Mathematics of gravitational lensing: multiple imaging and magnification

The mathematical theory of gravitational lensing has revealed many generic and global properties. Beginning with multiple imaging, we review Morse-theoretic image counting formulas and lower bound results, and complex-algebraic upper bounds in the case of single and multiple lens planes. We discuss recent advances in the mathematics of stochastic lensing, discussing a general formula for the global expected number of minimum lensed images as well as asymptotic formulas for the probability densities of the microlensing random time delay functions, random lensing maps, and random shear, and an asymptotic expression for the global expected number of micro-minima. Multiple imaging in optical geometry and a spacetime setting are treated. We review global magnification relation results for model-dependent scenarios and cover recent developments on universal local magnification relations for higher order caustics.

[1]  Random complex zeroes, I. Asymptotic normality , 2002, math/0210090.

[2]  J. Milnor.  Dynamics in one complex variable , 2000 .

[3]  Brian Rider,et al.  A mathematical theory of stochastic microlensing. II. Random images, shear, and the Kac-Rice formula , 2008, 0807.4984.

[4]  The Theory of Caustics and Wave Front Singularities with Physical Applications , 1999, gr-qc/9906065.

[5]  M. Atiyah,et al.  A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications , 1968 .

[6]  D. Clayton,et al.  Placing the Sun and Mainstream SiC Particles in Galactic Chemodynamic Evolution , 1997 .

[7]  Israel Kovner.  Fermat Principle in Arbitrary Gravitational Fields , 1990 .

[8]  P. Schneider,et al.  The two-point-mass lens: detailed investigation of a special asymmetric gravitational lens , 1986 .

[9]  A. Petters,et al.  A universal magnification theorem for higher-order caustic singularities , 2008, 0811.3447.

[10]  Joe W. Harris,et al.  Principles of Algebraic Geometry: Griffiths/Principles , 1994 .

[11]  R. McKenzie.  A gravitational lens produces an odd number of images , 1985 .

[12]  N. Katz,et al.  Random scattering approach to gravitational microlensing , 1986 .

[13]  A gravitational lens need not produce an odd number of images , 1994, gr-qc/9402038.

[14]  R. Blandford,et al.  Fermat's principle, caustics, and the classification of gravitational lens images , 1986 .

[15]  Wenbo V. Li,et al.  On the expected number of zeros of a random harmonic polynomial , 2008 .

[16]  N. Dalal.  The Magnification Invariant of Simple Galaxy Lens Models , 1998, astro-ph/9809061.

[17]  C. Keeton.  Gravitational lensing with stochastic substructure: Effects of the clump mass function and spatial distribution , 2009, 0908.3001.

[18]  V. Perlick.  On Fermat's principle in general relativity. I. The general case , 1990 .

[19]  C. Herdeiro,et al.  Stationary Metrics and Optical Zermelo-Randers-Finsler Geometry , 2008, 0811.2877.

[20]  P. Schneider,et al.  The gravitational lens equation near cusps , 1992 .

[21]  A. Petters,et al.  Fixed points due to gravitational lenses , 1998 .

[22]  V. Perlick.  Criteria for multiple imaging in Lorentzian manifolds , 1996 .

[23]  R. Blandford,et al.  Multiple imaging of quasars by galaxies and clusters , 1984, Nature.

[24]  N. Evans,et al.  Are there sextuplet and octuplet image systems , 2001, astro-ph/0108374.

[25]  No glory in cosmic string theory , 1993 .

[26]  Samuel,et al.  Fermat's principle in general relativity. , 1992, Physical review. D, Particles and fields.

[27]  G. Gibbons,et al.  Applications of the Gauss–Bonnet theorem to gravitational lensing , 2008, 0807.0854.

[28]  P. Marshall,et al.  An Atlas of Predicted Exotic Gravitational Lenses , 2009, 0904.1454.

[29]  Infinite dimensional Morse theory and Fermat’s principle in general relativity. I , 1995 .

[30]  M. Atiyah,et al.  A Lefschetz Fixed Point Formula for Elliptic Complexes: I , 1967 .

[31]  M. Deakin.  Catastrophe theory. , 1977, Science.

[32]  V. Perlick.  Global Properties of Gravitational Lens Maps¶in a Lorentzian Manifold Setting , 2000, gr-qc/0009105.

[33]  R. Jones.  Gravitational Curvature – An Introduction to Einstein's Theory , 1979 .

[34]  V. Perlick.  On Fermat's principle in general relativity. II. The conformally stationary case , 1990 .

[35]  S. Mao,et al.  On the Minimum Magnification between Caustic Crossings for Microlensing by Binary and Multiple Stars , 1995 .

[36]  A. Petters.  Morse theory and gravitational microlensing , 1992 .

[37]  W. L. Burke.  Multiple Gravitational Imaging by Distributed Masses , 1981 .

[38]  A Lefschetz fixed point theorem in gravitational lensing , 2007, math-ph/0703050.

[39]  P. J. Forrester,et al.  Exact statistical properties of the zeros of complex random polynomials , 1999 .

[40]  P. Madau,et al.  Compound Gravitational Lensing as a Probe of Dark Matter Substructure within Galaxy Halos , 2001, astro-ph/0108224.

[41]  H. Witt.  Investigation of high amplification events in light curves of gravitationally lensed quasars. , 1990 .

[42]  D. Freedman.  The General Case , 2022, Frameworks, Tensegrities, and Symmetry.

[43]  A. Petters.  Arnold’s singularity theory and gravitational lensing , 1993 .

[44]  V. I. Arnol’d.  Evolution of singularities of potential flows in collision-free media and the metamorphosis of caustics in three-dimensional space , 1986 .

[45]  R. Adler,et al.  Random Fields and Geometry , 2007 .

[46]  Mario Wschebor,et al.  On the distribution of the maximum of a Gaussian field with d parameters , 2005 .

[47]  J. Stachel,et al.  The Origin of Gravitational Lensing: A Postscript to Einstein's 1936 Science Paper , 1997, Science.

[48]  Stephen Smale,et al.  Complexity of Bezout's Theorem: III. Condition Number and Packing , 1993, J. Complex..

[49]  B. Carter,et al.  Optical reference geometry for stationary and static dynamics , 1988 .

[50]  J. Stewart,et al.  Characteristic initial data and wavefront singularities in general relativity , 1983, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[51]  N. Evans,et al.  Lensing Properties of Cored Galaxy Models , 2002, astro-ph/0204206.

[52]  B. Gaudi,et al.  Identifying Lensing by Substructure. I. Cusp Lenses , 2002 .

[53]  Volker Perlick,et al.  Ray optics, Fermat's principle, and applications to general relativity , 2000 .

[54]  P. Schechter,et al.  The Mean Number of Extra Microimage Pairs for Macrolensed Quasars , 2002, astro-ph/0208439.

[55]  On the magnification relations in quadruple lenses: a moment approach , 1999, astro-ph/9906323.

[56]  Joe W. Harris,et al.  Principles of Algebraic Geometry , 1978 .

[57]  Probing Dark Matter Substructure in Lens Galaxies , 2001, astro-ph/0109499.

[58]  Boris Tsirelson,et al.  Random complex zeroes, III. Decay of the hole probability , 2003 .

[59]  Gravitational lensing by stars in a galaxy halo: Theory of combined weak and strong scattering , 1984 .

[60]  Alberto M. Teguia,et al.  A Mathematical Theory of Stochastic Microlensing I. Random Time-Delay Functions and Lensing Maps , 2008, 0807.0232.

[61]  J. M. Boardman,et al.  Singularties of differentiable maps , 1967 .

[62]  P. Schneider,et al.  Evidence for substructure in lens galaxies , 1997, astro-ph/9707187.

[63]  A. Petters.  Multiplane gravitational lensing. I. Morse theory and image counting , 1995 .

[64]  A. Petters.  Multiplane gravitational lensing. III. Upper bound on number of images , 1997 .

[65]  N. Evans,et al.  The Milky Way Galaxy as a strong gravitational lens , 2006, astro-ph/0611134.

[66]  G. Gibbons,et al.  Universal properties of the near-horizon optical geometry , 2008, 0809.1571.

[67]  B. Gaudi,et al.  Identifying Lenses with Small-Scale Structure. I. Cusp Lenses , 2002, astro-ph/0210318.

[68]  Sun Hong Rhie.  n-point Gravitational Lenses with 5(n-1) Images , 2003 .

[69]  A. Petters,et al.  A universal magnification theorem. II. Generic caustics up to codimension five , 2009, 0904.2236.

[70]  A. Masiello,et al.  A Morse theory for light rays on stably causal lorentzian manifolds , 1998 .

[71]  A. Petters,et al.  A universal magnification theorem. III. Caustics beyond codimension 5 , 2009, 0909.5235.

[72]  Dmitry Khavinson,et al.  On the number of zeros of certain rational harmonic functions , 2004, math/0401188.

[73]  Robert Gilmore,et al.  Catastrophe Theory for Scientists and Engineers , 1981 .

[74]  Geometry of universal magnification invariants , 2009, 0904.0630.

[75]  V. I. Arnol'd,et al.  Normal forms for functions near degenerate critical points, the Weyl groups of Ak, Dk, Ek and Lagrangian singularities , 1972 .

[76]  S. Rhie.  Infimum microlensing amplification of the maximum number of images of n-point lens systems , 1995, astro-ph/9508123.

[77]  Antal Majthay.  Foundations of catastrophe theory , 1985 .

[78]  M. Lombardi,et al.  Gravitational lenses: odd or even images? , 1999 .

[80]  M. Golubitsky,et al.  Stable mappings and their singularities , 1973 .

[81]  P. Schechter,et al.  Quasar Microlensing at High Magnification and the Role of Dark Matter: Enhanced Fluctuations and Suppressed Saddle Points , 2002, astro-ph/0204425.

[82]  K. Subramanian,et al.  ‘Missing image’ in gravitational lens systems? , 1986, Nature.

[83]  N. Dalal,et al.  Magnification relations in gravitational lensing via multidimensional residue integrals , 2000, astro-ph/0009002.

[84]  Stable singularities of wave-fronts in general relativity , 1998, gr-qc/0108012.