Weak Gravitational Lensing and Cosmic Shear

Every faint galaxy in a deep image has crossed billions of light-years of lumpy universe, its light threading a gravitational field that was never quite uniform. The tidal part of that field — the difference in deflection across the light bundle — magnifies the image slightly and stretches it into a shallow ellipse. When the stretch is a percent rather than the giant arcs of strong lensing, we are in the weak-lensing regime, and the coherent pattern that large-scale structure imprints on millions of background galaxy shapes is called cosmic shear. The power of the effect is that it is honest about mass. Lensing maps a source position to an observed one through a Jacobian $A$ departing from the identity by a scalar convergence $\kappa$ and a two-component shear $(\gamma_1,\gamma_2)$ ; the convergence equals the projected surface mass density in critical units, so weak lensing weighs all matter — dark and baryonic — with no assumption about its dynamical state or whether it shines. Because any one galaxy's intrinsic shape dwarfs the $\sim1\%$ signal, cosmic shear is irreducibly statistical: you recover the shear only by averaging enormous numbers of shapes. The central prediction is compact. The two-point statistics of the shear field are a line-of-sight projection of the 3D matter power spectrum $P_\delta(k,z)$ , weighted by survey geometry and the source redshift distribution and collapsed by the Limber approximation into a single integral over comoving distance. Cosmic shear thereby ties galaxy shapes to the growth of structure and the expansion history, constraining $S_8=\sigma_8(\Omega_m/0.3)^{1/2}$ — a pillar, and a live controversy, of the $\Lambda$ CDM concordance model.

Key takeaways

Weak lensing is the small, coherent distortion of galaxy images by the tidal field of foreground matter, split into the convergence $\kappa$ (isotropic magnification) and the two-component shear $\gamma$ (tangential stretch).
The convergence equals the projected surface mass density in critical units, $\kappa=\Sigma/\Sigma_{\rm cr}$ , so lensing measures total mass directly — its power for mapping dark matter.
Each galaxy's intrinsic ellipticity ( $\sigma_e\approx0.3$ ) is $\sim30\times$ the $\sim1\%$ shear, so cosmic shear is a big-numbers measurement: $\langle e\rangle\to\gamma$ over many sources.
The observable is the shear two-point function $\xi_\pm(\theta)$ , or the convergence power spectrum $C_\ell^{\kappa\kappa}$ — a Limber-weighted line-of-sight projection of the 3D matter power spectrum $P_\delta(k,z)$ .
Cosmic shear is chiefly sensitive to $S_8\equiv\sigma_8(\Omega_m/0.3)^{1/2}$ , a low-redshift test of $\Lambda$ CDM; surveys report $S_8\approx0.76$ against Planck's $0.83$ — the $S_8$ tension.
Realising the signal demands control of intrinsic alignments, photo- $z$ errors, PSF modelling and nonlinear-plus-baryonic corrections to $P_\delta$ — the systematics that dominate DES, KiDS, HSC, Euclid and Rubin/LSST.
Key idea: the lensing convergence is the projected matter surface density in units of a critical density, and its power spectrum is a Limber-weighted line-of-sight integral of the 3D matter power spectrum — so averaging millions of tiny galaxy-shape distortions yields a direct map of total mass and a measurement of $S_8=\sigma_8(\Omega_m/0.3)^{1/2}$ .

The core insight

The lensing convergence is the projected matter surface density measured in units of a critical density; its power spectrum is a Limber-weighted line-of-sight integral of the 3D matter power spectrum, so averaging millions of tiny galaxy-shape distortions yields a direct map of total mass and a measurement of $S_8=\sigma_8(\Omega_m/0.3)^{1/2}$ .

Where this lives today

Cosmic shear is now a headline probe of precision cosmology, and the story is the $S_8$ tension. Stage-III surveys — the Dark Energy Survey (DES), the Kilo-Degree Survey (KiDS) and Hyper Suprime-Cam (HSC) — each measure $S_8$ to a few percent from tomographic $\xi_\pm(\theta)$ , clustering near $0.76$ , some $2$ – $3\sigma$ below Planck's $0.83$ . Whether this is new physics in late-time growth or an unmodelled systematic is the central open question, and it is why the effort moved to Stage-IV: Euclid (launched 2023) and the Vera C. Rubin Observatory's LSST will image billions of galaxies until systematics — intrinsic alignments, photo- $z$ , PSF and baryonic feedback — dominate. The same physics maps dark matter: Kaiser–Squires inversion turns shear into a convergence map of filaments and voids, cluster lensing calibrates mass–observable relations, and cross-correlating shear with CMB lensing (Planck, ACT, Simons Observatory) and galaxy positions — the "3×2pt" analysis — breaks degeneracies and self-calibrates the systematics that limit shear alone.

The physical picture

Picture a grid of intrinsically round galaxies behind a concentration of mass. The mass makes each image slightly larger — the convergence, an isotropic focusing — and squashes each into a shallow ellipse whose long axis points tangentially around the mass — the shear. The two are the trace and trace-free part of one distortion matrix. Because the foreground mass is smoothly correlated on the sky, neighbouring galaxies are sheared in nearly the same direction; that coherence alone separates the cosmic-shear signal from the random jumble of intrinsic shapes. The first figure freezes this: round becomes tangentially-elliptical, and the two eigenvalues carry the whole story.

Figure

A grid of intrinsically round galaxies seen through a foreground mass concentration — convergence κ = Σ/Σcr enlarges every image isotropically while shear γ stretches them tangentially around the peak, so the local distortion eigenvalues are 1 − κ − γ across and 1 − κ + γ along the shear

Two facts make cosmic shear demanding. First, the signal is tiny: a galaxy's intrinsic ellipticity ( $\sigma_e\approx0.3$ ) outshouts the $\sim1\%$ shear thirtyfold, so the shear emerges only from an average over thousands of galaxies per patch and $10^8$ – $10^9$ across a survey, the scatter falling as $\sigma_e/\sqrt{N}$ . Second, the signal is a projection: the convergence weighs every gram of matter to the sources, but with a geometric efficiency kernel that peaks midway in comoving distance — so cosmic shear is naturally a low-redshift probe of the recently-assembled cosmic web, as the second figure shows for sources at $z=1$ : it crests near $z\approx0.5$ and falls to zero at the source.

Figure

Lensing efficiency W as a function of redshift for sources fixed at z = 1 in a flat Ωm = 0.3 universe — the kernel vanishes at the observer and at the source and peaks near z ≈ 0.5, roughly midway in comoving distance, so cosmic shear weighs intermediate-redshift structure most heavily

Finally, the signal marries geometry and growth. The distances in the kernel depend on the expansion history and hence dark energy; the amplitude of $\delta$ depends on how much structure has grown, hence $\Omega_m$ and $\sigma_8$ . Slicing sources into redshift bins — tomography — separates the geometric factors from the growth factors, turning cosmic shear into a simultaneous probe of expansion and the growth rate.

The core equations

The starting point is the lens mapping. A source at true position $\boldsymbol\beta$ is seen at a shifted position $\boldsymbol\theta$ ; for a small source the two are related linearly by a Jacobian $A$ measuring how a circle maps to an ellipse. Everything in weak lensing is a statement about $A$ .

\delta\boldsymbol{\beta} = A\,\delta\boldsymbol{\theta}, \qquad A_{ij} = \frac{\partial \beta_i}{\partial \theta_j} = \delta_{ij} - \frac{\partial^2 \psi}{\partial \theta_i \partial \theta_j} \qquad (1)

where: $\boldsymbol{\beta}$ — true (source-plane) position; $\boldsymbol{\theta}$ — observed (image-plane) position; $A_{ij}$ — lensing Jacobian; $\psi$ — 2D lensing potential. Reading: distortion is the identity minus the second derivatives of $\psi$ , so a flat potential leaves images undistorted. Splitting the symmetric $\psi_{,ij}$ into trace and trace-free parts names the two distortions: the trace is the convergence $\kappa$ (isotropic focusing); the trace-free part is the shear, packaged as a complex $\gamma=\gamma_1+i\gamma_2$ carrying strength and orientation.

A = \begin{pmatrix} 1-\kappa-\gamma_1 & -\gamma_2 \\ -\gamma_2 & 1-\kappa+\gamma_1 \end{pmatrix} \qquad (2)

where: $\kappa=\tfrac12\nabla^2\psi$ — convergence; $\gamma_1=\tfrac12(\psi_{,11}-\psi_{,22})$ — shear along the axes; $\gamma_2=\psi_{,12}$ — shear along the diagonals; $\gamma=|\gamma|e^{2i\phi}$ — complex shear. Reading: the eigenvalues $1-\kappa\mp|\gamma|$ stretch the image along and across the shear, and the factor $2\phi$ marks shear as spin-2 — unchanged by a $180^\circ$ rotation. For a single thin lens the convergence is the projected surface mass density in units of a critical surface density set purely by geometry. Where $\Sigma$ reaches $\Sigma_{\rm cr}$ lensing is strong; cosmic shear lives at $\Sigma\ll\Sigma_{\rm cr}$ .

\kappa(\boldsymbol{\theta}) = \frac{\Sigma(\boldsymbol{\theta})}{\Sigma_{\rm cr}}, \qquad \Sigma_{\rm cr} = \frac{c^2}{4\pi G}\frac{D_S}{D_L D_{LS}} \qquad (3)

where: $\Sigma$ — projected surface mass density; $\Sigma_{\rm cr}$ — critical surface density; $D_L,D_S,D_{LS}$ — angular-diameter distances to lens, source, and lens-to-source; $c,G$ — speed of light and Newton's constant. Reading: $\Sigma_{\rm cr}$ is smallest (lensing most efficient) for a lens midway between observer and source — the geometric root of the kernel's mid-way peak. For a line of sight through large-scale structure there is no single lens; the 2D potential is a weighted line-of-sight integral of the 3D Newtonian potential $\Phi$ sourced by every fluctuation along the way.

\psi(\boldsymbol{\theta}) = \frac{2}{c^2}\int_0^{\chi_S} \mathrm{d}\chi\,\frac{f_K(\chi_S-\chi)}{f_K(\chi_S)\,f_K(\chi)}\,\Phi\big(f_K(\chi)\boldsymbol{\theta},\chi\big) \qquad (4)

where: $\chi$ — comoving distance; $\chi_S$ — comoving distance to source; $f_K(\chi)$ — comoving angular-diameter distance ( $f_K=\chi$ when flat); $\Phi$ — Newtonian potential of large-scale structure. Reading: $f_K(\chi_S-\chi)/f_K(\chi_S)$ is the same geometric efficiency as in $\Sigma_{\rm cr}$ , now integrated over all lens distances. Taking the 2D Laplacian of $\psi$ and inserting the Poisson equation converts $\Phi$ into the density contrast $\delta$ , so the convergence becomes a projection of the density field, weighted by the efficiency kernel $W(\chi)$ .

\kappa(\boldsymbol{\theta}) = \int_0^{\chi_H}\!\!\mathrm{d}\chi\; W(\chi)\,\delta\big(f_K(\chi)\boldsymbol{\theta},\chi\big), \quad W(\chi)=\frac{3H_0^2\Omega_m}{2c^2}\frac{f_K(\chi)}{a(\chi)}\!\int_\chi^{\chi_H}\!\!\mathrm{d}\chi'\,n(\chi')\frac{f_K(\chi'-\chi)}{f_K(\chi')} \qquad (5)

where: $W(\chi)$ — lensing efficiency kernel, peaks midway to sources; $\delta$ — 3D matter density contrast; $n(\chi)$ — normalised source distribution; $a(\chi)$ — scale factor; $\chi_H$ — horizon distance. Reading: the prefactor $3H_0^2\Omega_m/2c^2$ is the Poisson coupling — why $\kappa$ 's amplitude scales with $\Omega_m$ . The workhorse of the field is the angular power spectrum of $\kappa$ . The Limber approximation collapses the double distance integral to a single one, mapping each multipole $\ell$ to a transverse wavenumber $k=(\ell+1/2)/f_K$ .

C_\ell^{\kappa\kappa} = \int_0^{\chi_H} \mathrm{d}\chi\,\frac{W(\chi)^2}{f_K(\chi)^2}\,P_\delta\!\left(k=\frac{\ell+1/2}{f_K(\chi)},\,\chi\right) \qquad (6)

where: $C_\ell^{\kappa\kappa}$ — angular convergence power spectrum; $P_\delta(k,\chi)$ — 3D nonlinear matter power spectrum; $\ell$ — angular multipole; $k=(\ell+1/2)/f_K$ . Reading: shear and convergence share the same power spectrum, $C_\ell^{\gamma\gamma}=C_\ell^{\kappa\kappa}$ , since both are second derivatives of one potential. Surveys measure not $C_\ell$ directly but the real-space shear correlation functions $\xi_\pm(\theta)$ — Hankel transforms of $C_\ell^{\kappa\kappa}$ , robust to the survey mask and gaps.

\xi_\pm(\theta) = \frac{1}{2\pi}\int_0^\infty \mathrm{d}\ell\,\ell\,C_\ell^{\kappa\kappa}\,J_{0,4}(\ell\theta) \qquad (7)

where: $\xi_+,\xi_-$ — observable shear correlations (using $J_0$ and $J_4$ ); $J_{0,4}$ — Bessel functions; $\theta$ — angular separation of galaxy pairs. Reading: together they separate the physical E-mode signal from the systematics-flagging B-mode. A telescope records ellipticity, not shear. Averaging over random intrinsic orientations makes the mean ellipticity an unbiased estimator of the reduced shear $g$ , which equals $\gamma$ in the weak limit.

\langle e_i \rangle = g_i \approx \gamma_i \;(\kappa\ll1), \qquad g=\frac{\gamma}{1-\kappa} \qquad (8)

where: $e_i$ — measured complex ellipticity; $g$ — reduced shear (the true observable); $\langle\cdot\rangle$ — ensemble average over random orientations. Reading: only $\gamma/(1-\kappa)$ is observable — the origin of the mass-sheet degeneracy, since a uniform $\kappa$ leaves $g$ unchanged. The constraint cosmic shear delivers most sharply is not $\sigma_8$ or $\Omega_m$ alone but their nearly-degenerate combination $S_8$ .

S_8 \equiv \sigma_8\left(\frac{\Omega_m}{0.3}\right)^{1/2} \qquad (9)

where: $\sigma_8$ — RMS linear density contrast in $8\,h^{-1}\mathrm{Mpc}$ spheres; $\Omega_m$ — present matter density; $S_8$ — the best-constrained combination. Reading: $C_\ell^{\kappa\kappa}$ scales roughly as $\sigma_8^2\Omega_m^{\,\sim1.5}$ , so $\sigma_8\Omega_m^{1/2}$ is pinned while the orthogonal direction is not.

Derivation AFrom bent light rays to convergence and shear

The goal is to build the distortion Jacobian $A$ and its clean split into a convergence $\kappa$ and a two-component shear $\gamma$ from one physical input: source light is deflected by the gradient of a 2D lensing potential $\psi$ . We linearise the mapping around a small image, split the matrix of second derivatives into trace and trace-free parts, identify each physically, and read off the eigenvalues that turn a round source into an ellipse — the matrix on which everything downstream hangs.

Step 1. Write the lens equation $\boldsymbol{\beta}=\boldsymbol{\theta}-\nabla_\theta\psi(\boldsymbol{\theta})$ : the observed position differs from the true one by the reduced deflection, the sky-gradient of $\psi$ . Why: a gradient deflection makes $\psi_{,ij}$ symmetric, which is what allows the clean scalar-plus-spin-2 split.

Step 2. Linearise over a small source: $A_{ij}=\partial\beta_i/\partial\theta_j=\delta_{ij}-\psi_{,ij}$ . Why: a circle maps to an ellipse because the mapping is locally affine; higher-order flexion terms are negligible for cosmic shear.

Step 3. Decompose the symmetric $\psi_{,ij}=\kappa\,\delta_{ij}+\gamma_1\sigma_3+\gamma_2\sigma_1$ , with $\kappa=\tfrac12\nabla^2\psi$ , $\gamma_1=\tfrac12(\psi_{,11}-\psi_{,22})$ , $\gamma_2=\psi_{,12}$ . Why: the trace is the only rotational scalar, so it is the isotropic magnification; the two-component traceless remainder is the spin-2 shear.

Step 4. Assemble $A=\mathbb{1}-\psi_{,ij}$ and read the eigenvalues $1-\kappa\mp|\gamma|$ : positive $\kappa$ magnifies and enlarges isotropically, while $\gamma$ stretches the image into the tangential ellipse seen around every overdensity. Why: this coherent tangential alignment of shapes around mass peaks is exactly what surveys detect statistically. *Note: $\gamma$ transforms as $e^{2i\phi}$ , confirming its spin-2 character — unchanged by a $180^\circ$ rotation.*

Go deeper · Derivation B · 1 minThe convergence power spectrum via the Limber approximation

Derivation BThe convergence power spectrum via the Limber approximation

The goal is to turn the convergence — a line-of-sight projection of the 3D density contrast with kernel $W(\chi)$ — into its angular power spectrum $C_\ell^{\kappa\kappa}$ , and to show that a long projection collapses the calculation to a single distance integral over the matter power spectrum at one wavenumber per multipole. This is the master formula of cosmic shear, connecting a catalogue of galaxy shapes to $P_\delta(k,z)$ and hence to $\sigma_8$ , $\Omega_m$ , growth and expansion. We correlate two convergences, Fourier-expand the density, and apply the Limber collapse.

Step 1. Form the two-point function $\langle\kappa(\boldsymbol{\theta})\kappa(\boldsymbol{\theta}')\rangle=\int\mathrm{d}\chi\,\mathrm{d}\chi'\,W(\chi)W(\chi')\langle\delta\,\delta'\rangle$ by inserting the projected-density $\kappa$ at two sky positions. Why: the signal is statistical, so the object of interest is a correlation; the deterministic kernels come outside the ensemble average.

Step 2. Fourier-expand $\delta$ and use statistical homogeneity, $\langle\delta(\mathbf{k})\delta^*(\mathbf{k}')\rangle=(2\pi)^3\delta_D(\mathbf{k}-\mathbf{k}')P_\delta(k)$ . Why: homogeneity forces the two-point function to depend only on separation — the statement that its transform is a power spectrum, the quantity theory predicts.

Step 3. Apply Limber: over a long projection only transverse modes survive, so $k_\parallel\to0$ , $k_\perp=(\ell+1/2)/f_K(\chi)$ , and the sharply-peaked radial correlation collapses $\chi'\to\chi$ , killing one integral. Why: modes with $k_\parallel\neq0$ oscillate along the line of sight and average away, projecting the 3D field onto a 2D map with a local $k$ -to- $\ell$ mapping.

Step 4. Collect the single $\chi$ -integral: $C_\ell^{\kappa\kappa}=\int\mathrm{d}\chi\,(W^2/f_K^2)\,P_\delta((\ell+1/2)/f_K,\chi)$ ; for $\ell\gtrsim300$ most signal is nonlinear, so $P_\delta$ carries baryonic-feedback corrections. Why: the weighting $W^2/f_K^2$ favours intermediate-redshift, small-scale structure — exactly where nonlinear gravity and baryons matter, the field's modelling challenge. *Note: the flat-sky Limber result is recovered for $\ell\gtrsim10$ , and shear and convergence share the same power spectrum, $C_\ell^{\gamma\gamma}=C_\ell^{\kappa\kappa}$ .*

Go deeper · Derivation C · 1 minWhy cosmic shear measures S8 and why it is a percent-level effect

Derivation CWhy cosmic shear measures S8 and why it is a percent-level effect

The goal is to estimate the typical shear amplitude from the master formula, show it lands at the percent level far below any single galaxy's intrinsic ellipticity, and expose why the emerging constraint is the degenerate $S_8=\sigma_8(\Omega_m/0.3)^{1/2}$ rather than $\sigma_8$ and $\Omega_m$ separately. This ties the projection integral to two facts of experimental life: cosmic shear must be measured over billions of galaxies, and its natural output is one number, not two. We estimate the kernel, variance and scaling in turn.

Step 1. Estimate the kernel for sources at $z\sim1$ ( $\chi\sim c/H_0$ ): $W\sim(3H_0^2\Omega_m/2c^2)\,\chi$ , a dimensionless prefactor of order $\Omega_m/2\approx0.15$ . Why: the Poisson factor $3H_0^2\Omega_m/2c^2$ is the only explicit amplitude dependence, so the signal scales linearly with $\Omega_m$ from geometry alone.

Step 2. Estimate the variance $\langle\kappa^2\rangle\sim\int(\mathrm{d}\ell\,\ell/2\pi)\,C_\ell^{\kappa\kappa}$ ; with $\ell^2 C_\ell/2\pi\sim10^{-4}$ near $\ell\sim10^3$ , $\langle\kappa^2\rangle^{1/2}\sim$ a few $\times10^{-2}$ , so $\gamma_{\rm rms}\sim1$ – $3\times10^{-2}$ . Why: the coherent shear is $\sim1\%$ , set by how much matter has clustered ( $\sigma_8$ ) times how efficiently geometry lenses it ( $\Omega_m$ ).

Step 3. Compare to shape noise $\sigma_e\approx0.3\gg\gamma_{\rm rms}\approx0.02$ : recovering the shear needs $N\sim(\sigma_e/\gamma)^2\sim2\times10^2$ galaxies per element, hence $10^8$ – $10^9$ per survey. Why: buried thirty deep in shape noise, cosmic shear is a big-numbers experiment with a floor falling only as $\sigma_e/\sqrt{N}$ .

Step 4. Track the scaling: $C_\ell^{\kappa\kappa}\propto\sigma_8^2$ (from $P_\delta$ ) times $\Omega_m^{\,\sim1.5}$ (kernel plus growth), so the constraint runs along $\sigma_8\Omega_m^{\,\sim0.5}=$ const, i.e. $S_8=\sigma_8(\Omega_m/0.3)^{1/2}$ . Why: the two parameters enter in a fixed product, so one 2-point analysis pins $S_8$ and leaves the orthogonal direction to tomography. *Note: with $\sigma_8\approx0.8$ and $\Omega_m\approx0.3$ one finds $S_8\approx0.8$ , and the predicted $\xi_+\sim10^{-4}$ at arcminute scales matches observations.*

Limiting cases

The weak-lensing formalism is a controlled expansion, and its regimes are exactly the approximations that make the master formula tractable:

Case	Result	Meaning
Weak-field, small deflection	$\vert\kappa\vert,\,\vert\gamma\vert\ll1$	Jacobian a small perturbation of the identity; $g\approx\gamma$ . The entire cosmic-shear regime, distinct from the $\kappa\gtrsim1$ arcs of strong lensing.
Born approximation	$\psi\simeq\int\Phi\,\mathrm{d}\chi$ along the straight ray	Integrate along the unperturbed line. Valid since deflections are $\lesssim$ arcminutes; lens–lens and geodesic corrections are second order and sub-percent.
Limber / flat-sky	$\ell\gtrsim10$	Spherical projection becomes a single line-of-sight integral at $k=(\ell+1/2)/f_K$ . Excellent on the small scales that dominate; breaks down at the largest scales.
Linear vs nonlinear power	$P_\delta\to P_\delta^{\rm NL}(k,z)$	For $\ell\gtrsim300$ most signal is nonlinear, so use a nonlinear $P_\delta$ (HALOFIT, emulators) plus baryonic-feedback suppression.
Single-source thin lens	$\kappa=\Sigma/\Sigma_{\rm cr}$	Collapse the projection to one screen; the cluster- and galaxy–galaxy-lensing limit of the same equations.

Common trapThe measured ellipticity of a galaxy is its shear

It is not, and treating it so is the classic beginner's error. Observed ellipticity is dominated by the unknown intrinsic shape, whose scatter $\sigma_e\approx0.3$ is thirty times the $\sim1\%$ shear; one galaxy carries essentially no shear information, and the signal appears only in the average $\langle e\rangle\to\gamma$ over many randomly-oriented sources. Three further traps. Intrinsic alignments: galaxies formed in the same tidal field are physically aligned, mimicking a lensing correlation, and must be modelled. Mass-sheet degeneracy: only reduced shear $g=\gamma/(1-\kappa)$ is observable, so a uniform convergence sheet is invisible to shear alone. B-modes: to leading order lensing makes only curl-free E-mode shear, so any B-mode flags residual systematics (PSF leakage, intrinsic alignments), not cosmology.

Worked examples

1 · The critical surface density (fully worked)

A lens at $z_L=0.3$ lenses sources at $z_S=1.0$ in a flat universe with $\Omega_m=0.3$ , $H_0=70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ . Find $\Sigma_{\rm cr}$ and say why cosmic shear is weak.

Step 1. Compute comoving distances $\chi=(c/H_0)\int_0^z\mathrm{d}z'/E(z')$ with $E=\sqrt{\Omega_m(1+z)^3+\Omega_\Lambda}$ and $c/H_0=4283\ \mathrm{Mpc}$ : $\chi_L=1194$ , $\chi_S=3304\ \mathrm{Mpc}$ . Flat, so $f_K=\chi$ and $D=\chi/(1+z)$ : $D_L=919$ , $D_S=1652$ , $D_{LS}=(\chi_S-\chi_L)/(1+z_S)=1055\ \mathrm{Mpc}$ .

Step 2. Insert into $\Sigma_{\rm cr}=(c^2/4\pi G)(D_S/D_LD_{LS})$ . Converting to metres ( $1\ \mathrm{Mpc}=3.086\times10^{22}\ \mathrm{m}$ ) gives $\Sigma_{\rm cr}=\mathbf{5.9\ kg\,m^{-2}}\approx\mathbf{2.8\times10^{3}\ M_\odot\,pc^{-2}}$ .

Step 3. Interpret. That is what a projected mass must reach for strong lensing ( $\kappa\sim1$ ); the smooth cosmic web projects far less, so $\kappa\sim10^{-2}$ — the percent-level shear. $\Sigma_{\rm cr}$ is smaller (lensing more efficient) for a mid-way lens, the reason the kernel peaks midway. $\Sigma_{\rm cr}\approx5.9\ \mathrm{kg\,m^{-2}}\approx2.8\times10^3\ M_\odot\,\mathrm{pc^{-2}}$ ; the cosmic web sits far below it, so cosmic shear is a $\sim1\%$ distortion.

2 · How many galaxies to see a 1% shear (fully worked)

Shape noise is $\sigma_e=0.3$ per component. To measure $\gamma\approx0.02$ at $3\sigma$ in one patch, how many galaxies are needed, and what survey does that imply?

Step 1. The error on the mean shear is $\sigma_\gamma=\sigma_e/\sqrt{N}$ . A bare detection ( $\sigma_\gamma=\gamma$ ) needs $N=(\sigma_e/\gamma)^2=(0.3/0.02)^2=\mathbf{225}$ galaxies.

Step 2. For $3\sigma$ demand $\sigma_\gamma=\gamma/3$ , i.e. $N=9\times225\approx\mathbf{2\times10^3}$ per patch. No single galaxy, and no handful, suffices — hence binning.

Step 3. Scale up: a Stage-III survey reaches $n_{\rm gal}\approx10\ \mathrm{arcmin^{-2}}$ over thousands of square degrees ( $\sim1.3\times10^7\ \mathrm{arcmin^2}$ per $1000\ \mathrm{deg^2}$ ), so $\sim10^8$ galaxies map $\xi_\pm(\theta)$ across many angular and redshift bins. About $2\times10^3$ galaxies per element for a $3\sigma$ shear, hence $10^8$ – $10^9$ per survey.

3 · The S8 tension as a number

KiDS-1000 cosmic shear reports $S_8=0.759\pm0.021$ ; Planck 2018 CMB infers $S_8=0.834\pm0.016$ . How significant is the discrepancy?

Step 1. The difference is $\Delta S_8=0.834-0.759=0.075$ . Treating the measurements as independent, add errors in quadrature: $\sigma_\Delta=\sqrt{0.021^2+0.016^2}=\mathbf{0.0264}$ .

Step 2. The tension is $\Delta S_8/\sigma_\Delta=0.075/0.0264\approx\mathbf{2.8\sigma}$ — a late-time amplitude a few percent lower than the CMB prediction, as if structure grew slightly less than $\Lambda$ CDM extrapolated from $z\approx1100$ requires.

Step 3. A $\sim2$ – $3\sigma$ offset is intriguing but not decisive: possibly new physics (suppressed growth from dark-energy or neutrino effects) or a systematic (baryons, intrinsic alignments, photo- $z$ bias). The KiDS-versus-Planck gap is $\approx2.8\sigma$ — the headline " $S_8$ tension" motivating Stage-IV surveys.

Physics of the portrait

Cosmic shear is one of the youngest tools in cosmology built on one of the oldest ideas — that mass distorts distant images predates any means of measuring it by sixty years.

Origins · 1 mincosmic shear and how this was discovered

cosmic shear

the coherent one-percent distortion of galaxy shapes by large-scale structure · weak gravitational lensing and cosmology

The idea that galaxies weigh the universe through the distortion of background light traces to Fritz Zwicky, who in 1937 — the year he inferred dark matter in Coma from galaxy velocities — noted that nebulae and clusters would act as gravitational lenses, a direct way to measure total mass. The thought lay dormant for decades because the effect on ordinary galaxies is a percent-level distortion swamped by intrinsic shapes. The theory matured in stages: Kristian and Sachs analysed image distortions in an inhomogeneous universe in the 1960s; Gunn worked out the statistics of deflection by matter fluctuations in 1967; and in the early 1990s Blandford, Miralda-Escudé, Kaiser and others recast the effect as a statistical signal of large-scale structure. Nick Kaiser's 1992 prediction of the shear power spectrum — the projection of $P_\delta$ derived above — made cosmic shear a defined observable tied to $\sigma_8$ and $\Omega_m$ . The measurement arrived in spring 2000, when four independent teams (Bacon, Refregier & Ellis; Kaiser, Wilson & Luppino; Van Waerbeke et al.; Wittman et al.) announced detections within months of one another — a genuine multiple discovery. The field has since scaled to the hundred-million-galaxy catalogues of DES, KiDS and HSC, completing what Zwicky began: mass, not light, mapped directly across cosmic time.

History

1937

— Zwicky argues galaxies and clusters act as gravitational lenses that weigh their total mass

1966

— Kristian and Sachs describe image distortion (shear) in an inhomogeneous universe

1992

— Kaiser predicts the shear power spectrum imprinted by large-scale structure

2000

— four teams independently announce the first detections of cosmic shear in blank fields

2021

— KiDS-1000 measures S₈ = 0.759, sharpening the tension with the Planck CMB

Going further

Texts. Dodelson, Modern Cosmology, 2nd ed., ch. 11 · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., §29 · Ryden, Introduction to Cosmology, 2nd ed., ch. 8–9 · Mo, van den Bosch & White, Galaxy Formation and Evolution, §6 · Bartelmann & Schneider, Weak Gravitational Lensing, Phys. Rep. 340 (2001) 291 · Kilbinger, Rep. Prog. Phys. 78 (2015) 086901. Key results used throughout: the Jacobian splits into $\kappa,\gamma_1,\gamma_2$ ; $\Sigma_{\rm cr}=c^2D_S/(4\pi G D_LD_{LS})\approx5.9\ \mathrm{kg\,m^{-2}}\approx2.8\times10^3\ M_\odot\,\mathrm{pc^{-2}}$ for $z_L=0.3$ , $z_S=1.0$ ; the kernel carries the prefactor $3H_0^2\Omega_m/2c^2$ and peaks mid-way at $z\approx0.5$ ; the Limber power spectrum is $C_\ell^{\kappa\kappa}$ with $k=(\ell+1/2)/f_K$ ; and the best-constrained combination is $S_8=\sigma_8(\Omega_m/0.3)^{1/2}$ . Survey values (KiDS $0.759\pm0.021$ , Planck $0.834\pm0.016$ , $\approx2.8\sigma$ tension) are drawn from the DES-Y3, KiDS-1000, HSC-Y3 and Planck 2018 papers.

Prerequisites. Density Perturbations and the Density Contrast · Transfer Function and the Matter Power Spectrum · Linear Growth of Structure · The ΛCDM Concordance Model. Leads to. Dark Energy and the Cosmological Constant · Cold Dark Matter · The Primordial Power Spectrum.

Ask the tutor: Ask me to walk through the Limber projection line by line, work a tomographic two-bin $S_8$ forecast, or show exactly how intrinsic alignments bias a cosmic-shear analysis.

What is the difference between strong and weak lensing?

It is a matter of degree, set by $\kappa$ . Where the projected mass approaches critical, $\kappa\gtrsim1$ , lensing is strong: multiple images, giant arcs and Einstein rings, analysable one system at a time. Where it is far below critical, $\kappa\ll1$ , lensing is weak: each image is distorted by about a percent, too little to see in one galaxy, so the signal is extracted statistically from the coherent alignment of many shapes. Cosmic shear is the weak signal produced not by one object but by the diffuse large-scale structure threading the whole line of sight.

Why does cosmic shear constrain S8 rather than σ8 and Ωm separately?

Because over the range a shear survey probes, the two parameters enter in a nearly fixed combination. The amplitude of the convergence power spectrum scales as $\sigma_8^2$ (from $P_\delta$ ) times roughly $\Omega_m^{1.5}$ (from the kernel's Poisson factor and growth), so a survey measures $\sigma_8\Omega_m^{\sim0.5}$ well while the orthogonal direction stays nearly flat. Packaging this as $S_8=\sigma_8(\Omega_m/0.3)^{1/2}$ gives one tightly-constrained number. Splitting $\sigma_8$ from $\Omega_m$ needs extra information — tomography, or external data like the CMB, BAO or redshift-space distortions.

What is tomography and why does it help?

Tomography divides the sources into photometric-redshift bins and measures the shear correlations within and between them. A higher-redshift source is lensed over a longer path, so watching the shear grow with source distance separates the two things that create it: geometric distance factors (sensitive to dark energy) and growth of the density field (sensitive to $\Omega_m$ , $\sigma_8$ and the growth rate). A single-redshift measurement gives one degenerate amplitude; several bins turn cosmic shear into a simultaneous probe of expansion and growth, and help isolate redshift-dependent systematics like intrinsic alignments.

What are intrinsic alignments and why are they dangerous?

Cosmic shear assumes galaxies are intrinsically randomly oriented, so any coherent alignment must be lensing. But galaxies formed in the same tidal field can be physically aligned, correlating their true shapes and mimicking shear. Two effects appear: a correlation between the intrinsic shapes of nearby galaxies (the "II" term), and one between a foreground galaxy's intrinsic shape and a background galaxy's shear (the "GI" term), which is anti-correlated and survives even between well-separated redshift bins. Intrinsic alignments are a leading systematic and must be modelled and marginalised in the fit, not assumed to average away.

Is the S8 tension real evidence for new physics?

It is a genuine, repeated offset but not yet a discovery. Multiple independent Stage-III surveys land near $S_8\approx0.76$ , a few percent below Planck's $\approx0.83$ , at the $2$ – $3\sigma$ level — enough to take seriously, not to claim a breakdown of $\Lambda$ CDM. It could be real physics suppressing late-time growth (some dark-energy or modified-gravity models, or a larger neutrino mass), or a systematic: baryonic feedback, residual intrinsic alignments, or photometric-redshift bias. Distinguishing them is the mission of Stage-IV surveys, whose larger samples help only if the systematic modelling keeps pace with the shrinking statistical errors.

Why must weak-lensing modelling include baryons and nonlinear gravity?

Because most of the signal comes from small scales where matter is nonlinear. For $\ell\gtrsim300$ — much of the statistical weight — the wavenumbers are nonlinear, so a nonlinear prescription (HALOFIT or a simulation-trained emulator) is mandatory, not the linear power spectrum. On top of that, baryonic feedback — gas cooling, star formation and especially AGN — suppresses $P_\delta$ by up to $10$ – $20\%$ on scales of a few $h^{-1}\mathrm{Mpc}$ . Ignoring this biases $S_8$ , so analyses model the suppression, marginalise over feedback, or discard the most affected small scales.