The image separation distribution of strong lenses: Halo versus subhalo populations

0.38 MB·English

by  Masamune Oguri

Mon.Not.R.Astron.Soc.000,1–11(2004) Printed5February2008 (MNLATEXstylefilev2.2)
The image separation distribution of strong lenses:
Halo versus subhalo populations
Masamune Oguri1,2⋆
1Princeton UniversityObservatory, Peyton Hall, Princeton, NJ 08544, USA.
2Department of Physics, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo113-0033, Japan.
5February2008
6
0
ABSTRACT
0
We present a halo model prediction of the image separation distribution of strong
2
lenses. Our model takes account of the subhalo population, which has been ignored
n
in previous studies, as well as the conventional halo population. Halos and subhalos
a
are linked to central and satellite galaxies by adopting an universal scaling relation
J
between masses of (sub-)halos and luminosities of galaxies. Our model predicts that
6 10%−20%oflensesshouldbecausedbythesubhalopopulation.Thefractionoflensing
by satellite galaxies (subhalos) peaks at ∼ 1′′ and decreases rapidly with increasing
2
image separations. We compute fractions of lenses which lie in groups and clusters,
v
8 and find them to be ∼ 14% and ∼ 4%, respectively: Nearly half of such lenses are
2 expected to be produced by satellite galaxies, rather than central parts of halos. We
5 alsostudy mass distributions oflensing halos and find that evenatimage separations
8 of∼3′′ thedeviationoflensmassdistributionsfromisothermalprofilesislarge:Ator
0 beyond ∼3′′ image separations are enhanced significantly by surrounding halos. Our
5 model prediction agreesreasonablywell with observedimage separationdistributions
0 from galaxy to cluster scales.
/
h
Key words: cosmology:theory — dark matter — galaxies:elliptical and lenticular,
p
cD — galaxies: formation — galaxies: halos — galaxies: clusters: general — gravita-
-
o tional lensing
r
t
s
a
:
v
1 INTRODUCTION will be extremely useful for understanding the assembly of
i
X structures.
Image separations between multiple images play an impor-
r tant role in the statistics of lensed quasars: The image sep- In theoretical sides, there have been many attempts to
a
calculatethefullimageseparationdistribution.Historically,
aration is mainly determined by the potential wall depth
the image separation distribution is computed from either
of the lens object, thus the image separation distribution
the galaxy luminosity function (Turneret al. 1984) or the
reflectshierarchicalstructureintheuniverse.Thisfact sug-
massfunctionofdarkhalos(Narayan & White1988).While
gests that lensed quasars may categorized roughly into two
populations; small-separation lens (θ 1′′) that are pro- theformerapproachaccountsfortheobservedlensingprob-
∼ ability and itsdistribution at θ<3′′ fairly well, it has diffi-
duced by a single galaxy, and a large-separation lens (θ &
10′′) thatarecaused byclusters ofgalaxies. Thusfarabout cultyinexplainingtheexistenceofthelarge-separation lens
for which dark matter dominates the lens potential. There-
80 lensed quasars are known: Most are small-separation
fore in this paperwe take thelatter approach.
lenses and only one lens, SDSS J1004+4112 (Inada et al.
2003; Oguriet al. 2004), has the image separation larger However,modelingtheimageseparationdistributionon
than10′′.Thecurrentlargest homogeneous lenssurvey,the thebasisofdarkhalosisnotsimple.Ithasbeenfoundthat
Cosmic Lens All Sky Survey (CLASS; Myerset al. 2003; none of models that consider only one population for lens-
Browne et al. 2003), contains small separation only, thus it ing halos are consistent with current observations. Correct
may not be suitable for studying the image separation dis- modelsneedtoincludeacharacteristicmassaroundMcool
∼
tribution from galaxy to cluster scales. However, ongoing 1013M⊙, where the density distribution inside dark halos
lens surveys such as that in the Sloan Digital Sky Survey changes(Porciani & Madau2000;Kochanek & White2001;
(SDSS) are expected to unveil the full distribution, which Keeton 2001; Sarbu et al. 2001; Li & Ostriker 2002, 2003;
Oguri 2002; Ma 2003; Huterer& Ma 2004; Kuhlen et al.
2004; Chen 2004; Zhang 2004; Chen 2005; Zhang et al.
⋆ E-mail:[email protected] 2005): Large mass halos M > Mcool (correspond to groups
(cid:13)c 2004RAS
2 M. Oguri
or clusters) havecooling time longer than theHubbletime, We compute the image separation distributions in 4, and
§
thus the internal structure of dark halos is not strongly af- show the results in 5. Finally, we summarize and discuss
§
fected by baryon cooling and is well described by that of our results in 6. Throughout the paper we assume a con-
§
dark matter (Navarro et al. 1997). On the other hand, in cordance cosmology with the mass density Ω = 0.3, the
M
small mass halos M < M (correspond to galaxies) the cosmological constant Ω = 0.7, the dimensionless Hubble
cool Λ
baryoncoolingisveryefficientandthemassdistributionbe- parameter h=0.7, and the normalization of the mass fluc-
comesstronglycentrallyconcentrated,whichiswellapprox- tuation σ =0.9.
8
imated by the Singular Isothermal Sphere (SIS). Although
this two-component model is successful in explaining the
image separation distribution to some extent, clearly it is
2 MODELING DARK HALOS AND
not sufficient. First of all, this model considers only galax-
SUBHALOS
ies that lie at the center of isolated dark halos, and thus
does not take galaxies in groups and clusters into account. 2.1 Dark Halos
Put another way, this model neglects substructures (sub-
For the density profile of original dark halos, we adopt the
halos) which should corresponds to satellite galaxies. Actu-
spherical NFW profile(Navarro et al. 1997):
ally, it has been speculated that lens systems in such dense
ρ
environments is quite common (Keeton et al. 2000). In ad- ρ(r)= s . (1)
dition, sometimes lens systems are significantly affected by (r/rs)(1+r/rs)2
the mass distribution outside the lens object. A good ex- The characteristic density ρ is computed from the nonlin-
s
ampleisQ0957+561(Walsh et al.1979);itslensobjectisa ear overdensity ∆ . The scale radius r is related to the
vir s
galaxyinacluster,andtheimageseparation ismuchlarger concentration parameter c=r /r , where r is the virial
vir s vir
thanexpectedfromtheluminosityofthelensgalaxybecause radius.Themedianconcentrationparameterdependsonthe
theimageseparationisboostedbytheclusterpotential.In- massM andredshiftzofthedarkhalo:Weadoptthefitting
deed, it has been shown theoretically that at intermediate- form derived by Bullock et al. (2001):
separation regime (θ=3′′ 7′′) lensgalaxies in denseenvi-
ronments are quite commo−n (Oguri et al. 2005). Therefore 10 M −0.13
c¯= , (2)
thecorrect model needsto account for such externalmass. 1+z M∗
(cid:16) (cid:17)
To construct a model that is based on dark halos and where M∗ is determined by solving σM = δc at z = 0 (σM
subhalos, we have to model the relation between mass of is the standard deviation of linear density field smoothed
dark halos (subhalos) and luminosities of galaxies inside with a top-hat filter enclosing mass M, and δ 1.68 is
c
them. There are several models to link them: Conditional thethresholdlinearoverdensityabovewhichther≈egion col-
luminosity function (CLF) which is defined by the lumi- lapses).Wealsoincludethescatterintheconcentrationpa-
nosity function in a halo of given mass (Yanget al. 2003; rameter, which is well described by a log-normal distribu-
van den Bosch et al. 2005), universal mass-luminosity rela- tion:
tion between halo/subhalo mass and hosted galaxy lumi-
1 (lnc lnc¯)2
nosity(Vale& Ostriker2004),andthemixtureofthesetwo p(c)= exp − , (3)
models(Cooray & Milosavljevi´c2005;Cooray & Cen2005). √2πσlncc (cid:20)− 2σl2nc (cid:21)
There models are calibrated to match observed galaxy lu- where thestandard deviation is σ =0.3.
lnc
minosity functions, clustering of galaxies as a function of In this paper, we consider the modification of mass
galaxy luminosities, andtheluminosity function of galaxies distribution in dark halos due to baryon cooling. Tra-
in groups or clusters. ditionally, the response of dark matter to baryon infall
In this paper, we study the image separation distribu- has been calculated by the model of adiabatic contraction
tion with a particular emphasis on the different contribu- (Blumenthal et al. 1986). The model assumes the spherical
tionsfromcentralandsatellitegalaxies.Forcentralgalaxies, symmetryandcomputestheresponsebyimposing thecon-
we consider baryon cooling in a dark matter halo as done servation of angular momentum between before and after
by Kochanek & White (2001) and Keeton (2001). Satellite baryoninfall. Recently,Gnedin et al. (2004) studiestheva-
galaxies, whichhavebeen neglected inpreviousstudies,are lidity of the model using high-resolution numerical simula-
linkedtodarkhalosubstructures.Webasicallytaketheap- tions,and found that themodel can beimprovedbytaking
proach proposed by Vale & Ostriker (2004) to relate mass theeccentricorbitsofparticlesintoaccount.Theyalsopre-
ofdarkhalosandsubhaloswithluminositiesofgalaxies,but sented a series of analytic fitting functions which make it
wemodifyittoaccountfortherelativelysmallabundanceof much easier to implement the modification. We adopt this
satellitegalaxiesingalactichalos(seeCooray & Cen2005). modified adiabatic compression model to derive the total
For satellite galaxies, theexternal mass from thehost halo, mass distribution. For the final mass distribution of the
which is shown to beimportant at large-image separations, cooled baryonic component, we assume Hernquist (1990)
is taken into account. In addition to the difference between profile:
central and satellite galaxies, we also pay particular atten-
M 1
tion to how lens objects make the transition from SIS to ρ (r)= b , (4)
b 2π (r/r )(r +r)3
NFW as halo masses increase. b b
This paper is organized as follows. In 2, we introduce whichisknowntodescribethestellardensityprofileofellip-
§
modelsofdarkhalosandsubhalosthatareusedtocompute ticalgalaxies well. Notethattocomputethefinalmassdis-
lensingprobabilities.Section3aredevotedtomodelthere- tributionweneedtospecifythecooledmassM (orequiva-
b
lation between (sub-)halo masses and galaxy luminosities. lentlymassfractionf M /M)andthecoreradiusr (or
b b b
≡
(cid:13)c 2004RAS,MNRAS000,1–11
The image separation distribution of strong lenses 3
equivalently the ratio of the core radius to the scale radius
x r /r ), in addition to halo parameters such as c and
b b s
≡
M.
For the mass function of dark halos, we use the form
proposed by Sheth& Tormen (1999):
dn ρ(z)dlnσ−1
= M (5)
dM M2 dlnM
2a σ2 p δ aδ2
A 1+ M c exp c , (6)
× rπ (cid:20) (cid:18)aδc2(cid:19) (cid:21)σM (cid:20)−2σM2 (cid:21)
where ρ(z) is the mean matter density at redshift z, A =
0.322, a = 0.707, and p = 0.3. The mass function is the
modification of Press & Schechter (1974) to account for el-
lipsoidalnatureofgravitationalcollapseincosmological sit-
uations, and is known to reproduce the mass distribution
and its redshift evolution in numerical simulations (e.g.,
Reed et al. 2003).
2.2 Subhalos
WeconsidersubhaloswhichlieinadarkhalowithmassM.
Themassof thesubhaloisdenotedbym and thedistance Figure 1. The comparison of the cumulative number distribu-
s
from the halo center by r. We always adopt an SIS for the tionsofsubhalosderivedfromananalyticmodelofOguri&Lee
(2004)withthefittingfunction(obtainedbyintegratingequation
densitydistributionofgalaxies associated withsubhalos for
simplicity.Indeedithasbeenknownthatthecentralpartof (7)overms).Theformerisshownbyhistograms,andthelatter
is plotted by a solid line. We plot the distributions of the ana-
thedensityprofileoftotal(darkmatterpluscooledbaryon)
lyticmodelforthreedifferentmassesofthehosthalos.Notethat
matter in a dark halo after baryon infall is quite close to
thecumulativemassdistributionderivedfromthefittingfunction
an SIS. For the mass and spatial distributions of subhalos, does not depend onthe mass of thehost halowhen itis plotted
we use a model constructed by Oguri & Lee (2004). In the asafunctionofms/M.
paper,theyshowedthattheanalyticmodelreproduceswell
theresultsofhigh-resolutionnumericalsimulations.Inreal-
ity,weusethefollowingfittingformulaeofmassandspatial with the model of Oguri & Lee (2004). They agree reason-
distributions to speed up the computation. ably well with each other for a range of parameters we are
The fitting function of the numberdistribution of sub- interested in.
structures was given byVale & Ostriker (2004):
−β
dN 0.18 m m 1
= s exp s , (7) 3 LINKING DARK HALOS AND SUBHALOS
dms γΓ(2 β) γM −γM γM
− (cid:18) (cid:19) (cid:18) (cid:19) TO GALAXIES
where β = 1.91 and γ = 0.39. The mass function has a
WeconsideracentralgalaxywithluminosityLwhichresides
power-law form at m M, and has a sharp cut-off at
s
≪ inadarkhalowithmassM.Weadoptthemass-luminosity
m M. In addition, we assume that dN/dm = 0 at
s s
∼ (b band) relation proposed by Vale& Ostriker(2004):
m > M. We compare this fitting function with the re- J
s
sultofOguri& Lee(2004)inFigure2,confirmingthegood Mp
accuracy of the fittingfunction. L(M)=5.7×109h−2L⊙ 11 1/s, (9)
Ithasbeenshownthattheradialnumberdensitydistri- q+M(p−r)s
11
butions of subhalos are less centrally concentrated relative h i
where p = 4, q = 0.57, r = 0.28, s = 0.23, and M
tothedarkmatterparticlecomponents.Moreover,theyde- 11
pendonmassesofsubhalos:Moremassivesubhalosarepref- M/(1011h−1M⊙). Basically this relation asymptote to L ≡
M4 at low mass end and L M0.28 at high mass end∝.
erentially located in theexternal regions of their host halos
∝
Vale& Ostriker (2004) claimed that the same relation can
(DeLucia et al. 2004; Oguri & Lee 2004; Gao et al. 2004).
be applied to dark halo substructures (with mass m ) and
Wefindthatthespatialdistributionofsubhalosisdescribed s
satellite galaxies (with luminosity L ):
byanNFWprofile(eq.[1])withdifferentconcentrationpa- s
rSapmeceitfiecra(llwy,hitchhedceopnecnedntorantimons)pfarroammettheartoofftthheerhaodsitalhdalios-. Ls(ms)=5.7×109h−2L⊙ mp11 1/s, (10)
tribution of subhalos is fitted by q+m(p−r)s
11
c∗ =c 1+ 3ms/M 0.2 −1, (8) wherem11 ≡m/(1011h−1Mh⊙)andmistihemassofsubhalos
10−7 before the tidal stripping of the outer part of subhalos. We
" (cid:18) (cid:19) # adopt theapproximation relation m=3m ,which was also
s
where c is the concentration parameter of the correspond- assumed in Vale & Ostriker (2004), throughout the paper
ing host halo. Figure 2 shows the comparison of this fitting for simplicity.
(cid:13)c 2004RAS,MNRAS000,1–11
4 M. Oguri
Figure2.Thecumulativeradialdistributionsofsubhalosinananalyticmodel(Oguri&Lee2004)arecomparedwiththoseofanNFW
fitwithconcentrationparametersgivenbyequation(8).Theformerdistributionsareplottedwithhistograms,whilethelatterareshown
bycurves.Fromlefttorightpanels,wechangethemassofthe hosthalo.Differentlinesindicatedifferentmassesofsubhalos.
While Vale & Ostriker (2004) adopted monotonic one- M & 1014h−1M⊙ the efficiency function is almost always
to-one correspondence between halo/subhalo masses and unity.
residentgalaxy luminosities,Cooray & Milosavljevi´c (2005) We neglect the redshift evolution of the relations pre-
showed that it is essential to assign log-normal scatter to sented in this section, since most lens galaxies are at z <1
the mass-luminosity relations to recover the tail of the ob- where theevolution is expected to be mild.
served luminosity function. Following them, we assume the
log-normal luminosity distribution of
1 log [L/L(M)]2
p(LM)= exp 10 , (11)
| √2πln(10)ΣL − 2Σ2 4 COMPUTING THE IMAGE SEPARATION
(cid:26) (cid:27)
DISTRIBUTION
with Σ = 0.25, and the same distribution with mean of
Ls(ms) for Ls. Tocomputelensingcrosssections,wehavetoconvertgalaxy
Although this model is successful in explaining many luminosities to any quantities which characterize the mass
observations such as the occupation number, the luminos- distributionofgalaxies.Todoso,weusethescalingrelations
ity function of clusters and the group luminosity function, betweenthegalaxyluminosityL,velocitydispersionσ,and
it cannot reproduce the strong dependence of the faint effectiveradiusR .Specifically,weadoptthosederivedfrom
0
end slope of the CLF on the halo mass: Observationally, 9,000early-typegalaxiesintheSDSS(Bernardi et al.2003):
it changes from 1.2 at the cluster scale to 0 at
galactic scales. Sin∼ce−the mass function of subhalos∼at low L = σ 4.0, (14)
mmaassss-liusmdinno/sdimtys r∝elatmio−snβLwh(mere)β ∼m1η.9yi(esledes§t2h.2e),CtLhFe 1010.2h−702L⊙ (cid:16)102.197rms−1(cid:17)
Ls−1−0.9/η. Therefore thesabovse m∝ass-lsuminosity relation L = R0 1.5, (15)
i∝mplies the faint end slope of ∼ −1.2. It is also found that 1010.2h−702L⊙ (cid:18)100.52h−701kpc(cid:19)
evenifwemodifythevalueofηwecannotmakethefaintend
where L is g-band galaxy luminosity and we defined h
slope shallower than 1.Hencetoaccount for theobserved 70 ≡
− h/0.7. Note that the effective radius is related to the core
trend we need extra mechanism. Following Cooray & Cen
radius of Hernquist profile (eq. [4]) by r = 0.551R . For
(2005), we introduce an efficient function to suppress the b 0
simplicity,weneglect tinydifferenceof b -bandandg-band
numberof satellite galaxies in galactic halos: J
luminositiesandregardLintheaboveequationsasthesame
f(ms)=0.5 1+erf logmσ−mlogmc , (12) lcuamnindoesscitriiebseasthtehocsoerrinela§t3i.oAnltmhoorueghtitghhetlfyu,nwdeamaednotpatlpthlaensee
whereσ =h1.5andt(cid:16)hecutoffmassm(cid:17)idependsonthehost relations between two observables because they are much
m c
halo mass M: easier tohandle.Wealso convertluminosities of galaxies to
stellar masses assuming the universal stellar mass-to-light
2
mc =1011h−1M⊙exp"−(cid:18)5×101M3h−1M⊙(cid:19) #. (13) ratioOonfeΥo=f t3h.0eh7m0oMst⊙/imL⊙p.ortant ingredient in computing
lensing probabilities is magnification bias which is deter-
For less massive (M . 1013h−1M⊙) halos, the efficiency mined by the source luminosity function. Below we assume
function behaves as f(ms) 0 when ms 1011h−1M⊙ a power-law Φ(L) L−ξ, which makes the computation of
and f(ms) 1 when ms →1011h−1M⊙. A≪t cluster scales magnification bias∝easier.
→ ≫
(cid:13)c 2004RAS,MNRAS000,1–11
The image separation distribution of strong lenses 5
4.1 Dark Halo Component 1 1+ 2 arctanh 1−x (x>1)
= 1−x2 − √1−x2 1+x (23)
sAiosndimscoudsesleodfiGnn§2e.d1i,nweetuasle.(t2h0e0i4m)ptroocvoemdpaudtiaebtahteictoctoamlpmraesss-  x21−1h1− √x22−1arctan xxq−+11 i (x>1).
distribution of a galaxy plus dark halo and hence to derive h q i
Fromthese, we compute the image separation distribution
itslensingcrosssection.First,wederivetheluminosityofa
as
given halo with mass M using the mass-to-luminosity rela-
dP cdt dn dN
tnioornm(a9l)i,zeadndcoersetirmadaituestxhbefcrooomletdhebastryelolanrfmraacstsio-tno-flbigahntdratthioe dθs = Z dzl(1+zl)3dzl Z dMdM Z dmsf(ms)dms
Υ and equation (15), respectively. From the mass distribu-
dκ p(κ )σ δ(θ θ(κ )). (24)
tion,wecomputeanimageseparationθandbiasedcrosssec- × ext ext lens − ext
tion σ : We use approximations proposed by Oguri et al. Z
lens
As in 4.1, the final probability distribution is obtained by
(2002) to compute these quantities. Then the image sepa- §
averagingequation(24)overthePDFsoftheconcentration
ration distribution can beobtained byintegrating thecross
parameterofhosthalos(eq.[3])andthemass-to-luminosity
section:
relation(eq.[11]).Wesettheupperlimitoftheintegralover
dP cdt dn
dθh = dzl(1+zl)3dzl dMdMσlensδ(θ−θ(M)). (16) 0κ.e9x)tstuob0h.a9l,obseschaouusledifnallthinesiedveenthtsebcreiytoicnadlrtahdeiulismoiftt(hκeexhtos&t
Z Z
The final probability distribution is obtained by averag- haloandthereforetheyshouldbeincludedinlensingbydark
ingequation(16)overtheprobabilitydistributionfunctions halos computed in 4.1.
§
(PDFs) of the concentration parameter and the mass-to- Wenotethattheinclusion of theefficient function(eq.
luminosity relation. [12])impliesthatapartoflow-masssubhalosremainsdark,
i.e., does not harbor a galaxy. We neglect lensing by these
dark subhalos because their contribution to total lensing
4.2 Subhalo Component
probabilitydistributionsisrathersmallunlesstheinnerpro-
fileofsubhalosisverysteep(Ma2003;Li & Ostriker2003).
The projected mass distribution of a subhalo is approxi-
Thetotalimage separation distribution issimplygiven
mated as the sum of an SIS and external convergence κ
ext
bya sum of equations (16) and (24):
originates from the mass associated with the host halo. We
only consider the external convergence and neglect the ef- dP dP dP
t = h + s. (25)
fect of the external shear, since it has been shown that the dθ dθ dθ
external convergence is more effective to modify the im-
age separation distribution (Oguri et al. 2005). From the
velocity dispersion σ, the image separation θ and the bi- 5 THE IMAGE SEPARATION DISTRIBUTION
ased cross section σ can be computed analytically (see
lens In this section, we compute image separation distributions
Keeton & Zabludoff 2004, for the dependenceon κ ):
ext forhalos(centralgalaxies) andsubhalos(satellitegalaxies).
2θ 8π(σ/c)2 D Inthespecificexamplesbelow, wefixthesourceredshift to
θ= E = ls, (17)
(1 κext) (1 κext)Dos zs = 2.0 which is typical for lensed quasars. The magnifi-
− −
cation bias is computed assuming a power-law luminosity
2ξ(1 κ )−2(ξ−1)
σlens=π(DolθE)2 −3extξ , (18) function with ξ = 2.1 which is the luminosity function of
the CLASS survey (Myers et al. 2003). First we will show
−
whereD istheangulardiameterdistancefromtheobserver the image separation distributions for dark halos and sub-
os
to the source, etc. The external convergence is computed halos separately, and next we see the contribution of each
from the mass distribution of the host halo after baryon component to thetotal distribution.
cooling ( 2.1). Since the external convergence is written as
§
afunctionoftheprojectedradiusR,thePDFoftheexternal
5.1 Dark Halo Component
convergence becomes
dκ Before going totheimage separation distribution, wecheck
p(κ )dκ =f (R) extdR, (19)
ext ext R dR the image separation as a function of halo masses to see
theeffectofbaryoninfall onlensing,whichisshowninFig-
where f (R) is the PDF of subhalos projected along line-
R ure 3. Two special cases are also shown for reference: One
of-sight,whichweadopttheprojectedNFWprofilewithits
is neglecting baryon infall and leaving halos as NFW, and
concentration parameter given by equation (8):
the other is adopting an SIS approximation with the ve-
1 π/2 c∗R/r c∗ locity dispersion inferred from theluminosity of thecentral
f (R)dR= dφg vir dR, (20)
R h(c∗) sinφ r galaxy.Itisfoundthattheimageseparationsagreewellwith
Zφmin (cid:18) (cid:19) vir those of SIS at low-mass ends, which is consistent with ob-
φ =sin−1 R (21) servations(e.g.,Rusin et al.2003;Rusin & Kochanek2005).
min rvir Withincreasing masses, image separations begin todeviate
g(x)= (1+xx(cid:16))2, (cid:17) (22) (frMom&SI3S fr1o0m14hθ−∼1M1⊙′′)(iMmag∼e s1e0p1a2rha−ti1oMns⊙)a.reAqtuθite&si1m0i′-′
×
lar to those of NFW. This figure justifies to some extent
x
thetwo-populationmodelinwhichSISandNFWlensesare
h(x)= dyg(y)
considered, but we note that at M 1013 1014h−1M⊙
Z0 ∼ −
(cid:13)c 2004RAS,MNRAS000,1–11
6 M. Oguri
Figure 3. The image separations of lensing by dark halos after
baryon cooling are plotted as a function of halo masses. In this
Figure 5. The probability distribution of “effective” external
plot,wedonottakethePDFsofthemass-to-luminosityrelation
convergence κeff (eq. [26]) which is a measure of the deviation
and concentration parameter into account. The lens redshift is
of the halo mass distribution from an SIS approximation of the
fixed to zl = 0.5. For comparison, we also show cases that we correspondingcentralgalaxy.Wederivethedistributionsforsev-
neglect baryon infall (NFW; dashed line) and we adopt an SIS eral representative image separations: θ = 0′.′3 (upper left), 1′′
approximationforthemassdistributionofthehalosafterbaryon (upper right),3′′ (lower left),and6′′ (lower right).
infall(SIS;dotted line).NotethatthevelocitydispersionforSIS
isderivedfromthescalingrelationofequation(14).
probabilities are close to those of SIS and NFW, respec-
tively,but at 3′′ .θ.10′′ it is difficult to describe lensing
probabilitiesaseitherSISorNFWalone.Thelensingprob-
abilities also differ from the sum of those of SIS and NFW.
Therefore, we conclude that the image separation distribu-
tioncannotbedescribedbyasimpletwo-populationmodel:
We need to model the transition of halo mass distributions
from SIStoNFWtopredictcorrectlythedistributionfrom
small- to large-image separations.
We explore the transition from SIS to NFW more in
terms of an “effective” external convergence definedby
θ
κ 1 SIS, (26)
eff ≡ − θ
where for each halo θ is computed from the luminosity
SIS
of the central galaxy via equation (14). The case κ = 0
eff
means that the Einstein radius of the halo is exactly the
same as that inferred from the luminosity of the central
galaxy.Thedefinitionismotivatedbythedescriptionofthe
image separation in the SIS plus external shear model (eq.
[17]). We compute the PDFs of κ for several represen-
eff
tative image separations, which are shown in Figure 5. As
expected,atθ.1′′lenshalosareonaverageclosetoanSIS,
Figure 4.Theimageseparationdistributionfromthedarkhalo
component (eq.[16]).LinesaresameasFigure3. being consistent with observations (e.g., Rusin et al. 2003;
Rusin & Kochanek 2005). However, at θ & 3′′ they clearly
have larger image separations than those of corresponding
things are too complicated to be described by such simple SISlenses.ThereforethisFigurerepresentsanothercasefor
two-populationmodel:Centralgalaxiesaremainlyresponsi- the deviation from simple SIS lenses at that image separa-
blefor lensing, buthalo masses outside thelensing galaxies tion region: At or beyond θ&3′′ the image separations are
boosttheimageseparationsignificantly,makingtheSISap- enhanced significantly by surroundingdark halos.
proximation inaccurate. We also point out that κ has rather broad distri-
ext
Weshowtheimageseparation distributioninFigure4. butions even at small-separations θ . 1′′, which comes
As in Figure 3, at small- and large-separation regions the from the scatters of the concentration parameter (eq. [3])
(cid:13)c 2004RAS,MNRAS000,1–11
The image separation distribution of strong lenses 7
Figure 6. The image separation distribution from the subhalo
component (eq. [24]) is plotted by a solid line. We also calcu-
latethedistributionwithoutincludingκext,whichisshownbya sFuibghuarleop7o.pTuhlaetidoinst.rTibhuetiPoDnFosf feoxrtesrenvaelraclodnivffeerrgeenntciemκaegxetsfeopratrhae-
dottedline. tionsareshown:θ=0′.′3(upper left),1′′ (upper right),3′′ (lower
left),and6′′ (lower right).
and the mass-to-luminosity relation (eq. [11]) around their
medians. This scatter around SIS approximations may be
able toexplain possible heterogeneous natureof lens galax- genceonaverage thancentralgalaxies. Ourresultindicates
ies inferred from the combination of gravitational lensing thatitisveryimportanttoincludetheexternalconvergence
and stellar dynamics (Treu & Koopmans 2004) or time de- of satellite galaxies, which originates from the host halo, to
laymeasurementsbetweenmultipleimagesoflensedquasars make reliable predictions of the image separation distribu-
(Kochanek et al. 2005). tion for thesubhalo component.
It is interesting to compute the posterior PDF of κ
ext
tosee expected environmentsof lensed satellite galaxies, as
5.2 Subhalo Component donebyOguri et al.(2005).InFigure7,weshowthePDFs
forseveraldifferentimageseparations.Asexpected,theyare
Weshowtheimageseparationdistributionfromthesubhalo
strong function of the image separation: Larger-separation
component in Figure 6. We find that the distribution has a
lensestendtolieinmoredenseenvironments.Forinstance,
peakataroundθ∼0.5′′,andrapidlydecreaseswithincreas- at θ . 1′′ only 10% of lensing by satellite galaxies have
ing θ. It is also seen that the shape of the curve changes at large external c∼onvergence κ > 0.1, while at θ 3′′
around θ 5′′ over which the probability decreases less ext ∼
∼ more than half of lenses are accompanied by such large ex-
rapidly. We attribute this behavior to the contribution of ternal convergence. Lenses with θ 6′′ are dominated by
very large external convergence (κext & 0.5), because the those with very large κ , which m∼ean that lensing of such
probabilitiesaredominatedbysuchextremeevents(seebe- ext
large image separations occurs only when satellite galaxies
low). This means that the result is somewhat sensitive to
lie close to the center of their host halos on the projected
the upper limit of κ which we set 0.9. However, as we
ext two-dimensional lens plane.
will see in the next subsection, this does not cause any sig-
nificant problems since at that image separation region the
fraction of subhalo lensing in the total lensing probabilities
5.3 Total Distributions
are rather minor.
As shown in Oguri et al. (2005), the external conver- Inthissubsection,weputtogethertheresultsintheprevious
genceaffectstheimageseparationdistributionsignificantly, subsections to derive the total image separation of strong
particularly at the tail of the distribution θ & 3′′. To show lenses (eq. [25]).
thisexplicitly,inFigure6wealsoplotthedistributionwith- Figure 8 plots the total image separation distribution
outincludingκ .Itisfoundthattheexternalconvergence as well as distributions for the halo and subhalo popula-
ext
enhancesthelensingprobabilitiesatθ&1′′ wheretheprob- tions.Wefindthat thehalo population dominates thelens-
abilitiesaredecreasingfunctionofθ.Thisisconsistentwith ing probability at all image separations. However, the con-
the finding by Oguriet al. (2005). The enhancements are tribution of the subhalo population is also significant: The
quite significant; factors of 6 at θ = 2′′ and 80 at fractiontakesthemaximumvalueof 0.2ataroundθ=1′′,
θ = 3′′. Note that these num∼bers are much larger t∼han the anddecreasesrapidlyatθ&3′′.Ther∼efore,weconcludethat
results of Oguri et al. (2005), because here we concentrate the subhalo population should not be ignored for the accu-
onsatellitegalaxiesthatshouldhavelargerexternalconver- rate prediction of the image separation distribution. Since
(cid:13)c 2004RAS,MNRAS000,1–11
8 M. Oguri
Table1.Thefractionoflensingprobabilities,integratedoverthe
image separation. The entry “total” indicates a sum of lensing
probabilitiesforhaloandsubhalopopulations.
type galaxy group cluster
([h−1M⊙]) (<1013) (1013−1014) (>1014)
halo 0.74 0.08 0.02
subhalo 0.08 0.06 0.02
total 0.82 0.14 0.04
the differences may be ascribed to the different definitions
of “group” and “cluster”).
How well does the total image separation distribution
derivedhereaccountforobservedimageseparationdistribu-
tions?Wecompare ourmodel prediction with thefollowing
two observed distributions:
All 75 lensed quasars discovered to date. This sam-
•
ple includes both optical and radio lenses and are quite
heterogeneous, but it has an advantage of the large num-
beroflensesincludingseveralintermediate-separationlenses
(3′′ .θ.7′′)aswellasonelarge-separationlenswhosesep-
aration is θ 15′′.
∼
22 radio lenses discovered in the CLASS (Myers et al.
Figure8.Upper:ThetotalimageseparationdistributiondPt/dθ •
(eq.[25])isplottedbyasolidline.Thecontributionsofhaloand 2003). Although this sample contains smaller number of
subhalo components are also shown by dashed and dash-dotted lenses (and all lenses have θ < 5′′), it is much more homo-
lines, respectively. Lower: The ratio of the distribution for the geneous and has well-defined selection functions. Although
subhalopopulationtothetotal distribution,(dPs/dθ)/(dPt/dθ), amongthe22lenses13wereselectedforastatistically well-
isplottedasafunctionoftheimageseparation. definedsubsample(Browne et al.2003),weuseall22lenses
simply to increase statistics.
moststronglenseshaveimageseparationsof 1′′ 2′′,our The results are shown in Figure 10. We find that our
∼ − modelexplainsbothobserveddistributionsreasonablywell.
model predicts that 10% 20% of lenses are produced by
− An exception is that for all lenses the observed number of
satellite galaxiesand80% 90% iscaused bycentralgalax-
− sub-arcsecond (θ < 1′′) lenses appears to be smaller than
ies.
expected. However, this is clearly due to a selection effect
Anadvantageofourhaloapproachistheability toob-
thatsub-arcsecondlensesaredifficulttofindparticularlyin
tain further insight into the image separation distribution
opticallenssurveys.Indeed,fortheCLASSlenses, which is
by computing the contribution of the image separation dis-
alenssurveyinaradioband,thediscrepancyislesssignifi-
tributionfromdifferentmassintervals.Forthispurpose,we
cant.Thusweconcludethatourmodelwellaccountsforthe
consider the following three types of halos, “galaxy” (de-
fined by (host-)halos with masses of M < 1013h−1M⊙), observed image separation distributions.
“group” (1013h−1M⊙ < M < 1014h−1M⊙) , and “clus- It should be noted that the theoretical predictions is
ter” (1014h−1M⊙ < M), and explore the contribution of based on many assumptions. The most important ingredi-
ent of our model is the mass-to-luminosity relation defined
eachtype.WeshowtheresultinFigure9.Wefindthatthe
inequations(9)and(10).Toseehowtheresultisdependent
distributions of group and cluster have tails toward small-
on the adopted relation, in Figure 11 we show the depen-
separations because of subhalo populations. Each type has
denceofthetotalimageseparationdistributiononthemass-
quite similar distributions of the subhalo population, while
to-luminosity relations. As seen in the Figure, the shape
those of the halo population are very different with each
(ratherthantheoverallnormalization) oftheimagesepara-
other. We also integrate the distributions over the image
tiondistributionisquitesensitivetothemass-to-luminosity
separation to estimate the fraction of each component to
relation. This indicates that precise measurements of the
the total lensing probability, which is summarized in Table
imageseparation distributionofferquiteusefulprobeofthe
1. Again, the fractions are very different for halo popula-
mass-to-luminosity relation.
tions,butthedifferencesaremuchsmallerforsubhalopop-
ulations.Asaresult,nearlyhalfoflenseswhichlieingroups
and clusters are caused by satellite galaxies (subhalos), in
contrastwithlensesingalactichaloswhicharedominatedby
6 SUMMARY AND DISCUSSIONS
thehalopopulation. Finally,wecomputefractions of lenses
in lie in groups and clusters and find that they are 14% Inthispaper,wehaveconstructedamodeloftheimagesep-
∼
and 4%, respectively. These are roughly consistent with aration distribution. Our model is based on dark halos and
∼
Keeton et al. (2000) who predicted that 20% and 3% subhaloswhicharelinkedtocentralandsatellitegalaxiesre-
∼ ∼
of lenses lie in groups and clusters, respectively (a part of spectively via simple mass-to-luminosity relation. For dark
(cid:13)c 2004RAS,MNRAS000,1–11
The image separation distribution of strong lenses 9
Figure 9. The contributions of different types of halos on the image separation distribution. We consider the following three types:
“galaxy”whichisdefinedby(host-)haloswithmassesofM <1013h−1M⊙,“group”by1013h−1M⊙<M <1014h−1M⊙,and“cluster”
by1014h−1M⊙<M.Left:Thetotal(haloplussubhalopopulations)imageseparationforeachtypeisplotted.Thesumofdistributions
of three types is shown by thick solidline. Right: For each type, contributions of halo (dashed) and subhalo (dash-dotted) populations
areshownseparately.
Figure10.Observedimageseparationdistributions(histograms)arecomparedwiththetheoreticalpredictionpresentedinFigure8.We
considertwoobserveddistribution:(i)Thedistributionofall75lensedquasarsdiscoveredtodate,and(ii)thatof22lensesdiscoveredin
theCLASS(Myersetal.2003).Theobserveddistributionswereshiftedverticallytomatchtheirnormalizationstothetheoreticalone.
halos and central galaxies, we haveconsidered baryon cool- beproducedbysatellitegalaxies(subhalos).Thefractionof
ing in a dark matter halo usingan improved adiabatic con- the subhalo population takes the maximum at θ 1′′, and
∼
tractionmodelofGnedin et al.(2004).Forsatellitegalaxies, becomes smaller with increasing image separations.
wehaveconsidered themass associate with theirhost halo.
The mass distributions of lensing halos at small (θ .
Our primary interest is to quantify the contribution of the 3′′•) and large (θ & 10′′) image separation regions are close
subhalo population which has been ignored in all previous tothoseofSISandNFW,respectively.At3′′ .θ.10′′ the
studies. Ourresults are summarized as follows:
massdistributionsarecomplicatedandcannotbedescribed
byeither SIS or NFW.
Wepredict that 80% 90% of lenses should be caused
• −
by central galaxies (halos) and 10% 20% of lenses should For both halo and subhalo populations, already at
− •
(cid:13)c 2004RAS,MNRAS000,1–11
10 M. Oguri
Figure 11.Thedependence of thetotal imageseparation distributionon themass-to-luminosityrelations L(M)(eq. [9])andLs(Ms)
(eq. [10]). Left: We change the overall normalizations of L(M) and Ls(Ms) by multiplying numerical factors to them. From upper to
lowerlines,wemultiply100.5,100.25,1,10−0.25,and10−0.5.Right:WeshiftthecharacteristicmassscaleM11 andm11 inthemass-to-
luminosityrelationsbymultiplyingthesamefactorstobothM11 andm11 (upper linescorrespondtothecasesthatsmallerfactorsare
multiplied).
θ 3′′ the effect of dark halos becomes significant, mak- Thefractionoflensingbythesubhalopopulation,whichhas
∼
ingasimpleSISapproximationworse.Thismeansapplying beenderivedforthefirsttimeinthispaper,isimportantfor
isothermalprofilestotheselensesresultsinbiasedestimates notonlyanaccuratepredictionoftheimageseparationdis-
of parameters such as velocity dispersions and the Hubble tribution but also interpreting results of mass modeling of
constant from time delays. individual lens systems since central and satellite galaxies
Our model predicts that lensing halos of small sepa- may show rather different lensing characteristics.
•
ration lenses are heterogeneous: Halo-by-halo differences of
theshapeofrotationcurvesandthefractionofdarkmatter We note that there are several ways to improve our
at image positions are large. This is a natural consequence modeling.First,wehaveconsideredearly-typegalaxiesonly.
ofthescattersoftheconcentrationparameterandthemass- Althoughthiscanbejustifiedbecausemostoflensinggalax-
to-luminosity relation around their medians. iesare early-type,theveryexistenceof lensing bylate-type
Incomputingtheimage separation distribution for the galaxiesindicatesthatthecorrectmodelneedstotakeboth
sub•halo component, it is important to take account of the the two galaxy types into account. Since the lensing by
external mass which comes from their host halos. This also spiral galaxies has quite small image separation θ . 1′′
implies very strong dependence of environments of lensing and also is inefficient (Keeton & Kochanek 1998), we ex-
satellite galaxies on image separation distributions. pect that the inclusion of late-type galaxies decreases the
Haloand subhalopopulations havequitedifferent con- number of sub-arcsecond lenses caused by central galax-
tri•butions of image separation distributions from different ies. In addition, we have neglected the scatters of the re-
(host-)halomass intervals. Wepredict that 14% of lenses lations between galaxy luminosities, velocity dispersions,
lie in groups, and 4% in clusters. Almost∼half of lenses in and effective radii. The scatters could affect the quanti-
groups and cluster∼s are produced bysatellite galaxies (sub- tative results, as the scatter in the mass-to-luminosity re-
halos), rather than central galaxies (halos). lation is important. The redshift evolutions of relations
Wehavecomparedourmodelpredictionswithobserved are also neglected. However, if we assume the evolution of
im•age separation distributions and found that they are in early type galaxies at z . 1 is purely passive, it implies
reasonable agreements with each other. Since the shape of that the mass distribution of lensing halos does not change
the image separation distribution is rather sensitive to the across the redshift. Therefore we expect the effect of red-
mass-to-luminosityrelation,precisemeasurementsoftheim- shiftevolutionisrathersmall.Moreimportantly,wehaveas-
age separation distribution in well-defined statistical lens sumedsimplesphericalhalos.Inreality,darkhalosarequite
samplesofferpowerfulprobeoftheconnectionbetweendark triaxial (Jing & Suto 2002) and the triaxiality enhances
halos and galaxies. lensing probabilities by a few factors at large-separations
(Oguriet al. 2003; Oguri & Keeton 2004; Hennawi et al.
In summary, we have constructed a realistic model 2005a,b).Ontheotherhand,small-separation lensingprob-
which predicts lensing probabilities from small to large im- abilities are hardly affected by the ellipticity of lens galax-
age separations. The model is quite useful in understand- ies: Hutereret al. (2005) showed that the change is less
ing lens populations as a function of the image separation. than a few percents unless quasars are very bright. Large-
(cid:13)c 2004RAS,MNRAS000,1–11