ASTR 420 · Lecture 12

Lecture 12: Weak Lensing and Cluster Mass

Cosmology

Lensing measures total projected mass, including dark matter, but inversion has a mass-sheet degeneracy and line-of-sight structure.

Objectives and route

Use shear and convergence as projected mass probes and identify degeneracies.
Define and distinguish lensing geometry, convergence, and mass mapping.
Derive or audit one quantitative relation with units and assumptions.
Connect the calculation to an observational test and a limitation.

Terms for This Lecture

Convergence

Definition: The dimensionless projected surface-density field that describes isotropic gravitational-lensing magnification.

Why it matters here: In Weak Lensing and Cluster Mass, Convergence is needed to reason about lensing geometry, convergence, and mass mapping rather than merely label the phenomenon.

Relationship: It is interpreted alongside Shear and the lecture relation below; identify which quantities are observed, calibrated, or model-derived.

Limitation or common confusion: The value is dimensionless, but translating shear to a mass map requires source redshifts and an inversion convention.

\(\\kappa=\\Sigma/\\Sigma_{\\rm crit}\)

Motivating evidence

Lensing measures total projected mass, including dark matter, but inversion has a mass-sheet degeneracy and line-of-sight structure.

Cosmological claims are indirect: an observable is calibrated, placed in a model, and compared against alternatives. Keep those layers separate throughout the lecture.

lensing geometry, convergence, and mass mapping. The axes, labels, and geometry carry the physical inference rather than serving as decoration.

Physical framework

Start from the governing relation and define its domain:

\(\\kappa=\\Sigma/\\Sigma_{\\rm crit}\)

State the coordinate choice, units, boundary conditions, and approximation before substituting numbers. The same symbol can mean a different quantity in Newtonian, special-relativistic, and expanding-space contexts.

Read the visual

Use the figure introduced on the evidence slide as an argument: identify the measured axis, the inferred axis, the comparison model, and the parameter that controls the conclusion.

The paired lab asks you to reproduce the visual from a supplied data extract and test one modeling choice.

Worked calculation

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.

Estimate the scale first, then check dimensions and comparative direction. The displayed value is generated from shared ASTR420 constants and data.

Evidence versus inference

Raw data are not the same as calibrated observables.
A fitted parameter is conditional on the likelihood, prior, and selection function.
Independent probes can agree because the model is useful, or disagree because an assumption is hidden.

Boundary condition

Limit of the model: The value is dimensionless, but translating shear to a mass map requires source redshifts and an inversion convention.

Ask what happens if the dominant assumption fails. In cosmology, approximation honesty matters because distances, times, and abundances accumulate model dependence.

Active reasoning

Prompt: Compute convergence for a projected surface density of 2.0e14 solar masses per square Mpc and a stated critical density.

Predict the direction before calculating. Then name one observation that could discriminate between two explanations.

Connection to the seven-unit arc

Earlier tools: ASTR310 radiation, dynamics, and statistical mechanics.
Paired data activity: use the corresponding CSV and reproduce the calculation with a perturbation test.
Next step: combine this probe with an independent epoch or distance measure.

Takeaways

Use shear and convergence as projected mass probes and identify degeneracies.
A cosmological parameter is meaningful only with its model and uncertainty context.
State whether a number is live-verified, a published value awaiting spot-check, or an instructor-provided model value.
Carry the limitation forward into the lab and problem set.

Reading and provenance

OpenStax Astronomy 2e, Chapter 28, section 28.4; embedded Figures 28.20-28.24.

OpenStax Astronomy 2e Chapter 29 embedded figure numbers were checked against the extracted text; the source index has chapter-label offsets near the Chapter 29 boundary, so the adopted PDF remains the final exercise-number authority.

Additional sources and dataset status are recorded in reference-log.md. Standard published values not re-fetched this session are level 2 and require human spot-checking.