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(Ωm/0.3)0.5. All measurements are from ground-based weak lensing surveys unless noted. Planck 2018 gives S8 = 0.832 ± 0.013 for reference.

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:

\[\xi_\pm^{ij}(\theta) = \langle \gamma_t^i \gamma_t^j \rangle(\theta) \pm \langle \gamma_\times^i \gamma_\times^j \rangle(\theta)\]

These are related to the convergence power spectrum via Limber integration:

\[C_\kappa^{ij}(\ell) = \int_0^{\chi_H} \frac{q_i(\chi)\,q_j(\chi)}{\chi^2}\, P_m\!\left(\frac{\ell+1/2}{\chi}, z(\chi)\right)\, \mathrm{d}\chi\]

where qi(χ) is the lensing efficiency kernel:

\[q_i(\chi) = \frac{3H_0^2\Omega_m}{2c^2} \frac{\chi}{a(\chi)} \int_\chi^{\chi_H} n_i(\chi')\,\frac{\chi'-\chi}{\chi'}\,\mathrm{d}\chi'\]

The projection means cosmic shear is sensitive to a combination σ₈(Ωm/0.3)0.5 ≡ S8 rather than σ₈ and Ωmseparately. 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± optimised to separate E-modes (lensing signal) from B-modes (null test); used in KiDS-1000 (Asgari+2021).

  • Bandpower spectra Cℓ: Fourier-space estimator; used in HSC (Hikage+2019, Li+2023). More naturally connected to theory predictions; less affected by survey boundary effects than ξ±.

  • Aperture mass statistics Map: 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 Cℓ.

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:

\[P_{\rm II}(k,z) = F^2(z)\,P_\delta(k,z), \quad P_{\rm GI}(k,z) = -F(z)\,P_\delta(k,z)\]

where F(z) = -AIA C₁ ρcr Ωm / 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 Cℓby 10–30% at ℓ > 1000. Parameterised through HMcode2020 (Abary, ηbary) or the OwlS/BAHAMAS simulation grid. Most analyses apply angular or k-scale cuts to mitigate this uncertainty, at the cost of information.

Code: 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

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, neff ~ 22 arcmin⁻²); Cℓ estimator; S8 = 0.800

2021

KiDS-1000 (Asgari+2021, Heymans+2021): 1006 deg², 5-bin tomography; three statistics (ξ±, COSEBIs, Cℓ) 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, Cℓ + 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

Survey

Area (deg²)

Nbins

Nsrc

neff (arcmin⁻²)

zsrc

Shear

Statistic

CFHTLenS (Heymans+2012)

154

6

4.2 M

11

0.2–1.3

lensfit

ξ+/ξ-

CFHTLenS (Kilbinger+2013)

154

1

4.2 M

11

0.2–1.3

lensfit

ξ+/ξ- (2D)

KiDS-450 (Hildebrandt+2017)

450

4

15 M

8.5

0.1–0.9

lensfit

ξ+/ξ-

DES Y1 (Troxel+2018)

1321

4

26 M

5.9

0.2–1.3

IM3SHAPE+METACAL

ξ+/ξ-

HSC Y1 (Hikage+2019)

136

4

11 M

22

0.3–1.5

HSMRegauss

Cℓ

KiDS-1000 (Asgari+2021)

1006

5

21 M

6.2

0.1–1.2

lensfit (selfcal)

ξ±, COSEBIs, Cℓ

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+/ξ-

DES Y3 (Secco+2022)

4143

4

100 M

5.6

0.0–2.0

METACALIBRATION

ξ+/ξ- (scale cuts)

HSC Y3 (Li+2023)

416

4

25 M

15

0.3–1.5

HSMRegauss

Cℓ

Summary figure

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.

Directory

Survey

Area (deg²)

Nsrc

zsource

S8 constraint

Cites

Heymans2012_CFHTLenS

CFHTLenS

154

4.2 M

0.2 – 1.3

0.748+0.058-0.044

~700

Kilbinger2013_CFHTLenS

CFHTLenS

154

4.2 M

0.2 – 1.3

0.79+0.08-0.07 (2D)

~600

Hildebrandt2017_KiDS450

KiDS-450

450

15 M

0.1 – 0.9

0.745+0.039-0.038

~700

Troxel2018_DES_Y1

DES Y1

1321

26 M

0.2 – 1.3

0.782+0.027-0.027

~600

Hikage2019_HSC_Y1

HSC Y1

136

11 M

0.3 – 1.5

0.800+0.029-0.028

~400

Asgari2021_KiDS1000

KiDS-1000

1006

21 M

0.1 – 1.2

0.763+0.019-0.017

~600

Heymans2021_KiDS1000

KiDS-1000 (3×2pt)

1006

21 M

0.1 – 1.2

0.766+0.020-0.014

~600

Amon2022_DES_Y3

DES Y3

4143

100 M

0.0 – 2.0

0.772+0.018-0.017

~300

Secco2022_DES_Y3

DES Y3 (scale cuts)

4143

100 M

0.0 – 2.0

0.759+0.025-0.023

~250

Li2023_HSC_Y3

HSC Y3

416

25 M

0.3 – 1.5

0.776+0.032-0.033

~120

CFHTLenS

Canada-France-Hawaii Telescope Lensing Survey: 154 deg² in 5 CFHT MegaCam fields; lensfit shear; PSF modelled from stars using lensfit.

Directory

Statistic

Shear

Reference

Heymans2012_CFHTLenS

ξ+/ξ- tomographic (6 bins)

lensfit

MNRAS 427, 146 (arXiv:1210.0032)

Kilbinger2013_CFHTLenS

ξ+/ξ- 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.

Directory

Statistic

Shear

Reference

Hildebrandt2017_KiDS450

ξ+/ξ- tomographic (4 bins)

lensfit

MNRAS 465, 1454 (arXiv:1606.05338)

Asgari2021_KiDS1000

ξ+/ξ-, COSEBIs, bandpowers (5 bins)

lensfit (selfcal)

A&A 645, A104 (arXiv:2007.15633)

Heymans2021_KiDS1000

3×2pt: ξ+/ξ- + γt + 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.

Directory

Statistic

Shear

Reference

Troxel2018_DES_Y1

ξ+/ξ- tomographic (4 bins)

IM3SHAPE + METACAL

PhRvD 98, 043528 (arXiv:1708.01538)

Amon2022_DES_Y3

ξ+/ξ- tomographic (4 bins)

METACALIBRATION

PhRvD 105, 023514 (arXiv:2105.13543)

Secco2022_DES_Y3

ξ+/ξ- 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.

Directory

Statistic

Shear

Reference

Hikage2019_HSC_Y1

Cℓ power spectrum (4 bins)

HSMRegauss

PASJ 71, 43 (arXiv:1809.09148)

Li2023_HSC_Y3

Cℓ power spectrum (4 bins)

HSMRegauss

PhRvD 108, 123518 (arXiv:2304.00702)

Shear estimator glossary

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