Gravitational lensing by point masses on regular grid points

0.83 MB·English

by  Jin H. An

J.A:Gravitationallensingbypointmassesonregulargridpoints
Gravitational lensing by point masses on regular grid points
Jin H. An⋆
MITKavliInstituteforAstrophysics&SpaceResearch,MassachusettsInstituteofTechnology,77MassachusettsAvenue,CambridgeMA02139,USA
MonthlyNoticesoftheRoyalAstronomicalSociety
ABSTRACT
7
Itisshownthatgravitationallensingbypointmassesarrangedinaninfinitelyextendedregular
0
latticecanbestudiedanalyticallyusingtheWeierstrassfunctions.Inparticular,wedrawthe
0
criticalcurvesandthecausticnetworksforthelensesarrangedinregular-polygonal–square,
2
equilateraltriangle,regular hexagon– grids. From this, the mean numberof positive parity

n images as a function of the average optical depth is derived and compared to the case of
a the infinitely extendedfield of randomlydistributed lenses. We find that the high degree of
J
the symmetryin the lattice arrangementleads to a significantbias towards cancelingof the

1 shearcausedbytheneighboringlensesonagivenlenspositionandlensingbehaviourthatis
3 qualitativelydistinctfromtherandomstar field. We also discusssomepossible connections

tomorerealisticlensingscenarios.
2
v
2
9
1
1 INTRODUCTION icalcurvesandthecausticnetworks.InSect.4,westudythemean
2
number of positive parity images, which is compared to the case
1 Beyond the Galactic and Local-Group (in particular, Magellanic
oflensingbyaninfiniterandomstarfield.InSect.5,weconsider
6 Clouds and M31) microlensing experiments (Alcock et al. 2000;
0 theeffectoftheexternalpotentialbymeanofaddingexternalshear
Afonsoetal.2003;CalchiNovatietal.2005;deJongetal.2006),
/ (andconvergence),anddiscusssomepossibleconnectionstomore
h the main interest on the point-mass lenses lies in the effect of
realisticlensingscenarios.
p the individual stars in a lensing galaxy (Chang & Refsdal 1979,
Wearguethat,whilethesystemconsideredheremaybesome-
- 1984; Nityananda & Ostriker 1984; Irwin et al. 1989; Witt et al.
o whatartificialinitsconstructandratherabstractinitsnature,the
1995; Schechter & Wambsganss 2002; Keeton et al. 2006). For
r study presented in this paper can lead us to some insight on mi-
t these cases, each point-mass lens cannot be treated individually
s and their collective effects usually differ drastically from that of crolensingathighopticaldepth.
a
: the simple linear superposition of them. The usual approach to
v this“high-optical-depthmicrolensing”problemis“inverseraytrac-
i
X ing”(Kayseretal.1986;Schneider&Weiss1987;Wambsgansset 2 LENSEQUATION
al.1990,1992),thatis,examiningstatisticalpropertiesoflensing
r Inmostastronomicalsituations,gravitationallensingisdescribed
a observablesusingMonte-Carlorealizationsofthelensingsystem.
However,asthenumberoftheindividualpoint-masslensestore- by the lens equation y = x ∇ψ, which is derived from the
−
lowest-order nontrivial approximation to the path of light propa-
producetherealisticscenariowellapproachesthenumberofstars
inthegalaxy,thisbecomesveryexpensiveratherquicklyinterms gation(Schneideretal.1992; Pettersetal.2001;Kochanek etal.
2005).Here, yand xarethe(2-dimensional)vectorsrepresenting
oftherequiredresources.Anothercomplementaryapproachtothe
problem is from the random walk and the probability theory to- thelinesofsighttoward,respectively,theangularpositionsofthe
getherwiththethermodynamicapproximation(thatis,effectively sourceundertheabsenceoflensingandthelensedimage.Thelens-
consideringthelimitattheinfinitenumberofthestars),whichhas ingpotentialψisthegravitationalpotential(equivalently,thegrav-
itational‘Shapiro’timedelay)integratedalongthelineofsightup
beenquitesuccessfulincertainwell-definedproblems(Nityananda
&Ostriker1984;Schneider1987;Schneideretal.1992). toascaleconstant(Blandford&Narayan1986).
Inthefollowing,weconsideradifferentapproachtotheprob- Inparticular,gravitationallensingbyapointmassisdescribed
bythepotentialψ(x)=θ2ln x l ifthepoint-masslensislocated
lem: seeking complete analytic solutions. In many complex sys- E | − |
alongthelineofsightindicatedbythevector l(Paczyn´ski1986).
tems,imposinghighlevelofsymmetrycanreducetheproblemto
asimplerone,whichmaybeotherwisetoocomplicatedtoanalyze. Here,thescaleconstantθEisknownasthe‘Einstein(ring)radius’
anddeterminedbythemassofthelens(θ2 M)andthedistances
Ina similar spirit,in thispaper we consider gravitational lensing E ∝
by equal point masses that are arranged in an infinitely extended among thesource, thelens, and theobserver. Withthispotential,
wefindthelensequationforpoint-masslensing,
regular lattice. After we derive the appropriate lens equation in
Sect.2,theyareexaminedinSect.3tofindthecorrespondingcrit- x l
y= x − (1)
− x l2
| − |
⋆ E-mail:[email protected] wheretheunitofangularmeasurementshasbeenchosensuchthat
c 2007RAS
(cid:13)
2 J. An
θ = 1.Theequationisgeneralizedtothecaseofmultiplepoint- 1972,orseealsoAppendix).Here,Γ Γ(x)isthegammafunc-
E x ≡
masslenseswithasmoothly-varying‘largescale’potential(Young tion,℘(z)istheWeierstrassellipticfunctionand℘(z)isitscomplex
′
1981)suchthat derivative.Wealsouseanabbreviatednotationfor(thepowerto)
thegammafunctionsuchthat[Γ(x)]n = Γn inordertoavoidpro-
y=x−Xi=N1 mMi|xx−−llii|2 −∇ψext. (2) ltpihefeenrdraeitxaio)lnhaanoldff-pκpaerrei=ondtπhoe/fs(e2tshωea)2nledimsbntrhiasecckoaetpittcsi.ccaIanlsedaxedopdftih℘tio-(fVnu,ineωctrt0iio&≈n(O1s.es8et5rAi4kpeisr-
⋆
Here,theunitofangularmeasurementsisnowgivenbyθ corre-
E 1983; Paczyn´ski 1986) of the lenses. If we consider the square
spondingtoafiducialmassscaleMandsubsequentlythe‘external’
cell given by the centres of each lensing lattice, it is straightfor-
lensingpotentialψ shouldberescaledappropriately.Inaddition,
ext wardtoseethatthereexistsalenscorrespondingtoeachcellwith
the masses of individual lenses m enter into the equation as the
i sidelengthof2ω.Hence,themeansurfacedensityofpointmasses
ratiotothefiducialscale.
is in fact given by σ = (2ω) 2, and therefore, the optical depth
−
Next,letusthinkofthecasethattheequalpoint-masslenses by κ = πσ = π/(2ω)2. Moreover, the 90 -rotational symme-
⋆ ◦
are located at the lattice points on an infinitely extended square
try of the system implies that the total shear due to all the other
grid.Inaddition,weignoretheeffectoftheexternalpotentialfor
lenses on any given lens cancels out to be zero [mathematically
themoment. Utilizingthecomplex-number notation(Bourassa&
this indicates the constant term in the Laurent-series expansion
Kantowski 1975; Subramanian et al. 1985; Witt1990; An2005),
of f (z) is null]. Consequently, we find that f (z) = ℘(z;g ,0)
thelensequation(2)thencanbewrittenas ′ ′ − 2
and f(z) = ζ(z;g ,0)+C.Here,ζ(z)istheWeierstrasszetafunc-
2
η=z f(z); tion(Abramowitz&Stegun1972) andC isanarbitrarycomplex
− constant.Ifwechoosethe‘centre’ofthesystemtocoincidewith
1 ∞ 1 (3) the lens at the coordinate origin, then we have C = 0. We note
f(z)= = ,
Xi z−λi m,Xn= z−2(m+ni)ω that,whiletheconstantCcannotbeindependentlydetermined,its
−∞ choiceisphysicallyinconsequential.Thesituationisanalogousto
whereη,z,andλarethecomplexifiedvariablescorrespondingtoy,
lensing by an infinitely extended mass screen. While a na¨ıve ex-
x,and l,respectively,andtheover-barnotationisusedtoindicate
pectation from the symmetry appears to indicate null deflections
complexconjugation.Wefurtherassumethatthelenslocationsare
along every lineof sight, detailed physical consideration leads to
given byλ = 2(m+ni)ωwherem andnareanypair of integers
theuniformfocusingwithrespecttoanarbitrarychoiceofthecen-
(bothrunningfromthenegativeinfinitytothepositiveinfinity)and
tre.Inthecurrent situation,thediscretetranslationsymmetryim-
ω is the half distance between adjacent grid points. Without any
pliesthearbitrarinessinthechoiceofthecentreandconsequently
lossofgenerality,ωcanberestrictedtobeapositivereal.Strictly
thatofC,whichresultsintheinfinitesuminequation(3)notbeing
speaking,the‘deflectionfunction’ f(z)givenformallyintermsof
well-determined.However,theconstant-deflectionterminthelens
theinfinitesuminequation(3)isnotwell-definedasitisnotcon- equation againonlyleads toaconstant offset betweenthesource
vergent.Thisisinfactanaturalconsequenceofhavinganinfinite and the image/lens plane, which has no observable consequence
totallensingmass, whichisunphysical. However, whatisimpor-
andthereforecanbeignored.
tantinourunderstandingofthelensingisnottheabsoluteamount
Someresultsareimmediatefromtheresultinglensequation
ofthedeflectionbutthedifferenceofthedeflectionsbetweenneigh-
bouringlinesofsights.Thatistosay,ifweaddanyconstanttothe η=z−ζ(z;g2,0). (6)
deflectionfunctioninthelensequation,theresultinglensequation
Forinstance,thepropertyofζ(z)suchthat
isreducedtotheequivalentonewithouttheconstantbyintroduc-
π
ingthe“offset”tothesourcepositiontocanceltheconstant.Two
ζ z+2(m+ni)ω;g ,0 =ζ(z;g ,0)+ (m ni) (7)
equationsthenareobservationallyequivalentbecausethereisnoa 2 2 2ω −
(cid:2) (cid:3)
prioriinformationregardingthesourcepositionintheabsenceof forarbitraryintegersmandnindicatesthattheverticesz,z+2ω,
thelens. Similarly,despitetheseemingglobal failureofequation z+2(1+i)ω,andz+2iωofasquareinthelensplanemaptothe
(3),wecanstillproceedbyfocusingonthelocalpropertiesofthe pointsonthesourceplane:η = z ζ(z),η+ ,η+(1+i) ,and
lensingdescribedbyequation(3). η+i where = 2ω π/(2ω),−and therefoℵre the mean iℵnverse
Forthis,wefirstwritedownthe(formal)second-order com- magnℵificationℵisgivenb−y[ /(2ω)]2 = [1 π/(2ω)2]2.Theresult
plexderivativeofthedeflectionfunction; confirms the expectation thℵat the sufficien−tly large beam of light
wouldseethesystemasifitwereauniformscreenofmasswitha
f′′(z)=m,Xn∞= [z−2(m2+ni)ω]3 (4) rceoanlvleirngee,necqeu(aotriotnhe(6o)ptmicaaplsdeepacthh)sκeg=mπenσt =betπw/e(e2nω)t2w.oAalodnjagcethnet
−∞
≡−℘′(z|ω,iω)=−℘′(z;g2,0) ltehnesreesaroentionfitnhietewnhuomlebereraolfliinmea,gaensd. Hthouwseivteirs, uenasleilsysaκrgu=ed1,thaaltl
⋆
where butafinitenumberofimageshavenegativeparity.
ω 4 Γ8
g = 0 = 1/4 κ2. (5)
2 (cid:18)ω(cid:19) 16π4 ⋆ 2.1 lensesontriangularandhexagonalgrid
Wefindthatthisisrelatedtothedefinitionofthespecialfunction Analogous to the preceding case, we can also set up the lens
knownastheWeierstrassellipticfunction1(Abramowitz&Stegun equation for the equal point-mass lenses located at the lattice
points on an infinitely extended equilateral-triangular grid. The
60 -rotational symmetry of the system again implies that the to-
◦
1 Throughoutthispaper,theargumentsoftheWeierstrassfunctionsfollowedbyaverticalbar– – talshear on anygiven lenscausedby theremainingset of lenses
|
denotethehalf-periodswhereastheellipticinvariantsareindicatedbythosefollowedbyasemicolon also cancels out. This fact, combined with the general definition
(;).Fordetails,seethelistedreferencesorAppendix.Wheneveritisappropriate,thesearguments
maybesuppressed,providedthatthereislittledangerofconfusion. of℘-functionindicatesthatthelensequationcanagainbewritten
c 2007RAS,MNRAS000,1–19
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Lensingbyregularpoint-masslattices 3
Figure1.Contourplotsoftheequipotentiallinesofthelensingpotentialforthelatticelenses.TopLeft:squarelattice.TopRight:triangularlattice.Bottom
Left:hexagonallatticewithequation(14).BottomRight:hexagonallatticewithalensbeingatthecenter.
downusingtheWeierstrassfunctions.Ifthelensesareplacedover convergenceisκ = π/(2√3ω2)(andthemeansurfacedensityσ
⋆
thegridgivenbyλ = 2ω(m+neπi/3)wheremandnareintegers, ofthepointmassesonthetriangulargridisgivenbyσ 1 =2√3ω2
−
thecorrespondinglensequationisgivenby where2ωisthesidelengthoftheunittriangularcell).
Wenotethatthetheprecedingtwocasesofthelensarrange-
η=z f(z); f(z)=ζ(zω,eπi/3ω)=ζ(z;0,g ) (8)
− | 3 mentcorrespondtotwoofthethreepossibleregulartessellationsof
thetwo-dimensionalplane.Itmaybeofsomeinteresttoconsider
where
the regular lens placement corresponding to the remaining regu-
ω 6 √3Γ6 3 lar tessellation – the hexagonal or honeycomb tiling. It turns out
g = 2 = 1/3 κ3 (9) thattheregularhexagonal gridcaseiscloselyrelatedtotheequi-
an3dω(cid:18)ω (cid:19)1.530is8tπh3erealh⋆alf-periodsoftheequiharmoniccaseof lfaotremraslatr(i3a0n◦g-urolatratgedri)dl.a2rgBeyrtrreimanogvuilnagrgerviedryofthsiidrdeloefngththeolefn2s√th3aωt
℘-func2ti≈on(seeAppendix)whereasκ = π/(2√3ω2)isthecorre- fromthebasetriangulargridwithsidelengthof2ω,theremaining
⋆ lensesformahexagonalgrid(ortheverticesofhoneycombcells).
spondingopticaldepth.Likewise,thequasiperiodicityofζ(z;0,g ),
3
π
ζ z+2(m+neπi/3)ω;0,g3 =ζ(z;0,g3)+ m+ne−πi/3 (10)
√3ω
(cid:2) (cid:3) (cid:0) (cid:1)
wheremandnarearbitraryintegers,indicatesthatthevertexpoints
z,z+2ω,andz+2eπi/3ωoftheequilateraltriangleinthelensplane
mapontothesourceplanepoints:η=z ζ(z),η+iandη+ieπi/3
wherei = 2ω π/(√3ω)sothatthem−eaninversemagnification 2 Notethattheyareinfactso-calleddualtilingofeachother.Thatis,ifweconsideragridwith
− vertexpointsatthecentresofeachtriangularcelloftheequilateraltriangulargrid,thentheresulting
is again given by (1−κ⋆)2 where the optical depth or the mean latticeformsaregular-hexagonal(honeycomb)gridandalsoviceversa.
c 2007RAS,MNRAS000,1–19
(cid:13)
4 J. An
Figure2.Critical curves(dottedlines)andcausticnetworks (solidlines) Figure3.SameasFig.2exceptforω=1.2[κ⋆=π/(2ω)2 0.545].
forlensesonasquarelatticewithω=1.4[κ⋆=π/(2ω)2 0.401]. ≈
≈
for the lensing potential for some purposes such as the time de-
Consequently,thecorrespondinglensequationmaybewrittenas lay.Althoughthedirectinfinitesumoftheindividuallogarithmic
potential of thepoint-mass lensisdivergent everywhere, one can
η=z d(z;g );
− 3 get around this difficulty through the known antiderivative of the
d(z;g3)=ζ(z|ω,eπi/3ω)−ζ(z|√3eπi/6ω,√3iω) (11) Weierstrasszeta function. First, wenote therelation between the
g realpotentialandthecomplexdeflectionfunction(An2005;An&
=ζ(z;0,g3)−ζ(cid:16)z;0,−273(cid:17) tEhvearnesa2l0p0o6te);nttihaelcisomgipvleenxbleynψs.eHqueareti,otnheisogpievreantobry∂η=indzi−ca2t∂esz¯ψthief
whereg isrelatedtoωthroughequation (9)providedthat 2ωis z¯
3 ‘Wirtingerderivative’(e.g.,Schramm&Kayser1995)withrespect
stillthesidelengthofthebasetriangulargrid(andalsothatofthe
toz¯,thatis,ifψ=ψ(x,y)andz=x+yi,then
unithoneycombcell).However,sinceeverythirdoflenshasbeen
removedfromthebasetriangularcell,theopticaldepthisreduced ∂ψ∂x ∂ψ∂y 1∂ψ i ∂ψ
∂ψ= + = + (13)
toκ⋆ =π/(3√3ω2),andthus z¯ ∂x ∂z¯ ∂y ∂z¯ 2∂x 2 ∂y
33/2Γ6 3 becausex=(z+z¯)/2andy=i(z¯ z)/2.Forthecurrentscenario,we
g3 = 24π13/3 κ⋆3. (12) have f(z)=2∂z¯ψ=2∂zψ¯ =2∂zψ−(notethatψ=ψ¯ sinceψisreal).
Icnomadpdleitxiodni,ffde′r(ezn;tgi3a)tio=n,−ih.e(.z,;tgh3e)d(tehreivpartiivmeedwistyhmrebsopleicntdtiocazt)e.sTthhee ∂Mcoψomrep=olevx0e.ra,Tnfha(lezyn)t,iic∂sfa(uψcnoc+mtiψo¯pnl)eψx=ca(∂nz)aψlsyut+cich∂ftuψh¯nact=tfio(∂zn)ψa=n+dψ∂s′co(ψzt)h=e=re∂fze(ψzx)ci,saatsnndda
z¯ c z c c z c z c z c z¯ c
functionh(z;g )isstudiedinAppendixA2.1.Itisagainstraightfor- thus,2ψ=ψ +ψ¯ ,thatis,ψ(x,y)= [ψ (x+yi)]uptoanadditive
wardtoestabl3ishthatthemeaninversemagnificationis(1 κ )2. constant(Anc2005c).Here, [f]istheℜrealcpartofacomplex-valued
− ⋆ ℜ
Unliketheprecedingtwocases,thesystemdescribedbylensequa- function f.Consequently, wefindthat thereal potential up toan
tion (11) does not have the lens at the centre, but the centre of additiveconstantfortheregularlensinglatticeisgivenby
thesystemcorrespondstothelocationonthelensplanesuchthat
d′(0)=h(0)=0.However,wenotethatthisisdeliberatelychosen ln|σ(z;g2,0)| (cid:3)
feonrcmeoatfhtehmeahtoicnaelyscimomplbicgitryid,afnrdomdotehsensoqtuiamreployratnryiainngtruilnasricgrdiidff.eIrn- ψ(x,y)=llnn|σ(zσ;0(z,;g03,)g|3) 7△ (14)
proeansrpttihececuthlatoorn,ewaynecyonlmoetnbestghlroeidc1a2eti0xo◦pn-er,roiwetanhtciiocehnsazalecsrtyoumaslmhlyeeatirrmyfproolfimetshtehthesayrtsetamenmayinwleiinntghs wherez=x+(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)σy(iza;n0d,−σg(3z/;2g72),g(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)3)istheWeierstrasssigmafunction
(Abramowitz&Stegun1972),whichcanbedefinedastheanti-log-
pointmasses.Inotherwords,forallthreecasesofthepointmasses
derivativeoftheWeierstrasszetafunction,thatis,(d/dz)lnσ(z)=
ontheregulargrid,thetotalshearfromalltheadjacentlensesex-
ζ(z).Here,wehavealsousedthepropertyofthecomplexlogarithm
actlycancelsoutatanylensposition.
functionsuchthat [lng] = ln g wheregisanycomplex-valued
ℜ | |
function. Note that an additional linear term may be required to
beaddedintheexpressionforthepotentialifthecentreofthesys-
2.2 lensingpotential
temischosendifferentlyfromthoseofthelensequationsdiscussed
Whilemostofthelensingpropertiesofthelatticelenscanbestud- earlier.Fig.1showscontourplotsfortheequipotentiallinesforthe
iedusingthelensequations,itisstillusefultohaveanexpression potentialgiveninequations(14).
c 2007RAS,MNRAS000,1–19
(cid:13)
Lensingbyregularpoint-masslattices 5
Figure4.Causticnetworksforasquarelatticelenswithω=1(κ⋆ 0.785;topleft),ω=0.8(κ⋆ 1.227;topright),ω=0.77(κ⋆ 1.325;bottomleft),and
≈ ≈ ≈
ω=0.73(κ⋆ 1.474;bottomright).
≈
3 CRITICALCURVESANDCAUSTICS Hence, the topography of the critical curves may be studied
from the lines of constant values of ℘(z;1,0) (square grid) and
Asusual,theJacobiandeterminantofequation(6),(8)or(11) | |
℘(z;0,1) (triangular and hexagonal grid). In particular, the criti-
| |
1 ℘(z;g ,0)2 (cid:3) calcurvesfor thesquare (ortriangular) lenslatticewiththehalf-
−| 2 | periodωarebasically thecurves of constant ℘(z;1,0) = ω2/ω2
Jisth=e1in−ve|rfs′e(zm)|2ag=nifi11ca−−ti||o℘hn((zzo;;gf03;t)hg|2e3)i|n2div7△idualimageatthelimi(t15o)f tc[houerrv|o℘ets(hzeo;rf0h,c1aon)n|ds=,tatωnhto2s|/℘eω(f22zo];r0ret,hs1ce)a|hl=eodnωeby2y/c(ao√mfa3bcωtlo)artstocicfaelω|ea/drωeb0yfo(ou√rn3|ωdω/f/ωrωo2m2).aOtnhdne0
rotatedby 30 .Since℘-functionsarewell-definedellipticfunc-
thepointsource.Inaddition,thecriticalcurvescanbefoundfrom ◦
−
f (z) = 1 or equivalently solving f (z) = e 2iφ for z(φ) withφ tions, for all three cases the critical curves are an infinite set of
′ ′ −
| | − nonintersectingclosedcurvesexceptwhenωassumesaparticular
beingarealparameter(Witt1990;Wambsganssetal.1992;An&
valueatwhichthecurvesaregivenbyinfinitelyextendedboundary
Evans2006),andthecausticsfromtheirimagesunderthemapping
linesthatdividethelensplane.
given by the lens equation. Since f (z) is elliptic (hence, biperi-
′
odic),thereexistsaninfinitesetofcriticalcurvesandcausticsal-
thoughtheycanallberecoveredfromthediscretegridtranslation
3.1 squarelattice
ofa‘unit’curve.
Thehomogeneityrelationof℘-functionindicatesthat Fπo/r(2tωhe2)squ0a.4re57g]r,iedacchasoef,tihfeωcrit<ica(lωcu0/rv√e2s)is≈cen1t.r3e1d1at[tih.ee,cκe⋆ntr>e
ω 2 ω 0 ≈
℘(z;g ,0)= 0 ℘ 0 z;1,0 ; ofasquarecelldefinedbythefouradjacentlenses(i.e.,kω+ipω
2 (cid:18)ω(cid:19) (cid:16)ω (cid:17) (16) wherekand pareoddintegers).Anyimageintheregion‘within’
℘(z;0,g )= ω2 2℘ ω2 z;0,1 thecriticalcurveshaspositiveparityandviceversa.Ontheother
3 (cid:18)ω(cid:19) (cid:16)ω (cid:17) hand,ifω > 2−1/2ω0 [κ⋆ < π/(2ω20)],thecriticalcurvesarecen-
andfromequation(A2)that tredaround eachlens location. For thiscase, the images‘within’
thecriticalcurvesnowhavenegativeparity.Finally,ifω=2 1/2ω
− 0
h(z;g )= e−2πi/3 ω2 2 ℘ eπi/6ω2z;0,1 −2. (17) [κ⋆ =π/(2ω20)],thecriticalcurvesaregivenbytwoinfinitesetsof
3 3 (cid:18)ω(cid:19) " (cid:18) √3 ω (cid:19)# paralleldiagonallines,andthewholelensplaneisevenlydivided
c 2007RAS,MNRAS000,1–19
(cid:13)
6 J. An
Figure5.Criticalcurves(dottedlines)andcausticnetworks(solidlines)for Figure6.SameasFig.5exceptforω=1.15[κ⋆=π/(2√3ω2) 0.685].
lensesonaequilateraltriangularlatticewithω=1.25[κ⋆=π/(2√3ω2) ≈
≈
0.580].
curves2ω(κ 1)increasewhiletheirsizescontinuallydecrease.
⋆−
Consequently,thenetworkeventuallyreducesanarrayofseparate
intochecker-liketiledregionsaccordingtotheparityoftheimages tilesoftheunitcaustics,whichexhibitnooverlapbetweenneigh-
inthem. boring caustics. The critical value at which the sizes of the unit
Forω , 2−1/2ω0,thelensequationmapseachconnectedand causticscoincide withtheseparations between them isfound ap-
closed ‘unit’criticalcurve toacausticthatisalsoconnected and proximatelytobeω 0.748(κ 1.405).
closed. Fig. 2 shows the critical curves and the caustic network One physical in≈terpretatio⋆n≈of the caustics in gravitational
for ω = 1.4. Like the critical curves, the whole caustic network lensing is that they are boundaries between regions inthe source
is composed of the lattice arrangement of ‘unit’ caustic curves, plane that produce different number of images. In addition, it is
but the unit curve exhibits self-crossing in contrast to the critical alsoknownthatapairofimagesofoppositeparityappearsordis-
curves.Theshapeoftheunitcausticcurve,whichisroughlysim- appearswheneverthesourcecrossesthecaustics,andthatthenum-
ilartoavariant of (irregular)octagram (i.e., 8/3-star-polygon3), ber of positive parityimages, if the source liesoutside any caus-
{ }
isactuallygeneric for thecasethat ω > 2−1/2ω0. The‘centre’ of tics, is one or nil depending on the characteristics of the system.
eachunitcausticcurveformsasimilarsquaregridtothelenslat- Hence,themaximumnumberofpositiveparityimagescanbede-
ticebutthesidelengthofthegridonthesourceplaneisgivenby terminedfromexaminationofthecausticsnetwork.Forexample,
2ω(1−κ⋆).Asωgetslarger(orequivalentlyκ⋆ getssmaller),the the quasi-octagram caustics for ω > 2−1/2ω0 allow at most four
sizeof thecausticsshrinksandtheyasymptoticallyreduce tode- positiveparityimagesintheregionaroundthecentresofcaustics.
generatepoints.Forω < 2−1/2ω0,thecausticnetworkcanstillbe It is easy to convince oneself that the minimum number of posi-
understoodasthelatticearrangement of‘unit’causticcurves,the tiveparityimagesforthecorresponding scenarioisonewhenthe
shapeofwhichissimplydescribedasbeingadiagonallystretched source liesoutside any caustic. As for the case ω 6 2 1/2ω , the
− 0
square.Again,theseparationbetweenthecentresoftheunitcaus- examinationofthecausticnetworkleadsustoconcludethat,while
ticsaregivenby2ω|1−κ⋆|.However,thisissmallerthanthe‘size’ themaximumnumberofthepositiveimagesapproachesinfinityat
loafpepaicnhguanmitocnagusthtiecnifeκig⋆h∼bo1rinsogtchaautstthicesn,ewtwhoicrhklceaandsexthoibaitroavtheerr- 0κ⋆.80=31.(ωi.e..,ω1.0=39π.1I/n2/o2th≈er0w.8o8r6d)s,,ittheisngurmeabteerrothfapnosfoituivreopnalyritiyf
complexnetwork(albeitregularthankstothesymmetryofthesys- imagesisboundedbyfourprovidedthatω&1.039(κ .0.727)or
⋆
tem).Thecriticalcurvesandcausticnetworkforω=1.2areshown thatω.0.803(κ &1.217).Thesourcepositionswithnopositive
⋆
inFig.3.Thetransitionofthecausticnetworkoverω = 2−1/2ω0 parityimagestarttoexistifω.0.776orequivalentlyκ⋆ &1.039.
canbeunderstoodasthecontactbetween(theverticesof)adjacent Finally, if ω . 0.748 (κ & 1.405), there is no overlap between
⋆
quasi-octagram caustics leading to their connection. The further neighboringcaustics,andthusthenumberofpositiveparityimages
evolution of thecaustic topography for ω < 2−1/2ω0 ispresented isoneifthesourceliesinthecaustic,andnilifotherwise.
inFig.4.Sincetheseparationbetweentheunitcausticsreducesto
nilasκ 1, thenetwork isthedensest around κ 1.Onthe
⋆ → ⋆ ≈
otherhand,asκ⋆increasespastunity,theseparationsbetweenunit 3.2 equilateraltriangularlattice
Thelenses onanequilateral triangular latticeproduce thecritical
curvesandthecausticnetworkinabasicallyconsistentpatternas
3 A{p/q}-star-polygon,withp,qpositiveintegers,isafigureformedbyconnectingwithstraight thesquarelatticelensalthoughmanydetailsarequitedifferent.For
lineseveryq-thpointoutofregularlyspaced ppointslyingonacircumference.Seehttp:
//mathworld.wolfram.com/StarPolygon.html. triangularlattices,theunitcriticalcurvesforω<(ω2/√32)≈1.214
c 2007RAS,MNRAS000,1–19
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Lensingbyregularpoint-masslattices 7
Figure7.Criticalcurves(dottedlines)andcausticnetworks(solidlines)for Figure8.SameasFig.7exceptforω=1.3[κ⋆=π/(3√3ω2) 0.358].
lensesonahoneycomblatticewithω=1.45[κ⋆=π/(3√3ω2) 0.286]. ≈
≈
3.3 regularhexagonallattice
[κ >π/(21/331/2ω2) 0.615]arelocatedineachoftheequilateral Inqualitativeterms,thecriticalcurvesofthehoneycomblatticeare
⋆ 2 ≈
triangularcelldefinedbythreeadjacentlenses.Ontheotherhand, basicallythe‘dual’ofthatofthetriangularcase.Thecriticalcurves
those for ω > 2 1/3ω [κ > π/(21/331/2ω2)] are centred at each forω < (22/33 1/2ω ) 1.402[κ > π/(31/224/3ω2) 0.308]are
− 2 ⋆ 2 − 2 ≈ ⋆ 2 ≈
lensposition.Notethat,incontrasttothesquarelattice,thenumber analogoustothoseforthetriangularlatticewithω > 2 1/3ω and
− 2
ofpositionswithzeroshearistwicethatofthepoles(i.e.,thelens locatedwithineachhexagonalcellcentredatthepositionsofnull
positions)sothatthereistwo-to-onecorrespondence betweenthe shear. On the other hand, if ω > 22/33 1/2ω , the critical curves
− 2
unitcriticalcurvesoftheformerandthatofthelatter.Atthecrit- arelikethoseforthetriangularlatticewithω < 2 1/3ω andcen-
− 2
icalvaluethatω = 2 1/3ω [κ = π/(21/331/2ω2)],thelensplane tredaroundeachlenspositions.Thedualcharacteristicfurtherin-
− 2 ⋆ 2
isdividedintotwoclassesbythecriticalcurves:triangularregions dicates that there are twice more poles than the locations of null
aroundthenullshearpositionswithinwhichtheimageshaveposi- shear(whichareinfactfourth-orderzeros),andthusthereisone-
tiveparityandhexagonalregionsaroundthelensescorresponding to-two correspondence between unit critical curves of the former
totheimagesofnegativeparity. tothelatter(formally,the‘windingnumber’oftheformeristwice
For ω > 2 1/3ω , the shape of the unit caustics may be de- thatofthelatter).
− 2
scribedassnowflake-likeoranirregularquasi-dodecagram( 12/5- On the other hand, the caustics of a honeycomb lattice are
{ }
star-polygon). Each unit caustic is arranged in the same equilat- qualitativelydistinctfromthoseofatriangularlattice.Fig.7shows
eral triangular grid with side length of 2ω(1 κ ) to form the anexampleforasparsehoneycomb latticewithω = 1.45,which
− ⋆
wholecausticnetwork.Fig.5showsanexampleofthis,thecaus- is in fact archetypal for ω > 22/33 1/2ω . The unit caustics are
− 2
ticnetworkforω = 1.25. Themaximumnumber ofpositivepar- givenbyasix-sidedclosedself-intersectingcurve,whichmaybe
ity images associated with these unit caustics is found to be six. labeledabi-triangle,andtheyarearrangedinasimilarhexagonal
If ω < 2 1/3ω , the unit caustics are given by stretched triangu- pattern as the latticeof thelens with side length 2ω(1 κ ). We
− 2 − ⋆
larcells,andthewholenetworkisgivenbytilingthesourceplane alsonotethatthemaximumnumber of positiveparityimagesas-
with these cells, which may overlap with one another. Here, the sociatedwiththesecausticsisthree.Foradensehoneycomb lens
centres of each cell actually form a hexagonal (honeycomb) grid lattice(ω < 22/33 1/2ω ), theunit causticsare found tobe inthe
− 2
with side length of 2ω 1 κ /√3, which is in fact the mapped shapeofastretchedhexagon(anexampleofwhichforω = 1.3is
| − ⋆|
image of the dual grid to the lens lattice in the lens plane, and giveninFig.8)andarrangedinatriangulargridwithsidelengthof
the adjacent cells alternate their orientation with their symmetry 2√3ω 1 κ .Theyalsoexhibittrendsofvaryingoverlapsimilar
| − ⋆|
maintained.Anexampleofthesecausticsnetworksispresentedin tothesquareortriangularlattice,asκ getslargerpasttheunity.We
⋆
Fig. 6 for ω = 1.15. Analogous tothe case of the square lattice, findthatthecriticalvaluefornooverlappingcausticsforthehon-
denseoverlappingofunitcausticsleadstoverycomplexnetwork eycomblatticeisapproximatelygivenbyω 0.622(κ 1.565).
≈ ⋆≈
around κ 1. It is found that that the critical value that corre- Despitetheirvariety,wealsonoticesomecommoncharacter-
⋆ ∼
spondstonooverlapbetweenadjacentcausticsisaboutω 0.791 isticsofthecausticnetworksofthreekindsofregularlensinglat-
≈
(κ 1.449).Thetriangularlenslatticewithsidelengthlessthan tices.Forexample,withanopticaldepththatisgreaterthanacer-
⋆ ≈
thisvalue(henceadenserlattice)producestriangularunitcaustics taincriticalvalue,theunitcurvesaremoreorlessinasimilarshape
arrangedonahoneycombgridwithnooverlap,andthenumberof asthelensinglatticeexceptforthefactthattheyaredistortedina
positiveparityimagesiseitheroneornildependingontheposition waystretchingtheveticesradiallyoutward.Theydenselyoverlap
ofthesourcerelativetothecaustics. withoneanotheraroundκ 1.However,asκ getslargerpastthe
⋆∼ ⋆
c 2007RAS,MNRAS000,1–19
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8 J. An
Figure9.Top:meannumberofpositiveparityimageforlensesonasquare
lattice(solidline),atriangularlattice(short-dashedline),ahexagonallattice
(long-dashedline),andfortherandomstarfield(dottedline).Alsoplotted
indot-dashedlinesisthemeantotalmagnificationhµi=(1−κ⋆)−2.Bottom:
meannumbernormalizedbythemeantotalmagnification.Thelinetypes
arethesame.
Figure10.Distributionoflensingshear,γ.Thelinetypesarethesameas
unity,theirsizeshrinkwhereastheseparationsamongthemgrow, Fig.9.
and so the networks eventually reduce to a simple array of tiles.
Likewise,towardstheloweropticaldepthsbelowthecriticalvalue,
thenetworksarecomprisedofstar-shapedunitcurveswhoseexact itive images reside and thus there are no positive images formed
characteristicsarerelatedtomeetingpatternofthelensinglattice ‘outside’ of any caustics. On the other hand, for ‘sparse’ lattices
atthecommonvertexpoints. thatproducethenetworkcomposedofagridsetofself-intersecting
‘star-shaped’curves,theself-crossingoftheunitstructuresrequires
someconsiderationonthemeaningof‘area’thatproperlyaccounts
fortheimagemultiplicity.Wearguethatthesolutionisrathersim-
4 MEANNUMBEROFPOSITIVEPARITYIMAGES
ple.Oncetheareacalculationisreducedtothelineintegralalong
The area in the caustic accounting for its multiplicitycan be un- itsboundary(seee.g.,An2005;An&Evans2006)usingthefun-
derstoodasthecoveringfractionofthesourceplanebytheregion damentaltheoremofmultivariatecalculus4,thecorrespondingline
with‘extraimage’pairsonceitisproperlynormalized.Ingeneral, integralalongthecausticactuallyresultsintheareacountedwith
however,thenormalizationisill-definedforlocalizedlenses.This multiplicity. That is to say, they can be considered as the ‘area-
difficultyisamelioratedwiththeintroductionofthelatticeofpoint excess’ or the ‘over-covering factor’ of the source positions that
massessincethewholeplaneisnowevenlydividedintoidentical producepositiveparityimages.Unlikethepreviouscase,thenor-
cells.Withtheregularlensinglattice,theareaundertheunitcaustic malizationcellinthelensplaneistheunitcellof thedual ofthe
canbeproperlynormalizedwithrespecttotheareaofthemapped lensinglattice,i.e.,thecelldefinedbytheadjacentnullshearloca-
image (on the source plane) of the unit cell comprising the lens- tions(whichisinfactthecentreofeachcellofthelensinglattice).
inglattice.Theresult,ifthemultiplicityisproperlyaccountedfor, The corresponding critical curves lie completely within this unit
isactuallythemeannumberofextraimagepairs,whichhasbeen normalizationcellandfurthermoreenclosetheregionsoftheim-
calculated for the case of the random star field using techniques ageplanewherenegativeparityimagesreside.Subsequently,there
fromtheprobabilitytheory(Wambsganssetal.1992;Granotetal. isatleastonepositiveparityimageforanygivensourceposition
2003). – that is, the source ‘outside’ caustics forms one positive parity
For‘dense’latticesthatproducenonintersecting(butpossibly image. Asa result, the mean number of extra image pairs isless
overlapping) unit caustics, the relevant area under the unit caus- thanthenumberofpositiveparityimagesbyone.Wenotethatthe
ticsisstraightforwardtocalculatesince,despiteoverlapping,they meannumberofpositiveparityimagesactuallyvariescontinuously
aresimplyreduced toaset of closedloops. Then,theareaunder overthecriticalvaluedividingthedenselatticefromthesparseone
one such loop normalized to the area of a unit cell in the source whilethemeannumberofextraimagepairsjumpsbyone.Infact,
plane, Aω(1−κ⋆)2 where Aω is the area under the unit cell de-
finedbytheadjacentlenspositionsinthelensplane,isthemean
numberbothofextraimagepairsandofpositiveparityimagesfor 4 ThetheoremisusuallyknownasGreen’stheoreminelementarymultivariatecalculuswhenrefer-
a given source position because the corresponding critical curves ringtotherelationbetweentwo-dimensionalintegraloveradomainintwo-dimensionalspaceand
one-dimensionalintegralalongtheboundaryofthedomain.Thisisalsoaspecialcaseofso-called
intheimageplanecompletely enclosetheregionwhereanypos- (generalized)Stoke’stheorem,thenameunderwhichthetheoremisusuallyknownas.
c 2007RAS,MNRAS000,1–19
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Lensingbyregularpoint-masslattices 9
designatingacertainimagepairasanextrainvolvessomedegrees valueofκ insteadofκ canbechosenasthenormalizationandthe
⋆ 0
ofarbitrarinesswhereascountingthenumberofpositiveparityim- resultingsimilarityrelationisvalidforallthreeregularlattices.
agesiswell-defined.Forthesereasons,henceforth,weshalldiscuss The resulting distributions for all three regular lattices are
themeannumberofpositiveparityimagesexclusively. showninFig.10togetherwiththesamedistributionfortherandom
Fig.9showstheresultingmeannumberofpositiveparityim- starfield(Nityananda&Ostriker1984; Schneider 1987; Wambs-
agesasafunctionofκ .Comparisontothecaseoftherandomstar ganssetal.1992),whichisgivenby
⋆
field(withzero external shear) (Wambsganss et al. 1992) reveals
κ γ 1 γ
some significant differences between the regular lensing lattices Pran(γ|κ⋆)= (κ2 +⋆γ2)3/2 = κ2 1+(γ/κ )2 3/2. (22)
andtherandomfield,particularlyinthebehaviourtowardκ . ⋆ ⋆ ⋆
⋆ →∞
Whilethemeannumberfallsoffasκ⋆−4fortherandomstarfieldand Unliketherandomfieldcase,w(cid:2)efindthatth(cid:3)ereisnonzeroproba-
triangularlensinglattices(albeitthenormalizationfortherandom bilitythatthelineofsightexperiencesnonetshearforthesquare
fieldbeinglargerbyafactorof 4),itsbehavioursforthelenseson latticelens.Bycontrast,thesamedistributionlinearlytendstozero
∼
squarelatticesandhoneycomblatticesarecharacterizedbyslower asγ 0forthetriangularlattice(andalsotherandomfield)andit
fsaolml-oewffshagtivceonunbtyer∼intuκ⋆−it3ivaen.dT∼hatκ⋆−i2s.,5,inretshpeecrtaivnedloym. Asttafirrsfite,ltdh,isthies rdeivlaetre→gdestoatsh∼efγa−c1t/t2hfaotrtthheezheoronseyocfo℘m(zb;lgat,ti0c)e,.℘T(hz;e0se,gbe)haanvdiohu(rzs;gare)
2 3 3
distance stotheclosestneighboring lensisdistributedaccording aresecond,first,andfourth-order,respectively;thatistosay,their
to Taylor-seriesexpansionsattheirzeroesaregivenby
P(s)ds=2πσse−πσs2ds (18) ℘(z;g2,0)≃−g42 (z−z0)2+O(|z−z0|6)
Twhheermeeσan=anκd⋆/thπeivsatrhiaenmceeaarnesguirvfeancebynuωm=be1r/d(2en√sσity)aonfdth[(e4/sπta)rs. ℘(z;0,g3)≃ig13/2(z−z0)+O(|z−z0|4) (23)
g
1]ω2 =[1−(π/4)]κ⋆−1.Inotherwords,asthedensityofstellarfiel−d h(z;g3)≃ 273 z4+O(z10)
increases, the random fluctuations around the mean decrease and
wherez isazeroofthecorrespondingfunction(notethatzeroofh
onewouldexpect thattherandomstellarfieldmayapproach toa 0
regulTarhfieealcdt.ualresult canbeunderstood usingthedistributionof iisndaitcza0te=s0th).aFtoArω|γ|0≪γP1(,γth′)eddγi′sc≃ussπio|zn−inzt0h|2ewprheecreedizn0gipsatrhaegrnauplhl
the(internal)lensingshearγduetostarsexperiencedbythegiven shearpositionandRztracesthelocationsatwhichtheshearisgiven
byγ.Bynoticingtheshearisgivenby ℘(z;g ,0), ℘(z;0,g ) and
lineof sight towards alensed image. For afieldof point masses, | 2 | | 3 |
h(z;g ) forsquare,triangular,andhoneycomblattices,wecanthen
oncethisdistributionisknown,themeannumberofpositiveparity | 3 |
findtheleading-orderbehavioursforthesheardistributionsasγ
imagesisrecoveredthrough(Wambsganssetal.1992)
→
0fromtheaboveTaylorexpansions5;
1
hN+i=hµiZ0 (1−γ2)P(γ)dγ. (19) P(cid:3)(γ|κ⋆)≃ κ1⋆ 2Γ681π/44 +O(cid:20)(cid:16)κγ⋆(cid:17)2(cid:21)≈ 0.2κ0⋆88
Withlensesonaregularlattice,wehaveshownthattheshearatthe
1 211π9 γ γ 4 0.2326
gtiiovne,nainmdahgeenpclea,ntehelodciasttiroinbuistiogniveonnbthyethwehWoleeieimrstargaesspelallnipetricedfuucnecs- P△(γ|κ⋆)≃ κ⋆ 33/2Γ118/3 κ⋆ +O(cid:20)(cid:16)κ⋆(cid:17) (cid:21)≈ κ⋆2 γ (24)
to that on the unit cell in the image plane. Then, the cumulative 1 24π9/2 κ 1/2 γ 0.1705
dwihsterriebuthtieonshoefasrhiesasrmisalfloeurntdhabnytthheegfirvaecntiovnaloufe,araenadincotnhseeuqnuietnctelyl,l P7(γ|κ⋆)≃ κ⋆ 33/4Γ91/3 γ⋆! +O(cid:16)κ⋆(cid:17)≈ κ⋆1/2γ1/2,
itsderivativegivesthedifferentialdistributionoftheshear.Further- whichagreewiththenumericalresultsshowninFig.10(notethat
more,thehomogeneityrelationof℘-functionimpliesthesimilarity κ = π for Fig. 10). In other words, the higher-than-linear-order
⋆
relationbetweenthedistributioncorrespondingtodifferentω.For (i.e.,degenerate) zerosoftheshear leadtononzerofiniteoreven
example,withasquarelensinglattice,wehavethat divergent probability for the zero net shear. Physically speaking,
accidental canceling of the net shear in the random star field is
γ 1
Z0 P(γ′|κ⋆)dγ′= AωAωh|℘(z;g2,0)|6γi mmeunchtolfesthselilkeenlsyest,ofoorcwcuhricthhatnheirnemthaeyhbigehsliygnsiyfimcamnettcrhicanacrerasntghea-t
1 ω ω2 π γ theimageislocatedatanexactlybalancedpositionwherethenet
= Aω0Aω0"(cid:12)(cid:12)℘(cid:16)ω0 z;1,0(cid:17)(cid:12)(cid:12)6 ω20 γ= 4ω20κ⋆# shearisnull.Thisisbasicallythereasonforslowerfall-offsofthe
κ0γ/κ⋆ (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) meannumber ofpositiveparityimagesindensesquareorhoney-
=Z0 P(γ′|κ0)dγ′ (20) cliokmebPla(tγti|κc⋆e)s.≃FroSmκ⋆−e1(qγu/aκt⋆io)nα(a2s1γ),/iκf⋆th→esh0ewarhdeirsetrSibuistioancboenhstaavnets,
α=(2/n) 1> 1istheasymptoticpowerindexforthesheardis-
− −
P(γ|κ⋆)= A1ω0 ddγ′Aω0(cid:20)(cid:12)(cid:12)(cid:12)℘(cid:16)ωω0 z;1,0(cid:17)(cid:12)(cid:12)(cid:12)6 κκ⋆0 γ′(cid:21)(cid:12)(cid:12)(cid:12)(cid:12)γ′=γ t(r1i9b)utaisonκ⋆an→dn∞isretdhuecoersdteorofthezerosoftheshear,thenequation
= A1ω0 ddγgddgAω(cid:12)0(cid:20)(cid:12)(cid:12)℘(cid:16)ωω0 z;1,0(cid:12)(cid:17)(cid:12)(cid:12)6g(cid:21)(cid:12)(cid:12)(cid:12)g=(cid:12)(cid:12)κ0γ/κ⋆ (21) hN+i= (1−1κ⋆)2 Z01dγ(1−γ2)P(γ|κ⋆) (25)
κ κ (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) S 1
= κ⋆0 P(cid:18)κ⋆0 γ κ0(cid:19) (cid:12) ≃ (κ⋆−1)2κ⋆α+1 Z0 dγ(1−γ2)γα.
where κ0 = π/(2ω0)2 and Aω[|℘(z;g2,0)| 6 γ] is thearea of the
unitcellwhere|℘(z;g2,0)|6γforasquarelensinglatticewithside 5 Itisactuallyeasiertoworkoutfromtheseriesfortheinversefunctions,whichareinfacthyper-
lengthofω(seeeq.[5]forg2asafunctionofωandκ⋆).Infact,any geometricfunction(seealsoAppendix).
c 2007RAS,MNRAS000,1–19
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10 J. An
Figure11.Locationsoflensedimageswithasquarelensinglattice withω = 1.35(κ⋆ 0.431).Thelocations ofthesourcewithrespecttothecaustic
≈
networksareindicatedinthetoprightpanel.Intheremainingthreepanels,thelocationsoftheindividuallensedimagesaremarkedbyopencircles(positive
parityimages)orcloseddots(negativeparityimages).Alsodrawnindottedlinesarethecriticalcurvesandthelocationsofthelensesaremarkedbycrosses.
ShNin+cie∼theκ⋆−la(αs+t3i)natesgκra⋆li→sa∞fin,iwtehcicohnsitsanctofnosrisαten>t−w1it,hwtehefinrdestuhlatst κb⋆e/hγav2i[oiu.er.,oPf(tth)e≃sht−e2arasdits→trib∞ut]i.oSnuapspγo/seκ⋆th→atp∞owtoerb-seePries(γe|xκ⋆p)an≃-
foundearlier(Fig.9). sionofP(t)att=∞isgivenbyP(t)≃t−2(1+ ∞n=1cnt−pn)(andit
Infact,thesheardistributionsforregularlensinglatticesare isuniformlyconvergentinanintervalcontainingPt= ).Then,for
quitedistinctfromthatoftherandomfield,anditissomewhatre- κ 0,wefindthat(assumingp,1/nforanypositi∞veintegern);
⋆ →
markablethatthebehaviours of N aregenerally similarfor all
+
csiacsaelslyatalneainsttefgorrastimonal(loκr⋆c.oTnhviosliusthiroenla)iteodftthoethsheefaarctdtihstartibhuNt+ioini,sabnad- Z01/κ⋆dtP(t)≃1−κ⋆−Xn∞=1 cpnnκ⋆p+n+11
therefore any ‘roughness’ in the shear distribution is ‘smoothed’
outin N+ .Inparticular,wenotethatthesimilarityrelationofthe 1/κ⋆dtt2 (t) 1 P ∞ cnκ⋆pn−1 (27)
sheardhistriibutionisessentiallytheresultofthescaleinvarianceand Z0 P ≃ κ⋆ − 0−Xn=1 pn−1
sisoashfuonucldtiobneogfenγe/rκic.,aTnhdenc,oPnse(qγu|κe⋆n)tl=yκw⋆−e1Pha(γv/eκ⋆)whereP(γ/κ⋆) hN+i 1 2κ +P κ2 + ∞ 2cnκ⋆pn+1
⋆ µ ≃ − ⋆ 0 ⋆ (pn 1)(pn+1)
h i Xn=1 −
1 1/κ⋆
hN+i= (1−κ⋆)2 Z0 dt(1−κ⋆2t2)P(t). (26) w(thhaetreis,Pf0arisfraomfintite=con)stbaenhtatvhiaoturwoefakly(t)d,eapnedndthsuosnNthe/glµobal
+
∞ P h i h i ≃
Furthermore, since theshear for thelinesof sight that passsuffi- 1−2κ⋆+O(κ⋆2)andhN+i=1+O(κ⋆2),regardlessofthehigher-order
cientlyclosetoanyofthelensesissimplygivenbyγ d 2where behaviourof (t),providedthat p > 1.Infact,wehave (t) =
≃ − P Pran
d istheseparationbetweenthegivenlineof sightandthattothe t(1+t2) 3/2 = t 2(1+t 2) 3/2 sothat p = 2for (t),and also
− − − − Pran
lens, it isstraightforward to establish that the generic asymptotic thepower-seriesexpansionsof℘-functionsattheirpolesandzeros
c 2007RAS,MNRAS000,1–19
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