Cosmology · Clusters and gravitational lensing

Strong and weak gravitational lensing

Use one weak-field point lens to compare image multiplicity, parity, flux magnification and small outer-image distortions.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

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After reading, you can

Distinguish strong image multiplicity from a strong gravitational field along the light path.
Calculate the two image branches of a specified point lens.
Separate signed image parity, positive flux magnification and local shear.

Lensing changes the relation between source and image direction

Gravitational lensing relates an apparent image direction to the direction a source would have without the foreground deflection model. Multiple images and small shape distortions can arise in different regions of that mapping.

Strong lensing commonly refers to conspicuous nonlinear mapping effects such as multiple images or critical curves. This usage does not require light rays to pass through a strong-field metric near a compact object.

Use a stationary thin point-mass lens in geometric optics. Require weak gravitational potential along the rays, with impact parameters much larger than the Schwarzschild radius. A point source and a fixed lens geometry are supplied.

The point lens is a teaching construction, not a galaxy-cluster mass profile or a photograph of an observed arc. Exact alignment and its ideal infinite point-source magnification are excluded from the displayed domain.

Two image branches solve one source position

Let θ_E be an arbitrary positive Einstein angular scale and define u = β/θ_E and x = θ/θ_E. β is the unlensed source angular offset, while θ is a signed image offset on the source–lens line.

The point-lens equation is u = x − 1/x for nonzero x. Its two roots are x_± = [u ± √(u²+4)]/2. For positive u, one image lies on the positive side and the other on the negative side.

If the scale is calculated physically, θ_E² = (4GM/c²)D_ls/(D_lD_s), with angular-diameter distances to lens, to source and from lens to source. D_ls is not generally obtained by subtracting D_l from D_s.

Display 0.25 ≤ u ≤ 4. All table angles are ratios to the same θ_E, so no measured angular separation, lens mass or redshift has been inserted.

Signed magnification describes orientation as well as area

At an image coordinate x, the radial and tangential eigenvalues of the source-to-image inverse response are λ_r = 1 + 1/x² and λ_t = 1 − 1/x². The signed magnification is μ = 1/(λ_rλ_t).

The positive branch has positive parity. The minor branch has negative signed μ because one local orientation is reversed. Negative parity does not mean a negative photon count or negative physical flux.

For an unresolved ideal point source, the total flux ratio is |μ_+| + |μ_-|. At u = 1 the image positions are about 1.618034 and −0.618034, with signed magnifications 1.170820 and −0.170820; the total is about 1.341641.

For the transparent static lens in this geometric-optics treatment, lensing conserves surface brightness while changing apparent image area and total flux. These are different observables, so flux amplification does not mean intrinsically brighter surface emission.

Supplied ideal point lens with u > 0; signed μ denotes parity and total flux magnification uses absolute values
u	x_+	x_-	Signed μ_+	Signed μ_-	Total flux ratio
0.25	1.132782	−0.882782	2.546573	−1.546573	4.093146
0.5	1.280776	−0.780776	1.591410	−0.591410	2.182821
1	1.618034	−0.618034	1.170820	−0.170820	1.341641
2	2.414214	−0.414214	1.030330	−0.030330	1.060660
4	4.236068	−0.236068	1.003115	−0.003115	1.006231

A small local distortion can coexist with a faint second image

Outside the ideal point mass, the local convergence κ is zero and tangential shear is γ_t = 1/x². Reduced shear g_t = γ_t/(1−κ) therefore equals γ_t there.

At outer image radii x = 2, 4 and 8, g_t is 0.25, 0.0625 and 0.015625. The corresponding signed magnifications are about 1.066667, 1.003922 and 1.000244.

The outer response becomes small at large x, while the full mathematical point-lens mapping still has a minor image for a positive source offset. That image may be too faint or too close to resolve in an actual observation.

This local shape description assumes an image small enough that the response does not vary strongly across it. Extended arcs, finite sources and a distributed cluster lens require a more complete spatial mapping.

A stationary thin point-mass lens is supplied in geometric optics with a point source. Rays are required to pass at impact parameters much larger than the Schwarzschild radius, so the potential along them is weak even though the mapping can produce multiple images. No observed arc, finite-source image, galaxy-cluster profile or angular calibration is provided. Define u = β/θ_E and signed x = θ/θ_E with one arbitrary positive Einstein angular scale. The equation u = x−1/x gives x_± = [u ± √(u²+4)]/2. Panel A uses linear axes with source u from 0.25 to four and image x from −1 to five. Solid blue circle samples mark the positive branch and dashed orange square samples the negative branch. Both branches use the same signed angular coordinate and no exact-alignment endpoint u = zero is included. At u = 0.25, 0.5, 1, 2, 4, positive images are 1.132782, 1.280776, 1.618034, 2.414214, 4.236068; negative images are −0.882782, −0.780776, −0.618034, −0.414214, −0.236068. Radial and tangential inverse-response eigenvalues are 1+1/x² and 1−1/x², so signed μ is their inverse product. At u = one these are signed magnifications 1.170820 and −0.170820. The negative sign records reversed parity; physical flux contributions are positive absolute values. Their total is about 1.341641. Across those five source samples the total flux ratios are 4.093146, 2.182821, 1.341641, 1.060660 and 1.006231. Surface brightness is conserved in this transparent static treatment; apparent area and flux change. Panel B is a different local outer-radius comparison. Its horizontal axis is image radius x from two to eight, not source u, and its linear dimensionless vertical axis is reduced shear from zero to 0.3. Outside the point mass κ = zero and g_t = γ_t = 1/x². Solid green diamonds at x = 2, 4, 8 give 0.25, 0.0625, 0.015625, with corresponding positive μ = 1/(1−x⁻⁴) about 1.066667, 1.003922, 1.000244. In particular x = eight corresponds to u = 7.875 and is outside panel A’s source interval. The local response assumes a small image relative to its spatial variation. The full ideal mapping still has a minor image for positive source offset, even when it becomes unresolved or too faint in practice. θ_E² = (4GM/c²)D_ls/(D_lD_s) would use angular-diameter distances; D_ls is not generally D_s−D_l. No strong-field propagation, exact alignment, fitted mass or observation is inferred. Original Ark teaching model.

A real lensing inference needs geometry and source information

Observed image positions, shapes, sizes and fluxes carry different combinations of lens and source information. The unlensed source position and shape are generally not given by one image alone.

Weak-shape analyses use populations and calibration because galaxies have intrinsic ellipticities, and instruments blur and distort their light. Source-redshift distributions and distance geometry also affect conversion to mass.

Multiple-image constraints can add information about a foreground lens, but they do not automatically supply one unique three-dimensional cluster mass. Model flexibility, projected matter and source assumptions remain relevant.

Keep image multiplicity, weak-field propagation, local shear and magnification distinct. The next lesson shows explicitly how different projected mass distributions can preserve the same reduced shear over a supplied field.

Worked reasoning

Solve the two images and their flux contributions

For u = 1, how are the two image locations and total ideal flux magnification obtained?

Solve x² − x − 1 = 0 to obtain x_+ = (1+√5)/2 and x_- = (1−√5)/2.
Use μ = 1/[(1+1/x²)(1−1/x²)] to obtain approximately 1.170820 and −0.170820.
Add absolute flux contributions: 1.170820 + 0.170820 ≈ 1.341641.

Two signed angular positions yield a positive total flux ratio about 1.341641; the negative minor-image sign records reversed parity.

Try the reasoning

Does a negative signed magnification mean negative observed flux?
Show the answer and reasoning
No.
Its sign describes parity; the flux contribution uses the absolute magnification.
May D_ls always be replaced by D_s − D_l in the Einstein-angle formula?
Show the answer and reasoning
No.
These are angular-diameter distances in a specified geometry, not generally additive Euclidean distances.
Does strong image multiplicity require rays to pass near the Schwarzschild radius?
Show the answer and reasoning
No.
The supplied example has multiple images while imposing weak potential and large impact parameters along the rays.

What to carry forward

Strong image mapping effects and strong gravitational fields are different uses of the word strong.
A point lens has two signed image branches; parity and physical flux must be distinguished.
Local shear, magnification, source properties and distance geometry enter different parts of mass inference.

References

Sources supporting the explanations, physical distinctions and examples in this lesson.