Principles of Gravitational Lensing: Light Deflection as a Probe of Astrophysics and Cosmology

292 Pages·2018·9.545 MB·English

by  Arthur B. Congdon, Charles R. Keeton

Arthur B. Congdon
Charles R. Keeton
Principles
of Gravitational
Lensing
Light Defl ection as a Probe
of Astrophysics and Cosmology
Springer Praxis Books
Astronomy and Planetary Sciences
Serieseditors
MartinA.Barstow
Leicester,UnitedKingdom
IanRobson
Edinburgh,UnitedKingdom
DerekWard-Thompson
Preston,UnitedKingdom
Moreinformationaboutthissubseriesathttp://www.springer.com/series/4175
Arthur B. Congdon • Charles R. Keeton
Principles of Gravitational
Lensing
Light Deflection as a Probe of Astrophysics
and Cosmology
123
ArthurB.Congdon CharlesR.Keeton
Monrovia,CA,USA Physics&AstronomyDepartment
RutgersUniversity
Piscataway,NJ,USA
SpringerPraxisBooks
ISSN2366-0082 ISSN2366-0090 (electronic)
AstronomyandPlanetarySciences
ISBN978-3-030-02121-4 ISBN978-3-030-02122-1 (eBook)
https://doi.org/10.1007/978-3-030-02122-1
LibraryofCongressControlNumber:2018960262
©SpringerNatureSwitzerlandAG2018
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
CoverImage:©LynetteCook/SciencePhotoLibrary
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Preface
Gravitationallensinghasmaturedintoathrivingareaofastrophysics,withapplica-
tions ranging from detecting extrasolar planets (microlensing), to constraining the
distributionofdarkmatteringalaxies(stronglensing),todeterminingcosmological
parameters (weak lensing). While some problems are best suited to one of the
three flavors of lensing (strong, weak, or micro), there are others for which it is
possibletocombineconstraintsfromthedifferentlensingregimes.Thedistribution
ofdarkmatterinclustersofgalaxiesoffersoneimportantexample.Forthisreason,
proficiencyintheentiresubjectisindispensabletothestudentandresearcheralike.
In their influential 1992 monograph, Gravitational Lenses, P. Schneider, J.
Ehlers, and E.E. Falco presented the theory of gravitational lensing in a rigorous
and systematic way and discussed the observations and applications then known.
During the following quarter century, though, advances in instrumentation have
made observations not possible at that time become almost routine today. The
qualityandthequantityofthedatathusgeneratedallowforsophisticatedstatistical
analysis,makingeventhesubtledistortionsintheobservedshapesofgalaxiesdue
tothelarge-scalestructureoftheuniversedetectable.Suchweaklensingwasinits
infancyintheearly1990s,andmicrolensinghadyettobeborn.
A recent textbook that incorporates current research methods and applications
of gravitational lensing at the undergraduate level is Gravitational Lensing by
Scott Dodelson. His aim is to present enough gravitational lens theory so that the
student can quickly confront the current literature and begin conducting research.
Weaimtotakeacomplementaryapproachandpresentathoroughdiscussionofthe
principlesofgravitationallensing.Asthecentenaryoftheobservationaldiscovery
ofgravitationallensingapproachesinMay2019,wecanonlyhopethatthepresent
effortisfittingtributetoresearchersinthefield:past,present,andfuture.
Itisourhopethatthisbookwillproveusefultostudentsandresearchersalike.
Forthosewithpriorexperienceinlensing,thisbookmayserveasreferencematerial
or as a supplement for researchers who wish to explore aspects of lensing outside
their own expertise. For the student, we envision this book as the basis for a one-
semester courseorseminar inlensingattheadvanced undergraduate orbeginning
v
vi Preface
graduatelevel.Withthisinmind,problemsareincludedattheendofeachchapter
(apartfromtheIntroduction)tobuildfamiliaritywithlensingcalculationsandshow
howtheyconnectwithastrophysicsresearch.
Thisbookisorganizedasfollow:WebegininChap.1withahistoricaloverview
to offer context and background to the development of gravitational lensing.
Gravitational lenses are introduced formally in Chap.2. For the sake of gaining
hands-on experience, we offer an intuitive, Newtonian presentation and make the
necessary relativistic correction on the fly. The properties of gravitational lenses
with circular symmetry, many of which generalize to lenses with asymmetry, are
thendiscussedinsomedetail.Chapter3derivesthefundamentalequationforlight
bending in its full, relativistic glory and introduces the homogeneous, isotropic
universe that is the backdrop for what follows in later chapters. We then turn to
thetheoryofmultipleimagingbyarbitrarymassdistributionsinChap.4.
The remaining chapters explore various applications of gravitational lensing.
Microlensing by stars and planets, in which multiple images cannot be spatially
resolved, is presented using complex variables in Chap.5. Strong lensing, where
multipleimagesareresolved,isdiscussedinthecontextofgalaxiesinChap.6and
clustersofgalaxiesinChap.7.Weaklensing,inwhichthegravitationalfieldistoo
weak to produce multiple images, is the subject of Chap.7, where it is applied to
clusters,andChap.8,whereitisappliedtolarge-scalestructure.Chapter9givesan
overviewoflensingofthecosmicmicrowavebackground,inwhichweextendthe
methodsdevelopedintheprecedingchapters.
Strictly speaking, we assume background only in multivariable calculus and
introductory physics. Familiarity with intermediate classical mechanics, electro-
dynamics, and quantum mechanics would be useful, as much of the mathe-
matics encountered in those subjects applies to lensing. For those without such
experience, we include appendices on several topics that come up in the book:
variational calculus (Appendix A), complex variables (Appendix B), orthogonal
functions(AppendixC),Fourieranalysis(AppendixD),andcomputationalmethods
(AppendixE).
Monrovia,CA,USA ArthurB.Congdon
Piscataway,NJ,USA CharlesR.Keeton
November2018
Acknowledgments
Thisbookultimatelyowesitsexistencetotheproductiveandenjoyablecollabora-
tion between PhD student, Arthur B. Congdon (ABC), and thesis advisor, Charles
R. Keeton (CRK), a decade ago. As the influence on astrophysical research of the
bending of light by gravity has only grown in the intervening years, we decided
to rekindle our partnership to bring the principles of gravitational lensing to as
broadanaudienceaspossible.Tothisend,wehavesoughtinputfromastronomers,
physicists,mathematicians,andstudentsalike.
Our own expertise has been supplemented in important ways by many people.
In addition to being a friend and mentor to ABC for over 20 years, Allan Moser
hasofferedmanyinsights,inspiredbothbyhisworkasaphysicistandbyhisclose
interaction with the engineering world. Carl Droms continually reminded us that
whatisconsideredrigoroustoaphysicistisoftenhand-wavingtoamathematician.
His suggestions helped us to clarify our own thinking, which has translated into
clearer, more concise prose throughout the text. He also contributed a number
of figures that illustrate ideas presented in the text. Ted Burkhardt offered many
helpfulpointersonmakingourbookaccessibletophysicistsaswellasastronomers
andresearchersaswellasstudents.Severalend-of-chapterproblemshesuggested
appear in the following pages. We greatly appreciate their input throughout the
durationofthewritingprocess.
Thefollowingpeoplereadoneormorechapterdraftsandpointedoutmathemati-
cal,scientific,stylistic,andtypographicalimperfections:AmirBabakAazami,Sean
Brennan,JerodCaligiuri,JohnCallas,TimJones,ErikNordgren,SeanPerry,Catie
Raney,andBarnabyRowe.Muchofthematerialinthisbookservedasthebasisfor
a graduate seminar on gravitational lensing taught by CRK at Rutgers University.
Wearegratefultothisinauguralgroupofstudents,whohelpedustranslateouridea
ofwhatalensingtextbookshouldbeintoonethatcanactuallyworkinpractice.
ABC wishes to thank his family for their ongoing support and encouragement
and Anna Göddeke for her steadfast friendship, both personal and intellectual,
throughout the course of the project. CRK acknowledges support from the US
NationalScienceFoundationthroughgrantAST-1716585.
vii
Contents
1 Introduction .................................................................. 1
1.1 LightandGravityinNewtonianPhysics.............................. 1
1.2 LightBendinginGeneralRelativity................................... 2
1.3 ConsequencesofLightBending....................................... 3
1.4 GravitationalLensingasanObservationalScience................... 4
1.4.1 StrongLensing................................................. 5
1.4.2 Microlensing................................................... 6
1.4.3 WeakLensing.................................................. 8
References..................................................................... 9
2 GravitationalLenseswithCircularSymmetry........................... 11
2.1 DeflectionAngle:NewtonianDerivation.............................. 11
2.2 TheoryofAxisymmetricLenses....................................... 15
2.2.1 ThinLensApproximation..................................... 15
2.2.2 LensEquation:GeometricDerivation........................ 16
2.2.3 ImageMagnification........................................... 18
2.3 AxisymmetricLensModels............................................ 20
2.3.1 PointMassLens ............................................... 20
2.3.2 SingularIsothermalSphere ................................... 22
2.3.3 NonsingularIsothermalSphere............................... 27
2.4 EinsteinRings .......................................................... 31
2.5 SupercriticalityandStrongLensing................................... 33
2.5.1 DifferentiableLensMappings ................................ 34
2.5.2 DiscontinuousLensMappings................................ 35
2.5.3 DivergentConvergenceFields ................................ 37
2.6 MagnificationandShear ............................................... 38
Problems....................................................................... 39
References..................................................................... 43
ix
x Contents
3 LightDeflectioninCurvedSpacetime..................................... 45
3.1 ReviewofSpecialRelativity........................................... 45
3.1.1 GalileanTransformations ..................................... 45
3.1.2 LorentzTransformations ...................................... 46
3.1.3 Four-VectorsinMinkowskiSpace............................ 47
3.2 GeodesicEquation...................................................... 51
3.2.1 ContravariantandCovariantVectors ......................... 51
3.2.2 MetricTensor.................................................. 53
3.2.3 PrincipleofStationaryInterval ............................... 55
3.3 SchwarzschildSolution ................................................ 58
3.3.1 GravitationalTimeDilation................................... 58
3.3.2 Spacetime Interval Outside a Static, Spherically
SymmetricMassDistribution................................. 59
3.3.3 CircularOrbits................................................. 61
3.4 LightPropagationintheSchwarzschildMetric....................... 65
3.4.1 DeflectionAngle............................................... 65
3.4.2 TimeDelay..................................................... 67
3.5 Friedmann-Robertson-WalkerCosmology............................ 71
3.5.1 Homogeneous,IsotropicUniverse............................ 71
3.5.2 Robertson-WalkerMetric ..................................... 74
3.5.3 Friedmann’sEquation ......................................... 75
3.5.4 CosmologicalDistances....................................... 79
Problems....................................................................... 82
References..................................................................... 84
4 MultipleImagingintheWeak-FieldLimit................................ 87
4.1 LensEquation .......................................................... 87
4.1.1 LensPotential.................................................. 87
4.1.2 Fermat’sPrinciple ............................................. 91
4.1.3 ConvergenceandShear........................................ 93
4.2 AmplificationTensor................................................... 95
4.2.1 MagnificationforConstantConvergenceandShear......... 96
4.2.2 GeneralCase................................................... 98
4.2.3 EigenvaluesandImageParity................................. 99
4.3 TimeDelayandParity ................................................. 100
4.4 Burke’sTheorem ....................................................... 101
4.5 CriticalCurvesandCaustics........................................... 102
4.5.1 Folds............................................................ 102
4.5.2 Cusps........................................................... 103
4.6 SurfaceBrightnessandExtendedSources ............................ 106
4.6.1 ConservationofSurfaceBrightness .......................... 106
4.6.2 MagnificationofanExtendedSource ........................ 107
4.7 DegeneraciesintheLensEquation.................................... 108
4.7.1 SimilarityTransformations.................................... 109
4.7.2 Mass-SheetTransformation................................... 109
Contents xi
4.7.3 SourcePositionTransformation .............................. 110
4.7.4 ConnectiontoElectrodynamics............................... 111
4.8 MultiplaneLensing..................................................... 112
Problems....................................................................... 114
References..................................................................... 115
5 MicrolensingWithintheLocalGroup..................................... 117
5.1 MicrolensingbyaPointMass ......................................... 118
5.1.1 LightCurves ................................................... 118
5.1.2 Parallax......................................................... 120
5.1.3 AstrometricMicrolensing..................................... 121
5.2 MicrolensingbyMultiplePointMasses............................... 121
5.2.1 ComplexFormulation ......................................... 122
5.2.2 BinaryLens.................................................... 125
5.3 MicrolensingofanExtendedSource.................................. 132
5.3.1 SingleLens..................................................... 132
5.3.2 FoldCaustic.................................................... 134
5.4 MicrolensingStatistics................................................. 135
5.5 Applications ............................................................ 138
5.5.1 ProbingDarkMatter........................................... 138
5.5.2 FindingPlanets ................................................ 139
5.5.3 CharacterizingCompactObjects.............................. 140
Problems....................................................................... 141
References..................................................................... 143
6 StrongLensingbyGalaxies................................................. 145
6.1 SingularIsothermalLensModels...................................... 146
6.1.1 SphericalLenseswithExternalShear ........................ 146
6.1.2 EllipticalLenses ............................................... 151
6.2 LenseswithaCoreofFiniteDensity.................................. 156
6.2.1 EmergenceofRadialCaustics ................................ 156
6.2.2 CentralImages................................................. 157
6.2.3 CausticMetamorphosis ....................................... 158
6.3 RingImagesofExtendedSources..................................... 162
6.4 PerturbationsDuetoSmall-ScaleStructure........................... 164
6.4.1 MillilensingbyDarkMatterSubstructure.................... 164
6.4.2 MicrolensingbyStars ......................................... 166
6.5 LensModeling.......................................................... 168
6.5.1 AnalyticDeterminationofLensParameters ................. 168
6.5.2 StatisticalDeterminationofLensParameters................ 170
6.5.3 ModelingLenseswithExtendedSources .................... 174
6.6 Applications ............................................................ 176
Problems....................................................................... 178
References..................................................................... 181