Your thesaurus co desare:
12 (12.07.1; 12.03.4; 12.03.3)
AND
ASTROPHYSICS
1.9.1995
Cosmological parameters and the redshift distribution of
gravitational lenses
PhillipHelbig ?
and RainerKayser ??
HamburgerSternwarte,Gojenb ergsweg112,D-21029Hamburg-Bergedorf,Germany
Received16January;accepted15July,1995
Abstract. Forknowngravitationallenssystemsthe
red-shiftdistributionof thelensesiscomparedwith
theoreti-calexp ectationsfor10 4
Friedmann-Lema^trecosmological
models,whichmorethancovertherangeofp ossiblecases.
Thecomparison isused forassigning a relative
probabil-itytoeachofthemodels.Theentirepro cedureisrep eated
fordi erentvalues oftheinhomogeneit yparameterand
the limiting sp ectroscopic magnitudem
lim
, which is
im-p ortant for selection e ects. The dep endence on these
two parameters is examined in more detail for the sp
e-cial cases
0
=0andk=0.
Previous results that this method is a b etter prob e
for
0
than
0
are con rmed, but it app ears that the
lowprobabilit yofmodelswithlarge
0
valuesfoundusing
similarmethodsisdue toa selectione ect.
Thep owerofthismethodtodiscriminateb etween
cos-mologicalmodelscanofcourseb eimpro vedifmore
grav-itationallens systems arefound. However,ournumerical
sim ulationsindicatethatareasonablenum b erofobserved
systems cannotdeliverinterestingconstraintsonthe
cos-mologicalparameters.
Key words: gravitationallensing{ cosmology: theory{
cosmology:observations
1. Introduction
It has recently b een suggested by man y authors (see,
forexample,Fukugitaetal.(1992 )andreferencestherein)
that gravitational lensing statistics can provide a means
of distinguishing b etween di erent cosmological models,
most e ectively concerning the value of the
cosmologi-cal constant. This is fortunate, since mostof the
classi-cal methodsfordeterminingcosmological parametersare
moresensitivetootherquantitiessuchasthedensity(
0 )
Sendo printrequeststo:P.Helbig
?
phelbig@hs.uni-hamburg.de
??
or deceleration (q
0
) parameters. It has even b een
sug-gested (Carrolletal.1992) that gravitational lens
statis-tics based on current observations already give the b est
upp er limitson
0
for world models withk >0,andare
themostpromising methodofdoingsofork=0.
Ko chanek(1992)hassuggestedamethodbasednoton
the total num b erof lens systems but rather on the
red-shiftdistribution ofknown lenssystems characterised by
observables suchas redshiftand imageseparation.Lo
ok-ing at a few di erent models, he concludes that at,
-dominatedmodelsare vetotentimeslessprobablethan
more `standard' models. The advantage of this method
is that it is not plagued by normalisation diculties as
are most schemes involving the total num b er of lenses.
The aim of this pap er is to extend this method to
arbi-traryFriedmann-Lema^trecosmologicalmodelsaswellas
to look at thein uence of observational bias concerning
thebrightnessofthelens. Inaddition, numericalsim
ula-tions are used to estimate the usefulness of the method
whenmore systemsareavailable.
It is importantto note that the method described in
this pap er treats
0 and
0
as indep endent parameters,
that is, they can in principle b e determined sim
ultane-ously. Also important is the fact that
0
is the global
value, i.e., determined by the contribution of all
com-p onents, regardless of degree of homogeneit y and so on.
This isb ecause thecosmological parameters mak e
them-selvesfeltthroughthecosmologicalmodel;mostmethods
of determining
0
will miss any matter homogeneously
distributedona scalelargerthanthat surveyed.
2. Theory
We mak e the `standard assumptions' that the Universe
can b e described by the Robertson-Walker metric and
that lensgalaxies canb emodelledasnon-evolving
singu-larisothermalspheres(SIS).Ifonedropsthe rst
assump-tion,thecosmologicalparameters
0 , 0 andH 0 losetheir
calcula-the attainable accuracy (see Krauss & White(1992) for
a discussion). In order to have a well-de ned
statisti-cal quantity, which is based on the optical depthd for
`strong'lensingevents, 1
thisdiscussionislimitedto
grav-itationallenssystemswithsourceswhicharem ultiply
im-aged(!imageseparation)byisolated(!negligible
clus-ter in uence) singlegalaxiesand with known sourceand
lensredshifts. Anadditionalrequirementisthat the
sys-temm usthaveb eenfoundwithoutanybiases concerning
the redshift of the lens. (See Ko chanek(1992) for a
dis-cussionoftheseselectioncriteria.)
MakinguseofthefactthattheSISpro ducesaconstant
de ection angle,i.e.,indep endent of the p osition of the
sourcewithrespecttotheopticalaxis(de nedaspassing
through observer and lens), one can de ne the angular
crosssectiona 2
ofasinglelensfor`strong'lensingevents
(Turneretal.1984): a 2 =16 3 v c 4 D ds D s 2 ; (1)
wherev is the one-dimensionalvelo city disp ersion ofthe
lens galaxy,c the sp eed of lightand D
ds (D
s
) the
angu-lar size distance b etween lens and source (observer and
source).Follo wingKo chanek(1992 ),onecan arriveat an
expressionfortheopticaldepthasfollo ws.
For a xed mass and mass distribution(!v ), world
model and z
s
, the di erential optical depth due to all
lenses of a given mass as a function ofz
d
is of course
prop ortional to the num b er of lenses of the given mass
p erz
d
-interval and to the cross section for strong
lens-ing events. In order to arriveat an expression ford for
a xed image separation, one needs to knowthe relative
num b er of lenses which, under the given circumstances,
can pro duce this image separation.This can b e doneby
using theSchechter luminosityfunction(Schechter 1976)
as well as the Fab er-Jackson and Tully-Fisher relations
(Fab er&Jackson1976,Tully&Fisher1977),which give
thedep endenceofthevelo citydisp ersion onthe
luminos-ityforellipticalandspiralgalaxies,respectively.Bringing
in the familiar parameters and dropping alltermswhich
are concernedonly withnormalisation, one arrives,after
some tediousbut trivialcalculations,at theexpression
d dz d = (1+z d ) 2 a a3 2 a a3 D s D ds 2 (1+ ) 2 D 2 d 1 p Q (z d ) exp 0 a a3 D s D ds 2 ! ; (2) wherea3 := 4 0 v3 c 1 (v 3 := v ofanL3galaxy ), is the
Fab er-Jackson/Tully-Fisherexp onent, theSchechter
ex-p onent,D
d
theangularsizedistanceb etweentheobserver
andthelensand
Q (z d ):=(1+z d ) 2 ( 0 z d +10 0 )+ 0 : (3) 1
SeeSchneideretal.(1992 )foraclari cationoftheconcept
Equation(2)is indep endentoftheHubbleconstantsince
the dep endences onH
0
in the angular size distances
and in the Fab er-Jackson/Tully-Fisher relation cancel.
In order to facilitate comparison with other authors, we
havechosenthe`standard values'01 : 1, 2.6,4, 144km/s
and 276km/s for the Schechter exp onent, the
Tully-Fisherexp onent,theFab er-Jacksonexp onent,v 3
spiral and
v 3
elliptical
, respectively. (Thevaluefor v 3
elliptical
includes
the factor (3= 2) 1
2
advo cated by Turneretal.(1984 ) and
so our ellipticalgalaxies correspond to thec =2 models
examinedbyKo chanek(1992).)
Theopticaldepth dep endsonthecosmologicalmodel
throughQ (z
d
) as well as through the angular size
dis-tances,b ecauseofthefactthatD
ij =D ij (z i ;z j ; 0 ; 0 ;).
The in uence of, which gives the fraction of
homoge-neously distributed, as opp osed to compact, matter is
felt only in the calculationof the angular size distances,
whereas the cosmological model in the narrower sense
mak esitsin uencefelthere aswell asthroughQ (z
d ).
Ingeneral,thereisnoanalyticexpressionfortheD
ij ;
they can b e obtained by the solution of a second-order
di erential equation. (See Kayser(1985) for the
deriva-tion of the di erential equation, also Linder(1988) for
a more general form ulation.Foran equivalentderivation
for
0
=0seeSchneideretal.(1992).Kayseretal.(1995)
giveageneraldiscussionandaneasy-to-usenumerical
im-plementation.)Ifonehasanecientmethodofcalculating
theangularsizedistances,itiseasytoevaluateEq.(2)for
variousworldmodelsdescribedbytheparameters
0 ,
0
and.
Worthyof note is theindependence of Eq.(2)on the
sourceluminosit yfunction(which ofcoursewillgenerally
itselfdep endonz
s
aswell),therelativenum b ersofgalaxy
typ es(thegalaxytyp eforaparticularlensisknown)and
thefractionofgalaxiesinclusters(themethodlooksonly
at eld galaxies); these factors haveto b e takeninto
ac-count whendoingstatistics basedonthetotalnum b erof
lenses. Also, Eq.(2) is insensitive to ner p oints of the
massmodelsuchas coreradiusandellipticity(cf.Krauss
&White(1992),Narayan&Wallington(1992 )).Themain
idea isto comparethe observed distributionof lens
red-shiftswiththeoreticalexp ectationsforvariousworld
mod-els;themethodisdescribedin thenextsection.
Equation2cantakeonappreciablevaluesat
interme-diate redshifts even though the lens galaxy would be too
faint to be seen at the redshift in question (m > m
lim ).
Inorder to correctforthis e ect,wehavecalculatedthe
redshiftat whichthe lensgalaxy wouldb ecome to o faint
to haveitsredshiftmeasuredfortheinvestigated
cosmo-logicalmodel andtruncatedd at this p oint. (Details in
thenextsection.)Itisimmediatelyobviousthatfailureto
correctforthefaintnessofthelensgalaxieswillarti cially
Table 1. Gravitational lens systemsused. For references seeRefsdal & Surdej(1994 ) and references therein. Note that 00
corresp ondstotheradiusoftheEinsteinringorhalf the imageseparation
name images 00 source m source z s lens m lens z l comments
0142-100 2 1.1 QSO B=17.0 2.719 EG R=19.0 0.493 `typical'multiply
B=19.1 imagedQSO
0218+357 2+ring 0.165 radiolobe 0.94 SG r20 0.6847 `radioring'
1115+080 4 1.15 QSO B=17.2 1.722 EG R=19.8 0.29
B=17.2
B=18.7
B=18.2
1131+0456 2+ring 1.05 EG,radiolobe 1.13 EG R=22 0.85 `radioring'
1654+1346 ring 1.05 radiolobe 1.74 EG R=18.7 0.254 `radioring'
3C324 3 1.0 AGN R=22.7 1.206 SG R=22.5 0.845
R=23.3
3. Calculations
The follo winggravitational lens systems meet our
selec-tioncriteria:0142-100(=UM673),0218+357,1115+080
(= TripleQuasar), 1131+0456, 1654+1346 and 3C324.
(SeeTable1forobservationaldataonthesesystems.)We
considered the follo wingranges of values for the
cosmo-logicalparameters: 010 < 0 < +10 0 < 0 < 10 0 < < 1
which, of course, are m uch larger than contemporary
knowledge demands. However, the history of cosmology
showsthattheknowledgeofto dayisoftenoutoffashion
tomorro w,sothatweprefertodevelopanapproach
capa-ble of dealingwitha widerange of cosmological models.
Alsoimportantisthefactthat itwouldb eanadditional,
thoughbynomeansnecessary,p ointinfavourofthe
valid-ityofthemethodifitassignsthehighestprobabilit ytoa
cosmologicalmodelwithinthepresentlyaccepted
canoni-calparameterspace.
Welookedat1002100modelsin the
0 -0 planefor = 0 : 0 ; 0 : 3 ; 0 : 5 ; 0 : 7 ; 1 : 0 for m lim =23 : 5 and m = 23 : 5 ; 24 : 5 ; 1 for =0 : 5
(JohnsonRmagnitudes).Inaddition,welookedat1002
100modelsinthe -0 andm lim -0
planesforthesp ecial
casesof
0
=0 andk=0.
Before one can examine the relative probabilit yof a
given cosmological model,one m ust rst see ifitis
com-patible with the observations. (Of course, this do es not
imply that the model is compatible with all
observa-tions, merely with the ones necessary for this analysis:
z l ,z s , 00
andgalaxytyp e.)Oneobviousrestrictionisthat
the largest source redshiftz
s; max
in the sample m ustb e
smallerthan z
max
,the maxim umredshiftp ossible in the
cosmologicalmodelinquestion.(See,e.g.,Stab ell&
Refs-dal(1966)orFeige(1992)foradiscussionofthese
cosmo-logical models.)Another restrictionconcerns the
bright-nessofthelens.Fromtheobservablesz
l ,z s , 00 andgalaxy
typ eone can useEq.(1)to calculatethevelo citydisp
er-sionv ,transformthistoanabsoluteluminosityusingthe
Fab er-JacksonorTully-Fisherrelationandthencalculate
theapparentmagnitudeforthegivencosmologicalmodel
(givenbytheangularsizedistanceuptop owersof(1+z
l )
andK-corrections 2
). Ifthiscalculatedlensbrightnessfor
2
Theapparent luminosityofthelensgalaxywascalculated
forthe JohnsonR-bandusingthe K-correctionsof Coleman,
Wu& Weedman(1980 ) andapplying astandardB0R
cor-rection(sincethe B-bandFab er-JacksonandTully-Fisher
re-lationswereused).SincetheseK-correctionsarebasedon
dis-placement ofstandardsp ectraatz=0whichextendintothe
UV-band,theyare givenonlyuptoz=2: 0,where
at least one lens is fainter than m
lim
then the model is
also incompatiblewith theobservations. Since a realistic
value for m
lim
is at least a magnitude fainter than any
m
lens
valueinTable1,thereisnodangerthattheactual
cosmologicalmodelwouldb eexcludedbythisrestriction,
even allo wingfor the uncertainty in calculating the lens
brightnessin thematterdescribed.
Inthesetwocasesweassignedthecorrespondingworld
modelaprobabilit yof0in ourplots,indicatedbywhite.
Sincethevalue0do esn'to ccur otherwise,allwhiteareas
are due to these two restrictions. Otherwise, to measure
therelative probabilit yofagivencosmologicalmodel,we
de nedthequantityf as follo ws:
0<f := R zl 0 d R z s 0 d <1 ; (4) wherez l
istheobserved lensredshiftfora particular
sys-tem. (z
d
is used to denote the variable corresponding to
lensredshiftasopp osedtothemeasuredvaluefora
partic-ularlens.) Thedistributionofthedi erentf values (one
foreachlenssysteminthesample)inbequally-sizedbins
in the interval ]0,1[ gives the relative probabilit yp of a
givencosmological model,with
p= b Y i =1 1 n i ! (5) wheren i
is the num b er of systems in thei-thbin. This
de nitionallo wsonlyafewdiscretevalues,ofcourse.The
variablebisa freeparameter;sincethemostinformation
isobtained whenb isequal tothenum b erof systems,we
adoptthis valueforb .
Were the other observables thesame for all systems,
theredshiftdistributionshouldb e givenbyEq.(2);since
this is notthe case, the quantity f is de ned, which
al-lowsoneto comparetheobservedwiththeexp ected
red-shiftsfordi erentobservablesandhencedi erentrelative
probabilit y curves(Eq.(2)) for each system. Ouransatz
is thus to exp ect that the f-values should b e uniformly
distributedforthecorrectcosmologicalmodel(barring
in-trinsicscatter,ofcourse).Simplecom binatorics(the
num-b erofwaystodistributeaobjectsinbbins) and
neglect-ingnormalisationthenleadstoEq.(5).Foragivenworld
model,the6f-values (oneforeachgravitationallens
sys-tem used) are calculated, and these values are used to
determinetherelativeprobabilit yviaEq.(5).
4. Results and discussion
Figure1 shows therelative probabilit y ofseveral
cosmo-logical modelsas given by Eq.(5). The grey scale in all
inmostcasesthegalaxyb ecomesto ofaintatmo destredshifts,
sotheassumptionofnoevolutionmadethroughoutthispap er
isprobablyjusti ed.
Table2.ValuesoftherelativeprobabilityintheplotsinFig.1
p 1 2 3 4 5 6 7 9 10 11 12 13 1 1 x 1 2 x x x x 1 4 x x x x x x x x x 1 6 x x x x x x x x x x x 1 8 x x 1 12 x x x x x x x x x x x 1 24 x x x x x x x x 1 36 x x x x x 1 48 x x x x x x x x x
plots is the same, regardless of the maxim um and
mini-m umini-mvaluesofeach individual plot.Theresolutionis 0.2
in
0
and 0.1 in
0
, thus giving 10.000di erent models.
The scale isat theright.(Because the relative
probabil-itycan only takeona few discretevalues, contourlevels
aren'tveryusefulasindicatorsoftherelativeprobabilit y,
sincetheywouldmerelyindicate theb oundaries b etween
areasofconstantprobabilit y.Inordertoindicatethe
val-uesdirectly,Fig.1plotstherelativeprobabilit yonagrey
scale. Although in themselves not important, the
inter-ested readercan read o theprobabilitiesdirectly in the
legend,wherethediscretevalueswhichactuallyapp earin
theplotshaveb eenmark ed.Inaddition,Table2givesthe
valuesoftherelativeprobabilit ywhicho ccurineachplot.)
All plotsexcept(9), (10),(12)and(13)arein the
0
-0
plane. Plot(8) gives some orientation in this plane. The
diagonalline from upp erleft tolower centre corresponds
tok =0; thesix curvesare, left toright,for ht
0 =4, 5, 6,8and10210 9 a(h:=H 0 1100 01 km 01 1s1Mp c)aswell
as theb orderto theso calledb ounce models,i.e.models
withnobigbangandthus amaxim umredshiftz
max ; the
line on theright corresponds to q
0
=05. The values of
the xed parameters and m
lim
are indicated on each
plot.Althoughsomeoftheareain thisplaneisde nitely
excluded 3
by simple arguments, a probabilit y has b een
computed foreach worldmodel,in keepingwiththe
sec-ondofourmotivationsmentionedab ove.Inthewhitearea
attherighttheprobabilit yis0b ecausetheseworldmodels
havea maxim umredshiftlowerthanthehighestredshift
in Table1;in plots(1), (2),(3) and(7) thereis an
addi-tionalwhitearea(p=0)separatedfromthe rstonebya
stripwhere p>0 duetothefact thatat leastonelens is
fainteratitsobservedredshiftthanm
lim
.(Thebrightness
of the lens was not used as an additionalconstraint due
mainlytothefactthatthecomputedbrightnessesareonly
correctto ab outa magnitudeorso(Ko chanek1992).)
3
Eventhoughnoteveryp ointintheplane,i.e.every
combi-nationof0 and0,corresp ondstoaworldmo delwhich
can-notb eexcludedbysimple arguments,neverthelesstheranges
of theindividual parametersareallowedassuming thelowest
l
W
-8.0
-4.0
0.0
4.0
8.0
2.0
4.0
6.0
8.0
m
W
21.0 22.0 23.0 24.0 25.0
2.0
4.0
6.0
8.0
l
W
-8.0
-4.0
0.0
4.0
8.0
2.0
4.0
6.0
8.0
l
W
-8.0
-4.0
0.0
4.0
8.0
2.0
4.0
6.0
8.0
l
W
-8.0
-4.0
0.0
4.0
8.0
2.0
4.0
6.0
8.0
l
W
-8.0
-4.0
0.0
4.0
8.0
2.0
4.0
6.0
8.0
l
W
-8.0
-4.0
0.0
4.0
8.0
2.0
4.0
6.0
8.0
h
W
0.2
0.4
0.6
0.8
2.0
4.0
6.0
8.0
l
W
-8.0
-4.0
0.0
4.0
8.0
2.0
4.0
6.0
8.0
l
W
-8.0
-4.0
0.0
4.0
8.0
2.0
4.0
6.0
8.0
l
W
-8.0
-4.0
0.0
4.0
8.0
2.0
4.0
6.0
8.0
l
W
-8.0
-4.0
0.0
4.0
8.0
2.0
4.0
6.0
8.0
l
W
-8.0
-4.0
0.0
4.0
8.0
2.0
4.0
6.0
8.0
m
W
21.0 22.0 23.0 24.0 25.0
2.0
4.0
6.0
8.0
l
W
-8.0
-4.0
0.0
4.0
8.0
2.0
4.0
6.0
8.0
h
W
0.2
0.4
0.6
0.8
2.0
4.0
6.0
8.0
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
(5)=0 : 0,m lim =23 : 5 (13) 0 =0,=0 : 5 (4)=0 : 3,m lim =23 : 5 (8)orientation (12) 0 =0,m lim =23 : 5 (3)=0 : 5,m lim =23 : 5 (7)=0 : 5,m lim =24 : 5 (11)=0 : 5,m lim =1 (2)=0 : 7,m lim =23 : 5 (6)Ko chanek (10)k=0,=0 : 5Fig. 1. Results for the systems in Table1. The relative probability fordi erent cosmological mo dels is plotted linearly on
thescaleshownat theright,where thediscretevalueswhichapp earintheplotsare marked.Fixed parametersareindicated
Lookingatplots(1),(2),(3),(4),(5),(7)and(11)the
general impression is that, apart from some small-scale
structure at the lower right of thep >0 area which
ap-p earstob esigni cant,thereisnotm uchstructure.Since
there are only discrete values ofp, there are some
dis-continuities. Apart from this large-scale-structure,
how-ever,thereisnotm uchchangefromworldmodeltoworld
model.Thecurvesalongwhichtheprobabilit yisconstant
are more vertical than horizontal, suggesting that this
method b etter prob es
0
than
0
. Although not of any
use statistically, for each plotin Fig.1 the Kolmogoro
v-Smirno vprobabilit y (see Sect.5) was also computed. In
the interest of brevity, we havenot included these plots
here;however,thestructureisqualitativ elysimilar
(with-out the sharp discontinuities, of course) and also hints
at thesigni cance ofthesmall-scalestructure mentioned
ab ove.
Plots(1), (2),(3),(4)and(5)showthee ectof
vary-ing.Asonecansee,thegeneralstructuredo esn'tchange
m uch, especiallyifone concentrates onthearea nearthe
roughly vertical thin strip (which would still b e allo wed
assuminghighervalues forH
0
and/ortheageofthe
uni-verse)for relatively low
0
values (
0
< 4,say).This is
compatiblewiththeresultofFukugitaetal.(1992)which
indicates only a weak dep endence of the lensing cross
section on. enters into the calculation only in the
computation of the angular size distances, whereas
0
and
0
also enter the calculation through the function
Q (z
d
)asde nedinEq.(3).Also,theparticularcom
bina-tion of the angular size distances involved in computing
the lensing crosssection, (D
d D
ds )=D
s
(see, e.g., F
ukugi-taetal.(1992)orKo chanek(1992);inEq.(2)theexplicit
dep endenceonD
d
hasb eenlostin thenecessaryvariable
transformationandintegration(cf.Ko chanek(1992))),is
relatively insensitive to . This is not true, for
exam-ple, for the com bination D
ds = (D d D s ), which is
impor-tant for computing H
0
from the time delay b etweenthe
di erent imagesof a m ultiplyimaged source(cf.F
ukugi-taetal.(1992)).
Plots(3),(7)and(11)showthee ectofvaryingm
lim .
Of course,the additionalwhitearea described ab oveb
e-comes smaller as m
lim
b ecomes fainter,disapp earing for
m
lim
!1 (thisconditionis sucient but notnecessary
foreverylensgalaxytob ebrighterthanm
lim
fortheworld
modelsexamined). Themain di erences, however,o ccur
inthesmall-scalestructureatthelowerrightofthep>0
area:thefainterm
lim
b ecomes,thelessprobablethe
mod-elsneartheb orderofthisareaapp ear.Forexample,ifone
compares the models near the Einstein-deSitter model
(
0
=0 and
0
=1) withthe modelsnear thedeSitter
model(
0
=1 and
0
=0), twomodelswhichhaveb een
examined rather extensively in the literature on lensing
statistics,especiallyinthedirection`perp endicular'tothe
curvesdividingdi erent probabilities,then onesees that
form
lim
=23 : 5and m
lim
=24 : 5those nearthedeSitter
probabilit ythanthosenear theEinstein-deSittermodel;
only for m
lim
= 1 is the situation reversed, those near
theEinstein-deSittermodelhavingaclearlyhigher
prob-abilit y.
This is easy to understand, since it is these models
near the deSitter model which have a maximum in d
at relativelylargeredshifts (cf.Ko chanek(1992 ));for
re-alistic values ofm
lim
, one cannotsee thelensgalaxies at
these redshifts.If one usesa realisticvalueform
lim , one
comparestheredshiftdistributionsforthedi erentworld
models (givenby Eq.(2)) at small redshifts, where they
don't di ervery m uch. Ifone takesm
lim
=1 , implying
that one could measure theredshifts of the lensesat all
redshifts,whatevertheir brightness,then itapp ears that
models with a large median redshift in Eq.(2), such as
those onthe righthand b order ofthep>0area
includ-ing themodelswitha largecosmological constant
exam-inedbyKo chanek(1992),areimprobable.However,thisis
merelya selection e ect.The relativelylow lens redshifts
in Table1don'tmean thatworld models with alarge
me-dianlensredshiftareimprobable;itmeansthatwecan'tsee
the lensesatthese redshifts.Comparingprobability
distri-butions assumingthat we could arti ciallyexcludes these
models in preferencetomodels like the Einstein-deSitter
modelwithasmallmedianlensredshift.Intro ducingm
lim
mak esthesituationmore realistic,but meanscomparing
the distributions at small redshifts. Thus, the p ower to
discriminateb etweenvariousworldmodelsisreduced.
For realisticvalues ofm
lim
,such as thosein plots (3)
and(7),theexactvalueofm
lim
isn'tveryimportant;what
do esmak eadi erenceisassuminga valuewhichism uch
to ofaint. Thisiseasytounderstand, since,nearthe
red-shiftwherealensgalaxytypicallyb ecomesfainterthana
realisticvalueform
lim
,thefunctionm(z
d
)israthersteep.
Thismeansthatachangeinm
lim
byamagnitudeortwo
corresponds toarelativelysmallchangeinz
d
andthusto
a correspondingly smallchange in thearea under thed
curveuptothisz
d
value;thus,thereislittlein uence on
thevalueoff as de nedin Eq.(4).This isillustratedin
Fig.2.
For comparison, we have also tested the method on
the systems used by Ko chanek(1992), usingm
lim = 1
und =1,b oth ofwhich heimplicitlyassumes. 4
The
re-sultsarein plot(6)wheretherelativeprobabilitiesare 1
6 ,
1
2
and 1 andcomparing thevarious modelsexamined by
Ko chanekcon rmhis conclusions.Forinstance, the
rela-tive probabilities of the Einstein-deSitter and deSitter
model are 1 and 1
6
, con rming his result that at,
-dominatedmodelsare5{10timeslessprobablethan
stan-dardones.(However,takingm
lim
intoaccountand/or
us-4
Of course, when one considers nite values form
lim , one
cannot include systems with lens redshifts which have b een
determinedbymeansotherthanmeasuredemissionredshifts,
z
0.3
0.6
0.9
1.2
1.5
1.8
2.1
2.4
2.7
z
m
0.3
0.6
0.9
1.2
1.5
1.8
2.1
2.4
2.7
20.0
21.0
22.0
23.0
24.0
25.0
26.0
27.0
28.0
Fig.2.Therelativedi erentialopticaldepth(thincurve)and
thecalculatedlens brightness m(thick curve)as functionsof
z
d
.Theworldmo delisthedeSittermo del(
0
=1: 0
0 =0: 0;
the value of do esn't matter since there is no matter) and
the observables are those for the gravitational lens system
01420100 (see Table1). The ordinate gives the magnitude
inJohnsonR
ing onlydirectly measured lens redshifts would pro duce
quite di erent results,as discussed ab ove.)This plot
ar-ti cially indicates a low probabilit y for models near the
deSittermodelforthesamereasonsasthosediscussedin
connectionwithplot(11).
Plots(9)and(10)examinethein uenceofandm
lim ,
respectively, for the sp ecial case k = 0. Plots(12) and
(13)examinethein uenceof andm
lim
,respectively,for
the sp ecial case
0
= 0. In plots(9) and (12) one can
easily seetheweakdep endenceon ,especially forsmall
values of
0
. Inplots(10)and (13)onecan see theeven
weakerdep endenceonm
lim
inthisrange.Therearehints
towardfainter magnitudes of a declining probabilit yfor
small
0
values, as discussed ab ove. The white area at
the left in plots(10) and (13) is due to the fact that at
least one lens is fainterat its observed redshift thanthe
corresponding m
lim
value. (Since these m
lim
values are
unrealistically small, these models are not incompatible
withtheobservations.)
At rst glance, the`oscillations' in the relative
prob-abilit y migh t app ear somewhat puzzling. According to
Eq.(5),ahigherprobabilit yisobtainedforamoreregular
distribution.Since Eq.(5)onlyallo wsdiscretevalues for
the relative probabilit y, a `jump' o ccurs when the
num-b er of systems in a certain bin changes (unless o setby
l
f
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
l
f
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
l
f
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
l
f
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
l
f
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
l
f
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
l
f
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
l
f
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
l
f
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
l
f
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
l
f
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
l
f
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
l
f
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Fig.3.`Oscillations'intherelativeprobability.Thethin
hori-zontallinesarethebinb oundaries.Thethickcurvesshow,from
top tob ottom, thef-valuesgiven by Eq.(4)forthe
gravita-tionallenssystems1131+0456,3C324,0218+357,01420100,
1115+080 and 1654+1346,resp ectively. Sincetwo systems
havesuchlargelensredshiftsthatd ispractically=0atthe
corresp ondingredshift,theirf-values(intheseworldmo dels)
arepractically=1andcannotb edistinguishedfromeachother
orfromthebinb oundaryatf=1intheresolutionoftheplot.
Table1showsthattheseare thetwohighestlensredshifts in
the sample.Beneaththe linesandcurves, showninthesame
wayasinFig.1,isthecorresp ondingrelative probability
in Fig.3,where for demonstrationpurp oses thef-values
given by Eq.(4) for each gravitational lens system used
are plotted as functions of
0
for k = 0 and = 0 : 3.
(That is, for the world models along the k = 0 line in
plot(4)inFig(1).)Itcanb eseenthat,although|dueto
thede nition|therelativeprobabilit ychangesby
notice-ableamoun tsb etweena few discretevalues,nevertheless
thef-valuesthemselvesaresmoothlyvaryingfunctionsof
theworldmodel.Figure3alsomak esthefollo winggeneral
conclusionsclearin thissp eci cexample.
{ Thefactthata coupleofsystemshavef-values which
are practically = 1 limits the maximum probabilit y,
sincetheseare alwaysin thesamebin.
{ `Oscillations' b etween di erent probabilities have no
physical signi cance, but rather are merely artifacts
of the particular lens redshifts. On the other hand,
extremelylowprobabilities,e.g.allsixsystems in the
same bin (p = 1
720
), would b e more indicative of a
low-probabilit ycosmological model.
in-8 Helbig&Kayser:Cosmologicalparametersandlensredshifts
indicate a probabilit y low enough to reject the
cor-responding cosmological models, were the parameter
space examined larger. Thus, the method migh t b e
abletoexclude`extreme'cosmologicalmodels.
TheinterestedreadercanuseFig.3togetherwithEq.(5)
toseehowtherelativeprobabilit yisarrivedat.
Thestatisticalsigni canceofallresultsinthissection
isdiscussedinthenextsection.
5. Numerical simulations
Numerical sim ulations were done for m
lim
= 23 : 5 and
for = 1. This value ofm
lim
= 23 : 5is the most
realis-tic basedon the present stateof observations and using
only one value form
lim
= 23 : 5 as well as for is
justi- edbasedontheweakdep endenceoftheresultsonthese
parameters,asdiscussedinSect.4.
Theobservables 00
(theradiusoftheEinsteinringor
half the imageseparation corresponding to thediameter
oftheEinsteinring),z
s
andgalaxytyp ewerechosen
ran-domly from anintervalroughlycorresponding tothe
ob-servedrangeofvalues inorder topro ducesyntheticdata
comparable to real observations. It is important to note
thatneithertheexactrangenortheshap eofthe
distribu-tion matters, sincethemethod looks at theredshift
dis-tributionofthelenseswiththeotherparameters xedby
observation. For convenience, a atdistributionwas
cho-sen foreach of the observables.For a given cosmological
model,thecorrespondinglensredshiftz
l
foreach system
wascalculatedfrom theobservables andarandomly
gen-eratedf through (numerical) inversion of Eq.(4). This
catalog was then used to determine a relative
probabil-ityforeachof thep ointsinthe
0
-0
planein thesame
manneras fortherealsystems.
Withtheprobabilit y givenby Eq.(5), basedon
sim-ple com binatorics, one cannot know to what degreethe
values foreach cosmological modelarein uencedby
sta-tistical uctuations. However,with such a small num b er
ofsystems,thereisreallynoothermethodofcomputinga
relativeprobabilit y.Weexp ectthatthedistributionofthe
f-values,barringstatistical uctuations,shouldb e atfor
thecorrectcosmologicalmodel.Soweneedatestto
com-parethis distributionwitha atprobabilit ydistribution.
TheKolmogoro v-Smirnov(K-S)testisofcourseawell
un-derstoo dmethodfortestingiftwodistributionsare
statis-ticallysigni cantlydi erent.(See,e.g.,Presset.al.(1992)
forageneraldiscussionandde nitionoftheK-S
probabil-ity.)However,thistestcan onlyb eused fordistributions
with more than 20 data p oints. For purp oses of
com-parison,forthesystemsinTable1notonlywasthe
prob-abilit yde nedinEq.(5)computed(shownin Fig.1)but
also the K-S probabilit y.The K-S probabilit y should of
coursenotb etakenseriouslyifthereareto ofewsystems.
Wehavedonesim ulationsfora varietyofworld
mod-elsandalsofornum b ersofsystemsb etween20and50.In
l
W
-2.50 -2.00 -1.50 -1.00 -0.50 0.00 0.50 1.00 1.50
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
Fig.4.Resultsbasedonacatalogueof50simulatedsystems.
(Notethedi erentscaleontheaxes.)Thecosmologicalmo del
used to generate the lens redshifts is the homogeneous
Ein-stein-deSittermo del( =1: 0,0 =0: 0,0 =1: 0).Thegrey
scaleisasinFig.1
shows the results derived from a catalogue of sim ulated
gravitational lens systems.Since, even with 50 systems,
no area can b e excluded basedon the K-S probabilit y{
the white area has p = 0 due to the fact that at least
one lens would b efainterthanm
lim
in these world
mod-els, as discussedab ove{ weconclude that,although one
canqualitativ elyunderstandthephysicswhichatleastin
part isresponsible fortheresultspresentedin Fig.1,the
actualrelative probabilitiesaremore indicativeof
intrin-sicscatterintheredshiftsofthelensesthanahintofthe
correctcosmological model.
6. Summary and conclusions
Inthispap erwehaveextendedthemethodoriginally
pro-p osed by Ko chanek (1992) for using the redshift
distri-bution of gravitational lenses to learn something ab out
the cosmological model. This method is particularly
at-tractive sinceitavoids thenormalisation diculties
nor-mally asso ciated with lensing statistics. First, we made
use of the equation derived by Kayser (1985 ; see also
Kayseretal.1995) to b e able to examine cosmological
modelsdescribedbyarbitraryvaluesof
0 ,
0
and.
Sec-ond, we looked at the in uence of observational bias by
intro ducingtheparameterm
lim
.(Thismeansthatwe
esti-Third, we used a more quantitative statistic to look at
relativeprobabilities.
Since the p otential reward from using the Ko chanek
formalism(settinglimitsoncosmologicalparametersb
et-terthanwithothermethodsusingonlyobservable
quanti-tiesandstandard assumptionswhosevalidit yinthis
con-text is undisputed) seemedlarge, our basic idea was to
extendthisformalismtoenableittolookatalarger
num-b er of cosmological models(arbitrary values for
0 and
0
as well as ) while correcting an obvious limitation
(selection e ects due to the brightness of the lens) and
using a variation of his statistic (Ko chanek(1992)
basi-cally usesourstatisticwith two bins) which allo wsmore
information ab out thedistribution of theredshift values
to b e used. That is, weintended to follo wtheformalism
of Ko chanek(1992) as closelyas p ossible. Unfortunately,
thefact thattherelativeopticaldepthgivenbyEq.(2)is
appreciably di erent only for those cosmological models
in which this di erence cannot b e seen due to the
selec-tion e ect rendersthe technique lessuseful than wehad
hop ed.
Wesawthatlittleextrauncertaintyinthederived
val-uesfor
0 and
0
isintro ducedbyletting b ea free
pa-rameter.Thesameistrueofm
lim
withtheexceptionthat
values which are unrealistically faint distort the results.
A comparisonwith theresults of Ko chanek(1992),
con- rmedwithourformalism,alsoshowtheconsequencesof
neglectingm
lim .
Aninteresting resultisthe degeneracyofthe derived
relativeprobabilit yalongcurvesroughlyparalleltocurves
of constant worldage in the
0
-0
plane. This indicates
that the method is more sensitive to
0
than to
0 , but
also showsthatdemonstrating theconsistencyof agiven
cosmological model with theobservations using this
sta-tisticalmethodalsoimpliesconsistencywithalargerange
ofothercosmologicalmodels.
Thedramaticdi erencecausedbynotneglectingm
lim
castsdoubt onthedegreeto which present observations,
basedonly ontheredshift distribution, 5
are ableto rule
outcertaincosmologicalmodels;inparticular, atmodels
withalargecosmologicalconstant, havingahighmedian
5
Moreinformation is available intheoryby lo oking at not
onlythe redshiftdistribution,i.e.,theshapeofthe curve,but
alsothenumb er oflenses,i.e.,theareaunderthecurve.This,
however,intro ducesadditionaluncertaintiesdueto
normalisa-tion.Additionally,onecouldconsiderthecompletenessofthe
sample,as p ointed outto theauthorsbythereferees.For
ex-ample,thenumb eroflenssystemswithunmeasuredredshifts
could inprinciple b e usedto exclude cosmological mo dels in
which the redshifts could have b een measured, or the other
way around. This requires information regardingthe reasons
astowhytheredshiftshaven'tb eenmeasured|thee ectsof
thecosmology shouldb eseparatedfromthe current
observa-tionalstand,whichrequiresadetailedanalysisoftheobserv
a-tionalliterature,oraseparateobservationalprogramme,b oth
exp ectedlensredshift,b ecomemoreprobablethrough
in-tro ducingm
lim .
Nevertheless,plots(6) and(11)givesanideaofwhat
couldb edone,ifonewereabletomeasuretheredshiftsof
thefaintestlensgalaxies.Foragivenimageseparation,the
calculatedbrightnessofthelensgalaxyhasaminim umat
some intermediateredshift;thisistypicallyatab out30 m
inR ,sothatlargertelescopesandadvancesinimage
pro-cessing will probably b e able to mak esome progress on
this front in the next several years. If one were able to
measurethelensredshiftattheminim umbrightness,this
would havetheside-e ect of eradicating thedep endence
onm
lim
.Ontheotherhand,probablymorewouldb elost
thangained,b ecauseitwouldnolongerb ep ossibleto
ne-glectevolutionarye ects. For thisreason,one could
sup-p osethatmoreprogress intheimmediatefuture(barring
a revolutionin theunderstandingofevolutionarye ects)
will probablycomefrom increasingthenum b erof usable
systems (through the discovery of more systems and/or
throughmeasuringmoreredshiftsinknownsystems)than
from pushingm
lim
to faintervalues.
The results of using the method on synthetic data,
however,cast doubt on thestatistical signi cance of our
results and onthe hop e of using this method to exclude
certaincosmologicalmodels,especiallythosewhichcannot
b eexcludedbyothertests.Thus,b eingconservativeinour
appraisalofwhatthestatisticsofredshiftsofgravitational
lensescantellus, weconclude thatat present andin the
foreseeable future this method will probably not giveus
anyusefulinformation.
Acknowledgements.It is a pleasure to thank S.Refsdal and
T.Schrammforhelpfuldiscussions.
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