Preparation: algebra, introductory mechanics, and the previous module's stated prerequisites. This is a 60-minute instructional module; use the slide timings as a guide and pause at the active check.
Terms Developed in Context
These terms organize the chain from recorded signal to calibrated observable, likelihood, and bounded cosmological claim.
- Weak lensing
Small coherent distortions of background images produced by foreground gravitational potentials. In this module it is used to answer
How can tiny image distortions reveal matter?
Observable/model connection: The term links
The DES Y3 paper is the cited source; the package uses only its published parameter summaries and a clearly synthetic geometry illustration, not shear catalogs or correlation-function vectors
to Cosmic shear and the quantitative relation \(gamma_t(R) approximately [barSigma(<R)-Sigma(R)]/Sigma_crit\). State which part is measured, calibrated, derived, or model-dependent.Limit: S8= sigma8 sqrt(Omega_m/0.3) compresses a degeneracy direction; it is not a direct measurement of sigma8 alone. That boundary must remain attached when the term is used in a cosmological claim.
- Cosmic shear
The two-point statistical pattern of weak-lensing distortions across source galaxies. In this module it is used to answer
How can tiny image distortions reveal matter?
Observable/model connection: The term links
The DES Y3 paper is the cited source; the package uses only its published parameter summaries and a clearly synthetic geometry illustration, not shear catalogs or correlation-function vectors
to Tangential shear and the quantitative relation \(gamma_t(R) approximately [barSigma(<R)-Sigma(R)]/Sigma_crit\). State which part is measured, calibrated, derived, or model-dependent.Limit: S8= sigma8 sqrt(Omega_m/0.3) compresses a degeneracy direction; it is not a direct measurement of sigma8 alone. That boundary must remain attached when the term is used in a cosmological claim.
- Tangential shear
The component of image distortion aligned tangentially around a foreground lens. In this module it is used to answer
How can tiny image distortions reveal matter?
Observable/model connection: The term links
The DES Y3 paper is the cited source; the package uses only its published parameter summaries and a clearly synthetic geometry illustration, not shear catalogs or correlation-function vectors
to Critical surface density and the quantitative relation \(gamma_t(R) approximately [barSigma(<R)-Sigma(R)]/Sigma_crit\). State which part is measured, calibrated, derived, or model-dependent.Limit: S8= sigma8 sqrt(Omega_m/0.3) compresses a degeneracy direction; it is not a direct measurement of sigma8 alone. That boundary must remain attached when the term is used in a cosmological claim.
- Critical surface density
The lens-source geometry factor converting projected mass contrast into shear. In this module it is used to answer
How can tiny image distortions reveal matter?
Observable/model connection: The term links
The DES Y3 paper is the cited source; the package uses only its published parameter summaries and a clearly synthetic geometry illustration, not shear catalogs or correlation-function vectors
to S8 and the quantitative relation \(gamma_t(R) approximately [barSigma(<R)-Sigma(R)]/Sigma_crit\). State which part is measured, calibrated, derived, or model-dependent.Limit: S8= sigma8 sqrt(Omega_m/0.3) compresses a degeneracy direction; it is not a direct measurement of sigma8 alone. That boundary must remain attached when the term is used in a cosmological claim.
- S8
The compressed amplitude combination sigma8 sqrt(Omega_m/0.3) aligned with a common lensing degeneracy. In this module it is used to answer
How can tiny image distortions reveal matter?
Observable/model connection: The term links
The DES Y3 paper is the cited source; the package uses only its published parameter summaries and a clearly synthetic geometry illustration, not shear catalogs or correlation-function vectors
to Weak lensing and the quantitative relation \(gamma_t(R) approximately [barSigma(<R)-Sigma(R)]/Sigma_crit\). State which part is measured, calibrated, derived, or model-dependent.Limit: S8= sigma8 sqrt(Omega_m/0.3) compresses a degeneracy direction; it is not a direct measurement of sigma8 alone. That boundary must remain attached when the term is used in a cosmological claim.
Physical and observational argument
Weak lensing shears background galaxy images through line-of-sight tidal fields. In the thin-lens limit, tangential shear traces the projected surface-density contrast divided by a geometry-dependent critical density. Cosmic shear correlates source shapes; galaxy-galaxy lensing cross-correlates lens positions and source shapes; clustering correlates lens positions. Combining them can calibrate biases but also couples nuisance parameters such as source redshift distributions, intrinsic alignments, shear response, galaxy bias and baryonic structure. S8 compresses a long posterior direction and should be interpreted with its model and covariance, not as sigma8 alone.
Lensing shear is a coherent distortion sourced by projected gravitational potential. A 3x2pt analysis combines cosmic shear, galaxy-galaxy lensing, and clustering, while source redshift distributions, intrinsic alignments, and calibration enter the inference.
The DES Y3 paper is the cited source; the package uses only its published parameter summaries and a clearly synthetic geometry illustration, not shear catalogs or correlation-function vectors.
Derivation and interpretation
\[gamma_t(R) approximately [barSigma(<R)-Sigma(R)]/Sigma_crit\]
Begin by naming observables, coordinate convention, and model assumptions; each algebraic transformation must preserve units and the distinction between measured and inferred quantities. Test a limiting case, then perturb one input and state which observable changes.
S8= sigma8 sqrt(Omega_m/0.3) compresses a degeneracy direction; it is not a direct measurement of sigma8 alone.
Worked numerical anchor
DES Y3 reports S8=0.776 +/- 0.017 and Omega_m=0.339(+0.032,-0.031) under its stated 3x2pt flat-LCDM analysis.
Reproduce the value with the course computation module, retain full precision in code, and round only for communication. Report units and all conditional assumptions with the result.
Evidence, systematics, and limits
S8= sigma8 sqrt(Omega_m/0.3) compresses a degeneracy direction; it is not a direct measurement of sigma8 alone.
The DES Y3 paper is the cited source; the package uses only its published parameter summaries and a clearly synthetic geometry illustration, not shear catalogs or correlation-function vectors.
Separate the directly recorded observable, its calibration, the derived intermediate quantity, and the model-dependent parameter. An uncertainty bar is not a complete likelihood when shared covariance or selection is omitted.
Study and retrieval practice
- Relate tangential shear to projected mass contrast and lens-source geometry.
- Define Sigma_crit and identify the redshift dependence.
- Explain the purpose and nuisance structure of a 3x2pt analysis.
Checkpoint: Discuss shear calibration, photo-z, intrinsic alignments, baryonic feedback, and survey masks.
Answer in words before calculating; then identify one observation or null test that could falsify your interpretation.