Weak Lensing

Why "weak" — and why the 1 % matters

Every photon that reaches our telescopes has crossed billions of light-years of mostly empty space — but not entirely empty. The galaxies, groups, clusters, and dark-matter filaments threaded across that path each deflect the photon by a tiny angle. By the time the light arrives, the apparent shape of the source galaxy has been distorted by the integrated mass it passed near. For most lines of sight, that distortion is small — about 1 % of the galaxy's apparent size, well below the source's intrinsic asymmetry — and unobservable in any single galaxy. Yet because the distortion is systematic (it points toward the mass), averaging the orientations of many galaxies behind the same overdensity recovers a signal that emerges crisply from the noise. That is the weak-lensing regime.

The contrast with strong lensing is sharp. When the surface mass density of a foreground object exceeds the critical density Σ_crit, the lens equation has multiple solutions for a single source — and the observer sees arcs, multiple images, or full Einstein rings. The brain pattern-matches these immediately; you don't need statistics to identify a strong lens. Weak lensing operates everywhere else: in every patch of sky outside the rare critical regions of clusters, where the distortion is just a slight, statistical alignment of galaxies tangent to the line connecting them to nearby mass. Roughly 99 % of the sky lenses weakly. Strong lensing tells you about individual mass concentrations; weak lensing tells you about the mass distribution of the universe.

Shear γ and convergence κ

The gravitational deflection of a light bundle between source and observer is described, in the thin-lens approximation, by a 2×2 distortion matrix called the Jacobian:

A = [[1 − κ − γ_1,        − γ_2],
     [       − γ_2,  1 − κ + γ_1]]

Two scalar fields fully describe the lensing for any line of sight. Convergence κ is the isotropic part: it magnifies a background image (makes it bigger and brighter) without changing its shape. Numerically, κ at a point equals the projected mass density at that point divided by Σ_crit, the lensing critical surface density set by the geometry of source, lens, and observer. Shear γ = γ_1 + i γ_2 is the anisotropic stretch: it elongates the image along one axis and compresses it along the perpendicular axis, changing the apparent ellipticity by 2γ in the linear regime.

The two fields are not independent. Both κ and γ are linear combinations of second derivatives of the same projected gravitational potential ψ:

κ   = (1/2)(∂²ψ/∂x² + ∂²ψ/∂y²)
γ_1 = (1/2)(∂²ψ/∂x² − ∂²ψ/∂y²)
γ_2 =        ∂²ψ/∂x ∂y

This algebraic link is the basis of the Kaiser-Squires inversion: given a measured shear field γ(x,y) across a patch of sky, you can solve for the underlying convergence κ(x,y) — that is, you can reconstruct the projected mass density from the shear pattern alone, up to an additive constant called the mass-sheet degeneracy.

The KSB algorithm: how shape becomes science

Turning a 1 % distortion in galaxy shapes into a measurement requires understanding everything else that might warp those shapes: the atmosphere, the optics of the telescope, the pixelisation of the detector, the asymmetry of the point-spread function (PSF). The 1995 paper of Nick Kaiser, Gordon Squires, and Tom Broadhurst — "A method for weak lensing observations" — was the breakthrough that made weak lensing a practical tool.

KSB models each galaxy's observed image as the intrinsic image, sheared by γ, then convolved with the PSF, then measured under photon noise. From the observed surface brightness I(x), it computes weighted quadrupole moments

Q_ij = ∫ I(x) W(x) x_i x_j d²x  / ∫ I(x) W(x) d²x

where W(x) is a Gaussian aperture matched to the galaxy's apparent size. From Q_ij an ellipticity estimator e_obs is constructed. KSB then derives, from the moments of the PSF and the galaxy together, a "pre-seeing shear polarisability tensor" P^γ such that

e_obs = P^γ · γ  +  (smear correction · PSF anisotropy)  +  noise

Inverting this on every detected galaxy gives an unbiased per-source shear estimator (in the linear regime). The catch is that "unbiased" depends on assumptions about the noise model and the higher moments of the PSF; in practice KSB-based pipelines (KSB+, RRG, im2shape) carry residual biases of a few percent. Modern surveys have largely shifted to model-fitting (lensfit, im3shape) or self-calibrating (metacalibration, BFD) approaches that absorb the bias-calibration step into the measurement itself. But the conceptual decomposition KSB introduced — observed shape = intrinsic shape + PSF distortion + lensing shear — is still how every weak-lensing pipeline thinks.

Tangential alignment around a cluster

The simplest weak-lensing measurement is the tangential shear profile around a known cluster. For each background galaxy at angular distance θ from the cluster centre, decompose its measured ellipticity into two components — γ_t (tangential to the cluster centre) and γ_× (rotated 45°). The cluster mass induces a tangential pattern with γ_t > 0 and γ_× = 0 (on average).

The radial profile of γ_t(θ) directly probes the cluster's projected mass distribution. For an isothermal sphere γ_t ∝ 1/θ; for the more accurate NFW profile, γ_t falls off more gently in the centre and steepens at large radii. Fitting γ_t(θ) is the cleanest mass measurement available for galaxy clusters at 0.2 ≲ z ≲ 1 — independent of any assumption about the gas being in hydrostatic equilibrium or the galaxies tracing the mass, both of which contaminate X-ray and dynamical estimates.

The orthogonal mode γ_×(θ) — radial rather than tangential — should be zero by parity for any real gravitational lens. A non-zero γ_× signal at the same level as γ_t is the canonical diagnostic of systematics: residual PSF anisotropy, charge-transfer inefficiency in the CCDs, or a bias in the shape measurement.

The Bullet Cluster: weak lensing sees dark matter

If lensing tracks all mass and the hot gas in X-rays tracks only the baryons, then a major cluster collision should separate them — the collisionless dark matter and stars passing through while the gas is slowed by ram pressure. The Bullet Cluster (1E 0657-56) at z = 0.296 is exactly this configuration: two clusters caught some 100 Myr after pericentre passage. Weak-lensing maps (Clowe et al. 2006) showed that the dominant mass peaks are offset from the X-ray gas — they lie spatially coincident with the galaxies, not the hot baryons.

This is direct empirical evidence that the dominant matter is collisionless and non-baryonic. No reformulation of the gravity law on its own (such as MOND in its simplest forms) can produce mass peaks offset from the baryonic centre, because in such theories the gravitational source is the baryons. The Bullet Cluster did not prove dark matter exists, but it strongly constrained the alternatives — and weak lensing is what made the measurement possible.

Cosmic shear and the σ_8 constraint

Behind every distant galaxy is some foreground mass; behind two distant galaxies that are angularly close on the sky is, on average, partially overlapping foreground mass. The lensing shears induced on the two should therefore be partially correlated. The two-point statistics of the shear field — its angular power spectrum, or equivalently the real-space correlation function ξ_+(θ) and ξ_-(θ) — encode the matter clumpiness of the universe along the entire line of sight.

The amplitude of the shear power spectrum scales primarily with two parameters of the underlying cosmological model:

C_ℓ^γγ ∝ σ_8²  ·  Ω_m^α       with α ≈ 1.3 – 1.7 (depending on ℓ and survey depth)

σ_8 is the rms density fluctuation in 8-megaparsec spheres (smoothed at late times). Ω_m is the total matter fraction today. Because the two enter in a strong degenerate combination, cosmic-shear surveys actually constrain the derived parameter S_8 = σ_8 √(Ω_m / 0.3) much more tightly than either parameter individually.

The survey landscape

Survey	Telescope	Area / sources	Years	Notes
COSMOS	Hubble ACS	2 deg² · 10⁶ gal	2003–	Pioneering deep HST shear (Massey, Heymans)
CFHTLenS	CFHT MegaCam	154 deg² · 6 × 10⁶ gal	2003–2012	First ground-based cosmic shear of cosmological precision
DES (Y3, Y6)	Blanco / DECam	5000 deg² · 100 × 10⁶ gal	2013–2019	Largest completed ground survey; Y3 published 2021
KiDS-1000	VST OmegaCAM	1000 deg² · 21 × 10⁶ gal	2011–2019	Drove the S_8-tension narrative
HSC-SSP	Subaru HSC	1100 deg² · 36 × 10⁶ gal	2014–2022	Deepest ground-based shear; sharpest seeing
Euclid (wide)	Euclid spacecraft	14 000 deg² · 1.5 × 10⁹ gal	2023–2029	Space-based optical+NIR; PSF stability is the killer feature
LSST (Rubin)	Rubin Observatory	18 000 deg² · 10⁹ gal	2025–2035	Ten-year survey; the definitive ground-based weak-lensing dataset
Roman Space Telescope	NASA Roman	2000 deg² (HLS)	2027–	Space-based, high-resolution shear over wide area

The trend is brutal: in 20 years source counts have grown from 10⁶ to 10⁹, a 1000× increase, and survey area from 2 deg² to 18 000 deg². The statistical noise on cosmic-shear measurements drops as 1/√N_gal, so by LSST and Euclid the dominant errors are systematic — shape-measurement biases, photometric-redshift errors, intrinsic alignments, and baryonic-feedback uncertainties on small-scale clustering — not the per-galaxy shape noise the field grew up worrying about.

Worked example: how thick is the foreground sheet?

How much mass is required to produce a typical 1 % shear behind a foreground structure at z = 0.3, lensing a source at z = 1? The critical surface density is

Σ_crit = (c² / 4πG) · (D_s / (D_l D_ls))
       ≈ 3.5 × 10³ M☉ / pc²    (for z_l = 0.3, z_s = 1, ΛCDM)

Within the linear regime κ ≈ γ ≈ 0.01, so a 1 % shear corresponds to a projected surface density of

Σ = κ · Σ_crit ≈ 0.01 × 3.5 × 10³ ≈ 35 M☉ / pc²

Across a 1 Mpc² aperture this is ≈ 3.5 × 10¹³ M☉ — comparable to a small galaxy group. So a typical weak-lensing measurement at 1 % corresponds to a foreground group with a total mass several × 10¹³ M☉, and a typical cluster lensing measurement at γ_t ≈ 0.05–0.10 corresponds to a halo mass of 10¹⁴–10¹⁵ M☉. The arithmetic is consistent with what is observed.

The S_8 tension

The Planck 2018 measurement of the cosmic microwave background, extrapolated forward to z = 0 under standard ΛCDM, predicts

S_8 (Planck, ΛCDM)  =  0.832 ± 0.013

Independent weak-lensing measurements from cosmic shear in DES, KiDS, and HSC find

S_8 (DES Y3)        =  0.776 ± 0.017
S_8 (KiDS-1000)     =  0.766 ± 0.020
S_8 (HSC Y3)        =  0.776 ± 0.030

Each is roughly 2σ lower than Planck individually; jointly the discrepancy reaches 2.5–3σ. Possible resolutions include: (i) a systematic in shear or photometric-redshift calibration that conspires to bias all three independent surveys low by similar amounts; (ii) baryonic feedback (AGN outflows, supernova-driven winds) that suppresses small-scale matter clustering more than current hydrodynamical simulations predict; (iii) genuine deviations from ΛCDM such as time-varying dark energy, sterile-neutrino contributions to the matter budget, or modified gravity on cosmological scales. The next data releases — DES Y6 (final), Euclid, LSST Y1 — are designed to push either to a definitive resolution or to a 5σ confirmation.

Where weak lensing can go wrong

• Intrinsic alignments. Galaxies that formed in the same tidal field of a large-scale-structure overdensity inherit correlated intrinsic ellipticities, mimicking a lensing signal. Modern surveys jointly model the cosmic shear and the intrinsic-alignment "II" and "GI" terms with non-linear-alignment templates, but residual IA contamination is a leading systematic at the 1 % level.
• Photometric redshifts. The shear weighting depends on the source redshift distribution n(z), and ground-based surveys rely on photometric redshifts (typical 3–5 % scatter, 1–2 % bias). A systematic shift of ⟨z_s⟩ by 0.02 translates into a few-percent shift in inferred S_8 — comparable to the entire reported tension.
• PSF anisotropy. Telescope optics, focus drifts, wind-shake on the dome — all leave the PSF slightly elongated, and that elongation transfers directly into shape measurements unless corrected. Space-based platforms (Euclid, Roman) sidestep most of this.
• Baryonic feedback on small scales. At wavenumbers k ≳ 1 h/Mpc the matter power spectrum is altered by gas physics that ΛCDM-only simulations do not capture. Mis-modelled feedback can absorb (or fake) an S_8 shift; modern analyses either marginalise over a feedback amplitude or chop off the small-scale modes.
• Mass-sheet degeneracy. The Kaiser-Squires reconstruction recovers κ only up to a uniform additive constant; an unobserved uniform mass sheet adds zero shear and is invisible to lensing alone. Combining with magnification information from galaxy number counts or with cluster member positions breaks the degeneracy.
• B-mode leakage. A real lensing signal is curl-free at first order — all B-modes should vanish. A B-mode detection of similar amplitude to E is a calibration red flag, not a discovery of exotic physics.

CMB lensing as the long-redshift backlight

The cosmic microwave background, photons released at z ≈ 1100, is itself weakly lensed by every overdensity it has traversed since. CMB lensing reconstructs the integrated convergence κ_CMB(n̂) across the sky from subtle smearing of the small-scale temperature and polarisation maps — a procedure quite different from galaxy shape measurement but recovering the same underlying field. The CMB-lensing kernel peaks at z ≈ 2 and extends out past z = 5, complementing galaxy lensing's z ≲ 1.5 lever arm.

The most important application is cross-correlation: galaxy shears measured by DES/KiDS/HSC are correlated with CMB-lensing maps from Planck, ACT, and SPT. Cross-correlations are insensitive to additive systematics that affect either dataset alone, since spurious shears are uncorrelated with spurious CMB-lensing reconstructions. The joint analyses sharpen S_8 constraints, calibrate the source-redshift distribution, and offer one of the cleanest avenues to closing the tension.

Related variants

• Cosmic-shear tomography. Slice the source galaxies into photo-z bins and measure shear correlations within and between bins. The growth of structure as a function of redshift falls out, constraining the dark-energy equation of state w(z).
• 3 × 2pt analysis. The DES, KiDS, and HSC "joint analysis" that combines (1) cosmic shear, (2) galaxy-galaxy lensing, and (3) galaxy clustering. Each two-point function has its own systematics; jointly fitting all three breaks degeneracies that any one alone cannot.
• Galaxy-galaxy lensing. The mean tangential shear induced by foreground "lens" galaxies of known redshift on background "source" galaxies. A clean halo-occupation probe that ties galaxy luminosity / colour / clustering bins directly to halo mass.
• Magnification lensing. Use the magnification κ rather than shear γ — count distant galaxies behind foreground overdensities and look for the predicted enhancement of bright sources / suppression of faint ones at fixed apparent magnitude. Complementary, but the signal-to-noise per galaxy is lower than shear.
• Cluster mass mapping. Combine weak lensing in the outskirts with strong lensing (arcs, multiple images) in the core to build a self-consistent NFW or composite mass profile spanning three decades in radius.

Common pitfalls

• Treating per-galaxy shear estimates as detections. A 1 % shear is fundamentally below the per-galaxy shape noise (~0.3 ellipticity rms). Any per-source claim has to be statistical.
• Forgetting the mass-sheet degeneracy. Reconstructed κ-maps are determined up to a constant. Quoting absolute masses from κ alone — without breaking the degeneracy through magnification, counts, or a calibrated boundary — gives biased numbers.
• Conflating shear γ with reduced shear g. What we actually observe is g = γ / (1 − κ); in the cluster regime where κ approaches 0.1 the difference is significant. Linear-theory cosmic-shear analyses use this approximation; cluster-mass analyses must not.
• Underestimating intrinsic alignments. IA contributes at the few-percent level on linear scales and is not optional in modern joint analyses. Older papers that ignored IA underestimated their systematic-error budget.
• Using KSB for sub-percent-precision modern surveys. KSB was the field's bootstrap; for DES, HSC, KiDS, and LSST-class precision, model-fitting or metacalibration pipelines are required to keep multiplicative shear bias under 0.5 %.