research30 min studyLesson 876 in Physics

Weak gravitational lensing and cosmic shear

Relate local image distortion to projected matter while tracking reduced shear, redshift kernels, and mass-sheet degeneracy.

Before you begin

On this page

The question for today

What you’ll understand

Calculate reduced shear and magnification from a lens Jacobian.
Explain the source-distance and matter-power dependence of cosmic shear.
Demonstrate a local mass-sheet degeneracy and identify calibration limitations.

Weak lensing uses correlated changes in distant galaxy images to probe intervening matter. The effect is gravitational, but the measured quantity is still an image after convolution with a telescope response, selection, and uncertain source distance. Moreover, shapes primarily constrain reduced shear rather than shear alone. We will connect a local distortion matrix with a statistical matter projection, then show why identical shape distortions need not determine an absolute projected mass. The numerical examples describe ideal local patches, not a full inversion of a real survey.

State the direction of the image mapping

Let the Jacobian A map a small displacement in observed image angle theta to displacement in unlensed source angle beta. Convergence kappa gives an isotropic focusing term and shear gamma gives a traceless distortion. In this convention, the eigenvalues for shear aligned with the axes are one minus convergence minus or plus the shear amplitude. Magnification is the reciprocal absolute determinant, while the weak positive-parity examples below have a positive determinant.

For suitable ellipticity conventions and an isotropically oriented source population, the mean shape response estimates reduced shear, gamma divided by one minus convergence. Approximating reduced shear by shear requires convergence small enough for the desired accuracy. Individual galaxies have unknown intrinsic shapes, so one cannot recover the local distortion exactly from one image. The estimator must also correct point-spread-function anisotropy, noise, blending, and selection responses that can mimic or bias the mean distortion.

A\equiv\frac{\partial\boldsymbol\beta}{\partial\boldsymbol\theta}=\begin{pmatrix}1-\kappa-\gamma_1&-\gamma_2\\-\gamma_2&1-\kappa+\gamma_1\end{pmatrix},\qquad g=\frac{\gamma_1+i\gamma_2}{1-\kappa},\qquad \mu=\frac{1}{|(1-\kappa)^2-|\gamma|^2|}

Kappa, gamma, reduced shear g, and magnification mu are dimensionless. A maps image to source; the inverse map expands image area by the magnification. Shape-response details depend on the chosen ellipticity estimator.↔ Scroll wide equations horizontally.

A shear spectrum integrates geometry and growth

Cosmic shear correlates distortions across many lines of sight. In a spatially flat background, a commonly used Born and Limber approximation projects the three-dimensional matter power through source-dependent lensing kernels. Each kernel includes the source redshift distribution and the efficiency of a lens between observer and source. Dividing sources into estimated redshift bins provides overlapping projections, not perfectly isolated slices of matter at the bin centers.

The matter power includes nonlinear gravitational evolution on sufficiently small scales, and baryonic feedback can alter it further. Accurate inference therefore needs a justified power prescription, scale cuts, and nuisance freedom. Photometric-redshift errors change the kernels, multiplicative shape calibration rescales amplitudes, and intrinsic alignments correlate galaxy orientations without lensing. These effects can mimic changes in clustering amplitude or matter density. A shear amplitude is not a model-independent direct reading of one cosmological parameter.

C_\ell^{ij}\simeq\int_0^{\chi_H}d\chi\,\frac{W_i(\chi)W_j(\chi)}{\chi^2}\,P_m\!\left(\frac{\ell+1/2}{\chi},z(\chi)\right),\qquad \kappa'=\lambda\kappa+1-\lambda,\quad\gamma'=\lambda\gamma\ \Longrightarrow\ g'=g

P m has volume units and W has inverse-length units in this kernel convention, giving dimensionless angular power. The projection assumes flat geometry and its stated approximations. Limber is appropriate at sufficiently high multipoles with broad, smoothly varying radial kernels; low multipoles or narrow kernels require the full radial projection. The mass-sheet transformation is written for one effective source plane and nonzero lambda.↔ Scroll wide equations horizontally.

Shapes alone leave an absolute focusing ambiguity

Scaling convergence and shear while adding a uniform sheet can preserve reduced shear. The source-plane scale changes, while corresponding image shape distortions remain the same. Magnification does change, so calibrated source sizes or number-count information can add constraints, with their own selection and population assumptions. Multiple source distances also provide extra information; the simple one-plane mass-sheet transformation should not be promoted unchanged into a claim about every tomographic or strong-lensing reconstruction.

A survey mask, spatially varying depth, and finite sky coverage affect shear correlations and their covariance. Intrinsic shape noise is not the only uncertainty: large-scale structure sample variance and modes larger than the survey can correlate estimated powers. Cross-correlating with galaxy positions or microwave-background lensing adds useful geometry and growth information but also introduces shared modes and nuisance parameters. Joint analyses need their cross-covariance rather than treating every available correlation as an independent experiment.

Work it through

From a local Jacobian to observable distortion

An ideal positive-parity patch has kappa = 0.2, gamma one = 0.1, and gamma two = 0. Find the diagonal Jacobian entries, determinant, reduced shear, and magnification. Compare gamma one directly with the reduced-shear value.

The image-to-source Jacobian is diagonal with entries $1-0.2-0.1=0.7$ and $1-0.2+0.1=0.9$ . Its determinant is $0.7(0.9)=0.63$ , so parity is positive.
The reduced shear is $g=0.1/0.8=1/8=0.125$ . Replacing it by gamma one = 0.1 would miss a 25 percent increase relative to gamma.
Magnification is $\mu=1/0.63=100/63\simeq1.587$ . This enlarges image solid angle relative to source solid angle while preserving surface brightness under transparent lensing.

The result

The Jacobian entries are 0.7 and 0.9, reduced shear is 0.125, and magnification is 100/63.

The living notebook

Explore the model

Interactive lab

Interactive lab · matrix

Distinguish map direction and area

Reset, set Diagonal scale a = 1 and Shear b = 0, and predict final determinant 1. For a fresh comparison Reset, set a = 0.5 and b = 0, and predict selected-final determinant 0.5 and singular values 1 and 0.5. Play or Step to inspect the deformation: solid geometry interpolates from identity, while readouts always describe the selected final matrix.

Interpolated geometryOriginal unit squareSelected final transform

Area scale · det A: 1.4
Singular values · σmax, σmin: 1.597, 0.876
Condition number · κ₂(A): 1.823

Diagonal scale a1.4Shear b0.6

t = 0 s

The selected matrix is A = [[a,b],[0,1]]. Playback interpolates geometry between the identity and A; readouts always describe A. Its determinant is a and its singular values come from AᵀA. Positive a makes every supported matrix invertible. This two-dimensional family is not a general eigensolver or a learned network. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forThe triangular matrix is a local mapping analogy. If treated as an image-to-source map its inverse area factor is two in the second case. It does not reconstruct lensing mass, source redshifts, or shear correlations.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

Both gamma and one minus kappa scale by lambda, so their ratio is invariant. The determinant scales by lambda squared, and magnification therefore changes.

Your turn

Try it yourself

Apply the mass-sheet transformation with lambda = 1/2 to the worked patch. Compute transformed kappa, gamma one, reduced shear, determinant, and magnification. Does matching reduced shear establish the same absolute focusing?

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Both diagonal Jacobian entries scale by lambda, while its determinant scales by lambda squared.

Compare with the solution

The transformed values are $\kappa'=0.5(0.2)+0.5=0.6$ and $\gamma_1'=0.05$ . Thus $g'=0.05/0.4=0.125$ . The entries become 0.35 and 0.45, giving determinant 0.1575 and $\mu'=400/63$ , four times the original magnification. The same reduced shear therefore does not fix absolute focusing for this single-plane idealization.

Keep the idea

Take it with you

Image shapes estimate a calibrated reduced-shear response.
Shear spectra mix geometry, growth, source distributions, and matter modeling.
Mass-sheet degeneracy and correlated systematics limit absolute mass inference.

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