Gravitational lensing: strong, weak, and microlensing — Einstein rings, dark matter maps, and exoplanet detection

Intuition Beginner

Albert Einstein's general relativity, published in 1915, says that mass curves space, and that light follows the curves. A star, a galaxy, or a galaxy cluster sitting between us and a distant light source acts like a magnifying glass: it bends the light rays passing near it, brightening and distorting whatever lies behind. This effect is gravitational lensing. Einstein himself doubted astronomers would ever observe it, because the alignments required are rare. He was wrong.

There are three regimes. Strong lensing, where a massive foreground galaxy cluster stretches background galaxies into arcs, rings, and multiple images. Weak lensing, where the foreground mass is too small to make a single image but subtly distorts thousands of background galaxies in a way that can be measured statistically — this is the most powerful method we have for mapping where the dark matter is. Microlensing, where a star passes in front of another star and briefly magnifies it, betraying any planets it carries.

In 1919 the British astronomer Arthur Eddington led an expedition to the island of Principe to photograph stars near the Sun during a total eclipse. The star positions had shifted by the amount Einstein predicted. The news made Einstein world-famous overnight. Today, lensing is cosmology's workhorse: it maps dark matter, weighs galaxy clusters, detects planets around stars thousands of light-years away, and has even produced the first image of a black hole.

Visual Beginner

The diagram shows the three lensing regimes side by side. On the left, a massive galaxy cluster bends the light of a distant galaxy into an Einstein ring and a pair of blue arcs. In the middle, a field of faint background galaxies is shown with each galaxy slightly stretched (sheared) along the radial direction toward an invisible foreground mass; the average elongation, too small for the eye to see on any one galaxy, reveals the dark-matter distribution when averaged over thousands. On the right, a foreground star crosses a background star; the background star brightens then fades along a symmetric Paczynski light curve, with a brief blip betraying a planet around the lens.

The same physics — light following the curvature of space — produces all three pictures. The difference is only the mass of the lens and how nearly the source sits directly behind it.

Worked example Beginner

The first gravitational lens ever discovered is the double quasar Q0957+561, found by Dennis Walsh, Robert Carswell, and Ray Weymann in 1979 and reported in Nature (volume 279, page 381). A single quasar about 9 billion light-years away appeared in the sky as two identical images separated by 6 arcseconds — about one-tenth the width of the full Moon.

Step 1. The lens is a giant elliptical galaxy about 3.3 billion light-years away, sitting almost exactly on the line of sight between us and the quasar. Its gravity splits the quasar's light into two paths that arrive at Earth from slightly different directions. The two images have identical spectra — the same emission lines at the same redshift — confirming they are one quasar, not two.

Step 2. Because the two light paths have different lengths, the two images flicker with a time delay. When the quasar brightens, one image responds first; the other follows about 417 days later. Measuring this delay gives the absolute distance to the lens, which fixes the Hubble constant — the expansion rate of the universe — independently of every other method.

Step 3. The 6-arcsecond image separation is far larger than the Sun could ever produce (about 0.0008 arcseconds). Only the combined mass of an entire galaxy — stars plus a huge halo of dark matter — can split light by that amount. The lens galaxy's visible stars account for only a small fraction of the bending; the rest is dark matter.

What this tells us: Q0957+561 confirmed Einstein's lensing prediction on a cosmological scale, sixty years after he published the theory, and turned gravitational lensing from a curiosity into a precision instrument for cosmology.

Check your understanding Beginner

Formal definition Intermediate+

Following Schneider, Ehlers, and Falco ^{[Schneider-Ehlers-Falco 1992]}, gravitational lensing is the systematic study of the mapping from a source position on the sky to the image positions produced by a foreground mass distribution. The relevant objects are the deflection angle, the thin-lens equation, the Einstein radius, the convergence and the shear, and the magnification.

Definition (deflection angle). In the weak-field, small-angle regime of general relativity, a point mass $M$ deflects a light ray with impact parameter $\xi$ by the angle

$$\hat{\alpha }(\xi )=\frac{4GM}{c^2\xi },$$

the Einstein deflection angle. For an extended, thin lens with projected surface mass density $\Sigma (\mathbf{\xi })$, the deflection is the two-dimensional integral

$$\hat{\mathbf{\alpha }}(\mathbf{\xi })=\frac{4G}{c^2}\int \Sigma ({\mathbf{\xi }}^{\prime })\frac{\mathbf{\xi }-{\mathbf{\xi }}^{\prime }}{|\mathbf{\xi }-{\mathbf{\xi }}^{\prime }{|}^2}{d}^2{\xi }^{\prime }.$$

Definition (thin-lens equation). Let $D_L$, $D_S$, and $D_{LS}$ be the angular-diameter distances from the observer to the lens, from the observer to the source, and from the lens to the source. The lens maps a source at angular position $\mathbf{\beta }$ on the sky to an image at angular position $\mathbf{\theta }$ according to

$$\mathbf{\beta }=\mathbf{\theta }-\mathbf{\alpha }(\mathbf{\theta }),\mathbf{\alpha }(\mathbf{\theta })=\frac{D_{LS}}{D_S}\hat{\mathbf{\alpha }}(D_L\mathbf{\theta }).$$

This is the lens equation. Its solutions $\mathbf{\theta }(\mathbf{\beta })$ are the image positions of the source. For a point-mass lens the reduced deflection simplifies to $\mathbf{\alpha }(\mathbf{\theta })={\theta }_E^2\mathbf{\theta }/{\theta }^2$, where the Einstein radius

$${\theta }_E=\sqrt{\frac{4GM}{c^2}\frac{D_{LS}}{D_L{D}_S}}$$

sets the angular scale of the lensing system.

Definition (convergence and shear). Locally, the Jacobian $A_{ij}=\partial {\beta }_i/\partial {\theta }_j$ of the lens map decomposes into an isotropic and a traceless part,

$$A=\left(\begin{array}{cc}1-\kappa -{\gamma }_1& -{\gamma }_2\\ -{\gamma }_2& 1-\kappa +{\gamma }_1\end{array}\right),$$

where $\kappa =\Sigma /{\Sigma }_{\mathrm{crit}}$ is the convergence (a dimensionless surface density with critical value ${\Sigma }_{\mathrm{crit}}=c^2{D}_S/(4\pi G{D}_L{D}_{LS})$) and $\gamma ={\gamma }_1+i{\gamma }_2$ is the complex shear. The convergence magnifies images isotropically; the shear stretches them into ellipses.

Definition (magnification). The scalar magnification of an image is the inverse determinant of the Jacobian,

$$\mu =\frac{1}{\det A}=\frac{1}{(1-\kappa {)}^2-|\gamma {|}^2}.$$

Images form wherever $\det A\ne 0$; the curves $\det A=0$ in the image plane are the critical curves, whose pre-images in the source plane are the caustics. Sources crossing a caustic produce or destroy pairs of images.

Definition (microlensing light curve). For a point-mass lens crossing in front of a point source, define the dimensionless angular separation $u(t)=\sqrt{u_0^2+((t-t_0)/t_E{)}^2}$, where $u_0$ is the impact parameter in units of ${\theta }_E$, $t_0$ is the time of closest approach, and $t_E={\theta }_E/{\mu }_{\mathrm{rel}}$ is the Einstein crossing time. The total magnification of the two unresolved images follows the Paczynski light curve

$$\mu (t)=\frac{u(t{)}^2+2}{u(t)\sqrt{u(t{)}^2+4}}.$$

This light curve is achromatic (the same in every band), symmetric in time about $t_0$, and has a single peak at ${\mu }_{\max }=(u_0^2+2)/(u_0\sqrt{u_0^2+4})$.

Counterexamples to common slips Intermediate+

• "A massive object always produces multiple images." Only if it is compact enough. A diffuse mass distribution with $\kappa <1$ everywhere (sub-critical) produces a single, weakly distorted image of every source; multiple images require that somewhere the surface density exceed ${\Sigma }_{\mathrm{crit}}$ and that a caustic be crossed. Most weak-lensing fields satisfy $\kappa \ll 1$ and produce no multiple images at all.

• "The deflection angle $4GM/(c^2\xi )$ is half the Newtonian value." The other way around: the general-relativistic deflection is twice the naive Newtonian estimate obtained by treating light as a corpuscle with speed $c$. Eddington's 1919 measurement distinguished the two predictions, ruling out the Newtonian value and confirming the relativistic one.

• "Microlensing detects planets by direct imaging." No. A planet orbiting the lens star introduces a brief, localized spike on top of the stellar Paczynski light curve when it crosses or perturbs one of the two unresolved images. The planet is never resolved; only its transient magnification signature is recorded. The OGLE and MOA surveys detect planets by scanning millions of light curves for these spikes.

• "Strong-lensing arcs are artifacts of telescope optics." They are not. The arcs in Hubble Frontier Fields images are the strongly lensed images of background galaxies, stretched along the tangential direction to the cluster centre by the cluster's tidal shear. Their uniform blue colours and their arrangement along an Einstein ring are the signatures.

• "Weak lensing measures the projected mass out to some fixed radius." It measures the integrated mass contrast along the entire line of sight, weighted by the lensing kernel $D_L{D}_{LS}/D_S$, which peaks when the lens sits midway between observer and source. Structures in front of and behind the main lens contribute noise that must be modelled — the line-of-sight projection.

• "The mass-sheet degeneracy makes all weak-lensing mass maps meaningless." It makes them degenerate up to one parameter, not meaningless. The transformation ${\kappa }^{\prime }=\lambda \kappa +(1-\lambda )$, ${\gamma }^{\prime }=\lambda \gamma$ leaves the reduced shear $g=\gamma /(1-\kappa )$ invariant, so a single lensing field cannot fix $\lambda$. Breaking the degeneracy requires magnification information, stellar kinematics, or X-ray gas in hydrostatic equilibrium — each a separate measurement.

Key theorem with derivation Intermediate+

Theorem (point-mass lens: Einstein ring, image positions, and total magnification). Consider a point source at true angular position $\beta \ge 0$ on the sky lensed by a point mass $M$ at angular-diameter distances $D_L$, $D_S$, $D_{LS}$. The lens equation reduces to $\beta =\theta -{\theta }_E^2/\theta$, with ${\theta }_E$ the Einstein radius defined above. The solutions are two images at

$${\theta }_{\pm }=\frac{1}{2}\left(\beta \pm \sqrt{{\beta }^2+4{\theta }_E^2}\right),$$

and the total magnification is

$${\mu }_{\mathrm{tot}}=\frac{{\beta }^2+2{\theta }_E^2}{\beta \sqrt{{\beta }^2+4{\theta }_E^2}}.$$

In the limit of perfect alignment $\beta \to 0$ the two images merge into the Einstein ring of angular radius ${\theta }_E$, and the magnification diverges.

Derivation. The deflection by a point mass is $\hat{\alpha }(\xi )=4GM/(c^2\xi )$ at impact parameter $\xi =D_L\theta$. Substituting into the lens equation $\beta =\theta -\alpha$ with $\alpha =(D_{LS}/D_S)\hat{\alpha }$ gives

$$\beta =\theta -\frac{D_{LS}}{D_S}\cdot \frac{4GM}{c^2{D}_L\theta }=\theta -\frac{{\theta }_E^2}{\theta },$$

where identifying

$${\theta }_E^2=\frac{4GM}{c^2}\cdot \frac{D_{LS}}{D_L{D}_S}$$

collects every distance and mass factor into a single angular scale. Multiplying the lens equation by $\theta$ yields the quadratic ${\theta }^2-\beta \theta -{\theta }_E^2=0$, with roots

$${\theta }_{\pm }=\frac{\beta \pm \sqrt{{\beta }^2+4{\theta }_E^2}}{2}.$$

The $+$ image lies on the same side of the lens as the source, the $-$ image on the opposite side; for $\beta \gg {\theta }_E$ the second image collapses toward the lens with ${\theta }_{-}\to -{\theta }_E^2/\beta$ and vanishing flux.

For the magnification, the Jacobian of the axisymmetric lens map is

$$A=\frac{d\beta }{d\theta }=1-\frac{d\alpha }{d\theta }=1-\frac{d}{d\theta }\left(\frac{{\theta }_E^2}{\theta }\right)=1+\frac{{\theta }_E^2}{{\theta }^2},$$

so each image carries magnification ${\mu }_{\pm }=(1+{\theta }_E^2/{\theta }_{\pm }^2{)}^{-1}$, negative for the $-$ image because ${\theta }_{-}<0$. Taking absolute values and summing,

$${\mu }_{\mathrm{tot}}=\sum _{\pm }{\left|1+\frac{{\theta }_E^2}{{\theta }_{\pm }^2}\right|}^{-1}.$$

Writing $u=\beta /{\theta }_E$ and using ${\theta }_{\pm }/{\theta }_E=(u\pm \sqrt{u^2+4})/2$, after the algebra of collecting the two terms and using ${\theta }_{+}{\theta }_{-}=-{\theta }_E^2$, the sum collapses to

$${\mu }_{\mathrm{tot}}=\frac{u^2+2}{u\sqrt{u^2+4}}=\frac{{\beta }^2+2{\theta }_E^2}{\beta \sqrt{{\beta }^2+4{\theta }_E^2}}.$$

In the limit $u\to 0$, ${\mu }_{\mathrm{tot}}\to 2/u\to \infty$ — perfect alignment produces an Einstein ring of radius ${\theta }_E$ and formally infinite magnification. With $\beta =u{\theta }_E$ and $u(t)=\sqrt{u_0^2+((t-t_0)/t_E{)}^2}$ this last formula is exactly the Paczynski light curve quoted in the Formal definition. $\square$

Bridge. The point-mass lens formula builds toward 28.04.05 large-scale structure, where the convergence $\kappa$ is the projected dark-matter density contrast consumed by Press-Schechter halo finding, and appears again in 28.05.04 exoplanet atmospheres, where the Paczynski light curve derived above is the discovery channel for the exoplanets whose atmospheres are then characterised spectroscopically. The foundational reason lensing is the cosmologist's instrument of choice is that the deflection $\hat{\alpha }=4GM/(c^2\xi )$ depends only on the total mass and not on its luminous or dark composition, and this is exactly why the Bullet Cluster's weak-lensing map separates the dark matter from the baryonic gas. Putting these together identifies the Einstein radius ${\theta }_E$ with the natural angular scale of every lensing phenomenon, from the arcsecond-scale arcs of galaxy clusters to the milli-arcsecond-scale Einstein rings of stellar-mass black holes, and the pattern generalises from the point-mass case treated here to extended axisymmetric lenses, to elliptical lenses that produce caustic networks, and ultimately to the continuous density fields of cosmic shear.

Exercises Intermediate+

Advanced results Master

Theorem 1 (Einstein 1915: light deflection in general relativity). Einstein's linearised treatment of the Schwarzschild metric predicts a null-geodesic deflection $\hat{\alpha }=4GM/(c^2\xi )$ for a light ray with impact parameter $\xi$ past a mass $M$ ^{[Einstein1915 Sitzungsber. Preuss. Akad. Wiss.]}. The factor of four — twice the Newtonian corpuscle prediction — is the signature general-relativistic result, arising because both the temporal and the spatial components of the metric contribute to the bending. Eddington's confirmation in 1919 measured the Solar deflection at $1.61\pm 0.40$ arcsec from Principe and $1.98\pm 0.16$ arcsec from Sobral, both consistent with $1.75$ arcsec and inconsistent with the Newtonian $0.87$ arcsec.

Theorem 2 (Zwicky 1937: galaxy clusters as lenses). Fritz Zwicky ^{[Zwicky1937 Phys. Rev. 51:290 and ApJ 86:217]} computed the deflection expected from a galaxy cluster and showed that (i) the deflection is large enough to produce observable multiple images and arcs of background galaxies, and (ii) the lensing mass inferred from galaxy velocity dispersions in the Coma cluster exceeds the luminous mass by a factor of order several hundred. Zwicky's prediction of cluster-scale strong lensing preceded its observational confirmation by four decades.

Theorem 3 (Walsh-Carswell-Weymann 1979: the first lens). The twin quasar Q0957+561 ^{[WalshCarswellWeymann1979 Nature 279:381]} appeared as two images separated by $6.1$ arcsec, with identical spectra at redshift $z=1.41$. A foreground giant elliptical galaxy at $z=0.36$ lies along the line of sight. The measured time delay of $\Delta t\approx 417$ days between the two images, combined with the lens model, gives a Hubble constant $H_0\approx 70$ km/s/Mpc, independent of the cosmic distance ladder.

Theorem 4 (Paczynski 1986: microlensing as a MACHO search). Bohdan Paczynski ^{[Paczynski1986 ApJ 304:1]} showed that compact halo objects in the mass range $10^{-7}$ to $30$ solar masses, lensing stars in the Large Magellanic Cloud, would produce detectable microlensing events with peak magnifications above $1.34$ and characteristic timescales of days to months. The MACHO, EROS, and OGLE collaborations subsequently monitored millions of stars; the inferred halo fraction in MACHOs is at most twenty percent, ruling out a fully baryonic dark halo.

Theorem 5 (Tyson-Valdes-Wen 1990 and Kaiser-Squires 1993: weak-lensing mass reconstruction). The first statistical detection of coherent weak distortions of background galaxies by a foreground cluster was reported by Tyson, Valdes, and Wen ^{[Tyson-Valdes-Wen 1990 ApJ 349:L1]}. Kaiser and Squires ^{[KaiserSquires1993 ApJ 404:441]} gave the inversion formula that recovers the convergence $\kappa$ from the measured shear field $\gamma$ via Fourier transform,

$$\hat{\kappa }(\mathbf{k})=\frac{k_1^2-k_2^2}{k^2}{\hat{\gamma }}_1(\mathbf{k})+\frac{2k_1{k}_2}{k^2}{\hat{\gamma }}_2(\mathbf{k}),$$

up to the mass-sheet degeneracy. This is the foundational equation of every modern weak-lensing mass map.

Theorem 6 (Falco-Gorenstein-Shapiro 1985: the mass-sheet degeneracy). The transformation ${\kappa }^{\prime }=\lambda \kappa +(1-\lambda )$, ${\gamma }^{\prime }=\lambda \gamma$, ${\mu }^{\prime }=\mu /{\lambda }^2$ leaves the reduced shear $g=\gamma /(1-\kappa )$ and all image shapes invariant. The degeneracy is exact for any lensing geometry and reflects the fact that adding a uniform mass sheet between observer and source rescales the angular-diameter distances without altering the observable image configuration. Breaking it requires a measurement that is not pure shear — magnification, stellar kinematics, or X-ray hydrostatic equilibrium.

Theorem 7 (Clowe et al. 2006: the Bullet Cluster). Clowe, Bradac, Gonzalez, Markevitch, Randall, Jones, and Zaritsky ^{[Clowe2006 ApJ 648:L109]} combined Chandra X-ray imaging with weak-lensing reconstruction of the Bullet Cluster 1E 0657$-$558, showing that the weak-lensing mass peaks are offset from the X-ray-emitting baryonic gas by hundreds of kiloparsecs. The offset empirically requires a collisionless dark component; no pure modified-gravity theory in which gravity is sourced by baryons alone can reproduce it.

Theorem 8 (EHT 2019, 2022: black-hole shadows via strong lensing). The Event Horizon Telescope collaboration produced the first direct image of a black-hole shadow, first for M87* ^{[EHT2019 ApJ 875:L1]} and then for the Galactic Centre source Sgr A* ^{[EHT 2022 ApJ 930:L12]}. The observed ring of emission at $1.3$ mm wavelength is the strongly lensed photon ring predicted by Kerr geometry: photons orbiting the black hole once or more before reaching the observer produce a bright crescent whose angular diameter fixes the black-hole mass and the geometry of the surrounding accretion flow.

Synthesis. Gravitational lensing is the foundational reason that cosmology can map the dark-matter distribution directly, and the central insight is that the deflection $\hat{\alpha }=4GM/(c^2\xi )$ depends only on the total projected mass and not on its composition. This is exactly the bridge from the visible baryonic universe to the invisible dominant component: putting these together identifies the Einstein radius ${\theta }_E$ as the universal angular scale of lensing across eighteen orders of magnitude in mass, from stellar-mass microlensing at $10^{-3}$ arcsec to galaxy-cluster Einstein rings at arcseconds to the cosmic shear of large-scale structure at degrees. The pattern generalises from the point-mass lens of the Key theorem to elliptical lenses producing quad configurations and caustic networks, appears again in 28.04.05 where the same projected-mass principle underlies the Press-Schechter halo mass function, and the bridge is from the strong-lensing arcs of Hubble Frontier Fields to the weak-lensing shear maps of DES, KiDS, HSC, and the forthcoming Rubin LSST and Euclid surveys that will measure the dark-energy equation of state to percent-level precision.

Full proof set Master

Proposition 1 (Einstein radius from the lens equation). For a point-mass lens $M$ at angular-diameter distances $D_L$, $D_S$, $D_{LS}$, the angular scale on which the lens produces significant magnification — the Einstein radius — is ${\theta }_E=\sqrt{(4GM/c^2)D_{LS}/(D_L{D}_S)}$.

Proof. Start from the lens equation $\beta =\theta -\alpha (\theta )$ with the reduced deflection for a point mass,

$$\alpha (\theta )=\frac{D_{LS}}{D_S}\hat{\alpha }(D_L\theta )=\frac{D_{LS}}{D_S}\cdot \frac{4GM}{c^2{D}_L\theta }=\frac{1}{\theta }\cdot \frac{4GM}{c^2}\cdot \frac{D_{LS}}{D_L{D}_S}.$$

Collecting the constant factor into ${\theta }_E^2=(4GM/c^2)D_{LS}/(D_L{D}_S)$ gives the canonical form $\alpha (\theta )={\theta }_E^2/\theta$. The lens equation becomes $\beta =\theta -{\theta }_E^2/\theta$; in perfect alignment ($\beta =0$) it reduces to ${\theta }^2={\theta }_E^2$, so $|\theta |={\theta }_E$: the image is the Einstein ring of radius ${\theta }_E$. For a Solar-mass lens at cosmological distances, ${\theta }_E$ is of order a milli-arcsecond, producing the micro-arcsecond astrometric shifts detectable by Gaia; for a $10^{14}{M}_{\odot }$ cluster at $z\approx 0.3$, ${\theta }_E$ is of order tens of arcseconds, producing the giant arcs visible in Hubble images. $\square$

Proposition 2 (mass-sheet degeneracy). The transformation ${\kappa }^{\prime }=\lambda \kappa +(1-\lambda )$, ${\gamma }^{\prime }=\lambda \gamma$, ${\mu }^{\prime }=\mu /{\lambda }^2$ leaves the reduced shear $g=\gamma /(1-\kappa )$ invariant for every $\lambda >0$.

Proof. By direct substitution,

$$g^{\prime }=\frac{{\gamma }^{\prime }}{1-{\kappa }^{\prime }}=\frac{\lambda \gamma }{1-(\lambda \kappa +1-\lambda )}=\frac{\lambda \gamma }{\lambda -\lambda \kappa }=\frac{\lambda \gamma }{\lambda (1-\kappa )}=\frac{\gamma }{1-\kappa }=g.$$

For the magnification, recall $\mu =((1-\kappa {)}^2-|\gamma {|}^2{)}^{-1}$, so under the transformation

$${\mu }^{\prime }=\frac{1}{(1-{\kappa }^{\prime }{)}^2-|{\gamma }^{\prime }{|}^2}=\frac{1}{(\lambda -\lambda \kappa {)}^2-{\lambda }^2|\gamma {|}^2}=\frac{1}{{\lambda }^2((1-\kappa {)}^2-|\gamma {|}^2)}=\frac{\mu }{{\lambda }^2}.$$

Since the observable quantity in a weak-lensing survey is the reduced shear $g$ (the local shape distortion of background galaxies), the family $\{{\kappa }_{\lambda }{\}}_{\lambda >0}$ produces indistinguishable observations. The degeneracy corresponds geometrically to rescaling the unlensed source plane by a factor $\lambda$ while adding a uniform mass sheet — both operations leave the local image shapes fixed. Falco, Gorenstein, and Shapiro (1985) first identified this invariance in the context of strong-lensing time-delay analysis; Saha and Williams extended it to weak-lensing mass reconstruction. Breaking it requires observables sensitive to absolute flux rather than to shape, such as magnification bias in source counts, or external constraints from stellar kinematics. $\square$

Connections Master

• Large-scale structure and Press-Schechter halos 28.04.05. The dark-matter halos whose mass function Press and Schechter derived in 28.04.05 are precisely the mass concentrations that gravitational lensing maps: every weak-lensing convergence peak is a projected dark-matter halo, every strong-lensing cluster arc traces a massive halo near the high-mass exponential cutoff of $n(M)$, and the cross-correlation of galaxy positions with the weak-lensing shear field (galaxy-galaxy lensing) measures the halo occupation distribution that bridges the simulated dark-matter clustering of 28.04.05 with the observed galaxy clustering of 28.03.04.

• Exoplanet atmospheres and biosignatures 28.05.04. The microlensing channel derived in this unit is one of the four principal exoplanet discovery methods (along with radial velocity, transit photometry, and direct imaging). Surveys OGLE, MOA, and KMTNet have discovered over two hundred cold exoplanets via the brief planetary perturbations on Paczynski stellar light curves; a subset of these — those that transit or are imaged directly — feed into the atmospheric-characterisation programme of 28.05.04, where transmission spectroscopy searches for water, methane, oxygen, and other biosignature gases.

• Galaxy formation and evolution 28.03.04. The cosmic star-formation history and hierarchical merging traced in 28.03.04 are reconstructed observationally using gravitational lensing in two complementary ways: weak lensing measures the evolving halo mass function that galaxy-formation models predict, and strong lensing by foreground clusters (the Hubble Frontier Fields programme) magnifies the earliest galaxies at $z>10$ bright enough for JWST spectroscopy, revealing stellar populations that would otherwise be undetectable.

• FLRW cosmology, inflation, and the CMB 13.08.02. The angular-diameter distances $D_L$, $D_S$, $D_{LS}$ that fix the Einstein radius are determined by the FLRW expansion history of 13.08.02 and hence by the cosmological parameters ${\Omega }_m$, ${\Omega }_{\Lambda }$, $H_0$. Time-delay cosmography (as in Q0957+561) and cosmic-shear tomography invert this dependence to measure the cosmological parameters independently of the CMB. Cosmic inflation, treated in 13.08.03, also seeds the lensing of the CMB itself: the CMB photons are deflected by intervening large-scale structure by about $2.7$ arcminutes, a signal detected by Planck and SPT and now a leading probe of structure at $z\sim 2$.

Historical & philosophical context Master

Albert Einstein ^{[Einstein1915 Sitzungsber. Preuss. Akad. Wiss.]} in November 1915 published the field equations of general relativity and derived, in the same Sitzungsberichte cycle, the perihelion precession of Mercury and the prediction that a light ray grazing the Sun would be deflected by $4G{M}_{\odot }/(c^2{R}_{\odot })\approx 1.75$ arcseconds — twice the value obtained by treating light as a Newtonian corpuscle with speed $c$. Einstein himself had earlier published the half-deflection in 1911 and corrected it after completing the full theory. The experimental confirmation was the work of Frank Dyson, Arthur Eddington, and Charles Davidson ^{[DysonEddingtonDavidson1920 Phil. Trans. R. Soc. A 220:291]}, who organised two eclipse expeditions in May 1919 to Principe (Eddington) and Sobral (Davidson and Crommelin). The measured deflections, $1.61\pm 0.40$ arcsec at Principe and $1.98\pm 0.16$ arcsec at Sobral, agreed with Einstein and ruled out the Newtonian prediction, making Einstein an international celebrity.

Fritz Zwicky ^{[Zwicky1937 Phys. Rev. 51:290]} in 1937 proposed galaxy clusters as gravitational lenses, computing the expected deflection and pointing out that clusters were massive enough to produce observable multiple images and arcs; in the companion paper (ApJ 86:217) he used the virial theorem on Coma cluster galaxies to infer the existence of unseen mass exceeding the luminous matter by two orders of magnitude. Zwicky's lensing prediction was not confirmed observationally until 1979, when Dennis Walsh, Robert Carswell, and Ray Weymann ^{[WalshCarswellWeymann1979 Nature 279:381]} identified the twin quasar Q0957+561 as two lensed images of a single source at $z=1.41$, split by $6.1$ arcseconds by a foreground elliptical galaxy at $z=0.36$.

Bohdan Paczynski ^{[Paczynski1986 ApJ 304:1]} in 1986 proposed gravitational microlensing as a method to detect compact dark-matter objects in the Galactic halo, deriving the characteristic achromatic symmetric light curve that bears his name and motivating the MACHO, EROS, and OGLE surveys. The extension of lensing from compact objects to diffuse dark matter is due to J. Anthony Tyson, Francisco Valdes, and M. S. Wen, who in 1990 reported the first statistical detection of coherent weak distortions of background galaxies by a foreground cluster. Nick Kaiser and Gordon Squires ^{[KaiserSquires1993 ApJ 404:441]} gave the Fourier-space inversion that recovers the projected mass from the shear field, founding modern weak-lensing mass reconstruction. The cosmic-shear theory for using weak lensing to constrain dark energy was developed by Alexandre Refregier ^{[Refregier2003 ARAA 41:645]} and collaborators, and underlies the science cases of the Dark Energy Survey (DES), the Hyper Suprime-Cam (HSC), the upcoming Vera C. Rubin Observatory LSST, and the ESA Euclid mission. The Bullet Cluster result of Clowe et al. ^{[Clowe2006 ApJ 648:L109]} provided the most direct empirical evidence for the existence of dark matter. The Event Horizon Telescope's image of M87* ^{[EHT2019 ApJ 875:L1]} demonstrated strong-lensing geometry in the most extreme gravitational field yet observed, and the JWST era (2022 onward) has used cluster lenses as natural telescopes to identify galaxies at $z>10$, the deepest look into the early universe yet achieved.

Bibliography Master

@article{Einstein1915,
  author  = {Einstein, Albert},
  title   = {Die Feldgleichungen der Gravitation},
  journal = {Sitzungsberichte der K{\"o}niglich Preu{\ss}ischen Akademie der Wissenschaften},
  pages   = {844--847},
  year    = {1915},
}

@article{DysonEddingtonDavidson1920,
  author  = {Dyson, Frank W. and Eddington, Arthur S. and Davidson, Charles},
  title   = {A Determination of the Deflection of Light by the Sun's Gravitational Field, from Observations Made at the Total Eclipse of {M}ay 29, 1919},
  journal = {Philosophical Transactions of the Royal Society of London Series A},
  volume  = {220},
  pages   = {291--333},
  year    = {1920},
  doi     = {10.1098/rsta.1920.0009},
}

@article{Zwicky1937lens,
  author  = {Zwicky, Fritz},
  title   = {Nebulae as Gravitational Lenses},
  journal = {Physical Review},
  volume  = {51},
  pages   = {290},
  year    = {1937},
  doi     = {10.1103/PhysRev.51.290},
}

@article{Zwicky1937masses,
  author  = {Zwicky, Fritz},
  title   = {On the Masses of Nebulae and of Clusters of Nebulae},
  journal = {Astrophysical Journal},
  volume  = {86},
  pages   = {217--246},
  year    = {1937},
  doi     = {10.1086/143864},
}

@article{WalshCarswellWeymann1979,
  author  = {Walsh, Dennis and Carswell, Robert F. and Weymann, Ray J.},
  title   = {0957 + 561 A, B: Twin Quasistellar Objects or Gravitational Lens?},
  journal = {Nature},
  volume  = {279},
  pages   = {381--384},
  year    = {1979},
  doi     = {10.1038/279381a0},
}

@article{Paczynski1986,
  author  = {Paczynski, Bohdan},
  title   = {Gravitational Microlensing by the Galactic Halo},
  journal = {Astrophysical Journal},
  volume  = {304},
  pages   = {1--5},
  year    = {1986},
  doi     = {10.1086/164149},
}

@article{TysonValdesWen1990,
  author  = {Tyson, J. Anthony and Valdes, Francisco and Wenk, R. A.},
  title   = {Detection of Systematic Gravitational Lens Galaxy Image Alignments: Mapping Dark Matter in Galaxy Clusters},
  journal = {Astrophysical Journal Letters},
  volume  = {349},
  pages   = {L1--L4},
  year    = {1990},
  doi     = {10.1086/185636},
}

@article{KaiserSquires1993,
  author  = {Kaiser, Nick and Squires, Gordon},
  title   = {Mapping the Dark Matter with Weak Gravitational Lensing},
  journal = {Astrophysical Journal},
  volume  = {404},
  pages   = {441--450},
  year    = {1993},
  doi     = {10.1086/172297},
}

@article{FalcoGorensteinShapiro1985,
  author  = {Falco, Emilio E. and Gorenstein, M. V. and Shapiro, Irwin I.},
  title   = {On Model-Dependent Bounds on {H}$_0$ from Gravitational Images. {T}ime Delay and {H}$_0$ in the 0957+561 System},
  journal = {Astrophysical Journal},
  volume  = {289},
  pages   = {L1--L4},
  year    = {1985},
  doi     = {10.1086/184422},
}

@article{Clowe2006,
  author  = {Clowe, Douglas and Brada{\v c}, Maru{\v s}a and Gonzalez, Anthony H. and Markevitch, Maxim and Randall, Scott W. and Jones, Christine and Zaritsky, Dennis},
  title   = {A Direct Empirical Proof of the Existence of Dark Matter},
  journal = {Astrophysical Journal Letters},
  volume  = {648},
  pages   = {L109--L113},
  year    = {2006},
  doi     = {10.1086/508162},
}

@article{Refregier2003,
  author  = {Refregier, Alexandre},
  title   = {Weak Gravitational Lensing by Large-Scale Structure},
  journal = {Annual Review of Astronomy and Astrophysics},
  volume  = {41},
  pages   = {645--668},
  year    = {2003},
  doi     = {10.1146/annurev.astro.41.082801.095438},
}

@article{EHT2019,
  author  = {{Event Horizon Telescope Collaboration}},
  title   = {First {M87} Event Horizon Telescope Results. {I.} The Shadow of the Supermassive Black Hole},
  journal = {Astrophysical Journal Letters},
  volume  = {875},
  pages   = {L1},
  year    = {2019},
  doi     = {10.3847/2041-8213/ab0ec7},
}

@article{EHT2022,
  author  = {{Event Horizon Telescope Collaboration}},
  title   = {First {Sagittarius A*} Event Horizon Telescope Results. {I.} The Shadow of the Supermassive Black Hole in the Center of the Milky Way},
  journal = {Astrophysical Journal Letters},
  volume  = {930},
  pages   = {L12},
  year    = {2022},
  doi     = {10.3847/2041-8213/ac6674},
}

@book{SchneiderEhlersFalco1992,
  author    = {Schneider, Peter and Ehlers, J{\"u}rgen and Falco, Emilio E.},
  title     = {Gravitational Lenses},
  publisher = {Springer-Verlag},
  year      = {1992},
  edition   = {2nd printing 1999},
}

@article{NarayanBartelmann1999,
  author  = {Narayan, Ramesh and Bartelmann, Matthias},
  title   = {Lectures on Gravitational Lensing},
  journal = {arXiv e-prints},
  year    = {1999},
  eprint  = {astro-ph/9606001},
}