvD 98, 043528 (`arXiv:1708.01538 `_) * - `Amon2022_DES_Y3 `_ - ξ\ :sub:`+`\ /ξ\ :sub:`-`\ tomographic (4 bins) - METACALIBRATION - PhRvD 105, 023514 (`arXiv:2105.13543 `_) * - `Secco2022_DES_Y3 `_ - ξ\ :sub:`+`\ /ξ\ :sub:`-`\ tomographic, scale cuts - METACALIBRATION - PhRvD 105, 023515 (`arXiv:2105.13544 `_) .. note:: ``Amon2022_DES_Y3`` and ``Secco2022_DES_Y3`` use the same source catalog and are published simultaneously. Amon focuses on data calibration robustness; Secco on modeling uncertainty and scale cuts. ---- HSC --- Hyper Suprime-Cam (Subaru, i-band shape measurement); HSMRegauss shear estimator — **the same pipeline used in the BGS+HSC Y3 galaxy-galaxy lensing analysis in this project**; photo-z from DNNz. .. list-table:: :header-rows: 1 :widths: 28 18 9 45 * - Directory - Statistic - Shear - Reference * - `Hikage2019_HSC_Y1 `_ - C\ :sub:`ℓ`\ power spectrum (4 bins) - HSMRegauss - PASJ 71, 43 (`arXiv:1809.09148 `_) * - `Li2023_HSC_Y3 `_ - C\ :sub:`ℓ`\ power spectrum (4 bins) - HSMRegauss - PhRvD 108, 123518 (`arXiv:2304.00702 `_) ---- Shear estimator glossary ------------------------ .. list-table:: :header-rows: 1 :widths: 20 80 * - Estimator - Notes * - lensfit - Forward-model Bayesian shape measurement; used by CFHTLenS and KiDS * - lensfit (selfcal) - KiDS-1000 version with self-calibration of multiplicative bias from the data * - IM3SHAPE - Maximum-likelihood forward-model fitting; used in DES Y1 alongside METACAL * - METACALIBRATION - Self-calibrating shear estimator using image manipulations; used in DES Y3 * - HSMRegauss - Re-Gaussianization PSF correction (Hirata & Seljak 2003); used in SDSS, HSC