Gravitational Lensing Statistics
PHILLIPHELBIG
Hamburger Sternwarte 1Gojenbergsweg112
D-21029 Hamburg1 Germany
INTRODUCTION
Lensing statistics gives a metho d of testing the cosmological constant at
in-termediate redshifts|at low and high redshifts the p ossibilities for measuring
0
are limited and even at intermediate redshifts
0
often cancels out in observable
relations|usingwell-understoo dlensingtheoryandstandardastrophysical
assump-tions.
Ithasrecentlyb eensuggestedbymanyauthors(see,forexample,Fukugitaetal. 1
and references therein) that gravitational lensing statistics canprovidea means of
distinguishingb etweendi erentcosmologicalmodels,moste ectivelyconcerningthe
value of the cosmological constant. Ko chanek 2
has suggested a metho d based not
onthetotalnumb eroflenssystemsbutratherontheredshiftdistributionofknown
lens systems characterised by observables such as redshift and image separation.
Lo oking at a few di erent models, he concludes that at, -dominatedmodels are
ve totentimes lessprobablethanmore `standard' models. The advantage ofthis
metho d is that it is not plagued by normalisation dicultiesas are most schemes
involving thetotalnumb er oflenses.
SinceKo chanek wasapparentlyabletogetsomeinterestingresultsusing
statis-tics based on only four gravitational lens systems, I wanted to exlpore this more
thoroughlybylo oking atnotjust a few butall models characterised by
0
and
0
aswell asvaryingdegrees ofhomogeneity. Ialso lo oked atselection e ectsand did
somesimulations togeta handleon what theresultsmean.
THEORY
I make the `standard assumptions' that the Universe can b e described by the
Rob ertson-Walker metric and that lens galaxies can b e modelled as non-evolving
singularisothermalspheres(SIS).Thisleadsanequationfortherelativedi erential
opticaldepth 3 d dz = ( 1+z d ) 2 a a3 2 a a3 D s D 2 (1+ ) 2
D 2 d 1 p Q( z d ) exp 0 a a3 D s D ds 2 (1) wherea3:=4 0 v 3 c 1
( v3:=v of an L3galaxy ), istheFab er-Jackson/Tully-Fisher
exp onent, the Schechter exp onent, D
d
theangular sizedistance b etween the
ob-serverand thelens and
Q( z d ) :=(1+z d ) 2 ( 0 z d +10 0 )+ 0 : (2)
The optical depthdep endson the cosmologicalmodelthrough Q( z
d
) aswell as
throughtheangularsizedistances,b ecauseofthefactthatD
ij =D ij ( z i ;z j ; 0 ; 0 ;).
The in uence of , which gives the fraction of homogeneously distributed, as
op-p osedtocompact,matterisfeltonlyinthecalculationoftheangularsizedistances,
whereas thecosmological modelin the narrowersensemakes its in uence felt here
aswellasthrough Q( z
d
). The angularsizedistancescanb ecalculatedforarbitrary
cosmologicalmodelsbythepro ceduregiven inKayser&Helbig. 4
CALCULATIONS
The following gravitational lens systems meet my selection criteria: 0142-100
(UM673),0218+357,1115+080(TripleQuasar),1131+0456,1654+1346and3C324.
(Formore details on these systems seeRefsdal&Surdej. 5
)
Iconsidered thefollowingrangesof valuesforthe cosmologicalparameters:
010 < 0 < +10 0 < 0 < 10
In order to measure the relative probability of a given cosmological model, I
de nedthequantityf asfollows: p
0<f := R z l 0 d R zs 0 d <1 ; (3) wherez l
istheobserved lens redshiftfora particular system. ( z
d
is usedtodenote
the variable corresp onding to lens redshift as opp osed to the measured value for
a particular lens.) The distribution of the di erent f values (one for each lens
systeminthesample)inbequally-sizedbins intheinterval ]0,1[ givestherelative
probability pof agiven cosmological model, with
p= b Y i=1 1 n i ! (4) where n i
is the numb er of systems in the i -th bin. (This de nition allows only a
few discrete values, of course.) The variable b is a free parameter; since the most
My resultsarein Fig.1, plot b. (This isfor =0 : 5;the resultsdo notdep end
strongly on . 3
See Kayser & Helbig 4
for a discussion of this parameter.) One
can see basically that areas of equal probability o ccur in some fashion which is
not stochastic. Although there are only a few discreet values for the probability
asde ned in Eq. (4), nevertheless one sees a degeneracy|there is a widerange of
cosmological models for a given probability. Plot c shows the result of neglecting
theobservationalbias,e.g. assumingthatthelens couldhaveits redshiftmeasured
whateverthisredshift were. Asacomparisonwithplotbshows,this leadstoabias
againstmodels withahighmedianexp ectedlens redshift|thosenearthede Sitter
model.
Forcomparison,Ihavealsotestedthemetho donthesystemsusedbyKo chanek, 2
using m
lim
=1 und =1,b oth of which he implicitlyassumes. (Of course, when
oneconsiders nite values form
lim
,one cannot includesystems with lens redshifts
whichhaveb eendeterminedbymeansotherthanmeasuredemissionredshifts,such
as absorption lines (which assumes that the lens is also the absorber).) The
re-sultsareinplot d wheretherelativeprobabilities are0, 1
6 ,
1
2
and 1and comparing
the various models examined by Ko chanek con rm his conclusions. For instance,
the relative probabilities of the Einstein-de Sitter and de Sitter model are 1 and
1
6
,con rminghisresult that at, -dominatedmodels are 5{10timesless probable
thanstandardones. (However,takingm
lim
intoaccount and/orusing only directly
measured lens redshifts would pro duce quite di erent results, as discussed ab ove.)
Thisplotarti ciallyindicatesalowprobability formodelsnearthedeSittermodel
forthesamereasons asthose discussedin connectionwith plotc.
NUMERICAL SIMULATIONS
Forthenumericalsimulations,theobservables 00
(theradiusoftheEinsteinring
or half the image separation corresp onding to the diameter of the Einsteinring),
z
s
and galaxy typ e were chosen randomly from an interval roughly corresp onding
to the observed range of values in order to pro duce synthetic data comparable to
real observations. For a given cosmological model, the corresp onding lens redshift
z
l
foreachsystemwascalculatedfromtheobservablesanda randomlygeneratedf
through (numerical) inversion of Eq. (3). This catalog wasthen used todetermine
arelative probabilityforeachofthep ointsinthe
0
-0
plane inthesamemanner
asforthereal systems.
The Kolmogorov-Smirnov(K-S) test is of course a well understo o d metho d for
testing if two distributions are statistically signi cantly di erent. (See, e.g., Press
et.al. 6
fora generaldiscussionand de nition oftheK-S probability.) However,this
test canonlyb e usedfordistributions withmore than20 datap oints. Therefore
I plot inFig. 1 in plots b, c, and d the probability given by Eq. (4) and in plot e
theK-S probability.
I have done simulations for a variety of world models and also for numb ers of
systemsb etween20 and 50. Intheinterest of brevity, Ipresent only oneplot. Plot
z
a
0.6 1.2 1.8 2.4
z
m
0.6 1.2 1.8 2.4
20.0
22.0
24.0
26.0
28.0
l
b
W
-8. -4.
0.
4.
8.
2.
4.
6.
8.
l
c
W
-8. -4.
0.
4.
8.
2.
4.
6.
8.
l
d
W
-8. -4.
0.
4.
8.
2.
4.
6.
8.
l
e
W
-2.0 -1.0 0.0 1.0
0.5
1.0
1.5
2.0
Figure 1: a. Therelative di erential optical depth (thin curve) and the calculated lens
brightnessm(thickcurve) asfunctionsofz
d
. TheworldmodelisthedeSittermodel(
0 =
1 : 0,
0
=0 : 0)andtheobservablesarethoseforthegravitationallenssystem01420100.The
ordinategivesthemagnitudeinJohnsonR . Forrealisticlimitingsp ectroscopicmagnitudes
(24 m
)itisclearthatonecannotsampletheprobabilit ydistributionwithoutastrongbias.
b. Relativeprobabilit yforthesystemsmentionedinthetext,witharealisticvalueform
lim .
Here andin theother halftoneplots,the relativeprobabilit yincreases linearlyfrom white
to black. c. The same as b., but neglecting m
lim
, which, as in plot d.,mak esthemodels
near thede Sittermodelapp earmore probablethantheyare. d. Relativeprobabilit yfor
thesystems used byKo chanek. e. Resultsbasedon a catalogue of50 sim ulatedsystems.
(Notethe di erent scale on the axes.) The cosmological model used to generate the lens
redshiftsisthehomogeneousEinstein-de Sittermodel(=1 : 0,
0
=0 : 0,
0 =1 : 0).
probability|the white area has p=0 due tothe factthat at least onelens would
b e fainter than m
lim
in these world models, as discussed in Helbig & Kayser 3
|
I conclude that, although one can qualitatively understand the physics which at
least in part is resp onsible for the results presented in Fig. 1, the actual relative
probabilities are more indicative of intrinsic scatter in the redshifts of the lenses
thana hint ofthecorrect cosmologicalmodel.
SUMMARY AND CONCLUSIONS
Forknown gravitational lens systems theredshift distribution of thelenses was
compared with theoretical exp ectations for 10 4
Friedmann-Lema^tre cosmological
models,whichmorethancovertherangeofp ossiblecases. Thecomparisonwasused
forassigningarelative probabilitytoeachof themodels. However,mysimulations
indicate that a reasonable numb er of observed systems cannot deliver interesting
constraints on thecosmological parameters using this metho d. Therefore, it seems
that lensing statistics can tell us something ab out the cosmological model only if
onemakes useof all information, which meanscoming togrips withnormalisation
diculties.
REFERENCES
1.Fukugita,M.,K.Futamase,M.Kasai&E.L.Turner. 1992. ApJ393;1.
2.Ko chanek,C.S.1992. ApJ384;1.
3.Helbig,P.&R.Kayser. 1995. A&Asubmitted.
4.Kayser, R.&P.Helbig. 1995. A&Asubmitted.
5.Refsdal,S. &J.Surdej. 1994. Rep.Prog.Phys. 56;117.
