Learning objectives
- Use shear and convergence as projected mass probes and identify degeneracies.
- Explain lensing geometry, convergence, and mass mapping using the governing relation.
- Separate measurement, calibration, inference, and model limitation.
Terms Developed in Context
These terms form the lecture's reasoning chain from observable through model to a bounded physical conclusion.
- Convergence
The dimensionless projected surface-density field that describes isotropic gravitational-lensing magnification. In this lecture it supports the evidence-to-model chain for lensing geometry, convergence, and mass mapping.
Use and relationship: Read it together with Mass-sheet degeneracy and Shear. The governing relation \(\\kappa=\\Sigma/\\Sigma_{\\rm crit}\) shows where the concept enters quantitatively; the student should state whether its value is measured, calibrated, adopted, or inferred.
Boundary: The value is dimensionless, but translating shear to a mass map requires source redshifts and an inversion convention. This caveat prevents the term from being treated as a unique explanation or a context-free number.
- Shear
The anisotropic image distortion produced by gradients in the projected gravitational potential. In this lecture it supports the evidence-to-model chain for lensing geometry, convergence, and mass mapping.
Use and relationship: Read it together with Convergence and Critical surface density. The governing relation \(\\kappa=\\Sigma/\\Sigma_{\\rm crit}\) shows where the concept enters quantitatively; the student should state whether its value is measured, calibrated, adopted, or inferred.
Boundary: The value is dimensionless, but translating shear to a mass map requires source redshifts and an inversion convention. This caveat prevents the term from being treated as a unique explanation or a context-free number.
- Critical surface density
The lens-source geometry factor that converts physical projected mass density into convergence. In this lecture it supports the evidence-to-model chain for lensing geometry, convergence, and mass mapping.
Use and relationship: Read it together with Shear and Lensing kernel. The governing relation \(\\kappa=\\Sigma/\\Sigma_{\\rm crit}\) shows where the concept enters quantitatively; the student should state whether its value is measured, calibrated, adopted, or inferred.
Boundary: The value is dimensionless, but translating shear to a mass map requires source redshifts and an inversion convention. This caveat prevents the term from being treated as a unique explanation or a context-free number.
- Lensing kernel
The redshift-dependent weight describing how efficiently foreground matter lenses a chosen source population. In this lecture it supports the evidence-to-model chain for lensing geometry, convergence, and mass mapping.
Use and relationship: Read it together with Critical surface density and Mass-sheet degeneracy. The governing relation \(\\kappa=\\Sigma/\\Sigma_{\\rm crit}\) shows where the concept enters quantitatively; the student should state whether its value is measured, calibrated, adopted, or inferred.
Boundary: The value is dimensionless, but translating shear to a mass map requires source redshifts and an inversion convention. This caveat prevents the term from being treated as a unique explanation or a context-free number.
- Mass-sheet degeneracy
The invariance that allows a uniform convergence sheet and rescaled source plane to reproduce the same reduced shear. In this lecture it supports the evidence-to-model chain for lensing geometry, convergence, and mass mapping.
Use and relationship: Read it together with Lensing kernel and Convergence. The governing relation \(\\kappa=\\Sigma/\\Sigma_{\\rm crit}\) shows where the concept enters quantitatively; the student should state whether its value is measured, calibrated, adopted, or inferred.
Boundary: The value is dimensionless, but translating shear to a mass map requires source redshifts and an inversion convention. This caveat prevents the term from being treated as a unique explanation or a context-free number.
Lecture arc
Lensing measures total projected mass, including dark matter, but inversion has a mass-sheet degeneracy and line-of-sight structure. Begin with the observational puzzle and ask students what is directly measured. Introduce lensing geometry, convergence, and mass mapping only after the data-model boundary is explicit.
The visual is a reasoning object: students should narrate what each line, region, or axis means before seeing the equation.
Derivation and interpretation
Define the symbols, units, and assumptions. Explain why the relation is useful and where it fails. Relate the result to the paired lab's data record rather than treating the equation as a detached recipe.
Worked example
With Sigma=2.0e14 and Sigma_crit=3.0e15 solar masses Mpc^-2, kappa = 0.0666667.
The value is dimensionless, but translating shear to a mass map requires source redshifts and an inversion convention. Require students to estimate the answer, show units, and state whether the approximation is controlled, empirical, or merely illustrative.
Misconceptions and study guidance
Lensing measures total projected mass, including dark matter, but inversion has a mass-sheet degeneracy and line-of-sight structure. Common traps include confusing redshift with a unique velocity, treating a model-dependent parameter as a direct measurement, and overlooking covariance. Study by reproducing the calculation, changing one assumption, and comparing the result with the exact OpenStax anchor listed in the reference log.
Preparation for the problem set: write a one-paragraph evidence chain from observation to conclusion and identify the weakest link.