Math 2.0

Astronomy and astrophysics Definition

Weak gravitational lensing

Builds on Lens equationStrong gravitational lensing

Definition

Consider a thin gravitational lens with lens equation β=θ−α(θ)\beta = \theta - \alpha(\theta) for θ,β∈ℝ2\theta, \beta \in \R^2, and suppose α=∇ψ\alpha = \nabla \psi for a lensing potential ψ\psi. The Jacobian matrix of the lens map is written A(θ)=∂β∂θ=(1−κ−γ1−γ2−γ21−κ+γ1),A(\theta) = \frac{\partial \beta}{\partial \theta} = \begin{pmatrix} 1 - \kappa - \gamma_1 & -\gamma_2 \\ -\gamma_2 & 1 - \kappa + \gamma_1 \end{pmatrix}, which defines the convergence κ=12∇2ψ\kappa = \tfrac{1}{2} \nabla^2 \psi and the shear γ=(γ1,γ2)\gamma = (\gamma_1, \gamma_2), with γ1=12(ψ11−ψ22)\gamma_1 = \tfrac{1}{2}(\psi_{11} - \psi_{22}) and γ2=ψ12\gamma_2 = \psi_{12}. Weak gravitational lensing is the regime ∣κ∣≪1\abs{\kappa} \ll 1 and ∣γ∣≪1\abs{\gamma} \ll 1. There the lens map is one-to-one near θ\theta, and each image is a slightly magnified and sheared copy of its source.

Remarks

In this regime a small circular source of radius rr is imaged, to first order, as an ellipse with semi-axes r/(1−κ∓∣γ∣)r / (1 - \kappa \mp \abs{\gamma}). So the ellipticity of the image is close to the reduced shear g=γ/(1−κ)g = \gamma / (1 - \kappa), and averaging the shapes of many galaxies, whose intrinsic orientations are random, estimates gg. The convergence equals Σ/Σcr\Sigma / \Sigma_{cr}, the surface mass density in units of the critical density.

Example: an isothermal sphere lens with Einstein radius θE\theta_E has κ=∣γ∣=θE/(2∣θ∣)\kappa = \abs{\gamma} = \theta_E / (2 \abs{\theta}); at ∣θ∣=50 θE\abs{\theta} = 50 \, \theta_E both equal 0.010.01, which is weak lensing.

Non-example: images near the Einstein radius of the same lens have κ=∣γ∣=1/2\kappa = \abs{\gamma} = 1/2, which is not small; this is strong lensing, with large distortions and multiple images.

Reading path

To understand this page from the ground up, read these 52 pages in order. The foundations come first.

  1. Derivative concept
  2. Differential equation concept
  3. Distance measures in cosmology concept
  4. Euclidean space object
  5. Euclidean vector concept
  6. Function concept
  7. Geodesics in general relativity concept
  8. Interval concept
  9. Ordinary differential equation concept
  10. Parameterized post-Newtonian formalism concept
  11. Ricci curvature concept
  12. Strong gravitational lensing concept
  13. Velocity concept
  14. Fréchet derivative definition
  15. Ordinary differential equation definition
  16. Cross product definition
  17. Function of several variables concept
  18. Vector-valued function concept
  19. Lipschitz condition definition
  20. Classical mechanics concept
  21. Schwarzschild metric example
  22. Limit of a function of several variables concept
  23. Momentum concept
  24. Partial derivative concept
  25. Solution of an ordinary differential equation definition
  26. Inertial frame of reference concept
  27. Continuity in R^n definition
  28. Differentiable function of several variables definition
  29. Partial derivative definition
  30. Initial value problem definition
  31. Newton's laws of motion statement
  32. Differentiable on an open set definition
  33. Gradient definition
  34. Higher partial derivatives definition
  35. Picard–Lindelöf hypotheses definition
  36. Iterated derivative definition
  37. Picard operator definition
  38. C^k function characterised by continuous derivatives statement
  39. Integral equation form of an initial value problem statement
  40. An iterate of the Picard operator is a contraction statement
  41. Vector field definition
  42. Picard–Lindelöf theorem statement
  43. Conservative vector field definition
  44. Classical mechanics definition
  45. Momentum definition
  46. Potential energy definition
  47. Angular momentum definition
  48. Central force definition
  49. Newton's law of universal gravitation statement
  50. Gravitational deflection of light statement
  51. Lens equation statement
  52. Weak gravitational lensing definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Weak_gravitational_lensing
About this page
Tag9RCRCB5V3R
URIm2://9RCRCB5V3R
Kinddefinition
WritingW0 stub
MSC85A04, 85A40, 83C99
DefinesWeak gravitational lensing
UsesLens equation, Strong gravitational lensing
Changed3 November 2026

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