Gravitational Lens Time Delays: A Statistical Assessment of Lens Model Dependences and Implications for the Global Hubble Constant

0.81 MB·English

by  Masamune Oguri

PreprinttypesetusingLATEXstyleemulateapjv.10/09/06
GRAVITATIONAL LENS TIME DELAYS: A STATISTICAL ASSESSMENT OF LENS MODEL DEPENDENCES
AND IMPLICATIONS FOR THE GLOBAL HUBBLE CONSTANT
Masamune Oguri
KavliInstitute forParticleAstrophysicsandCosmology,StanfordUniversity,2575SandHillRoad,MenloPark,CA94025.
ABSTRACT
Time delays between lensed multiple images have been known to provide an interesting probe of
the Hubble constant, but such application is often limited by degeneracies with the shape of lens
potentials. We propose a new statistical approach to examine the dependence of time delays on the
complexity of lens potentials, such as higher-order perturbations, non-isothermality, and substruc-
7 tures. Specifically, we introduce a reduced time delay of the dimensionless form, and explore its
0
behavior analytically and numerically as a function of the image configuration that is characterized
0
by the asymmetryand opening angle ofthe image pair. In particularwe derive a realistic conditional
2
probability distribution for a given image configuration from Monte-Carlo simulations. We find that

n the probability distribution is sensitive to the image configuration such that more symmetric and/or
a smaller opening angle image pairs are more easily affected by perturbations on the primary lens po-
J
tential. On average time delays of double lenses are less scattered than those of quadruple lenses.

3 Furthermore, the realistic conditional distribution allows a new statistical method to constrain the
2 Hubble constant from observed time delays. We find that 16 published time delay quasars constrain
the Hubble constant to be H0 = 70 6kms−1Mpc−1, where the value and its error are estimated
2 ±
using jackknife resampling. Systematic errors coming from the heterogeneous nature of the quasar
v
sample and the uncertainty of the input distribution of lens potentials can be larger than the sta-
4
tistical error. After including rough estimates of the sizes of important systematic errors, we find
9 H = 68 6(stat.) 8(syst.)kms−1Mpc−1. The reasonable agreement of the value of the Hubble
6 0 ± ±
constant with other estimates indicates the usefulness of our new approach as a cosmological and
9
0 astrophysicalprobe, particularly in the era of large-scale synoptic surveys.
6 Subject headings: cosmology: theory — dark matter — distance scale — galaxies: elliptical and
0 lenticular, cD — gravitationallensing
/
h
p 1. INTRODUCTION lays are quite sensitive to mass distributions of lens ob-
-
o It has been known that time delays between multiple jects, which leads to strong degeneracies between lens
r images of strong gravitational lens systems offer an in- mass profiles and the Hubble constant. The most fun-
t damental, mathematically rigorous degeneracy is the
s teresting method to measure the Hubble constant H ,
0
a mass-sheet degeneracy (Falco et al. 1985); inserting an
the most fundamental cosmological parameter that gov-
: uniform sheet instead of decreasing the mass normal-
v erns the length and time scale of our universe (Refsdal
i 1964). A huge advantage of this method is that it ization of lenses changes estimated values of the Hub-
X ble constant while leaving unchanged the image observ-
does not rely on so-called distance ladder and can mea-
r sure the global Hubble constant independently of any ables. This degeneracy implies that the derived values
a ofthe Hubble constantalsodegenerateswithradialden-
local measurements. Motivated by this time delays
sity profiles of of lens galaxies (Refsdal & Surdej 1994;
have been measured in more than 10 lensed quasar sys-
Witt et al. 1995; Keeton & Kochanek 1997; Witt et al.
tems (see, e.g., Kochanek 2006). The situation as it
2000; Saha 2000; Tada & Futamase 2000; Wucknitz
presentsis,however,somewhatconfusingandcontrover-
2002; Treu & Koopmans 2002; Kochanek 2002, 2003;
sial. Kochanek (2002, 2003) claimed from the analysis
Oguri & Kawano 2003; Rusin et al. 2003; Schechter
of several lens systems that the Hubble constant should
be relatively low, H 50kms−1Mpc−1. The time 2005; Mo¨rtsell et al. 2005; Kawano & Oguri 2006). In
0
∼ addition, it may degenerate with the angular structure
delay of SDSS J1650+4251 also prefers the low Hub-
of lenses as well (Zhao & Qin 2003; Saha & Williams
ble constant (Vuissoz et al. 2007). On the other hand,
2006). The sensitivity of time delays on mass profiles
Koopmans et al.(2003)performedsystematicmassmod-
suggests that by assuming the Hubble constant, which
eling of B1608+656 using all available data from radio
can be determined by other methods (Freedman et al.
toopticalandfoundconstrainedthe valueoftheHubble
constant to be H = 75+7kms−1Mpc−1. The analysis 2001; Tegmark et al. 2006; Spergel et al. 2007), we can
0 −6 put constraints on mass distributions of lenses, in par-
of the smallest separation lens B0218+357 yields H =
0
78 3kms−1Mpc−1 (Wucknitz et al.2004). By combin- ticular radial density profiles (e.g., Oguri et al. 2004;
± Kochanek et al. 2006b; Dobke & King 2006).
ing time delays in 10 lensed quasar systems Saha et al.
(2006) obtained H =72+8 kms−1Mpc−1. Onewayto overcomethe degeneraciesis toadoptlens
0 −11 systemswhoselenspotentialscanwellbeconstrainedby
The large variation of derived values of the Hubble
manyobservationalconstraints. Anexampleofsuchcon-
constant from time delays reflects the fact that time de-
straintscomesfromlensedimagesofquasarhostgalaxies.
Electronicaddress: [email protected] ThelensedhostgalaxiesoftenformEinsteinrings,which
2 Oguri
accuratelyandindependentlydeterminesthestructureof requiremodelingofeachlenssystem. Thishasanadvan-
the lens potential as well as the shape of the lensed host tage that we can include lens systems that have too few
galaxy (e.g., Keeton et al. 2000; Kochanek et al. 2001; constraintsto determine the lens potential. Inthis sense
Koopmans et al. 2003). The small-scale structure of theapproachisextensionofstudybyOguri et al.(2002)
lensed quasars, such as radio jets and sub-components, in which only spherical halos are considered to compute
determine lens potentials in a similar way as Einstein time delay distributions. We note that the methodology
rings(e.g.,Cohn et al.2001;Wucknitz et al.2004). Fur- is similar to that adopted by Keeton et al. (2003, 2005)
thermore, the measurement of velocity dispersions of who deriveddistributions offlux ratios ofimage pairs to
lens galaxies serves as useful constraints on mass dis- identify small-scale structure in lens galaxies.
tributions, helping to break degeneracies between mass In addition to the measurement of the Hubble con-
models (e.g.,Treu & Koopmans2002). Lens systems for stant, the sensitivity of time delays on mass models,
which these strong additional constraints are available, which we explore in this paper, offers guidance on the
sometime referredas“goldenlenses”,havebeenthought usefulness of each time delay measurement as a cosmo-
to be an effective probe of the Hubble constant. logical or astrophysical probe: If time delays at some
Anotherpotentiallypowerful,butlessstudied,method image configuration is quite sensitive to detailed struc-
tomeasuretheHubbleconstantisastatisticalapproach. tureofthelenssuchashigher-ordermultipoletermswith
Evenifeachindividuallenslacksstrongconstraintsthat small amplitudes or substructures, which are difficult to
allowdetailedinvestigationsof the mass distribution,by beconstrainedevenforbest-studiedlenssystems,itisal-
combining many lens systems one can put tight con- most hopeless to use these time delays to extract either
straints on the Hubble constant. As mentioned above, radial mass profiles or the Hubble constant. Our result
this approach was in some sense demonstrated recently can be used to assess quantitatively which lens systems
by Saha et al. (2006) who combined 10 lensed quasar are more suitable for detailed studies, i.e., less sensitive
systems to constrain the Hubble constant. A caveat is to the complexity of the lens potential. There has been
that lensed quasars sometimes suffer from selection ef- several insightful analytic work (e.g., Witt et al. 2000;
fects. For instance, brighter lensed quasars with larger Kochanek 2002), but here we perform more systematic
image separations are more likely to lie in dense en- andcomprehensivesurveyofmodel dependences oftime
vironments such as groups and clusters (Oguri et al. delays by parameterizing image configurations of lensed
2005; Oguri 2006), thus the Hubble constant from those quasar systems using dimensionless quantities.
lensed quasars may be systematically higher than the This paper is organized as follows. In 2 we intro-
§
true value without any correction of the effect of dark duce several dimensionless quantities that are used to
matter along the line of sight. This indicates the impor- explore the model dependence of time delays. We study
tance of well-defined statistical sample of lensed quasars the sensitivity of time delays on the various lens poten-
for which we can quantitatively estimate and correct tials analyticallyin 3. Section 4 is devotedto construct
§
the selection effect. While the statistical sample has the conditional distribution of time delays from realistic
been very limited so far, containing 20 lenses even for Monte-Carlo simulations. We compare it with observed
∼
the largestlens sample (Myers et al.2003;Browne et al. time delays in 5, and constrain the Hubble constant
§
2003), largerlens surveyswillbe soonavailableby ongo- in 6. Finally discussion of our results and conclusion
§
ing/futurelenssurveyssuchasdonebytheSloanDigital are given in 7. Throughout the paper we adopt a flat
§
Sky Survey (Oguri et al. 2006), the Large Synoptic Sur- universe with the matter density Ω = 0.24 and the
M
veyTelescope(LSST)1,andthe Supernova/Acceleration cosmological constant Ω = 0.76 (Tegmark et al. 2006),
Λ
Probe(SNAP)2.Futurelenssurveyswillalsofindstrong although our results are only weakly dependent on the
lensing of distant supernovae (e.g., Oguri et al. 2003) specificchoiceofthecosmologicalparameters. TheHub-
for which time delays can easily be measured (but see bleconstantissometimesdescribedbythedimensionless
Dobler & Keeton 2006). Therefore the statistical ap- form h H /(100kms−1Mpc−1).
0
≡
proach is growing its importance.
In this paper, we study how time delays depend on 2. CHARACTERIZINGTIMEDELAYQUASARS
variouspropertiesofthelenspotentialandimageconfig-
Letus consideralens systeminwhicha sourceatu=
urations, which is essential for the determination of the
(u,v) is multiply imaged at the image positions x =
Hubble constant from time delays. Using both analytic i
(x ,y ). We also use the polar coordinates for the image
andnumericalmethods,weshowhowtimedelaysareaf- i i
positions, x = (x ,y ) = (r cosθ ,r sinθ ). We always
fectedbyseveralcomplexityofthelenspotential,suchas i i i i i i i
choose the center of the lens object as the origin of the
radialmassprofiles,externalperturbations,higherorder
coordinates. Time delaysbetweenthese multiply images
multipoles, and substructures. We then derive the ex-
are given by (e.g., Blandford & Narayan1986)
pected distributions of time delays by adopting realistic
lenspotentials. Thedistribution,inturn,canbe usedto
1+z D D
derivestatisticallythevalueoftheHubbleconstantfrom ∆t = l ol os
ij
observedtimedelays. Thisapproachdiffersfromthesta- 2c Dls
tistical argument by Saha et al. (2006) in the sense that (x u)2 (x u)2 2φ(x )+2φ(x ) , (1)
i j i j
theyfitimagepositionsofindividuallenssystemstocon- × − − − −
strainthe Hubble constantandthencombinedresults of wherez i(cid:2)stheredshiftofthelens,cisthespeedof(cid:3)light,
l
all the lens systems: Our new approach does not even and D , D , and D are angular diameter distances
ol os ls
from the observer to the lens, from the observer to the
1 http://www.lsst.org/ source,andfromthelenstothesource,respectively. The
2 http://snap.lbl.gov/ lens potentialφ(x) is relatedto the surfacemass density
Gravitational Lens Time Delays 3
of the lens Σ(x) by the Poissonequation: of surrounding dark matter that is larger for wider sep-
aration lenses (Oguri et al. 2005; Oguri 2006). Put an-
Σ(x)
2φ(x)=2κ(x)=2 , (2) other way, such correlation is naturally expected from
∇ Σcrit very different mass distributions and environments be-
tween small ( 1′′) and large (& 3′′) separation lenses.
with Σcrit = c2Dos/(4πGDolDls) being the critical sur- Indeedtheapp∼arentlackofthecorrelationbetweentime
facemassdensity(Gisthegravitationalconstant). Note
delays and the scaled time delays in Saha (2004) comes
that image positions and source positions are related by
mostlyfromthelargescatteramongdifferentimagecon-
the lens equation
figurations: The effect of external mass is hindered by
u=x φ(x ). (3) the largescatter his parametrizationinvolves. Therefore
i i
−∇
in this paper we propose equation (6) as useful quantity
Equation (1) involves unobservable quantities such as
to extract the mass dependences on time delays.
uandφ(x),indicatingthattimedelaysingeneraldepend
Previous analytic calculations of time delays sug-
on details of mass models. However, Witt et al. (2000)
gest that the model dependence of time delays is a
has shown that for generalized isothermal potential
strongfunctionofimageconfigurations(Witt et al.2000;
φ(x)=rF(θ), (4) Kochanek 2002). In this paper we characterize image
configurations by the following two parameters. One is
whereF(θ)is anarbitraryfunctionofθ, time delayscan the asymmetry of the images define by
be expressed in a very simple form involving only the
r r
observed image positions: j i
R − . (7)
ij
≡ r +r
1+z D D (cid:12) j i(cid:12)
∆tij = 2c l oDl os(rj2−ri2), (5) Again, Rij is dimensionless(cid:12)(cid:12) and do(cid:12)(cid:12)es not depend on the
ls size of the lens: R 0 m(cid:12)eans th(cid:12)e images are roughly
ij
whereri denotethedistanceofimageifromthecenterof atthe same distance∼fromthe lens galaxy,while Rij 1
thelensgalaxy. Motivatedbythisanalyticresult,inthis indicatesveryasymmetricconfigurationsthatoneim∼age
paper we consider the reduced time delay that is defined lies very close to the lens center and the other image is
by: far apart from the lens. The other parameter we use is
the opening angle of images:
∆t 2c D
Ξ ij ls x x
≡(cid:12)(cid:12)(cid:12)rj2−ri2(cid:12)(cid:12)(cid:12)1+zlDolDos θij ≡cos−1(cid:18) rii·rji(cid:19). (8)
=(cid:12)(cid:12)(cid:12)(cid:12)(xi−u)(cid:12)(cid:12)2−(xj −rj2u−)2r−i2 2φ(xi)+2φ(xj)(cid:12)(cid:12). (6) sIonuthcthehri,assθdimejfi∼enrigt1ii8no0gn◦,.iimfOatnhgeetsihmneeaoagtrehsceaurrshepadnaidnre,dcctfllooylsdoepcipmaotaasgisteterpoeapaihcrhes
(cid:12) (cid:12)
In this d(cid:12)efinition, the reduced time delay Ξ is a(cid:12)lways haveθ 0◦. NotethatbothR andθ areobservables
unity if (cid:12)the lens potential can be expressed by(cid:12)equa- in theisje∼nse that they can be deirjived wiijthout ambiguity
tion (4), but can deviate from unity if the lens potential foreachobservedlenssystemaslongasthelensgalaxyis
hasmorecomplicatedstructures. Inparticular,fromthe identified: We do not have to perform mass modeling to
analysisofKochanek(2002) we obtainΞ=2(1 κ ) at compute these quantity from observations. In summary,
−h i
the lowest order of expansion, where κ is the average our task of this paper is to explore model dependences
h i
surface density in the annulus bounded by the images. of reduced time delay Ξ as a function of image configu-
This indicates that the deviation from the isothermal rations parameterized by R and θ .
ij ij
massdistributionhasadirectimpactonthereducedtime
delay. Butaswewillseelateritisaffectedtosomeextent 3. ANALYTICEXAMINATION
by other factors suchas externalperturbations or small- In this section, we analytically examine the behavior
scalestructuresaswell. Thus wecanregardthe reduced of the reduced time delay Ξ (eq. [6]) for various lens po-
time delay as a measure of the complexity of the lens. tentials,beforeshowingthedistributionofΞforrealistic
In addition, equation (6) indicates that we can compute complicated mass distributions. For this purpose, it is
reduced time delays for observed lensed quasar systems convenienttostudyintermsofmultipoleexpansion: We
with measured time delays by assuming the value of the consider the lens potential of the following from
Hisuabkbelyeqcounasnttaitnyt.thInattrheipsrseesnesnet,stahelinrekdbuectewdeteinmmeedaeslauyreΞd φ(x)= cβnRE2−inβrβcosn(θ−θn), (9)
time delays of lens systems and theoretical lens models.
X
We point out that Ξ is dimensionless because time de- wherecnisthedimensionlessamplitudeandθnisthepo-
lays are proportional to the square of the size of a lens sition angle. The coefficients are chosen such that REin
(theEinsteinradius). Thisallowsustodirectlycompare becomes the Einstein radius of the system if the am-
values of reduced time delays for different lens systems plitude of the monopole term is c0 = 1. Note that an
that have different sizes. externalshear (e.g., Kochanek 1991; Keeton et al. 1997)
We note that Saha (2004) adopted similar but differ- can be described by this expression as β = n = 2 and
ent dimensionless quantity to explore the dependence of cn = γ. For this potential the reduced time delay is
−
time delays on mass models. In the paper, the parame- given by (Witt et al. 2000):
ter essentially same as equation (6) was also considered,
φ(x ) φ(x )
butitwasdiscardedbecauseofthecorrelationwithtime Ξ= 1+ 2(1 β) j − i . (10)
delays. However, the correlation just reflects the effect (cid:12)(cid:12) X − rj2−ri2 (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
4 Oguri
This can be rewritten as
2−β
2R
Ein
Ξ= 1+ c X(R ,θ )cos(nθ δ) ,
n ij ij n
(cid:12)(cid:12) X (cid:18)rj +ri(cid:19) − (cid:12)(cid:12)
(cid:12) (1(cid:12)1)
(cid:12) (cid:12)
wher(cid:12)e δ and X(Rij,θij) are defined by (cid:12)
rβsinnθ rβsinnθ
tanδ j j − i i, (12)
≡ rβcosnθ rβcosnθ
j j − i i
1 β 1
X(R ,θ ) − (1+R )2β +(1 R )2β
ij ij ij ij
≡ 2β R −
ij
2(1 R2(cid:2))βcosnθ 1/2. (13)
− − ij ij
Hereweassumedrj >ri withoutloss(cid:3)ofgenerality. Note
that R and θ were defined in equations (7) and (8),
ij ij
respectively.
The above expression of Ξ (eq. [11]) has several im-
portant implications. First, θ comes only in the last
n
cosine term, thus assuming n= 0 and c is small Ξ> 1
n
6
and Ξ < 1 occurs equally if θ is uncorrelated with the
n
image configuration (as we will see later, however, this
is not necessarily the case). This indicates that we can
reduce the effect of multipole terms by averaging many
lens systems, suggesting the usefulness of our statistical
approach. Second,since2R /(r +r ) 1inmostcases,
E j i
∼
the dependence of the potential (β) and image configu-
rations (R and θ ) on the deviation from Ξ = 1 is
ij ij
encapsulated in X(R ,θ ), aside from the overall am-
ij ij
plitude c . In what follows, we use X(R ,θ ) to study
n ij ij
the behavior of Ξ.
First, we consider quite symmetric cases (R 0)
ij
→
for which images are nearly the same distance from the
lens center. Depending on the opening angles, limiting Fig. 1.—ThebehaviorsofX(Rij,θij)(eq. [13])asafunctionof
behaviors are given by theasymmetryparameterRij. Fromthicktothinlines,theradial
slopeβischangedfrom0.5to4.0. Fromtoptobottompanels,the
1 β (cosnθij =1) openingangleθij isfixedtocosnθij =−1,0,and1,respectively.
−
1 β This does not depends on θ , thus the opening angle is
− (cosnθ =0) ij
X(R 0,θ )  √2βR ij (14) no longer important in this situation. However it shows
ij → ij ≈ 1β−Rβ ij (cosnθij =−1). strFoinngaellrydwepeepnldoetnXce(Ronijt,hθeij)rafdoiralvaslroiopuesβp.arameter val-
ij ues in Figures 1 and 2. In Figure 1, X(R ,θ ) is plot-
Therefore,X(Rij,θij)divergesunlesscosnθij =1. More ted as a function of the asymmetry Rij. Iijt girjows quite
0ri:gorously,weshouldtakethelimitofRij →0andθij → raalspoidslhyowassRstirjonagpeprrosaecnhseitsivtiotyzeornotfhoer scloospneθβija6=t l1a.rgeItr
R . As is clear in Figure 2, the opening angle θ be-
nθij 2 1/2 coimj es more important for more symmetric lensesi.j All
X(R 0,θ 0) (1 β) 1+ ,
ij → ij → ≈ − " (cid:18)2βRij(cid:19) # tphreesseenbteehdaavbioorvsea.reconsistentwithanalyticalarguments
(15)
indicating that the divergence at the symmetric limit 4. TIMEDELAYSINREALISTIC LENSMODELS
can be avoided if θ R , or more appropriately
ij ≪ ij Inthis section,we considermore realisticsituationsto
1 cosnθ R2. Since close image pairs are always
− ij ≪ ij inordertostudyexpectedspreadofthereducedtimede-
near the circumference of the critical curve that has a
lay Ξ as a function of image configurations. Specifically
nearly circular shape centered on the lens galaxy, such
we adopt theoretically and observationally determined
pairs in general have R θ , indicating the diver-
ij ij distributions of lens potentials such as ellipticities, ex-
≪
gence cannot be avoided for two close images. On the
ternal shear, substructures, and multipole components
other hand, the divergence may not occur for opposite
to makepredictions onrealisticprobabilitydistributions
◦
images (θ 180 ), but only if n is even number.
ij ofΞ. ThemethodologyissimilartothatinKeeton et al.
∼
Inversely, if images are very asymmetric (R 1),
ij (2003,2005)whostudiedcuspandfoldrelationstoiden-
→
X(R ,θ ) reduces to the following simple form
ij ij tify lenses with small-scale structure.
2β−1(1 β)
X(R 1,θ ) − . (16) 4.1. Input Models
ij ij
→ ≈ β
Gravitational Lens Time Delays 5
Rusin & Kochanek (2005) obtained α = 0.94 0.17 by
±
combiningtheEinsteinradiiofmanylensedquasarswith
thefundamentalplanerelationofellipticalgalaxies. The
slope of the galaxy density profile α = 1.09 0.01 con-
±
strainedfromthe faint thirdimageofPMN J1632 0033
−
is consistent with the nearly isothermal density pro-
file (Winn et al. 2004). Detailed mass modeling of
B1933+503alsoindicatesnearlyisothermalprofileofthe
lensgalaxy(Cohn et al.2001). Byusingmeasuredveloc-
ity dispersions of lens galaxies of several lensed quasars,
Treu & Koopmans (2004, see also Hamana et al. 2007)
derived α = 1.25 0.2. Koopmans et al. (2006) put
±
tighter constrains α = 0.99+0.03, but the results are
−0.02
derived from a sample of much lower redshift lens sys-
tems than typical time delay quasars. From these re-
sults, in this paper we adopt the Gaussian distribution
α = 1 0.15 as a conservative input distribution of the
slope.3±For the ellipticity, we use the Gaussian distri-
bution with median e = 0.3 and dispersion 0.16 that
is consistent with observed distributions of isodensity
contour shapes of elliptical galaxies (Bender et al. 1989;
Saglia et al.1993;Jorgensen et al.1995;Rest et al.2001;
Sheth et al. 2003).
Toallowmorecomplexmassdistributionofthegalaxy,
we add higher order multipole terms to the potential:
1
φ (x)= R2−αrα (1 m2)A cosm(θ θ ). (20)
M α Ein − m − m
m
X
The factor 1 m2 is inserted such that A denotes
m
−
the standard parametrization for the deviation of the
mass density from an ellipsoid. We include only m = 3
and 4 terms, because m 5 perturbations have gen-
≥
Fig. 2.— Same as Figure 1, but X(Rij,θij) is plotted as a erally not been reported. For both A3 and A4, the
functionoftheopeningangleθij whilefixingRij =0.1(toppanel), amplitudes are distributed by the Gaussian with mean
0.5(middle panel), and0.9(bottom panel). zero and dispersion 0.01 that is roughly consistent with
reported distributions (Bender et al. 1989; Saglia et al.
As primary lens galaxies we only consider elliptical
1993; Rest et al. 2001). It is also compatible with the
galaxies because most (> 80%) of quasar lenses are
levelofthedeviationinferredfromindividualmodelingof
caused by massive elliptical galaxies (e.g., Turner et al.
lensedquasarsystems(Trotter et al.2000;Kawano et al.
1984;Kochanek2006;Mo¨ller et al.2007). We modelthe
2004;Kochanek & Dalal2004;Congdon & Keeton2005;
lens galaxy as a power-law elliptical mass distribution.
Yoo et al.2006). Weassumethatpositionanglesofthese
The surface mass distribution is given by
multipole perturbations and that of an ellipsoid are un-
2−α correlated with each other. In addition, we neglect the
α R
κG(x)= Ein , (17) correlation of e and A4 (Keeton et al. 2003) for simplic-
2 "r 1 ǫcos2(θ θe)# ity. Notethatthesemultipoleperturbationsdonotaffect
− −
Ξ if the lens galaxyis isothermal,α=1, but canchange
where R is the Epinstein ring radius and θ is the po-
Ein e quantitativeresults when combinedwith non-isothermal
sition angle of ellipse. The case α = 1 corresponds to
profiles and/or other small-scale structures. Since the
the standardsingular isothermal mass distribution. The
effect of A itself is small, we expect the effect of the
ellipticity e, which is defined by e = 1 q with q being 4
the axis ratio of the ellipse, is related to−ǫ by correlation between e and A4, which we have neglected
here, is also small.
1 (1 e)2 Wealsoincludeexternalperturbationsthatareknown
ǫ= − − . (18) to be important for individual mass modeling (e.g.,
1+(1 e)2
− Keeton et al. 1997). The potential of lowest order ex-
The corresponding lens potential can be described by
3 The adopted distribution does not include the mean value of
φ (x)= 1R2−αrαG(θ), (19) Treu&Koopmans(2004)withinits1-σuncertainty. However,we
G α Ein notethattheresultofTreu&Koopmans(2004)weredrawnfrom
only five lensed quasar systems and therefore scatter should be
with G(θ) being the complex function of θ, but notice large. Moreover,manyoflenssystemsusedbyTreu&Koopmans
that G(θ)=1 if e=0. (2004)appeartoresideindenseenvironments,whichmayexplain
their large value of α. Indeed, Rusin&Kochanek (2005) used a
Many previous work has shown that lens galax-
largersampleof22lensestoderivetheslopethatismoreconsistent
ies indeed have nearly isothermal mass distribution. withourinputdistribution.
6 Oguri
Fig. 3.—DependencesofthereducedtimedelayΞonseverallenspotentialsasafunctionofasymmetryRij (leftpanels)oropeningangle
θij (rightpanels). Ineachpanel,reducedtimedelaysof500doublelenses(filled squares)and500quadruplelenses(open circles)obtained
by the Monte-Carlo simulations are plotted. Magnification bias is not considered at this stage. From top to bottom panels, we consider
theexternal shear φE2 (eq. [21]),thirdorderexternal perturbation φE3 (eq. [22]),subhalos φS (eq. [24]),andnon-isothermalityα6=1in
the primarylens potential φG (eq. [19]). Besides the bottom panels, we adopt an isothermal elliptical lens as primary lens galaxies (eq.
[19],α=1)andignoretheotherperturbations(e.g.,inthesecondrowweonlyconsiderφE3 andignoretheexternalshearandsubhalos),
thus the effect of each potential can be measured by the deviation from Ξ= 1. In all simulations higher order multipoles (eq. [20]) are
included.
ternal shear is given by σ = γ2 and dispersion 0.2 dex. The adopted amplitude
γ is roughly consistent with values obtained from mass
φ (x)= r2cos2(θ θ ). (21)
E2 −2 − γ modeling of individual lens systems (e.g., Kawano et al.
2004). We allow small mis-alignment of the position an-
We adopt a log-normal distribution with median γ =
gle θ , which could happen when the external perturber
0.05 and dispersion 0.2 dex for the distribution of shear σ
is not spherical, by adopting the Gaussian distribution
amplitude. It is consistentwith expected shear distribu-
◦
around θ with dispersion 10 .
tionfromN-bodysimulations(Holder & Schechter2003; γ
Finally we consider substructures in the lens galaxy.
Dalal & Watson 2004). In addition to external shear,
A significant fraction of substructures (subhalos) in
weconsiderthirdorderexternalperturbation(Kochanek
galaxies have been predicted as a natural conse-
1991; Bernstein & Fischer 1999)
quence of cold dark matter cosmology (Moore et al.
σ
φ (x)= R−1r3[cos(θ θ ) cos3(θ θ )]. (22) 1999; Klypin et al. 1999). Anomalous flux ratios ob-
E3 4 Ein − σ − − σ
served in many gravitational lens systems indicate
Here we assumed the external perturber is a singular
thatsubstructuresareindeedpresent(Metcalf & Madau
isothermal object to relate the amplitudes and position
2001;Chiba2002;Dalal & Kochanek2002;Bradaˇc et al.
angles of n = 1 and n = 3 terms. In general we have
2002; Keeton et al. 2005; Kochanek & Dalal 2004;
σ γ2 (Bernstein & Fischer 1999), thus for the am-
Metcalf et al. 2004; Chiba et al. 2005). Note that small
≈
plitude we adopt a log-normal distribution with median
Gravitational Lens Time Delays 7
Fig. 4.— Contours of constant conditional probability p(Ξ|Rij) (left panels) and p(Ξ|θij) (right panels) computed from Monte-Carlo
simulations of realistic lens potentials. The contours are drawn at the 68% (solid lines), 95% (dashed lines), and 99.7% (dotted lines)
confidence levels. Upper panels show the conditional probability for double lenses, whereas lower panels are for quadruple lenses. For
double lenses we show p(Ξ|θij) only at θij & 110◦ because we have too few image pairs with θij . 110◦ to construct the probability
distribution.
perturbations may also come from small halos along the to b through b /R (v/V)2. We distribute subhalos
k k E
∝
line-of-sight(Keeton2003;Chen et al.2003;Oguri2005; randomly with an uniform spatial density in the pro-
Metcalf 2005). Although time delays are thought to be jected two-dimensionalplane in order to take accountof
lesssensitivetosmall-scalestructurethanfluxratiosthat the suggested anti-bias of the subhalo spatial distribu-
aredeterminedbythesecondderivativeofthetimedelay tion(De Lucia et al.2004;Mao et al.2004;Oguri & Lee
surface, they might be affected to some extent particu- 2004). Theresultingmassfractionofsubhalosataround
larlywhentwoimagesareclosetoeachother. Wemodel the Einstein radius is 0.5%, being consistent with the
∼
eachsubhalobypseudo-Jaffe(truncatedsingularisother- expectation from N-body simulations and analytic cal-
mal) profile. The lens potential of this profile is culation (e.g., Mao et al. 2004; Oguri 2005). The effect
of subhalos is described by the sum of the lens potential
φ (x)=b r r2+a2 of each subhalo:
PJ,k k − k
(cid:20) q φ (x)= φ (x x ) 1r2κ¯ . (24)
−a2k ln(cid:12)(cid:12)prr22++aa2k2k−+aakk(cid:12)(cid:12)+aklnr#, (23) Here wSe subtrXakctedPJt,hke c−onvseurbg,kenc−e2averasugbed over all
where a is a truncatio(cid:12)n radius and b(cid:12) is a mass nor- subhalos,κ ,toconservethetotalmassandradialpro-
k (cid:12)p k(cid:12) sub
malization that coincid(cid:12)es the Einstein(cid:12) radius for suf- fileofthelensgalaxy. Weincludeonly100mostmassive
ficiently large a . We adopt a = √b R assuming subhalosmainlyforcomputationalreason,butsmallsub-
k k k E
the truncation radius of the tidal radius of the sub- halosareexpectedto haveonlyverysmalleffectontime
halo (see, e.g., Metcalf & Madau 2001). For the veloc- delays.
ity distribution we assume N(> v) = (10v/V)−2.7 in-
4.2. Simulation Method
side three times the Einstein radius of the lens galaxy,
where v and V are velocity dispersions of the subhalo Using the model described above, we perform a large
and halo, respectively. The velocities can be converted Monte-Carlo simulation containing > 106 lensed image
8 Oguri
Fig. 5.—Contourplotofmedian(leftpanels)andscatter(rightpanels)oftheconditionalprobabilityp(Ξ|Rij,θij)intheRij-θij plane.
Here the scatter is defined by the 68% confidence interval width inunits of logΞ. The probability distributionfor double (upper panels)
andquadruple(lower panels) lensesareshownseparately. Thicksolidlinesindicate thelimitbeyond whichthenumber ofimagepairsin
Monte-Carlosimulationsistoosmalltoconstructtheconditionalprobability.
pairs. The simulation is done by first generating a lens From the ensemble of image pairs, we compute condi-
potential according to the distributions summarized in tional probability distribution functions of the reduced
4.1. All lengths are scaled by the Einstein ring radius time delay Ξ for given values of the asymmetry R
ij
§
R : Since our results do not depend on adopted length and/or opening angle θ , i.e., p(ΞR ), p(Ξθ ), and
E ij ij ij
| |
scales, we fix R to unity in the simulations. We gener- p(ΞR ,θ ). Theprobabilitydistributionsarecomputed
E ij ij
|
ate 10000different lens potentials in total. Foreachlens separately for double and quadruple lenses to see how
potential,weplacerandomsourceswithanuniformden- different distributions they exhibit. In what follow we
sity of 100R−2 in the source plane. We use a public ignore central faint images that are unobserved in most
∼ E
softwarelensmodel (Keeton2001) to solvethe lens equa- cases.
tion and compute time delays between multiple images.
The uniform sampling in the source plane indicates that 4.3. Contribution of Each Potential
eachlenspotentialisautomaticallyweightedbythelens-
ingcrosssection(seeKeeton et al.2003). Toaccountfor Before presenting results that include all potentials, it
magnification bias as well, for each source we compute is useful to see how each potential affects the distribu-
the total magnification factor µ , and when construct- tion of Ξ. To see this we adopt isothermal elliptical lens
tot
ing probability distributions of reduced time delays (see potential plus multipole terms (φG +φM, α = 1), and
below) we include a weight of µq−1, where q is a power- add each lens potential: Since the isothermal galaxy al-
tot ways yields Ξ = 1 (see 2) the effect of each potential
law slope of the luminosity function of source quasars. §
term can be estimated by the deviation from Ξ=1. We
We adoptq =2.1 that is relevantfor lenses identified by
also study the effect of non-isothermality (α = 1). The
the CLASS (Myers et al. 2003). 6
results are summarized in Figure 3. First, the effect of
Gravitational Lens Time Delays 9
Fig. 6.— Same as Figure 4, but observed values of Ξ with error-bars (see Table 1) are plotted on the contours by filled triangles with
error-bars. TheHubbleconstant ofh=0.73isassumedwhencomputingΞfromobservedtimedelays.
external shear appears as a scatter around Ξ=1, which consider non-isothermal lens galaxies. Such systematic
isconsistentwithdiscussionabove( 3)aswellasthatof shiftswerenotpredictedinouranalyticargumentsin 3,
§ §
Witt et al.(2000). BoththedivergingscatteratR 0 thus invite carefulconsideration. We find that the selec-
ij
→
and the modest increase of the scatter at smaller θ are tion effect can explain these systematic shifts. For very
ij
expectedfromour analyticexamination. The scatterin- asymmetric pairs, such configurations are possible only
troduced by the third order external perturbation and when the lens galaxy has a profile steeper than isother-
subhalos is smaller than that of external shear, but it is mal for which inner critical curve is degenerated at the
stillnoticeableparticularlyforsymmetricconfigurations. center. Ifthedensityprofileisshallowerthanisothermal,
This means that even if we can estimate external shear any images (expect central faint images that we ignore
and its orientation accurately for individual lens system in this paper) are outside the inner critical curve. This
by detailed mass modeling, these third order perturba- means that very asymmetric configuration,in which one
tions or subhalos can shift time delays, although these image lies very close to the center of the lens potential,
smallperturbationsmainlyaffectimagepairswith small never occurs for the profile shallower than isothermal.
opening angles and only increase the scatter rather than Since shallower profiles correspond to smaller Ξ, this in-
shiftthe mean. We commentthatthe use ofimagepairs creases the mean Ξ at R 1. The effect of external
ij
∼
with small opening angles is also limited by the larger shear is more complicated, but can be explained as fol-
uncertainties of their time delay measurements. As ex- lows. Consider a close pair of two images. If we add
pected, non-isothermality has a large impact on Ξ, but external shear whose position angle direction is perpen-
the size of its scatter is less dependent on image config- dicular to the segment that connects the two images, it
urations compared with other potentials. separatestwoimages. Onthe otherhand,externalshear
It is worthnoting that there are systematic deviations that is parallel to the segment makes the two images
from Ξ = 1 for a few specific image configurations. For closer. Therefore, combined with the steep dependence
instance, at small θ external shear preferentially pro- of the frequency of close image pairs on the opening an-
ij
duces image pairs with time delays smaller than the gle,this resultsinthe situationthatatfixedsmallopen-
isothermal case. In addition, very asymmetric image ing angle the direction of external shear is more likely
pairs (R 1) tend to have larger time delays when we to be parallel to the segment than perpendicular. The
ij
∼
10 Oguri
discussion in 3 indicates that the parallel shear yields small opening angles, just as our theoretical model pre-
§
smallertime delaysΞ<1 forfixedimage positions,thus dicts. We may need more quasar time delays to confirm
we canconclude that close image pairs have statistically this trend observationally.
smallertimedelaysthanthoseexpectedwithoutexternal More directly, observed time delays should be com-
shear. paredwiththeprobabilitydistributionforgivenR and
ij
θ , i.e., p(ΞR ,θ ). In Figure 7 we compare observed
ij ij ij
4.4. Conditional Probability Distributions Ξ with the p|robability distribution of Ξ expected from
We are now ready to compute conditional probabil- corresponding image configuration of each image pair.
ity distributions of the reduced time delay Ξ with all We find that the distributions agree with observed val-
lens potentials presented in 4.1. Figure 4 shows con- ues on average, similarly as Figure 6. The reduced time
tours of constant conditiona§l probability projected to delays of B0218+357 and SBS0909+532 have larger er-
one-dimensional surface, p(ΞR ) and p(Ξθ ). The be- rorsbecause the positions of the lens galaxiesare poorly
ij ij
haviors of contours can well |be understood| from discus- determined. The large errorof HE0435 1223AC comes
sions in 3 and 4.3. In particular, systematically small from small r2 r2, whereas that of B1−422+231 is sim-
§ § j − i
ΞatsmallopeninganglesandlargeΞforveryasymmet- ply because of the large time delay measurement errors.
ric image pairs, which we discussed in 4.3, are clearly We comment that RXJ0911+0551 and Q0957+561 has
§
seen. It appears quadruple lenses have larger scatter significantly smaller Ξ compared with our model predic-
on average: This is because strong perturbations (exter- tion. This is clearly because of the cluster convergence:
nalshear,thirdorderperturbationandsubhalos),which The two lenses lie in near the centers of clusters and
causethe scatteraroundΞ=1,alsoenhance the lensing therefore the observed time delays are pushed down by
cross section for quadruple images. theconvergencecomingfromdarkmatterinthe clusters
To study the conditional probability distribution as a (see 6). The reduced time delays of B1608+656 and
§
function of both R and θ , p(ΞR ,θ ), in Figure 5 SBS1520+530 are largely offset from the predicted val-
ij ij ij ij
|
we plot contours of median and 68% confidence inter- ues, and this is probably because of satellite galaxies in
val of the probability distribution in the R -θ plane. the lenssystemsthatsignificantlyaffectthe time delays.
ij ij
The features we have discussed, such as the divergence ThelargetimedelaysofRXJ1131 1231ABandACwere
−
of scatter at R 0, smaller median time delays for also noted by Morgan et al. (2007): Our result indicates
ij
→
small opening angle pairs, larger median time delays for thatthebroadenedtheoreticaldistributionsduetosmall
asymmetric pairs, can be seen in this Figure as well. In perturbations are enough to explain the observed high
addition, the scatter presented here is useful to check valuesofthetimedelays. ThelargeoffsetsofB1422+231
whichlenssystemsarelessdependentonlenspotentials: and SDSS J1650+4251 may come from the large uncer-
Our result indicates that asymmetric image pairs that tainties of measured time delays (see 6).
◦ §
are collinear with their lens galaxies (θ 180 ) have
ij
the least scatter and therefore are more su∼itable for the 6. IMPLICATIONSFORH0
determinationoftheHubbleconstant. Itisinterestingto In this section, we turn the problem around and con-
note that in the statistical sense double lenses are more strain the Hubble constant using the conditional prob-
valuablethanquadruplelensesbecauseitshowsasmaller ability distribution constructed from the Monte-Carlo
sensitivitytovariouslenspotentials. Inaddition,double simulations. Although the current sample of observed
lenses have advantages of easier measurements of time time delays summarized in Table 1 is somewhat hetero-
delays in observations and smaller fractional errors (be- geneous and may not be appropriate for the statistical
causedoublelenseshavelongertimedelaysthanquads). study, we do this to demonstrate how we can constrain
This is in contrastto individual mass modeling in which the Hubble constant from the probability distribution.
quadruple lenses are more useful because of much more We basically take all lens systems listed in Table 1, but
observational constraints on mass models. adopt the following setup to reduce systematic errors.
5. COMPARISONWITHOBSERVEDTIMEDELAYS
We do not use the AB time delay of SDSS
We now examine if the conditional distribution com- • J1004+4112. It is a lens system caused by a mas-
puted in 4 is consistent with the observed distribution siveclusterofgalaxies,thusourinputdistribution,
§
oftimedelays. SinceweneedtoassumetheHubble con- whichisdesignedforgalaxies,doesnotrepresenta
stant in order to convert observed time delays to reduce fair distribution of lens potentials. Moreover, the
timedelaysΞ,inthissectionweadopth=0.73thatwas center of the lens potential appears to be offset
obtained from the combined analysis of the cosmic mi- from the position of the brightest cluster galaxy
crowavebackground(CMB)anisotropyandclusteringof (Oguri et al. 2004), which makes it inaccurate to
galaxies (Tegmark et al. 2006). We derive reduced time estimate important parameters such Ξ, R , and
ij
delaysforthe 17publishedtime delayquasars(41image θ .
ij
pairs), which is summarized in Table 1.
First, we compare these reduced time delays with the BothRXJ0911+0551andQ0957+561areknownto
•
conditional probability plotted in Figure 4. We show resideintheclusterenvironment,thustheyaresig-
p(ΞR ) and p(Ξθ ) with observed values overplotted nificantly affected by the cluster convergence κ .
ij ij clu
| |
in Figure 6. It appearsthat the data are roughlyconsis- Since the mass-sheet degeneracy says Ξ ∆t
∝ ∝
tent with our probability distribution from simulations. 1 κ , we divide reduce time delays Ξ for these
clu
−
The large scatter of very symmetric lenses is also seen two systems by 1 κ in order to deconvolve the
clu
−
for observed time delays. It is interesting that observed effect of the cluster convergence. As the values of
timedelaysappeartoexhibitthedeclineoftimedelaysat κ ,weadoptκ =0.3 0.04forRXJ0911+0551
clu clu
±