Sky Survey Radio-Sele ted Gravitational Lens Statisti s K.-H. Chae, 1 A.D. Biggs, 1 R.D. Blandford, 2 I.W.A. Browne, 1 A.G. de Bruyn, 3 C.D. Fassna ht, 4 P. Helbig, 1 N.J. Ja kson, 1 L.J. King, 5 L.V.E.Ko opmans, 2 S.Mao, 1 D.R. Marlow, 6 J.P. M Kean, 1 S.T.Myers, 7 M.Norbury, 1 T.J. Pearson, 2 P.M. Phillips, 1 A.C.S. Readhead, 2 D.Rusin, 8 C.M. Sykes, 1 P.N.Wilkinson, 1 E. Xanthop oulos, 1 andT. York 1 1
Universityof Man hester,JodrellBank Observatory,Ma les eld,Cheshire,SK119DL, UK
2
California Institute of Te hnology, Pasadena, CA 91125
3
NFRA, Postbus 2, 7990 A A Dwingeloo, The Netherlands
4
Spa eTeles ope S ien eInstitute,3700 SanMartin Drive,Baltimore, MD 21218
5
University of Bonn, Auf dem Hugel, 69, D-53121 Bonn, Germany
6
Dept.ofPhysi sandAstronomy,UniversityofPennsylvania,209S.33rd StreetPhiladelphia,PA 19104
7
NationalRadio Astronomy Observatory, P.O. Box 0, So oro,NM 87801
8
Harvard-SmithsonianCenter for Astrophysi s,60GardenSt., MS-51,Cambridge, MA02138
(Dated: January15,2005)
Wederive onstraintson osmologi al parametersandtheprop ertiesofthelensinggalaxies from
gravitationallensstatisti sbasedonthe nalCosmi LensAllSkySurvey(CLASS)data. Fora at
universewitha lassi al osmologi al onstant,we ndthatthepresentmatterfra tionofthe riti al
density is m =0:31 +0:27 0:14 (68%) +0:12 0:10
(systemati ). Fora atuniverse witha onstantequation of
statefordarkenergyw=p
x (pressure )= x (energydensity),we ndw< 0:55 +0:18 0:11 (68%).
PACSnumb ers:98.80.-k,98.80.Es,98.80.Cq,98.62.Py
Gravitational lensing, in whi h a ba kground sour e
b ehind a foreground obje t is seen as distorted and
(de)magni ed image(s), is well-understo o d by the
sim-ple physi s of light de e tions in a weak gravitational
eld. Hen e, the analysis of lens statisti s an provide
ate hnique[1℄ for onstraining osmologi alparameters
thatisb othp owerfuland omplementarytoother
meth-o ds[2℄. However,forlensstatisti stob euseful,itisvital
to havean unbiasedstatisti alsample ofsour es thatis
omplete withinwell-de ned observational sele tion
ri-teria[3℄ andto prop erlytake intoa ount allfa torsin
astatisti allensingmo del[4℄. Thestatisti alprop erties
ofgravitationallensing inasamplearethetotallensing
rate,theimageseparations,thelensredshifts,thesour e
redshifts, andtheimagemultipli ities. Theseprop erties
dep end notonlyon osmologi alparameters butalsoon
the prop erties of the lensing galaxies and the
distribu-tionsof the sour es in redshift (z) and inluminosityin
theobservationalsele tionwaveband(e.g.radiohere)[4℄.
Re entadvan esinobservationsp ermitusto doa
re-liable analysis of lens statisti s to obtain limits on
os-mologi alparameters. First, the largest radio-sele ted
gala ti mass-s ale gravitational lens sear h proje t to
date,theCosmi LensAllSkySurvey(CLASS),hasnow
b een ompleted,resultinginthelargestsample suitable
for statisti al analyses [3℄. Se ond, urrently available
indep endent data setson thedistributionsof ( at sp
e -trum) radio sour es in ux density and in redshift are
on ordant[3, 4℄. Third, re ent large-s aleobservations
of galaxies, in parti ular the Two Degree Field Galaxy
Redshift Survey (2dFGRS) and the Sloan Digital Sky
totalgalaxyluminosityfun tion(LF;i.e.,distributionof
galaxy numb er density in luminosity), and there exist
fairlyreliableobservationalresultsthat p ermitusto
ex-tra tfromthetotalLFmorphologi altyp e-sp e i LFs,
i.e.,anearly-typ e(i.e.ellipti alsandS0galaxies)LFand
alate-typ e(i.e.spiralsandothers) LF[4,5℄. Inthis
Let-ter, we rep ort the main results on osmologi al
param-eters from a likeliho o d analysis of lens statisti s based
onthese ru ialsets ofdata. Thedetailed pro edure of
the analysis will b e des rib ed in a later publi ation [4℄
inwhi hextendedanalyseswillb epresentedforbroader
rangeofparametersandwithanemphasisontheglobal
prop ertiesofgalaxyp opulations.
TheCLASSstatisti alsamplesuitableforouranalysis
ontains8958radio sour es outof whi h 13 sour es are
multiply-imaged[3℄.Thelenssear hlo okedfor
multiple-lensingimage omp onents of the entral ompa t radio
ore in the ba kground sour es. The CLASS
statisti- al sample is de ned so as to meet the following
ob-servational sele tion riteria [3℄: (1) the sp e tral index
b etween 1.4 GHz and 5 GHz is atter than 0:5, i.e.
0:5 with S / where S
is ux density
mea-suredinmilli-jansky(mJy;1mJy10 29 Wm 2 Hz 1 );
(2) the total ux density of ea h sour e is 30 mJy
at 5 GHz; (3) the total ux density of ea h sour e is
20mJyat8.4GHz;(4)theimage omp onentsinlens
systemsmusthaveseparations0:3ar se andtheratio
of the ux densities of the fainter to the brighter
om-p onentindouble-imagesystemsmustb e0:1. Numb er
ounts for sour es at = 5 GHz are onsistent with a
rela-TABLE I: Gravitational lens systems in the well-de ned
CLASSstatisti alsample. zs andzlaresour eandlens
red-shifts,resp e tively;forsour eswithunmeasuredredshifts,we
takez
s
=2,whi histhe meansour e redshift fortheentire
sample of CLASS lenses. The maximum image separation
( ) is given in ar se . Image multipli i ty is listed under
N
im
. When the image splitting is due to multiple galaxies,
theimageseparationandtheimagemultipli ityarenotused
in our mo del tting. The image separation of 2045+265 is
notusedb e auseofun ertaintiesinthenatureofthelensing
s enario. Probable lensgalaxytyp eidenti ati on(G-typ e)is
o dedas: eforanearly-typ eandsforaspiral-typ e.
Sour e zs zl Nim G-typ e
0218+357 0.96 0.68 0.334 2 s 0445+123 - - 1.33 2 -0631+519 - - 1.16 2 -0712+472 1.34 0.41 1.27 4 e 0850+054 - - 0.68 2 -1152+199 1.019 0.439 1.56 2 -1359+154 3.235 - 1.65 6 -(3Gs) 1422+231 3.62 0.34 1.28 4 e 1608+656 1.39 0.64 2.08 4 e(2Gs) 1933+503 2.62 0.755 1.17 4 e? 2045+265 - 0.867 1.86 4 -2114+022 - 0.32/0.59 2.57 2? e(2Gs) 2319+051 - 0.624 1.36 2 e /(S=S 0 ) with=2:070:02(1:970:14)forSS 0 (S S 0 ) and S 0
= 30 mJy. These results are
onsis-tentwiththepredi tionoftheDunlop&Pea o k(1990)
free-formmo delnumb er5radioluminosityfun tionthat
is onsistent with redshift data at S > 1 mJy [3, 4℄.
Redshiftmeasurementsforarepresentative CLASS
sub-sampleandthepredi tionoftheaforementionedDunlop
& Pea o k (1990) mo del onsistently suggest that the
redshiftdistributionfortheCLASSunlensedsour es an
b eadequatelydes rib edbyaGaussianmo delwithmean
redshift,hzi=1:27anddisp ersion0.95[3,4℄. The
prop-ertiesofthe13multiply-imagedsour es aresummarized
inTable1.
The luminosityfun tion of galaxies is assumed to b e
well-des rib edbyaS he hter fun tion[5℄oftheform
dn d(L=L ) =n L L exp ( L=L ); (1)
where is a faint-end slop e and n
and L
are
har-a teristi numb er density and hara teristi luminosity,
resp e tively. FromEq.(1)anintegratedluminosity
den-sity j is givenbyj = R 1 0 dLL(dn=dL)=n L ( +2).
The luminosity of a galaxy is orrelated with its
line-of-sightstellar velo ity disp ersion ( ) via the empiri al
relations, L L 10 0:4(M M) = ; (2) where M and
are resp e tively hara teristi
ab-sion, and orresp onds to the Fab er-Ja kson
(Tully-Fisher) exp onent for an early-typ e (late-typ e) p
opula-tion [6℄; throughout we take
Fab er Ja kson
= 4:0 and
Tully Fish er
=2:9[6℄. The2dFGRSteamandtheSDSS
teamindep endentlydeterminetheS he hter parameters
for galaxies at z <
0:2 summedover all gala ti typ es
andtheirresultsare onsistentwithea hother[5℄. The
onvergingresult anb eexpressedintheb
J photometri systemasM 5log 10 h= 19:690:04(hereafter his
theHubble onstantH
0 inunits of100kms 1 Mp 1 ), = 1:220:02,andn =(1:710:07)10 2 h 3 Mp 3 ,
whi h gives an integrated luminosity density of j =
(2:020:15)10 8 h L Mp 3
. From the total LF
we extra tthetyp e-sp e i LFs based onmeasured
rel-ativetyp e-sp e i LFskeeping thetotalluminosity
den-sity xed. The typ e-sp e i LFs are required b e ause
theearly-typ eandthelate-typ ep opulationsare
dynam-i ally di erent and ontribute to the multiple imaging
in distin tively di erent ways, namely that the
early-typ ep opulation ontributesmoretothetotallensingrate
andonaveragegenerates largerimagesplittings[4℄. We
use the relative LFs obtained from the Se ond
South-ern Sky Redshift Survey (SSRS2), whi h is based on
5404 lo al (z 0:05) galaxies. We obtain the
follow-ingearly-typ e(e)andlate-typ e(s)LFsexpressed inthe
b
J
photometri system: for the early-typ e p opulation,
M (e) 5log 10 h= 19:63 +0:10 0:11 , (e) = 1:000:09,and n (e) =(0:640:19)10 2 h 3 Mp 3
; forthe late-typ e
p opulation, (s) = 1:220:02,M (s) = 19:690:04, and n (s) = (1:200:11)10 2 h 3 Mp 3 . To break
adegenera y we have hosen (s) = and M (s) =M
sin ethetotalLFisdominatedbylate-typ egalaxies.For
alternativetyp e-sp e i LFsobtainedfromthe2dFGRS,
thereaderisreferredto[4,5℄. Whenthealternativetyp
e-sp e i LFsareused, ourderivedlimitson osmologi al
parametersareessentiallyun hanged.
The multiple imaging ross se tion of a galaxy an
b e related to and al ulated from the line-of-sight
ve-lo itydisp ersion of thegalaxy. However,the relationis
notuniquebutdep endsonthemasspro leandintrinsi
shap eof the galaxy. This dep enden e willb e absorb ed
intoa\dynami alnormalization"fa tor(f)b elow. Itis
knownthatproje tedsurfa edensitiesoftheinner
ylin-dri alregions of galaxies along the line-of-sight prob ed
bygravitationallensingarewellapproximatedbya
sin-gularisothermalmasspro leorapro lesimilartoit[4℄.
Inthis Letter, we adopt a singularisothermal ellipsoid
(SIE) as a galaxy lens mo del, whose proje ted density
anb e expressed inunits oflensing riti alsurfa e
den-sity r = 2 D OS =(4 GD OL D LS )as (x;y )= r r 2 p f(f) p x 2 +f 2 y 2 ; (3)
mo del,and r r =4 2 D OL D LS D OS (4)
isthe riti alradius forf =1. D
OL , D LS andD OS are
angular diameter distan es b etween observer (O),
lens-ing galaxy(L), and sour e (S). Cal ulationsof
dynami- alnormalizationsforaxisymmetri ases(i.e.oblateand
prolate) anb e foundin[4℄. Forthe ase ofequal
num-b ersofoblatesandprolates,j(f) 1j <
0:06forf >0:4.
Lets
m
(f)b ethelensing rossse tionforimage
multi-pli ity m (= 2;4) due to the surfa e density given by
equation (3) in units of r 2
r
(Eq. 4). Then, we have
s m (f)=[(f)℄ 2 ^ s m (f)wheres^ m
(f) anb eevaluated
nu-meri ally [4℄. The di erential probability for a sour e
withredshiftz
s
and uxdensityStob emultiply-imaged
with image multipli itym and image separation to
+d( ),duetoadistributionofinterveninggalaxies
oftyp eg (g=e,s)atz toz+dzmo deledbySIEsand
des rib ed byatyp e-sp e i S he hter LF,isgivenby
d 2 p (g ) m dzd( ) = s m (f)8 2 dt dz (1+z) 3 D OL D LS D OS 2 n 4 1 ( + +2)=2 exp " =2 # B m (z s ;S); (5) where B m (z s
;S) is amagni ation bias fa tor (i.e., an
over-representation of multiply-imaged sour es b e ause
of uxampli ations)whi hdep ends onamagni ation
probabilitydistribution and jdN =dSj and an b e
al u-lated numeri ally[4℄. InEq. (5),
isa hara teristi
image(angular)separationgivenby
=(f)8 D LS D OS 2 : (6)
Thedi erentialprobabilitywith to +d( ) an
b e obtainedfromEq.(5):
dp (g ) m d( ) = Z z s 0 d 2 p (g ) m dzd( ) dz: (7)
Thedi erentialprobabilitywithany>
min dueto galaxiesat ztoz+dz isgivenby dp (g ) m dz = s m (f)16 2 +1+ 4 dt dz (1+z) 3 n 4 D OL D LS D OS 2 B m (z s ;S) Z min d 2 p (g ) m dzd( ) d( ): (8)
Finally,theintegratedprobabilityisgivenby
p (g ) m = Z zs 0 dp (g ) m dz dz: (9)
For the sample of CLASS sour es ontaining N
L
multiply-imaged sour es and N
U
unlensed sour es, the
likeliho o dLofobservingthenumb erandtheprop erties
of themultiply-imaged sour es (Table1) and the
num-b er of theunlensed sour es giventhe statisti al lensing
mo del,isde ned by lnL= NU X i=1 ln " 1 X m=2;4 p (all) m (i) # + NL X k =1 lnÆp (one) m (k ); (10)
where the integrated probability (Eq. 9) p (all) m (i) = p (e) m (i)+p (s) m
(i) and the di erential probability (Eqs. 5,
7,or8) Æp (one) m (k )isgivenby Æp (one) m (k )=w (e) (k )Æp (e) m (k )+w (s) (k )Æp (s) m (k ) (11) with w (e) (k )+w (s)
(k ) = 1. If the k -th lensing galaxy
typ eisknowntob eanearly-typ e(late-typ e),w (e)
(k )=1
(w (s)
(k )=1). Ifthek -thlensinggalaxytyp eisunknown,
weusew (g ) (k )=Æp (g ) m (k )=[Æp (e) m (k )+Æp (s) m (k )℄ifthelens
redshiftisknown,andweusew (e)
(k )=0:8andw (s)
(k )=
0:2otherwise. FromEq.(10),a\ 2 "isde nedas 2 = 2lnL: (12) We al ulate the 2
(Eq.12) over gridsof the mo del
free parameters (see b elow; inea h gridp ointthe 2
is
minimizedoverthenuisan e parameters)anddetermine
theb est- tvalues anderrorsofthefreeparameters. We
allow most of the riti al parameters to b e free while
we x the parameters that are well-determined by
ob-servations or to whi h the 2
is insensitive. The free
parameters to b e determined by minimizingthe 2
are
as follows: present matter density
m
, present va uum
density
or darkenergy density
x
(all expressed as
fra tionsofthepresent riti aldensity)withits onstant
equation of state w = p x = x (where p x and x are its
isotropi pressure and uniform energy density),
early-typ eandlate-typ e hara teristi velo itydisp ersions (e)
and (s)
, and the meanapparent axialratioof galaxies
f. A riti albut xedparameteris theearly-typ e
har-a teristi numb er density n (e)
. However, whi hever of
the presently available hoi es of typ e-sp e i LFs we
use,thederived onstraintson osmologi alparameters
turnout tob e virtually thesame. One key assumption
wemakeisthatthe(relatively)lo allydetermined
early-typ enumb erdensityn (e)
along withthefaint-endslop e
(e)
have notevolvedsin e z 1. This app earsto b e a
valid assumptionbased on urrent eviden e and
under-standing[7℄. However,ifthereisanysigni antredu tion
FIG.1: Con den elimits inthe
m
- plane.
with the present ep o h, our presently determined
mat-ter density (dark energy density) will b e overestimated
(underestimated).
Wepresentourresults based onthetyp e-sp e i LFs
derivedusingtheSSRS2LFsandadynami al
normaliza-tionassumingequalfrequen ies of oblatesand prolates.
Figure 1 shows likeliho o dregions ina matter plus
va -uum osmology: we ndthe ombination
1:2 m = 0:40 +0:33 0:56 (68%)( +0:52 1:57
at95%).Figure2showsthe
likeli-ho o dinthew
-m
planewiththepriorthattheuniverse
is at; we nd w < 0:55 +0:18
0:11
(68%). This limitis in
agreement with other re ent results [8℄. For a at
uni-verse witha osmologi al onstant(i.e. forw= 1),we
nd m = 0:31 +0:27 0:14 (68%) ( +0:75 0:23 at 95%) +0:12 0:10
(sys-temati ). Theidenti ed systemati s aredue to the
un- ertainties in theredshift distribution and the
di eren-tialnumb er- uxdensity relationofthesour es. For the
at w = 1 ase, other tted values of theparameters
are: (e) =198 +53 37 kms 1 , (s) =117 +45 31 kms 1 , and f <0:83,allat 95%.
In on lusion,thelikeliho o dfun tion(or, 2
;Eq. 12)
of the CLASS data has well-de ned on den e regions
intheparameterspa e de nedby our osmologi aland
gala ti parameters. Our onstraintinthe
m
- plane
based solely on lens statisti s agrees with that from
Typ e Ia sup ernovae observations [2℄. Our determined
value of
m
in at osmologies is also in go o d
agree-mentwithseveralre entindep endentdeterminations[2℄.
Comparedwithprevious works inlensstatisti s [4℄,our
results imply not only p ositive but higher dark energy
density. Therefore, ourresults, whi h are based on
dif-ferent physi s, assumptions and astronomi al relations,
and indep endent sets ofdata, add tothe eviden e fora
universe with low matter density and high dark-energy
density.
FIG.2: Con den elimits inthew
-m
planein at
osmolo-gies.
[1℄E.L. Turner, Astrophys. J. Lett., 365, L43 (1990);
M.FukugitaandE.L.Turner,Mon.Not.R.Astron.So .,
253,99(1991).
[2℄See,e.g.,A.G.Riess,etal.,Astron.J.,116,1009(1998);
S.Perlmutter,etal.,1999,Astrophys.J.,517,565(1999)
(Typ e IaSup ernovae); A. Ja e, etal., Phys.Rev. Lett.,
86,3475(2001)(CMB);W.J.Per ival,etal.,Mon.Not.R.
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(mat-ter p ower sp e tra); S.M. Molnar, M. Birkinshaw and
R.F.Mushotzky,Astrophys.J.,570,1(2002)(S-Z).
[3℄S.T.Myers, etal., (to b epublished ); I.W.A. Browne, et
al., (tob e published); For thedistribution s ofsour es in
redshift and in ux density, seeD.R. Marlow, etal.,
As-tron. J., 119, 2629 (2000); J.P. M Kean, et al., (to b e
published); I. Waddington, et al., Mon. Not.R. Astron.
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Not.R.Astron.So .,247,19(1990).
[4℄K.-H.Chae,(to b epublished) and referen estherein; in
parti ular, P.Helbig, etal., Astron.Astrophys.(Suppl.),
136,297(1999)(asubsampleofCLASS);C.S.Ko hanek,
Astrophys. J., 466, 638 (1996) (primarily
opti ally-sele ted sample).
[5℄For total LFs, seeP. Norb erg, etal., Mon. Not. R.
As-tron. So ., 328, 64 (2002) (2dFGRS); M. Blanton, et
al., Astron.J., 121, 2358 (2001); N. Yasuda, etal.,
As-tron.J.,122,1104(2001)(SDSS). Fortyp e-sp e i LFs,
see R.O.Marzke, et al., Astrophys. J., 503, 617 (1998)
(SSRS2); D.S. Madgwi k, et al., Mon. Not. R. Astron.
So .,333,133(2002)(2dFGRS).
[6℄G.de Vau ouleurs and D.W. Olson, Astrophys.J.,256,
346(1982);M.Bernardi,etal.,astro-ph/0110344 (SDSS)
(Fab er-Ja kson); R.B.Tully andM.J.Pier e,Astrophys.
J.,533,744(2000)(Tully-Fisher).
[7℄See,e.g.,P.J.E.Peebles,astro-ph/0201015,andreferen es.
[8℄See,e.g.,S.Perlmutter,M.S.Turner,andM.White,Phys.
