Weak Gravitational Lensing

The idea: shapes the sky should not have

Take a deep image of the sky and look at a few hundred faint, distant galaxies. Their orientations should be random — there is no reason for a galaxy a billion light-years away to point any particular direction. Yet if a lump of mass lies between them and us, the galaxies behind it will, on average, appear very slightly stretched tangentially around that lump, like a faint set of brushstrokes circling an invisible centre. That coherent, gentle combing of galaxy shapes is weak gravitational lensing.

It is the same light-bending that produces the spectacular arcs and Einstein rings of strong lensing — but in the weak regime the deflection is tiny. No single galaxy looks obviously distorted. A galaxy that is already an ellipse becomes an imperceptibly more elongated ellipse, by about one part in a hundred. You cannot see it in any one object. You can only see it by treating the universe as a giant statistical experiment: measure the shapes of millions of galaxies, average out their random intrinsic orientations, and what remains is the faint fingerprint of all the mass their light has passed. Crucially, that mass is mostly dark — gravity does not care whether matter shines — so weak lensing is the most direct way to weigh and map the invisible dark matter of the cosmos.

The physics: convergence and shear

Light from a distant source is deflected by the gravitational potential of everything along the line of sight. To first order, the effect of all that intervening mass is to remap the true (source-plane) position of a point onto a slightly shifted observed (image-plane) position. The local relationship between the two is captured by the 2×2 Jacobian (lensing distortion) matrix, conventionally written

A = [ 1 − κ − γ₁     −γ₂      ]
    [   −γ₂       1 − κ + γ₁ ]

Two quantities appear. The convergence κ is the isotropic part: it magnifies or demagnifies the image and changes its size and brightness without changing its shape. The shear γ = γ₁ + iγ₂ is the traceless, anisotropic part: it stretches a circular source into an ellipse. A circle of radius R is mapped to an ellipse with axis ratio approximately (1 − κ − |γ|) : (1 − κ + |γ|), so in the weak limit the induced ellipticity is essentially the shear itself.

The convergence is just the projected mass surface density measured in natural lensing units:

κ(θ) = Σ(θ) / Σ_cr ,   Σ_cr = (c² / 4πG) × (D_s / (D_l D_ls))

where Σ is the projected mass density of the lens, and Σ_cr is the critical surface density set by the angular-diameter distances to the lens (D_l), to the source (D_s), and between lens and source (D_ls). Strong lensing — arcs and multiple images — happens where κ ≳ 1; weak lensing is the κ ≪ 1 regime everywhere else. A beautiful identity makes lensing a clean mass probe: the shear and convergence are not independent but are two derivatives of the same projected potential, so the tangential shear averaged on a circle of radius θ around a mass equals the mean convergence inside that circle minus the convergence on the circle:

γ_t(θ) = κ̄(<θ) − κ(θ) = ΔΣ(θ) / Σ_cr

This is the engine of "galaxy–galaxy lensing": stack the tangential shear of background galaxies around many foreground lenses and you read off ΔΣ, the excess projected mass profile, directly.

The key numbers

Weak lensing lives or dies on a single fact: the signal is much smaller than the noise per galaxy, and only statistics rescue it.

• Cosmic shear amplitude: γ ≈ 0.01–0.03 for typical large-scale structure between us and galaxies at redshift z ≈ 0.5–1. Around a massive cluster the tangential shear can reach γ_t ≈ 0.05–0.3 just outside the strong-lensing core.
• Intrinsic ellipticity scatter: the dispersion of real galaxy shapes is σ_e ≈ 0.25 per component — roughly 8–25 times larger than the lensing signal. This "shape noise" is the fundamental limit.
• Galaxy number densities: ground-based surveys reach n ≈ 5–10 usable galaxies per square arcminute; space-based imaging (Euclid, Roman) reaches 30+; deep Hubble fields exceed 100.
• Distances: the lensing efficiency Σ_cr⁻¹ peaks when the lens sits roughly halfway (in comoving terms) between observer and source. For sources at z_s ≈ 1, the most efficient lenses are at z_l ≈ 0.3–0.4.
• Cosmological yield: the combination S_8 = σ_8 √(Ω_m/0.3) is measured to ~2–3% precision; current lensing values cluster near S_8 ≈ 0.76 with Ω_m ≈ 0.30.

How it is measured

The measurement pipeline is an exercise in beating down systematic errors that are themselves comparable to the signal.

1. Image and deconvolve. Each galaxy image is blurred by the telescope and atmosphere — the point-spread function (PSF). The PSF must be measured from stars and removed, because an anisotropic PSF mimics shear. A PSF ellipticity error of just 0.001 can swamp the cosmological signal.
2. Measure ellipticities. Algorithms such as KSB (Kaiser–Squires–Broadhurst, 1995), model-fitting methods like lensfit, and forward-modelling shear measurement (e.g. metacalibration) estimate each galaxy's two-component ellipticity and the multiplicative/additive biases of the estimator.
3. Photometric redshifts. Spectroscopy of every faint galaxy is impossible, so redshifts are estimated from multi-band photometry. Galaxies are sorted into tomographic redshift bins, since lensing efficiency depends strongly on the source distance.
4. Average and correlate. The two-point shear correlation function ξ±(θ) — how the shear at one point correlates with the shear an angle θ away — or its harmonic-space cousin, the shear power spectrum C_ℓ, is the headline statistic. Its amplitude and shape constrain Ω_m, σ_8, and the dark-energy equation of state.
5. Map the mass. The Kaiser–Squires inversion turns the observed shear field directly into a convergence (projected mass) map, revealing dark-matter clumps with no light at all.

A vital internal check is the decomposition of the shear field into "E-modes" and "B-modes". Gravitational lensing, to leading order, produces only curl-free E-mode patterns. Any significant B-mode (curl) signal flags residual systematics — PSF leakage, astrometric errors — rather than real lensing.

Weak versus strong lensing — and the cousins

Regime	Convergence κ	Visible effect	Detected	Maps
Strong lensing	κ ≳ 1	Arcs, Einstein rings, multiple images	Per object, by eye	Cluster/galaxy cores
Cluster weak lensing	0.01 – 0.3	Tangential alignment around clusters	~10²–10³ galaxies / cluster	Individual halo masses
Galaxy–galaxy lensing	~10⁻³–10⁻²	Stacked tangential shear	Stack 10⁵–10⁶ lenses	Mean halo profiles ΔΣ
Cosmic shear	~10⁻²	Shear–shear correlations	10⁸–10⁹ galaxies	Growth of structure, S₈
CMB lensing	~few × 10⁻²	Smeared CMB hot/cold spots	Whole-sky deflection field	All mass to z ≈ 1100
Microlensing	point-mass	Transient brightening (no resolved distortion)	Time-domain photometry	Stars, planets, compact objects

Weak lensing, cosmic shear, galaxy–galaxy lensing and CMB lensing are the same effect probed on different scales. Microlensing is the time-domain limit, where the source and lens are unresolved and you see only a brightening as alignment changes — see gravitational microlensing.

Worked example: how many galaxies to detect a cluster?

Suppose a massive cluster produces an average tangential shear γ_t ≈ 0.02 in an annulus around its centre. The shape noise per galaxy is σ_e ≈ 0.25. We want to detect the shear at, say, 5σ significance. The uncertainty on the mean shear from N galaxies is

σ(γ̄) = σ_e / √N

A 5σ detection requires the signal to exceed five times this uncertainty:

γ_t ≥ 5 σ_e / √N
0.02 ≥ 5 × 0.25 / √N
√N  ≥ 5 × 0.25 / 0.02 = 62.5
N   ≥ 3,900 galaxies

So roughly four thousand background galaxies in the annulus are needed to confidently detect this cluster's weak-lensing signal. At a ground-based density of n ≈ 10 galaxies per square arcminute, that is an area of about 390 square arcminutes — a circle of radius ~11 arcminutes, comfortably matching a rich cluster at z ≈ 0.3. Now scale up to cosmic shear, where the signal is only γ ≈ 0.01 and we want sub-percent precision on the mean shear across the survey: reaching σ(γ̄) ≈ 0.001 demands N ≈ (0.25/0.001)² ≈ 6 × 10⁴ galaxies per measurement, and mapping the full sky in many tomographic bins drives the total to the hundreds of millions that Euclid and Rubin will deliver. The arithmetic is why weak lensing is fundamentally a big-data, big-survey science.

Discovery, missions, and people

The theoretical groundwork is old: Albert Einstein's 1915 general relativity predicted light deflection, confirmed by Arthur Eddington's 1919 eclipse expedition (1.75 arcseconds at the Sun's limb). Fritz Zwicky pointed out in 1937 that galaxies and clusters would act as lenses and could weigh dark matter — the same Zwicky who had inferred "dunkle Materie" in the Coma Cluster in 1933.

• 1990 — first cluster weak lensing. Tony Tyson, Francisco Valdes and Rich Wenk detected the coherent tangential alignment of background galaxies behind clusters, the first true weak-lensing measurement.
• 1993 — the inversion. Nick Kaiser and Gordon Squires published the algorithm to turn a measured shear field into a projected mass map; the KSB shape-measurement method (Kaiser, Squires, Broadhurst) followed in 1995.
• 2000 — cosmic shear detected. Four independent groups (Bacon, Refregier & Ellis; Kaiser, Wilson & Luppino; Van Waerbeke et al.; Wittman et al.) announced the first detections of lensing by large-scale structure itself.
• 2006 — the Bullet Cluster. Douglas Clowe and collaborators used weak lensing to show the mass of merging cluster 1E 0657-558 is offset from its X-ray gas — direct empirical evidence for collisionless dark matter.
• 2012 — CFHTLenS, then KiDS (Kilo-Degree Survey, VST), the Dark Energy Survey (DES) from Cerro Tololo, and Subaru's Hyper Suprime-Cam (HSC) delivered precision cosmic-shear cosmology and the first hints of the S_8 tension.
• 2023 — Euclid launched. ESA's Euclid (1.2 m space telescope) began surveying 14,000–15,000 deg² to measure shapes of ~1.5 billion galaxies. NASA's Nancy Grace Roman Space Telescope and the Vera C. Rubin Observatory's LSST will push the precision further this decade.

Variants and related probes

• Cosmic shear. Pure shear–shear correlation from large-scale structure with no identified lens; the flagship cosmological statistic. See cosmic shear.
• Galaxy–galaxy lensing. Stacking the tangential shear of background galaxies around foreground galaxies to measure their average dark-matter halo profile ΔΣ(θ).
• Cluster weak lensing. Weighing individual clusters out to their virial radius, calibrating the mass–observable relations used in cluster cosmology.
• CMB lensing. The cosmic microwave background is itself lensed by all the mass back to z ≈ 1100; Planck and ground-based experiments (ACT, SPT) map the deflection field over the whole sky. Cross-correlating CMB lensing with galaxy shear is a powerful systematics check.
• Magnification bias. Convergence κ also changes source brightness and number counts, an independent handle on the mass that does not require shape measurement.
• 3×2pt analysis. The modern gold standard combines three two-point functions — galaxy clustering, galaxy–galaxy lensing, and cosmic shear — to break degeneracies between mass and galaxy bias.

Common misconceptions and subtleties

• "You can see the distortion on a single galaxy." You cannot. The shear is ~1% while a galaxy's own ellipticity is ~25%. Weak lensing is invisible per object and only emerges in the statistical average over many.
• "We measure the shear directly." We measure ellipticity, which is intrinsic shape plus shear. The estimator is unbiased only if intrinsic orientations are random — which is why intrinsic alignment (galaxies coherently aligned by shared tidal fields) is the most dangerous astrophysical systematic and must be modelled and marginalised.
• "Shear measures κ uniquely." Adding a uniform sheet of mass changes κ everywhere but leaves the observable reduced shear unchanged — the mass-sheet degeneracy. Breaking it requires magnification information or an absolute calibration at the field edge.
• "It only probes the mass of the lens." Cosmic shear integrates the matter density along the entire line of sight weighted by lensing efficiency; it constrains the growth of structure and hence dark energy, not just a single lens.
• "Baryons don't matter for a dark-matter map." Feedback from supernovae and active galactic nuclei redistributes gas and suppresses the matter power spectrum at small scales by several percent — a leading systematic for high-precision S_8 measurements that must be modelled with hydrodynamic simulations.
• "B-modes are real lensing." Pure gravitational lensing makes essentially only E-modes. A detected B-mode signal is a red flag for residual PSF or astrometric systematics, used as a built-in null test.