Statistics of gravitational lenses in the clumpy Universe

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by  G. Covone

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed2February2008 (MNLATEXstylefilev2.2)
Statistics of gravitational lenses in the clumpy Universe
G. Covone,1 ⋆ M. Sereno2,3 and R. de Ritis†
1 Laboratoire d’Astrophysique de Marseille, traverse du Siphon, 13012 Marseille, France
2 Universit`adegli Studi di Napoli “Federico II,” ViaCinthia, Complesso Universitario Monte S. Angelo, 80126 Naples, Italy
3 INAF-Osservatorio Astronomico di Capodimonte, Salita Moiariello, 16 80131 Naples, Italy
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2February2008
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n
a ABSTRACT
J

4 We evaluate the effect of small scale inhomogeneities on large scale observations
2
within the statistics of gravitationally lensed quasars.At this aim, we consider a cos-

mological model whose large scale properties (dynamics, matter distribution) are the
1
sameasinFriedmann-Lemaˆıtremodels,butwhosematterdistributionislocallyinho-
v
mogeneous.We use the well known Dyer-Roderdistances to allow a simple analytical
9
0 expression of the optical depth τ, and pay particular attention on the different role
5 playedbythedifferentnotionsofdistances(filledbeamangulardiameterdistanceand
1 Dyer-Roderdistances) when calculating this quantity,following the prescriptionfrom
0 Ehlers & Schneider for a coherent formalism. We find that the expected number of
5 gravitationallylensedquasarsis adecreasingfunctionofthe clumpiness parameterα.
0
/ Key words: cosmologicalparameters–cosmology:observations–gravitationallens-
h
ing
p
-
o
r
t
s
1 INTRODUCTION this has to be taken into account when fitting the FLRW
a
: parameterstoobservations.However,cosmological observa-
v One of the major task in modern cosmology is the precise
tions are usually fitted just using relationships derived for
i determination of the parameters which characterize the as-
X homogeneous models.
sumed cosmological model. In this direct approach (accord-
r ing to Ellis’ (1995) terminology) a theoretical description In recent years, several authors have addressed this
a probleminthecontextoftheobservationsofdistantTypeIa
of the space-time is postulated, the Friedmann-Lemaˆıtre-
supernovae(e.g., Holz1998, Holz&Wald1998, Kantowski,
Robertson-Walker (FLRW) model, and its parameters are
Kao&Thomas2000,Serenoetal.2002,Wang,Holz&Mun-
determined by fittingthe observational data.
shi2002, Pyne&Birkinshaw2004). Ithasbeenshownthat
The corner stone of the observational support to the
thenoiseduetoweaklensingmagnificationfromsmallscale
FLRW model is the existence and the high isotropy of the
matterinhomogeneitiesyieldslargeerrorsontheluminosity
relic Cosmic Microwave Background Radiation (CMWB).
measurement of high-z supernovae.
Ehlers,Geren&Sachs(1968)showedthatifthebackground
radiation appears to be exactly isotropic to a given family In this paper we aim at investigating the possible sys-
of observers then the space-time is exactly FLRW. There- tematic errors due to neglecting the effects of the local in-
fore, together with the Copernican Principle, we can prove homogeneities in the distribution of matter when evaluat-
the Universe to be FLRW just from our own observations ing the cosmological constant Λ from gravitational lenses
of the CMWB. Stoeger, Maartens & Ellis (1995) extended statistics. Statistics of gravitationally lensed multiply im-
this result to the case of an almost isotropic background agedquasarshasbeensincealongtimeconsideredanuseful
radiation, which implies an almost FLRW space-time. This tool toconstrain thecosmological parameters, in particular
important results is a firmer ground for the assumption of the cosmological constant (Turner, Ostriker& Gott 1984;
theFLRWmodeltodescribethelargescalestructureofthe Fukugitaet al. 1992; Kochaneck 1996), and the properties
Universe,butitalsomakesclearthatweneedtounderstand of the lensing galaxies (e.g. Maoz & Rix 1993, Kochanek
thedeparturesfromaspatiallyhomogeneousmodelwhenin- 1993). Recently, Chae (2003) has shown that the observed
terpreting observational data. Indeed,departures form per- gravitational lensing rate in the CLASS radio survey yields
fect homogeneity change the distance-redshift relation, and strong support to a flat cosmological model dominated by
vacuumenergy,withΩm 0.3andΩΛ 0.7.Theprecision
≃ ≃
oftheseresultsislimitedatthemomentbytheuncertainty
⋆ E-mail:[email protected] on the knowledge of the luminosity function of the lens-
† Deceased. inggalaxy population, theirdensity profileand their evolu-
2 G. Covone et al.
tion since z 1 (e.g., Mao 1991, Chae 2003). For instance, the gravitational lenses rate, and in Sect. 4 we discuss its
≃
Cheng&Krauss(2000)haveshownthatconstraintsoncos- behaviorasafunctionofα,consideringdifferentgravitatio-
mological parameter are strongly dependent on the choice nal lens models. Finally, systematic effects on the estimate
of galaxies parameters (see also discussion in Kochanek et ofthecosmological constantarediscussedinSect.5,andin
al. (1999)). Sect.6 we sum up our results.
Another major limit is the fact that only few syste-
matic surveys for multiply imaged quasars have been com-
pleted upto date(Claeskens & Surdej2001);today in fact,
2 ROLE OF COSMOLOGICAL DISTANCES
the statistical uncertainties on ΩΛ are still dominated by
thePoisson errors from thesmall numberof gravitationally While cosmological models which are homogeneous at all
lensedquasars.Inthenearfuture,themostpromisingsource scales are very successful in describing the overall dynam-
for new lensed quasars will be wide field surveys (Kuhlen, ics and evolution of the Universe, they do not allow a de-
Keeton&Madau2004)andtargetedfollowupsofnewlydis- tailed description of the lensing phenomena. As a fact, all
coveredquasars(Morgan 2002, Morgan et al.2004). Forin- thegravitational lensing phenomena (bendingof light rays,
stance,theSloanDigitalSkySurvey1willalmostdoublethe deformation of images, and formation of multiple images)
numberofknowngravitationallenses.Nextconsiderablein- areonlypossibleinaclumpyuniverse,(see,e.g.,thediscus-
crease in the number of gravitational lenses is expected by sion by Krasin´sky (1997)). Therefore, a coherent approach
new telescopes like the VLT Survey Telescope2, which will needsan inhomogeneous model.
allow very wide and deep optical surveys in the Southern However, in the statistical analysis of gravitational
hemisphere.SeealsoKuhlenetal.(2004)foradiscussionof lenses, as well as in analysis of other astronomical obser-
other gravitational lenses surveys to become operational in vations, perfect homogeneity is often assumed (see, e.g.,
thenext future. discussion in Buchert (2000) about this point). An impor-
While these large observational projects will consider- tant reason for this choice is the fact that in homogeneous
ably improve the precision of the results and the impor- space-times we have simple relations between the proper
tanceofthetool,theyalsomakenecessarytoconsidermore distance and the cosmological distances, i.e. the luminosity
realistically the detail of the light propagation through the distance and the angular diameter distance (e.g., Kayser,
observed not homogeneous universe. As pointed out by El- Helbig & Schramm 1997). In inhomogeneous cosmological
lis (1995), small scale inhomogeneities in matter distribu- models, these relations are much more complicated, and
tion have a considerable effect on both observations (Dyer Friedmann-Lemaˆıtre(FL)distancesarenotgenerallyagood
& Oattes 1988) and dynamics (Russ et al. 1997) at large approximationtobeusedinthedeterminationofcosmolog-
scale.Moreover,sincethelensingeffectsofthesmallinhomo- ical parameters from a given set of observational data.
geneities on thepropagation of light change theangular di- The relevant distance in gravitational lensing, the an-
ameterdistance-redshiftrelation(Schneider,Ehlers&Falco gular diameter distance D, is operationally defined by the
1992),wefocushereonthisspecificproblem.Manydifferent square root of the ratio of the area dA of a celestial
approaches have been developed to study the gravitational body to the solid angle dΩ it subtends at the observer
lensingininhomogeneouscosmologicalmodels,butthesim- (Schneideret al. 1992):
plestonefroman analyticalpointofview,andyetefficient,
istheoneproposedbyDyer&Roeder(1972,1973),inwhich dA
D . (1)
the effect of the local inhomogeneities along the light bun- ≡ dΩ
r
dles are described by the so-called clumpiness parameter α
In a homogeneous universe, the angular diameter distance
(seedefinitioninnextsection).Inthefollowing,weallowthe
can be derived from the proper distance Dp using the fol-
clumpinessparametertobeadirection-dependentquantity,
lowing relation
which isfunction both ofthelineof sight tothesource and
its redshift (see, e.g. Wang (1999)). D(z)= Dp(z). (2)
The statistical properties of a sample of gravitational 1+z
lensesincludethefrequencyofmultiplyimagedquasars,the This relation does not hold anymore in a clumpy universe.
distribution of the lenses and source redshifts, of the angu- The basic reason lies in thefact that the properdistance is
lar separation distribution and of theimage multiplicity. In relatedtotheglobalgeometryoftheuniverse,whiletherela-
this work we will focus on the discussion of the total lens- tionbetweentheangulardiameterdistanceandtheredshift
ing probability, leaving a detailed discussion of the other is determined mainly by thelocal matter distribution.
statistical properties for a following paper. Moreover, as we In this paper we assume a cosmological model which
focus our attention on an effect which is independent from is locally inhomogeneous, but homogeneous at very large
ourpresentdayknowledgeofthegalaxyluminosityfunction scale, according to some density averaging rule (see, e.g.,
andthedarkmattervelocitydispersion,wedonotconsider Krasin´ski(1997)).Inotherwords,weassumethattheover-
these aspects in detail. alldynamicsdoesnotdifferfromthedynamicsofahomoge-
The paper is organized as follows. In Sect. 2 we define neouscosmological model. Thisapproximation iswelljusti-
thecosmologicalmodelanddiscusstherelevantdistancesin fied:byaveragingtheFriedmannequation,Russetal.(1997)
thestudyofthepropagationoflight.InSect.3wecalculate showed that the influence of small scale clumpiness on the
globalexpansionfactorisnegligible(so,forinstance,theage
of the Universe can be evaluated using the FLRW relation
1 http://www.sdss.org/ with theHubbleconstant, with negligible errors), while the
2 http://www.na.astro.it/ distance-redshiftrelationissignificantlyaffected(sothatthe
Statistics of gravitational lenses in the clumpy Universe 3
measurements of the Hubble constant via measurements of
standardcandlesmagnitudeshavenotnegligiblesystematic
errors).
Inordertodescribethedegreeofinhomogeneity inthe
local distribution of matter, we use a generalized notion of
the so-called clumpiness parameter (e.g., Dyer & Roeder
1972, Schneider et al. 1992), introduced by Wang (1999).
The clumpiness parameter α was first introduced by Dyer
&Roeder(1972,1973,hereafterDR),whenwritingadiffer-
ential equation for the angular diameter distance in locally
clumpycosmological models,anditwasdefinedasthefrac-
tionofhomogeneouslydstributedmatterwithinagivenlight
cone.
Starting from the equations for scalar optics (e.g., Zel-
dovich (1964), Kristian & Sachs (1965)), DR derived a se-
condorderdifferentialequationforthediameterofthelight
ray bundle propagating far away from any clumps (i.e. in
regions where α < 1), assuming a negligible shear. For a
pressure-lessuniversewithcosmological constantΛ,theDR
equation reads:
d2D
(1 + z) Ωm(1+z)3+ΩK(1+z)2+ΩΛ dz2 +
7(cid:8) (cid:9) dD
2Ωm(1+z)3+3ΩK(1+z)2+2ΩΛ dz
(cid:26) (cid:27) Figure 1. The Dyer-Roeder distance r(z) for two values of the
+3αΩm(1+z)2D=0, (3) cosmologicalconstant(ΩΛ=0.0(continuouscurves),0.7(dashed
2 curves))intheemptybeamcase(upper curves)andinthefilled
′ beamcase(lowercurves).Space-timeisflat.
with initial conditions D(0) = 0 and D(0) = c/H0. Here
ΩK 1 Ωm ΩΛ, since we neglect the contribution from
≡ − −
any relativistic fluid or radiation. For α = 1 (filled beam
case) we recover the angular diameter distance, while for
α=0 we havethe well-known empty beam approximation. angular diameter and luminosity distances (Fig. 1), since
Foradetaileddiscussionofthesolutionsoftheequation(3) thereasmallercontributionfromtheRicciconvergence.On
within quintessence cosmological models we refer to Sereno the other side, along light beams characterized by α > 1
et al. (2000). Forthefollowing, It is useful tointroduce the (i.e.,propagatinginoverdenseregions)agulardiametersdis-
dimensionless distance r: tances are lower than in the filled beam case. For this rea-
sonalargeamountoflocalclumpinesscanresult inalower
r(z,Ωm,ΩΛ,α) H0 D(z,H0,Ωm,ΩΛ,α), (4) valueforΛwhenfittingtheobservationaldata.InFig.2we
≡ c
show theratio of theDR distancefor theemptybeam case
and the symbol r1 for the dimensionless angular diameter relativelytothefilledbeamforafewvalueofthesourcered-
distance in thefilled beam case. shifts. Uptoredshift 1, thedifferencesarenot important
∼
Note, howeverthat theDRequation (3) is welldefined intheDRdistanceitself(although,theymaybenotnegligi-
foranyα>0,andinitsderivationthemassdensityisnever bleforseveralastronomicalobservablequantities).Therole
required to be uniform. This allows to consider the clumpi- of the local clumps becomes more and more important at
nessparameertasalocalvariable,asdoneinWang(1999)to higher z.
descibe the weak lensing magnification of distant standard Thoughtheassumedcosmologicalmodelisnotthemost
candles. Therefore, in the following we assume the clumpi- satisfactory to describe inhomogeneities, since it has not a
nessparametertobeafunctionbothofthelineofsightand firmtheoreticalbasisintheframeworkofGeneralRelativity
the redshift. Given a source at resdhift z in a specific inho- (i.e., it is not a solution of the Einstein field equations), it
mogeneous cosmological model, for any line of sight to the allows a simple and efficient description of the light propa-
observertheclumpinessparameterαiscalculatedviaequa- gationthroughaclumpyuniverse.Thismodelhasbeendis-
tion(3),wherethedistanceDisgivenbynumericalsimula- cussedindetailinSchneideretal.(1992)andSeitz,Schnei-
tions. Asa consequence, a complete description of thelight der & Ehlers (1994). Ehlers & Schneider (1986) haveintro-
propagation intheclumpyUniverseneedstheknowledgeof ducedaself-consistent formalism tostudythegravitational
theprobablitydistributionfunction(PDF)forvaluesofthe lensing in a clumpy Universe. In particular, they stressed
clumpiness parameter. Wereturn to this point in Sect.5 the different roles that the different notions of cosmologi-
Letusnowconsiderinmoredetailtheeffectofinhomo- caldistances haveinthis model.Namely,when we consider
geneitiesinobservationsmeanttomeasurethecosmological quantitieswhicharerelatedtotheglobalgeometryoftheas-
constant. The DR distance r is a strongly decreasing func- sumedcosmologicalmodel(suchasvolumes),itisnecessary
tion of α, at fixed redshift (Schneider et al. 1992): there- touse theFL angular diameter distance. Then, thevolume
fore, a larger fraction of matter in clumps partially masks element(i.e. thevolumeofashellwith properthicknessdl)
theeffectof alarger cosmological constant whenevaluating is
4 G. Covone et al.
plane for which multiple imaging occurs for sources behind
it. This quantity depends on the redshift z of the lens and
a set χ of astrophysical parameters which characterize the
gravitational lens model. For the most generally used mod-
els (point-like mass distributions and isothermal spheres)
the cross section depends on a particular combination of
distances:
σ(z,z ,χ)=f DdsDd,χ , (6)
s (cid:18) Ds (cid:19)
where Ds,Dd and Dds are the DR angular diameter dis-
tancesbetweentheobserverandthesource,theobserverand
the lens and between the lens and the source, respectively.
Weremarkthattheoveralleffectoftheclumpydistribution
of matter along the light rays on the lensing probability is
not duetoany changeof thevolumeelement dV, see equa-
tion (5), but it is only because of the dependence of the
stronglensingcross section σ on theclumpinessparameter.
In the following we will consider the projection of the
cross section on thesource plane (located at redshift z ):
s
r(z ,α) 2
σˆ(z,z ,χ)= s σ(z,z ,χ), (7)
s r(z,α) s
(cid:20) (cid:21)
wheretheDRdistancesareconsidered.Thisquantityallows
Figure 2. DR distance versus the clumpiness parameter, rela- amore clear and compact definition of thelensing probabi-
tively to the filled beam case at several value of the redshift. lityanditisthenaturalquantitytoconsiderintheassumed
ΩΛ=0.7inaflatspace-time. cosmological model (Ehlers & Schneider1986). It is impor-
tanttonotethatthequantityσˆisnotingeneralafunctionof
c 2 thethedistance combination DdsDd/Ds: as a consequence,
dV =4π r2dl. (5) thepoint-likemassandisothermalspherecross sectionsare
(cid:18)H0(cid:19) 1 functions of different distances combinations (see Sect. 4).
Inotherwords, thevolumeelementdV doesnot dependon Wenowcalculate theprobabilityofstronglensingphenom-
the local inhomogeneity degree. This is consistent with the ena and evaluate its dependence on α, and then determine
factthatinthelocallyinhomogeneousmodel,onlargeangu- explicitly the distance functions for these two gravitational
lar scales (i.e., larger than θ 10′′, see Linder (1988)), the lens models.
∼
distance-redshiftrelationistheonecomputedintheFLRW
models, for any source redshift. This is also in agreement
withtheresultsfromN-bodynumericalsimulationsinCold
3 THE STATISTICS OF GRAVITATIONAL
Dark Matter scenarios (Tomita 1998), where it has been
LENSES
shown that the dispersion in values of α along the different
light paths becomes increasingly larger as the angular is as In this section, we derive the formulae for the statistics of
small as a few arcsec. We also note that equation (5) can gravitational lensing, following mainly the formalism dis-
bereadasthedefinitionoftheangulardiameterdistancein cussed in Ehlers & Schneider (1986), considering in partic-
homogeneous models (Schneideret al. 1992). ular the proper role of the two types of distances. Let us
Ontheotherside,thereasimplereasonwhy,whencon- consider a statistical ensemble of cosmological sources at
sidering gravitational lensing phenomena, the DR distance redshift zs and a set of comoving gravitational lenses with
hastobeused:lightdeflectionmodifiesthecrossareaofthe numberdensity n(z). Ifwe neglect gravitational lenses evo-
lightbundleandtheRiccifocusingtermisalinearfunction lutionandmergings,thecomovingnumberdensityoflenses
of the amount of matter within the ray bundle (see, e.g., isconserved:n(z)=n0(1+z)3,wheren0isthelocaldensity.
Schneider et al. 1992, sect. 3.4). As a consequence, all the The number of gravitational lenses in a shell with volume
distance dependent quantities which play a role in the de- dV is then
scription of thelensing phenomenaare functions of theDR
N(z) = n(z)dV
distance,and,in particular, thestronglensing cross section
dl
σ. = n(z)A(z) dz, (8)
dz
Let us consider in more detail the cosmological strong
lensingprobabilityτ.Thisisdefinedastheprobabilitythat where A(z) is thearea of the spherelocated at redshift z.
alightsourceatredshiftz ismultiplyimagedbyadeflector The probability dτ(z,z ,χ) that a quasar at z is mul-
s s s
at z <z . In the expression of τ(z ) there are two different tiply imaged by a gravitational lens in the redshift range
s s
physical quantities which are functions of combinations of (z,z+dz)isdefinedasthefractionoftheareaofthesphere
distances: the cross section for multiple images σ, and the at z = z (i.e., the fraction of the sky) covered by the gra-
s
volumeelement dV. vitational lenses cross sections σˆ(z,z ,χ). This definition
s
The cross section σ is defined as the area in the lens implicitly assumes that the projected cross sections do not
Statistics of gravitational lenses in the clumpy Universe 5
overlap, which is equivalent to state that dτ(z,z ,χ) 1.
s
≪
Thearea coveredbytheprojected cross sectionsofthegra-
vitational lenses in (z,z+dz) is therefore
dl
n(z)σˆ(z,z ,χ)A(z) dz. (9)
s dz
Accordingtothedefinition,thedifferentiallensingpro-
bability thenreads
A(z) dl
dτ(z,z ) = n(z)σˆ(z,z ,χ) dz
s s A(z )dz
s
= n(z)σ(z,χ) r(zs,α) 2 r1(z) 2 dldz.(10)
(cid:20) r(z,α) (cid:21) (cid:20)r1(zs)(cid:21) dz
In theparticularcase α=1, thedistanceratios factor-
ize, so that we obtain themore common expression
dl
dτ(z,z )=n(z)σ(z,z ,χ) dz, (11)
s s dz
i.e., the definition of differential lensing probability in cos-
mologies which arehomogeneous at all scales. Therefore, in
this case, the relevant distance combination is exactly the
ratio DdsDd/Ds which appears in equation (6), and does
notdependontheparticularchoiceofthegravitationallens
model.
Letusnowevaluatetheexplicitexpressionsforthelens- Figure3.Theprojectedcrosssectionsofapoint-likegravitatio-
ing probabilities in equation (11). The quantity dl can be nallensforα=0,0.5,0.8,normalizedtofilledbeamcase(α=1),
written in thefollowing way: versusthelensredshift.Sourceisatzs=5,andcosmologicalpa-
1 dz rametersareΩΛ=0.7andΩm=0.3
dl= cdt=c , (12)
− H(z)1+z
the dependence of the lensing probability τ on the clumpi-
where we considered the past light cone, and H(z) a˙/a,
≡ ness parameter. Asada (1998) has investigated the analyt-
a=1/(1+z)beingthenormalizedcosmologicalscalefactor.
ical properties of several combinations of DR distances as
In a FLRW cosmological model
functionsoftheclumpinessparameter,andhealsodeduced
H(z)=H0 Ωm(1+z)3+(1 Ωm ΩΛ)(1+z)2+ΩΛ.(13) some consequences on the observable quantities: the deflec-
− −
tionangle, thetimedelay,andthelensingprobability.Here
Hereaftper we focus our attention to flat cosmological
wefocuson thestudyofthequantitieswhich directlyenter
models, i.e., with ΩΛ +Ωm = 1, as these are preferred by theopticaldepth.Weconsiderthefollowingmodels:apoint
inflationary scenarios and strongly supported by many re-
mass distribution, singular isothermal sphere, and isother-
cent observational evidences (Wangat al. 2000). Then, the
mal sphere with a non zero core radius. In this section we
differential lensing probability reads
only consider theDR distances.
dτ(z,zs) = n0Hc0σ(z,zs,χ)(cid:20)rr((zzs))(cid:21)2(cid:20)rr11((zzs))(cid:21)2× 4.1 Point-like gravitational lens
(1+z)2
dz. (14) A point-like gravitational lens produces two images, what-
Ωm(1+z)3+ΩΛ ever the source position is; so, strictly speaking, the given
It isevidentthapt theproperties ofthefunctions dτ (andτ) definitionofstronglensingcrosssectiondoesnotapplyhere.
withrespecttoαaredeterminedonlybythestronglensing Anyway,itisnaturaltoassumesuchacrosssection(on the
cross section σˆ(z,z ). Thereby, it is necessary to consider lens plane) to be the disk with radius equal to Einstein ra-
s
only those functions of the DR distances which enter their diusrE. So, thecross section on the source plane reads:
expressions. In otherwords, thefunctions τ and σˆ havethe
same qualitative behaviors with respect to the clumpiness σˆ = Ds 2πr2
parameter. Then, our next point is to investigate the be- (cid:18)Dd(cid:19) E
havior of the quantityσˆ(α). = 4πGM DsDds , (15)
c2 Dd
whereM isthemassofthelens.Thisquantityisadecreas-
ing function of the clumpiness parameter α, as it is shown
4 PROPERTIES OF DR DISTANCES
in Fig. 3. Consequently, at smaller value of the clumpiness
COMBINATIONS
parameter,thelensingprobability ishigher,whenconsider-
Inthissectionweanalyzeindetailthecrosssectionsofsome ing the lensing cross section for a point-like distribution of
generalmodelsofgravitationallensesandtheinvolvedcom- mass.
binationsofDRdistances,inordertoevaluatequalitatively Consider that the distances ratio DdDds/Ds in the ex-
6 G. Covone et al.
pression of the Einstein radius rE is a decreasing function where we introduced the quantity C(β) σ/σ0. The pre-
≡
of the clumpiness parameter (Asada 1998), but it is not sence of a core radius lowers the lensing probability, but
theright function of DR distances which enters theexpres- does not change its qualitative properties with respect to
sion of the lensing probability in a DR cosmological model, the clumpiness parameter, see Fig. 4b. Note that the func-
as shown above. As a fact, considering this ratio is not tion C(β) is not a constant, since β is not, with respect to
coherent with the assumptions on the cosmological model the clumpiness parameter. The dimensionless core radius β
(Ehlers & Schneider1986), and would wrongly lead to pre- is a function of theredshifts z,z , and the cosmological pa-
s
dictthatthelensingprobabilitydecreasesfordecreasingval- rametersandtheclumpinessparameter,viathelengthunit
ues of theclumpiness parameter, as in Asada (1998). ξ0:
4.2 Isothermal spheres β(z,zs,ΩΛ,α)= 41πσc2v2ξc DdDDsds . (22)
Letusconsidernowthegravitationallensestobeisothermal It is interesting to note that, despite the fact that the
spheres,andletusstudythedependenceofthecrosssection cross sections of the considered gravitational lens models
on α. Many authors have shown that singular isothermal differ in the DR distances ratio, they are both decreasing
spheres(SIS)allow adetailed descriptionofthematterdis- functions of α. In other words, the monotonic properties
tribution in individual gravitational lenses (see, e.g., Rusin of the cosmological optical depth with respect to α are ge-
et al. 2002), and are therefore used in statistical analysis of neral. Finally, it is evident that neglecting α leads, in the
the lensing galaxies population (e.g., Chae 2003). SIS are theoretical predictions, to underestimate the probability of
characterized bythesurface mass density stronglensing,and,inthestatisticalanalysisofhighredshift
quasarscatalogs,tooverestimatethecosmological constant.
σ2 1
Σ(ξ)= v , (16) This is discussed in thenext section.
2Gξ
whereξ is theposition vectoron thelensplane, and σ the
v
line-of-sightvelocitydispersion.See,e.g.,Hinshaw&Krauss 5 ON THE EXPECTED NUMBER OF
(1987)andSchneideretal.(1992) foradetaileddescription GRAVITATIONAL LENSES
ofitslensingproperties.Theareainthelensplaneformul-
Inthis section we evaluatenumerically theoptical depthin
tiple lensing is
a flat cosmology as a function of the cosmological param-
σ0 =16π3σc4v4 (cid:20)DdDDsds(cid:21)2 . (17) ewteeranΩaΛlyaznedquoafntthiteactilvuemlypitnheesseffpaecratmdeuteertoα.neInglepcatrintigcutlhaer
localinhomogeneityincalculatingthelensingprobabilityin
Note that the distance combination which enters equation
a clumpy universe, along lines of sight with α<1. For this
(17) is the same as in the Einstein radius for a point-like
purpose, we consider the gravitational lenses to be singular
gravitational lens. But,when we consider theprojection on
isothermal spheres, because they allow a very simple and
thesource plane, we get
analytical treatment of theproblem,and havebeen used in
σ 4
σˆ0 =16π3 cv Dd2s. (18) avitvaatriioentyalolfenstsuesdi(esseeofCthhaeest(a2t0i0s3ti)caalnpdrorpefeerrteinesceosftthheereginra)-,
(cid:16) (cid:17)
As the angular diameter distance between two points at allowing a direct comparison of the results. Moreover, this
redshifts z1 and z2 > z1 is a decreasing function of α particular choice does not affect out qualitative results.
(Asada 1998), so is the cross section σ0 and, consequently, As we are mainly interested in the effects due to a va-
the optical depth τ. In Fig. 4a we plot the cross section riationofα,wecomparetheopticaldepthτ(zs,ΩΛ,α)with
σ0(z;α) relatively to the filled beam case, evaluated along τ0, the numerical value in the case with vanishing cosmo-
differentlineofsghtscharacterizedbydifferentvaluesofthe logicalconstantandlightpropagationthroughfilledbeams,
clumpiness parameter. so that our discussion is independent of the numerical pa-
Let us now consider an isothermal sphere with a rameter F 16π3 σv 4 c 3 which controls the lensing
non zero core radius ξc. The surface mass density is ≡ c H0
Σ(ξ) = σv2 1 , which leads to a non constant de- probability.InFig.(cid:0)5we(cid:1)p(cid:16)lott(cid:17)herelativeopticaldepthτ/τ0
2G√ξ2+ξc2 as a function of the source redshift, for six different cosmo-
flection angle. The cross section on the lens plane is logical models, in which ΩΛ = 0.6,0.8, considering lines of
(Hinshaw & Krauss 1987) sight with α = 0,0.5,1 for each case. As it is well known,
the strong lensing probability is very sensitive to the value
σ= σ0 1+5β− 21β2 − 12√β(β+4)3/2 β< 21 (19) of the vacuum energy (e.g., Fukugita et al. 1992). For any
( 0 h(cid:0) (cid:1) i β> 21 valueofα,theprobabilitythatasourceatgivenredshiftzs
where β is the core radius ξc in units of the natural length ismultiplyimagedgrowsbyafactorof∼2,ifΩΛ goesfrom
0.6 to 0.8.
scale ∼ ∼
Inordertodisentangletheeffectoftheclumpydistribu-
ξ0 =4π σv 2 DdDds. (20) tion of matter from that due to the cosmological constant,
(cid:16) c (cid:17) Ds we consider ∆τ/τ0, i.e. the relative variation of the lensing
The projected cross section for multiple imaging is, for β < probability with respect to the case α=1, as a function of
1/2, ΩΛ alonglines ofsightcharacterized byα<1.Weplotthis
σ 4 quantity,evaluatedatdifferentsourceredshifts,inFig.6.As
σ=16π3C(β) cv Dd2s, (21) shown above, for any zs and ΩΛ, the lensing probability is
(cid:16) (cid:17)
b
Statistics of gravitational lenses in the clumpy Universe 7
Figure4.Projectedcrosssectionsrelativelytothefilledbeamcaseforasingularisothermalsphere(left-handpanel)andanisothermal
sphere with core radius ξc = 10 pc (right-hand panel), versus the lens redshift. The source is located at zs = 5 and cosmological
parametersareΩΛ=0.7,Ωm=0.3.Forlensredshiftz≪1,thetwolensmodelshavethesamelimit.Thesamequalitativeresultsholds
foranyvalueofzs andΩΛ.
a decreasing function of theclumpiness parameter. Wealso value as the redshift increases (see, e.g., Fig. 3 in Wang
noticethat theeffect of theclumpiness parameterincreases 1999). In Fig. 7 we plot, for a flat spacetime with Ω =
m
with the redshift of the sources, if a given direction with 0.4, both the most likely value and the average values, as
α<1 is considered. calculated via the approximate analytic expressions of the
Themostimportantfeaturetonoteisthatthevariation PDF(α) given in Wang (1999).
becomesrapidlylessimportantatlargervaluesofthecosmo- The mean value of α is 1 at any redshift in all cosmo-
logical constant. The reason for this effect is twofold. First, logical models (if they are FLRW on average as assumed
for larger values of the cosmological constant, the influence here).Thisisindeedthesamepropriertywhichisknownto
ofallotherastrophysicalandcosmological parametersisex- hold for the PDF of the lensing magnification µ(z), whose
pected to be less important, since at high ΩΛ, the optical basic motivation is the flux conservation (Weinberg 1976).
depthisverysensibletoanysmallchangeinthevalueofthe Therefore we expect that the effect of inhomogeneoties on
cosmological constant. Second,α enterstheequation (3) as the lensing cross sections is reduced in large ensemble of
acoefficientofthematterdensityparameter,determined(in gravitational lenses, or very distant sources.
theflatcosmologicalmodels)bytherelationΩm =1 ΩΛ;so Foragravitationallensatz 1,themostlikelyvalueis
− ∼
it isrelatively less important for small valuesof thedensity α 0.85;thistranslatesinanunderestimatesofthelensing
∼
parameter. In the most commonly accepted range for the cross section for a SIS (using the fille beam distances and
cosmological constant (0.6 6 ΩΛ 6 0.85), the lensing pro- consideringasourceatzs &4)ofafactor 1.15,seeFig.4.
∼
bability increases (relatively to a completely homogeneous These systematic errors decrease for more distant gravita-
matterdistribution)byafactorofabout7%,17%,30%ifwe tional lenses or sources, but since the most likely value is
consider the clumpiness parameter α = 0.75,0.5,0 respec- alwayslowerthan1,andinarelativelysmallsampleofgra-
tively. vitational lenses the mean of a nonsymmetric probability
Asstatedabove,inordertohaveacoherentdescription distributionisnotlikelythebestestimator,sucherrorsmay
of the small scale clumpiness we need to consider the PDF becomenotneglibiblewhenevaluatingthecosmological pa-
for α, since this a direction-dependent quantity. The PDF rameters.
dependsonthebackgroundFLRWcosmological model,and While a detailed statistical analysis including an accu-
can be derived from the PDF of the lensing magnification ratedetermination for thePDFfor anycosmological model
µ(z),calculatedinnumericalsimulations(e.g.,Wambsganss isbeyondthescopeofthepresentwork,itisanywayimpor-
etal.1997,Holz&Wald1998,Tomita1998,M¨ortsell2002), tanttoestimatenowanupperlimittothepossibleerrorson
via therelation thepredictednumberofgravitationallenses.Wehavethere-
µ(z)= r1(z) 2 , (23) fhoarveecboenesnidoebresdervtehdeilnistthoeffo1l1l6ow3inlugmoipntoiucsalqsuuarsvaeryss3:.CwFhHicTh
r(z,α)
(cid:20) (cid:21)
andtheDRequation.AsshownbyWang(1999),ingeneral
the α PDF is peaked at α<1 for any z, but it tends to be 3 This catalog is available at the web address
moresymmetricandshowssmaller scatteraroundthepeak http://vela.astro.ulg.ac.be/themes/extragal/gravlens/bibdat
8 G. Covone et al.
Figure 6.Relative variation of the optical depth with respect to the filled beam case. The clumpiness parameter is α={0,0.5,0.75}.
Sourcesareatredshiftszs=3(lefthandpanel),5(righthandpanel),inaflatspace-time.ΩΛ=0.7
Figure 5. The optical depth as a function of the redshift for Figure7.Thepeak(continuosline)andthemeanvalue(dashed
differ values of α and ΩΛ relatively to τ0, the optical depth in - dotted line) of the clumpiness parameter in flat cosmological
the case ΩΛ = 0 and α = 1. Continuous and dashed curves are modelwithΩm=0.4,calculatedviatheanalyticapproximation
for ΩΛ =0.6,0.8 respectively. The clumpiness parameter is α= giveninWang(1999).
0,0.5,1.0,fromtheabovecurvetotheloweroneforeachvalueof
ΩΛ.GravitationallensesareSISandspace-timeisflat.
arebeyond z=3. Wehavecalculated therelative variation
in the expected numberof multiply imaged quasars consid-
(Crampton, McClure & Fletcher 1992), CFHT (Yee, Filip- ering different values for the clumpiness parameter, corre-
penko&Tang1993),HST(Maozetal.1993),NOT(Jaunsen sponding to peak values of its PDF for z 0.5 and 3.0.
≃
etal.1995).Thiscatalogcontains7confirmedgravitational The lensing galaxies are being modelled as SIS (note that
lenses, and its redshift distribution is plotted in Fig. 8: the including a small core makes the clumpiness effect slighlty
peakisatredshiftz 2,andonlyasmallfractionofsources larger). In Fig. 9, we plot the relative variation of the ex-
≃
Statistics of gravitational lenses in the clumpy Universe 9
Figure 8.Redshiftdistributionofthe1163luminousquasarsin Figure 9. Relative variation of the expected number of strong
thecatalogusedtocalculatetheeffectofaclumpinessparameter lensedquasarsinthegivensampleofhighluminosityquasarswith
α6=1.Seetextfordetails. respecttothefilledbeamcase,asafunctionofthecosmological
constant. The clumpiness parameter is α =0.75 (dashed curve)
andα=0.95(continous curve).LensesaremodelledasSIS.
pected number of lenses as a function of ΩΛ for the peak
values α = 0.75,0.95. this figure shows the upper limit for
the systematic errors that can be found for the predicted in Wang (1999). We have paid particular attention to dis-
numner fo gravitational lenses when adopting the simple entanglethedifferentrole playedbythedifferentnotionsof
filledbeamhypothesis;giventhepropertyα¯=1,thiscould distanceinthedefinitionoftheopticaldepthτ(α):thesmall
be further reduced in upcoming larger surveys. Note that, scale inhomogeneities along the line of sight do not change
sinceweareconsideringaflatspacetimeandtheeffectofthe thevolumeelementdV,butthestronglensingcrosssection,
localinhomogeneitiesincreaseswithΩm,thevariationofthe see equation (5) and Fig. 4.
expectednumberN ofmultiplyimagedquasarsisadecreas- Up to redshift z 3, the most probable value of the
ingfunctionofthecosmological constant.However,onlyfor clumpiness parameter∼is very different from the average
a very large value of the cosmological constant (ΩΛ &0.8), value (which is constrained to be 1 in a model homoge-
such a variation becomes rapidly small. This makes clear neous on a large scale), see Fig. 7, and the effect may be
the point that if in the evaluation of τ we use angular di- important in statistical analysis of relatively small sets of
ameter distances for a perfectly homogeneous cosmological gravitationally lensed sources.
model, we can underestimate the lensing probability and, Asada (1998) presented a similar calculation to the
consequently,overestimate ΩΛ. one done here in Sect. 4. He discussed the influence of the
clumpiness parameter on several gravitational lensing ob-
servable quantities, showing that in a clumpy universe de-
flectionanglesaresmallerandtimedelaysarelongerthanin
6 CONCLUSIONS
an homogeneous universe, given the same lens-source con-
In this work we have investigated whether the local depar- figuration. However, in contraddiction with Asada’s result,
turesfromacompletlyhomogeneouscosmologicalmodelcan wehavefoundthatthegravitationallensesrateisadecreas-
haveobservableeffectsinthestatisticalstudyofhigh-z gra- ing function of the clumpiness parameter. We have shown
vitationallenses. Following thework byEhlers&Schneider that in the empty and filled beam cases different angular
(1986),wederivedtheexpressionsforthecosmologicalopti- distancesratiosentertheopticaldepthexpression(Sect.3),
caldepthintheframeworkofacosmologicalmodelwhichis and this leads to an higher number of expected of gravita-
FLRW on very large scales (i.e., whose overall dynamics is tional lenses when light beams with α<1 are considered.
verywelldescribedbyFLRWmodels)andwhosematterdis- Whileadetailedstatisticalanalysisincludingtheeffect
tributionislocallyinhomogeneous.Thedirection-dependent of small scale inhomogeneities on the determination of the
clumpiness parameter α quantifiesthefraction of matterin cosmological constant is beyond the scope of the present
compactobjectsalongagivenlineofsight,anditspeakand work,wecanalready drawsomeimportantconclusions and
mean values (as a function fo the source redshift) are cal- compareourfindingswithpreviousworks.Sincetheαmean
culated via the analytical approximation of the PDF given value tends to be 1 when a sufficiently large number of dif-
10 G. Covone et al.
ferent lines of sight is considered, the effect described here ACKNOWLEDGMENTS
tends to be less important in large surveys for gravitation-
GC wishes to thank N.D. Morgan, N.R. Napolitano and
allylensedsources.Moreover,sincethepeakvaluealsotends
Y. Wang for stimulating discussions, and the referre for a
to 1 for z &5, statisical study of very high-redshift sources
carefulreadingandinsightfulremarks.SpecialthankstoA.
is less affected by the local clumpiness along the different
Pospieszalska-SurdejandtheGravitational Lensesgroup at
linesof sight.Onsmall set of lensed sources, sincethepeak
theInstitutd’Astrophysiqueet deG´eophysique(University
value of the clumpiness parameter is always lower than 1,
deLi`ege) forprovidingtothecommunitythecatalog ofthe
usingthefilledbeamangulardiameterdistancesleadstoan
quasars searched for multiple lensed images in optical sur-
overestimateofthenumberoftheexpectgravitationallenses
veys. GC is supported by the Euro3D RTN postdoctoral
(Fig.9),and,consequently,tooverestimatethecosmological
fellowship.
cosntant.
This work is dedicated to the beloved memory of Pro-
It is interesting to note that the same qualitative re-
fessor Ruggiero de Ritis.
sult(onthedeterminationofthecosmologicalconstant)has
been found in previous works in which the authors have
taken into accounts the effect on the distance-redshift rela-
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