through
Statistics of Redshifts of Lens Galaxies
Phillip King 3
Ham burgerSternwarte,Gojenb ergsweg 112, D-21029 Ham burg,Germany
Abstract: For known gravitational lens systems the redshift distribution of the lenses is compared
withtheoreticalexp ectationsfor10 4
Friedmann-Lema^trecosmologicalmodels,whichmorethancover
therange ofp ossible cases. The comparisonis usedforassigninga relative probability toeachof the
models. The entire pro cedure is rep eated for di erent values of the inhomogeneity parameter as
well as H
0
and the limiting sp ectroscopic magnitude, which are imp ortant for selection e ects. The
dep endence onthese three parameters is examinedinmore detailfor=0and k=0.
The previous resultof other authors thatthis metho d is a goo dprob e for
0
is con rmed,butit
app earsthatthelowprobabilityofmodelswithlarge
0
valuesrep orted bytheseauthorsmayb edue
toa selection e ect. The p owerof this metho d to discriminate b etween cosmological models canb e
improved dramaticallyifmore gravitational lens systemsare found.
1 Introduction
It has recently b een suggested by many authors (see, for example, Fukugita etal. (1992) and
references therein) that gravitational lensing statistics can provide a means of distinguishing
b etweendi erentcosmologicalmodels,moste ectiv elyconcerningthe valueofthecosmological
constant. This is fortunate, since most of the classical methods for determining cosmological
parametersaremoresensitivetootherquantitiessuchasthedensityordecelerationparameter.
It has even b een suggested (Carroll et al., 1992) that gravitational lens statistics based on
current observations already givethe b estupp er limitson
0
forworld models withk >0, and
are the most promising method ofdoing so for k=0.
Ko chanek (1992) has suggested a method based not on the total num b er of lens systems
but ratheronthe redshiftdistributionof knownlens systemscharacterisedby observablessuch
as redshift and image separation. Lo oking at a few di erent models, he concludes that at,
-dominatedmodelsare vetoten timesless probablethanmore`standard'models. Itwasm y
aim toextend thisformalism 1
toarbitraryFriedmann-Lema^tre cosmologicalmodels aswellas
to lo ok atthe in uence of observationalbiases.
3
e-mail: pking@hs.uni-hamburg.de
1
Aformalismwhichhastheadvantagesofb eing(almost)indep endentoftheHubbleconstantandnotb eing
plaguedbynormalisationdiculties asaremostschemesinvolvingthetotalnum b eroflenses.
Imak ethe`standardassumptions'thattheUniversecanb edescribedbythe Rob ertson-Walker
metricandthatlensgalaxiescanb emodelledasnon-evolvingsingularisothermalspheres(SIS).
If one drops the rst assumption, the cosmological parameters
0 ,
0
and H
0
lose their
signif-icance; the second assumption mak es for easy calculation but, more importantly, is probably
justi ed withinthe attainableaccuracy (see Krauss&White(1992) fora discussion). In order
to have a well-de ned statistical quantity, which is based onthe optical depth d for `strong'
lensing events, 2
this discussion is limited to gravitational lens systems with sources whichare
m ultiplyimaged( !imageseparation)byisolated(!negligibleclusterin uence) single
galax-iesandwithknownsourceandlensredshifts. Anadditionalrequirementisthatthesystemm ust
have b een found without any biasesconcerning the redshift ofthe lens. (See Ko chanek (1992)
for adiscussion of these selection criteria.)
Makinguseofthe factthattheSISpro ducesaconstantde ectionangle,i.e.,independentof
the p osition of thesource with resp ect tothe opticalaxis(de ned as passingthroughobserver
and lens), one can de ne the angular cross section of a single lens for `strong' lensing events
(Turner etal., 1984): a 2 =16 3 v c 4 D ds D s 2 (1)
where v is the one-dimensional velocity disp ersion of the lens galaxy, D
d
and D
ds
the
an-gular size distances b etween observer and lens and, resp ectiv ely, lens and source. Following
Ko chanek (1992), one can arriveatan expression for the opticaldepth as follows:
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 of z
d
is of course prop ortional to the
num b er of lenses p er z
d
-interval. In order to arrive at an expression for d for a xed image
separation, one needs to know the relativ e num b er of lenses which, under the given
circum-stances,canpro ducethisimageseparation. Thiscanb edonebyusingthe Schechterluminosity
function (Schechter, 1976) as well as the Fab er-Jackson and Tully-Fisher relations (Fab er &
Jackson, 1976, Tully & Fisher, 1977), which give the dep endence of the velocity disp ersionon
the luminosity for elliptical and spiral galaxies, resp ectiv ely. Bringing in the familiar
param-eters and neglecting allterms whichare concerned only with normalisation, one arrives, after
some tedious but trivial calculations,at the expression
d dz d =(1+z d ) 2 a a 3 2 a a 3 D s D ds 2 (1+ ) D 2 d 1 p P(z d ) exp 0 a a 3 D s D ds 2 ! (2) where a 3:=4 0 v 3 c 1
(v3:=v of anL3 galaxy), isthe Fab er-Jackson/Tully-Fisher exponent,
the Schechter exponent, D
d
the angular size distance b etween the observer and the lens
and P(z d ) := (1+z d ) 2 ( 0 z d +10 0 )+ 0
. The optical depth dep ends on the cosmological
model through P(z
d
) as well as through the angular size distances, b ecause of the fact that
D ij = D ij (z i ;z j
;cosmologicalmodel). In general, there is no analytic expression for the D
ij ,
which also dep end on ; they can b e obtained by the solution of a second-order di erential
equation. (SeeKayser(1985);foranequivalentderivationfor=0seeSchneideretal.(1992).)
If one has an ecien t method of calculating the angular size distances, it is trivial to
evaluate Eq. 2 for various world models described by the parameters
0 ,
0
and . (The
in uence of , which gives the fraction of homogeneouslydistributed, as opp osed to compact,
matter is felt only in the calculation of the angular size distances, whereas the cosmological
d
of note is the independence of Eq. 2 on the source luminosit y function (which of course will
generallyitselfdep end onz
s
aswell), the relativ enum b ersofgalaxytyp es(the galaxytyp e for
a particular lens is assumed tob e known) and the fraction of galaxies in clusters (the method
lo oks only at eld galaxies); these factors have to b e taken into account when doing statistics
basedonthe totalnum b erof lenses. Also,Eq.2isinsensitiv eto ner p ointsofthe massmodel
such as core radius and ellipticit y(Krauss& White, 1992, Narayan& Wallington, 1992). The
mainideaistocomparetheobserveddistributionoflens redshiftswiththeoreticalexpectations
for various world models; the methodis described inthe nextsection.
Although there is an exponential cuto towards z
d
= 0 and z
d = z
s
caused by the fact
thatthe requiredmassof thelensinggalaxyforpro ducing agivenimageseparationdivergesat
these p oints and the num b erof such galaxies declines exponentially inthe Schechterfunction,
d can nevertheless take on appreciable values at intermediate redshifts even though the lens
galaxy would be too faint to be seen at the redshift in question. In order to correct for this
e ect, I have calculated the redshift at whichthe lens galaxy would b ecome to o faint to have
its redshift measured for the investigated cosmological model and truncated d at this p oint.
(Details inthe nextsection.) It is immediately obviousthat failure tocorrectfor the faintness
of the lens galaxies willarti cially exclude cosmologicalmodels with ahigh medianredshift in
Eq. 2,which migh totherwise not b e excluded.
3 Calculations
The following gravitational lens systems 3
meet the selection criteria: UM 673, 0218+357,
1115+080(TripleQuasar),1131+0456,1654+1346and3C324. Iconsideredthefollowingranges
of values 4
for the cosmological parameters: 0 10 <
0 < +10, 0 < 0 < 10, 0 < < 1 and 30<H 0 <110 (H 0
isintheusualunitsof km
s1Mp c
). Keeping theothertwoparametersconstantat
theirdefaultvalues,Ilo okedat1002100 modelsinthe
0 -0 planefor =0: 0;0: 3;0: 5;0: 7;1: 0, H 0 =40;50;70;90 and m lim
=23: 5;24: 5;1 (Johnson R magnitudes). I used the following
de-fault values: =0: 5, H 0 =70 and m lim =23: 5 m
. In addition, I lo oked at1002100 models in
the -0 , H 0 -0 and m lim -0
planesfor the sp ecial cases of =0 and k =0.
To measurethe relativ eprobability of agivencosmologicalmodel, I de nedthe quantityf
as follows: 0<f := R z l 0 d R zs 0 d <1 wherez l
istheobserved lensredshiftforaparticularsystem. Thedistributionofthedi erentf
values (one for eachlens system inthe sample) inb bins in the interval ]0,1[ givesthe relativ e
probability p of a given cosmological model, with p = b Q i=1 1 n i ! (normally; if m gal > m lim or z s; max > z max then p := 0) where n i
is the num b er of systems in the i -th bin. The variable b
3
Forobservationaldataonthesesystems,see Surdej(1993).
4
Ofcourse, thesearem uchlarger thancontemporarywisdomdemands. However,I thinkthat thereareat
leasttworeasonsforusingsuchlargeranges:
Itwouldb e anadditional,thoughbynomeans necessary, p ointin favourofthevalidit yof themethod
if it assigns the highest probabilit y to a cosmological model within the presently accepted canonical
a b c d e f g h
l
orientation
W
-8. -4.
0.
4.
8.
2.
4.
6.
8.
l
orientation
W
-8. -4.
0.
4.
8.
2.
4.
6.
8.
l
orientation
W
-8. -4.
0.
4.
8.
2.
4.
6.
8.
l
orientation
W
-8. -4.
0.
4.
8.
2.
4.
6.
8.
l
orientation
W
-8. -4.
0.
4.
8.
2.
4.
6.
8.
l
orientation
W
-8. -4.
0.
4.
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2.
4.
6.
8.
l
orientation
W
-8. -4.
0.
4.
8.
2.
4.
6.
8.
l
(default)
W
-8. -4.
0.
4.
8.
2.
4.
6.
8.
l
h = 0.3
W
-8. -4.
0.
4.
8.
2.
4.
6.
8.
l
h = 0.7
W
-8. -4.
0.
4.
8.
2.
4.
6.
8.
l
H = 50
W
-8. -4.
0.
4.
8.
2.
4.
6.
8.
l
H = 90
W
-8. -4.
0.
4.
8.
2.
4.
6.
8.
l
m = 24.5
W
-8. -4.
0.
4.
8.
2.
4.
6.
8.
l
m = infinity
W
-8. -4.
0.
4.
8.
2.
4.
6.
8.
Figure 1: Plots of the relativ e probability in the
0
-0
plane. Deviations from the default
values are indicated.
is afree parameter, but it iseasily seen that the most information is obtained when b isequal
to the num b er of systems. The relativ e probability is thus 0 if the lens galaxy is to o faint to
have its redshift measured and/or if z
max
, the maxim umredshift p ossible in the cosmological
modelinquestion,issmallerthanthelargestsourceredshiftz
s; max
inthesample. Theapparent
luminosit y of the lens galaxy was calculated for the Johnson R-band using the K-corrections
of Coleman, Wu & Weedman (1980). (These are based on displacemen t of standard sp ectra
at z=0which extend intothe UV-band andare givenonly upto z =2: 0, where evolutionary
e ects would in any case have to b e considered. However, in most cases the galaxy b ecomes
to o faint atmodest redshifts, so the assumption of noevolutionis probablyjusti ed.)
4 Results and Discussion
Plot a gives some orientation in the
0
-0
-plane. The vertical line shows
0
=0, the slanted
line k = 0, the curve at the right z
max
=10, the other two curves corresp ond to t
0 h 0 = 5;7: 5 in units of 10 9 a (left to right, h 0 := H 0 1s1Mp c 1001km
); the area b etween the world age curve and the
z
max
curveisthe `allowedarea' basedonpresentknowledge. Taking basanexample plot, one
notices rst of all the region of relativ e probability 0 in the parameter space containing the
`bounce models' 5
and the fact that most of the `structure' o ccurs in the `allowedarea'. Since
the gradient runs more nearly parallel to the
0
-axis than to the
0
-axis, one can learn more
ab out
0
than ab out
0 .
Di erentvalues of (plots cand d)pro duce acontinuoustransformation ofthe plot
struc-ture,butinterestinglyverylittlechangesinthe`allowedarea'. Thefactthatplaysarelativ ely
unimportantr^ole inthe opticaldepthhas previouslyb een demonstratedinanother mannerby
Fukugita etal. (1992).
5
0
ues, a region of probability 0 covers part of the `interesting' parameter space, but this should
not b e taken to o seriously, resulting as it do es from the sudden `disapp earance' of a lens
galaxy|the calculated apparent magnitudes are probablyaccurate to only amagnitude or so
(Ko chanek, 1992). The factthat the structure inthe plots do es not severelyand/or
discontin-uously change hints at the fact that the in uence of the Hubble constant, whichmak es itself
felt onlyin the calculation of the apparent magnitudes ofthe lens galaxies, isnot to o severe.
A comparisonb etween bor gand hshowsthe e ect ofneglectingm
lim
; onthe otherhand,
there is relativ ely little di erence b etween b and g, so that the exact determination of m
lim
is not crucial and one sees that using the same m
lim
for all systems is go o d enough for a rst
approximation.
It can b e qualitatively understo o d why the value of H
0 or m
lim
do esn't exert a larger
in uence: around the redshift at which the lens galaxy b ecomes to o faint, the graph of m as
a function of z
d
isvery steep, 6
so that even a relativ ely large changein H
0
(movingthe entire
curve parallel to the m -axis) or m
lim
(changing the cuto value) corresp onds to just a slight
changeinz
d
. Since the graph of d as a functionof z
d
istypically not very steep atthis value
ofz
d
,asmallchangeinthe value ofz
d
atwhichthe probabilitydistributionistruncated mak es
little di erence asfar asthe integral upto this p ointis concerned.
Returning to the example plot, one notices that, at least in the `allowed area', roughly
oblong areas of constant relativ e probability run approximately parallel to curves of constant
world age.
5 Concluding Remarks
A comparison ofthe plots showsthat neglecting the limiting magnitude pro ducesmore
`struc-ture'. Thisarises fromthe factthatthe current samplecontainsmostly lensesof relativ elylow
redshift;one obtainsarelativ elylowprobabilityofworldmodelswithahighermedianredshift.
Correcting for this e ect means lo oking only at the di erence in distribution at low redshift,
whichmak esthe methodless able todiscriminateb etween di erentcosmologicalmodels.
Nev-ertheless, plot ggives an idea of whatcouldb e done, ifone were able tomeasurethe redshifts
of the faintest lens galaxies. For a given image separation, the brightness of the lens galaxy
has a minim um at some intermediate redshift; this is typically at ab out 30 m
in R, so that
larger telescopes and advances in image pro cessing will probably b e able to mak esubstantial
progress on this front in the next few years. If one were able to measure the lens redshift at
the minim umbrightness, this would havethe side-e ectof eradicating the dep endence onH
0 .
On the other hand, probably more would b e gained than lost, b ecause it would no longer b e
p ossible to neglect evolutionary e ects. For this reason, the most progress in the immediate
future (barring a revolution in the understanding of evolutionary e ects) will probably come
fromincreasing the num b erof usablesystems rather than frompushingm
lim
tofainter values.
The dramatic di erence caused by not neglecting m
lim
casts doubt onthe degree to which
present observations, based only onthe redshift distribution, are able to rule out certain
cos-mological models; in particular, at models with alarge cosmological constant, having a high
medianexpectedlensredshift, b ecomemore probable throughintro ducingm
lim .
7
It is
reassur-6
The apparent brightnessdecreases m uch faster as a functionof z
d
than in the `normal' case, b ecause in
thisz
d
-intervaltherequiredgalaxymass(!absolutebrightness)foragivenimageseparationtypicallydecreases
withincreasingz
d
;seeKo chanek(1992).
7
b e able toput limitson cosmologicalparameters comparable toother methods.
Atpresent,itisdiculttoquantifytheconclusions,sinceitisdiculttode ne,forexample,
a con dence region in the
0
-0
plane. The values of the relativ e probabilities are known, of
course, but it is to o simple to conclude that a quotient of, say, 5 b etween two areas means
that the one area is` vetimes morelik ely'than the other; the factthat there are only(afew)
discrete values of the relativ e probability mak es the situation a bit more complicated. The
b estmethodseems tob etocarryout sim ulationsusingarti cialsamplesand thentouse these
to de ne a useful con dence region. This will also allow the statistical uctuations (arising
b ecause of the small num b erof systems in the sample) tob e takeninto account. 8
To conclude, I nd thatthe methodoutlined here can probablyb e usedto setuseful limits
on
0
andp erhaps
0
,althoughatpresentthereareto ofewusefulsystemstoallowonetomak e
rmcon dence estimates,if onetakesm
lim
intoaccount, asseemstob e essential. Amethodof
usefully quantifyingthe estimates probably requires information derived fromsim ulations. As
the num b er ofsystems increasesand the interpretationofthe results b ecomesclearer,it migh t
proveuseful toestimatem
lim
foreachsystem individually,taking intoaccountallobservational
factors.
Acknowledgements
It is apleasure tothank S.Refsdal, R.Kayser and T. Schrammfor helpful discussions, Bernd
Feige for some source co deas wellas R.Schlotteand A.Dent forcomputational aid.
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uncertaintiesduetonormalisation. Itma yb ep ossibletoobtainmore informationbyconsideringthe
complete-nessofthesample,i.e.,thefractionofsystems withmeasuredlensredshiftsfrom thesampleofsystemswhich
areotherwisesuitable, assuggestedbyKo chanek(seetheDiscussion).
