## Singularity Theory and Gravitational LensingAstronomers do not do experiments. They observe the universe primarily through detect ing light emitted by stars and other luminous objects. Since this light must travel through space to reach us, variations in the metric of space affects the appearance of astronomical objects. These variations lead to dramatic changes in the shape and brightness of astronom ical sources. Because these variations are sensitive to mass rather than to light, observations of gravitational lensing enable astronomers to probe the mass distribution of the universe. With gravitational lensing observations, astronomers are addressing many of the most important scientific questions in astronomy and physics: • What is the universe made of? Most of the energy and mass in the universe is not in the form of luminous objects. Stars account for less than 1 % of the energy density of the universe. Perhaps, as much as another 3% of the energy density of the universe is in the form of warm gas that fills the space between galaxies. The remaining 96% of the energy density is in some yet unidentified form. Roughly one third of this energy density of the universe is "dark matter," matter that clusters gravitationally but does not emit light. Most cosmologists suspect that this dark matter is composed of weakly interacting subatomic particles. However, most of the energy density of the universe appears to be in an even stranger form: energy associated with empty space. |

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### Contents

Historical Highlights | 3 |

12 Detecting Gravitational Lens Effects | 10 |

Central Problems | 15 |

21 Two Basic Problems | 16 |

22 Mathematicians versus Physicists | 18 |

ASTROPHYSICAL ASPECTS | 23 |

Basic Physical Concepts | 25 |

31 Ingredients of Gravitational Lens Systems | 26 |

772 1Parameter Families of Lagrangian Caustics | 285 |

Classification and Genericity of Stable Lens Systems | 287 |

81 Some Cautionary Remarks | 288 |

82 Stability of Time Delay Functions and Lensing Maps | 291 |

83 Generalization of Thorns Theorem for Two Parameters | 295 |

832 Proof of Generalization | 298 |

84 Most Time Delay Families have Stable Lensing Maps | 310 |

842 Proof of Main Theorem | 311 |

311 General Relativity and Friedmann Cosmology | 27 |

312 Cosmic Distances | 37 |

313 Spacetime Geometry for Gravitational Lensing | 43 |

Density Perturbations | 46 |

315 Deflected Light Rays and Bending Angles | 58 |

316 Cosmic Light Sources | 64 |

32 Gravitational Lens Optics | 65 |

Time Delays | 66 |

322 Computing the Time Delay Function | 67 |

Ray Tracing SinglePlane Lens Equation | 77 |

324 Dimensionless Variables | 79 |

325 Magnification and Flux Conservation Definition of Magnification | 82 |

326 Critical Curves Caustics and Light Curves Critical Curves and Caustics | 88 |

327 Shear Convergence and Extended Sources | 95 |

33 Two Important Families of Lens Models | 101 |

332 Elliptical Lens Models | 105 |

333 Point Mass versus Elliptical Lens Models | 108 |

35 Weak Lensing by Large Scale Structures | 112 |

Physical Applications | 119 |

42 Dark Matter | 121 |

421 Detecting Compact Dark Objects and Planets | 122 |

422 Dark Matter in Galaxy Clusters | 126 |

423 Cluster Mass Reconstruction via Weak Lensing | 128 |

43 Structure of Quasars via Microlensing | 131 |

44 Time Delay and Hubbles Constant | 134 |

45 Statistical Lensing and Limits on Cosmology | 137 |

Cosmic Strings | 139 |

47 Outlook | 140 |

Observations of Gravitational Lensing | 143 |

51 Multiple Quasars | 146 |

The Double Quasar Q0957+561 | 147 |

Giant Luminous Arcs | 150 |

521 Fantastic Arcs in Galaxy Cluster CL0024+1654 | 152 |

531 The Einstein Ring 1938+666 | 155 |

54 Quasar Microlensing | 156 |

541 Microlensing in Quadruple Quasar Q2237+0305 | 157 |

From MACHOS to Planets | 159 |

56 Weak Lensing | 166 |

MATHEMATICAL ASPECTS | 169 |

Time Delay and Lensing Maps | 171 |

61 Gravitational Lens Potentials | 172 |

62 SinglePlane Lensing | 176 |

622 Time Delay Functions and Light Rays SinglePlane Time Delay Functions | 177 |

623 Local Geometry of Time Delay Surfaces | 179 |

624 Lensing Maps Lensed Images and Magnification Lens Equation Lensed Images and Lensing Maps | 180 |

63 Simple Models | 185 |

632 Singular Isothermal Sphere | 186 |

633 Point Mass | 187 |

634 A Gallery of Gravitational Lens Models | 192 |

64 Multiple Plane Lensing | 193 |

641 Multiplane Time Delay Functions | 195 |

642 Multiplane Lensing Maps | 198 |

65 Relating Time Delay and Lensing Maps | 203 |

Critical Points and Stability | 209 |

71 Jets and Transversality | 210 |

712 Trans versality and Multijet Transversality | 216 |

72 Mather Stability Theory | 226 |

722 Transverse and Global Stability Transverse Stability | 229 |

73 Singularity Manifolds | 231 |

732 Singularities Srs XY and Srs f | 235 |

74 Morse Theory | 238 |

742 Stability of Functions | 240 |

75 Whitney Singularity Theory | 244 |

752 Characterizations of Folds and Cusps Whitneys Forms for Folds and Cusps | 247 |

753 Stability of Maps between 2Manifolds | 260 |

76 Thorn Catastrophe Theory | 266 |

762 Elementary Catastrophes and Thoms Theorem | 272 |

77 Arnold Singularity Theory | 276 |

771 Lagrangian Maps and Generating Families | 277 |

Local Lensing Geometry | 327 |

91 Qualitative Features of Multiplane Lensing Near Caustics | 328 |

911 Lensing Near Folds | 329 |

912 Lensing Near Positive and Negative Cusps | 332 |

913 Magnification Lensed Caustics and Orientation | 338 |

92 Folds and Cusps via Taylor Coefficients of the Potential | 341 |

93 Local Convexity of Fold Caustics | 353 |

932 Local Convexity in the SinglePlane Case | 359 |

933 DoublePlane Lensing Maps Displaying Folds | 360 |

934 Violation of Convexity in DoublePlane Case | 364 |

94 Folds and Cusps via Directional Derivatives | 367 |

95 Caustic Metamorphoses in Lensing | 375 |

952 Equations for Caustics and Their Metamorphoses | 378 |

953 Illustrations of SinglePlane Caustic Metamorphoses | 383 |

954 Single versus DoublePlane Caustic Metamorphoses | 388 |

Morse Inequalities | 393 |

101 Betti Numbers | 394 |

1012 Precise Treatment of Betti Numbers | 400 |

102 Morse Inequalities A and B | 403 |

103 Proof of the Morse Inequalities | 407 |

1032 Relative Betti Numbers | 411 |

1033 Derivation of Morse Inequalities A and B | 414 |

Counting Lensed Images SinglePlane Case | 419 |

111 General SinglePlane Gravitational Lens | 421 |

112 Isolated Gravitational Lenses | 424 |

113 Nonsingular Isolated Lenses | 426 |

114 Point Masses with Continuous Matter and Shear | 429 |

1141 Subcritical Case | 431 |

1142 Strong Shear Case | 432 |

1143 Supercritical Case | 434 |

115 Upper Bounds on Number of Lensed Images | 436 |

1151 Resultants | 437 |

1152 Application to Point Masses | 439 |

116 Location of Lensed Images due to Point Masses | 441 |

Counting Lensed Images Multiplane Case | 445 |

121 General Multiplane Gravitational Lens | 446 |

1212 Genericity of Morse Boundary Conditions B | 451 |

122 Nonsingular Isolated Multiplane Lenses | 455 |

123 Point Masses in ThreeDimensional Arrays | 456 |

1231 LensedImage Counting Formulas and Lower Bounds | 457 |

1232 Upper Bound on Number of Lensed Images | 459 |

Total Magnification | 467 |

SinglePlane Case | 468 |

1311 General SinglePlane Gravitational Lens | 469 |

1312 Nonsingular Isolated Lenses | 470 |

1313 Point Masses with Continuous Matter and Shear | 471 |

Multiplane Case | 473 |

133 Magnification Cross Section for Multiplane Lensing | 475 |

1331 Coarea Formula for Magnification Cross Sections | 476 |

1332 Magnification Cross Section Near Folds and Cusps | 479 |

Computing the Euler Characteristic | 487 |

141 Locally Stable Maps from Surfaces into the Plane | 488 |

142 Projectivized Rotation Numbers | 489 |

143 Orientation of Critical Curves | 495 |

144 A Formula for the Euler Characteristic | 496 |

Global Geometry of Caustics | 503 |

151 Critical Points and Lens Equation in Complex Form | 504 |

152 Caustics of One and Two Point Masses | 509 |

1522 Binary Point Masses | 514 |

1523 Binaries as a Point Mass Plus ChangRefsdal Lens | 524 |

153 Caustics of Finitely Many Point Masses | 531 |

1532 Bounds on Number of Caustics and Metamorphoses | 535 |

1533 Bounds on Number of Cusps | 536 |

154 Curvature of Caustics | 544 |

1542 Linking Curvature and Obstruction Points | 549 |

561 | |

589 | |

593 | |

### Common terms and phrases

angular diameter distance arcs Betti numbers bound catastrophe map caustic curve caustics due compact continuous matter coordinates cosmic cosmological constant Crit critical curves cusp critical point defined delay family delay function denote diffeomorphism differentiably equivalent Einstein radius Einstein ring Figure finite fold and cusp fold caustic fold critical point galaxy cluster germ grad gravitational lens potential gravitational lensing Hence Hubble constant Jk(X k-plane Lagrangian Lemma lens equation lens plane lensed images lensing map light curve light rays light source plane locally stable manifold map-germ mathematical matter and shear microlensing Mmin models Morse inequalities multiplane neighborhood nondegenerate number of lensed observer obstruction points parameter perturbation point mass point mass lens positive quasar radius redshift relative saddle shear from infinity single-plane singularity singularity theory smooth space stars submanifold subset surface mass density total magnification total number vector weak lensing yields zero

### Popular passages

Page xix - For, as he had said already in an earlier letter, "if there is anything that can bind the heavenly mind of man to this dusty exile of our earthly home and can reconcile us with our fate so that one can enjoy living — then it is verily the enjoyment of ... the mathematical sciences and astronomy.

### References to this book

Einstein's Cosmos: How Albert Einstein's Vision Transformed Our ... Michio Kaku Limited preview - 2004 |