Gravitational lensing as observational evidence

Begin with images and their observing conditions

A lensing observation records radiation: image positions, shapes, brightness distributions, spectra and sometimes changes with time. A gravitational lens model relates those measurements to a source and to deflection along the light path. A reconstructed mass map is an inference through that relationship, rather than another photograph taken directly by the detector.

Multiple images of a variable source can provide relative positions, relative brightness and arrival-time differences. An ensemble of faint galaxies can provide statistical shape information. These are complementary observables with different dependencies; collecting them under one name does not make them interchangeable measurements of mass.

The geometry matters. In the thin-lens approximation, deflection is assigned to a narrow plane compared with the distances involved. Angular-diameter distance ratios connect the deflection with apparent source and image directions. Lens and source redshifts help specify that geometry under a distance model; the lens-to-source distance is not generally the difference of the two observer distances.

The point-mass lens in Foundations illustrates a restricted gravitational geometry. Real extended lenses need not be point masses, and the calculations below do not assume that they are. Lensing evidence can constrain a projected gravitational field under specified assumptions without identifying the microscopic nature of an unseen component.

Convergence and shear enter the same image mapping

Write the source angular coordinates as β and image angular coordinates as θ. The local Jacobian A = ∂β/∂θ describes how small image displacements map to source displacements. Its isotropic and anisotropic parts are conventionally expressed through convergence κ and two shear components γ1 and γ2.

Convergence is a dimensionless projected surface density relative to a critical density that depends on lens and source geometry within the adopted lens model. Shear describes anisotropic distortion. In axes aligned with the shear, choose γ2 = 0, so A is diagonal with entries 1 − κ − γ1 and 1 − κ + γ1.

Reduced shear is g = γ/(1 − κ), when that denominator is nonzero. It describes the ratio relevant to shape transformation, rather than measuring convergence and shear separately. Here γ means the aligned component γ1; no complex-component arithmetic is needed for this example.

Signed magnification is μ = 1/det A for a nonsingular local mapping. Its sign records parity; absolute flux magnification uses |μ| under the appropriate lensing and source assumptions. Near a critical curve det A approaches zero and this simple finite local numerical example would no longer describe a bounded magnification.

A supplied distortion does not fix the normalization

For an original mathematical example, stipulate one thin lens, one source-redshift geometry and locally constant coefficients κ = 0.20, γ1 = 0.10 and γ2 = 0. These are supplied model quantities, not estimates from real galaxies. Both eigenvalues below are positive and nonzero, so the example remains in positive parity.

Calculate g = 0.10/(1 − 0.20) = 0.125. The Jacobian has diagonal entries 0.70 and 0.90, giving det A = 0.63 and μ = 1/0.63 ≈ 1.587301587. Each quantity is dimensionless because the mapping relates angles to angles.

Now apply a mass-sheet transformation with λ = 0.80: κ′ = λκ + (1 − λ) = 0.36 and γ′ = λγ = 0.08. The transformed source coordinates scale as β′ = λβ. The corresponding A′ = λA has entries 0.56 and 0.72.

The reduced shear remains g′ = 0.08/0.64 = 0.125, but det A′ = 0.4032 and μ′ ≈ 2.480158730. Thus μ′/μ = 1/λ² = 1.5625. The same reduced shear is consistent with these two different convergence values when the unknown source scale can also change.

A galaxy image is not a direct shear meter

An individual unlensed galaxy can already be elliptical. Its observed ellipticity therefore combines intrinsic structure with lensing and the observing response. Bartelmann and Schneider distinguish ellipticity definitions and show why their transformations cannot all be replaced by one universal rule saying that ellipticity equals shear.

Under specific assumptions about intrinsic orientations, local lens constancy, source geometry and the ellipticity convention, an ensemble can estimate a reduced-shear quantity. That statistical statement does not say that each observed galaxy has the reduced shear as its shape. Intrinsic alignments, uncertain source redshifts and selection can alter the needed relationship.

The point-spread function also redistributes light. Its width can dilute shape information, while anisotropy can introduce an apparent preferred direction. Stars help characterize the response, but spatial and wavelength variation and differences between stellar and galaxy spectra remain relevant. Correcting a response is an inference with calibration uncertainty.

Measurement weights, noise, object detection and quality cuts matter as well. The ideal unweighted brightness-moment transformation is not automatically valid for weighted noisy measurements. A real weak-lensing analysis needs a calibrated estimator and selection treatment before translating a catalogue of shapes into gravitational constraints.

State which ambiguity each new measurement addresses

The mass-sheet transformation illustrates a precise identifiability limit: some image information remains unchanged while the inferred source scale and absolute magnification change. For a fixed source-redshift geometry, image positions and relative properties can survive the associated source-plane rescaling. More precise measurements of unchanged quantities alone do not fix λ.

An independently calibrated intrinsic luminosity or size could constrain an absolute magnification or source scale. Dynamical information, additional source geometries and environmental measurements can add constraints, but they bring their own calibration and modelling assumptions. None should be described as automatically removing every lens degeneracy.

Time-delay information also requires care. Under this transformation, the relevant modelled time-delay scale changes with λ. Converting observed delays into a distance or expansion parameter therefore depends on the lens model and other assumptions. No time-delay distance or expansion-rate fit is performed here.

This mathematical transformation does not assert that an infinite physical sheet has been observed, nor that the same freedom applies unchanged to arbitrary multiple source planes. Preserve the actual geometry, source information, image calibration, selection, degeneracies and external constraints in the evidence record. The next lesson applies a similar discipline to background-radiation measurements and component separation.

Worked examples

Try the reasoning

1. Does the ellipticity of a single galaxy directly reveal its reduced shear?

   Show the answer and reasoning

   No.

   Intrinsic shape and the observing response must also be accounted for.

2. Do the equal g values 0.125 force the two supplied convergence values to be equal?

   Show the answer and reasoning

   No.

   κ = 0.20 and 0.36 give the same g with correspondingly different shear.

3. What is μ′/μ for the stipulated λ = 0.80 transformation?

   Show the answer and reasoning

   1/0.80² = 1.5625.

   The determinant of a two-dimensional Jacobian scales as λ².

4. Why could independently known source luminosity help constrain this ambiguity?

   Show the answer and reasoning

   It can provide information about absolute magnification.

   That additional constraint still depends on its luminosity calibration, propagation and source assumptions.

What to carry forward

• Lensing constraints combine images, observing response, source information and geometry.
• Reduced shear alone does not identify a unique convergence or absolute magnification.
• Additional observables constrain specific degeneracies through stated assumptions; they do not by themselves identify an unseen particle.