A & A m anuscript no.
(w illbe inserted by hand later)
A ST R O N O M Y
AND
A ST R O PH Y SIC S
14.11.2016
Y our thesaurus codes are:
11.03.1 | 12.04.1 | 12.07.1 | 12.12.1
A rc statistics w ith realistic cluster potentials
IV . C lusters in di erent cosm ologies
M atthias B artelm ann 1, A ndreas H uss1, Jorg M .C olberg1, A drian Jenkins2,and Frazer R .P earce2
1
arXiv:astro-ph/9707167v1 15 Jul 1997
2
M ax-Planck-Institut fur A strophysik,P.O .B ox 1523,D -85740 G arching,G erm any
Physics D ept.,U niversity ofD urham ,D urham D H 1 3LE,U K
July 15,1997
A bstract. W e use num ericalsim ulations ofgalaxy clusters in di erentcosm ologiesto study their ability to form
large arcs.T he cosm ologicalm odels are:Standard C D M
(SC D M ; 0 = 1, = 0); C D M w ith reduced sm all-scale
power (param eters as SC D M , but w ith a sm aller shape
param eter ofthe power spectrum );open C D M (O C D M ;
= 0);and spatially at,low -density C D M
0 = 0:3,
( C D M ; 0 = 0:3,
= 0:7).A llm odels are norm alised
to the local num ber density of rich clusters. Sim ulating
gravitationallensing by these clusters,wecom puteoptical
depths for the form ation oflarge arcs.Forlarge arcsw ith
length-to-w idth ratio 10,the opticaldepth islargestfor
O C D M .R elative to O C D M ,the opticaldepth is lowerby
aboutan orderofm agnitudefor C D M ,and by abouttwo
ordersofm agnitude forS/ C D M .T hese di erencesoriginate from the di erent epochs ofcluster form ation across
the cosm ologicalm odels,and from the non-linearity ofthe
strong lensing e ect.W e conclude that only the O C D M
m odel can reproduce the observed arc abundance well,
w hile the other m odels failto do so by orders ofm agnitude.
K ey w ords: G alaxies: clusters: general | C osm ology:
dark m atter | C osm ology:gravitationallensing | C osm ology:large-scale structure ofU niverse
1.Introduction
G alaxy clusters form di erently in di erent cosm ological
m odels.T heir form ation history and their internalstructure are in uenced by the cosm ological param eters. In
dense m odel universes, 0 < 1, clusters form at signi cantly lowerredshiftsthan in low -density m odeluniverses
(e.g.R ichstone,Loeb,& Turner1992;B artelm ann,Ehlers,
Send o print requests to:M atthias B artelm ann
& Schneider 1993;Lacey & C ole 1993,1994).T he cosm ological constant has a fairly m oderate in uence on the
form ation tim escale.D elayed form ation is re ected in the
abundance ofclustersubstructure.M oreover,centralcluster densities are higher in clusters that form earlier.
It is currently unclear w hether the di erent degrees
of cluster substructure expected in cosm ologicalm odels
w ith di erent m ean densities lead to observationalconsequences that can signi cantly distinguish between highand low -density universes. W hile earlier studies found
cluster X -ray m orphologies and density pro les to di er
signi cantly between di erent cosm ologies (Evrard et al.
1993;C rone,Evrard,& R ichstone 1994;M ohretal.1995;
C rone,Evrard,& R ichstone 1996),m ore recentwork concluded that X -ray m orphologiesofclusters at the present
epoch are fairly sim ilar in di erent cosm ological m odels,rendering signi cant distinctions di cult (Jing et al.
1995).T he issue ofconstraining clustershapesusing their
X -ray em ission was also addressed in detailby B uote &
T sai(1995a,b).T he weak gravitationallensing e ect allow s a m easurem ent of the m orphology of the projected
m ass (W ilson, C ole,& Frenk 1996;Schneider & B artelm ann 1997)and constrainsclusterdensity pro les(C rone
et al.1997),but it rem ains to be show n w ith realistic num erical cluster m odels w hether this m ethod provides a
m ore sensitive toolto quantify cluster m orphologies than
that provided by X -ray em ission.
A n alternative tool is o ered by the strong gravitationallense ect.In orderto be strong lenses(i.e.in order
to produce appreciable num bers oflarge arcs from background sources),clusters have to satisfy severalcriteria.
First,they need to be com pact,that is,their centralsurface m ass densities need to surm ount the criticalsurface
m ass density for lensing.T he latter depends on redshift.
For background sources at redshifts zs
1, clusters at
redshifts 0:2 < zc < 0:4 are the m ost e cient lenses.If
arcs are to be produced in abundance,a su ciently large
num berofconcentrated clustersm ustbe in place atthose
redshifts.Second,strong lensing is a highly non-linear ef-
2
M atthias B artelm ann et al.:A rc statistics w ith realistic cluster potentials
fect. T his is m ainly because the num ber of strong lensing events depends sensitively on the num ber of cusps
in, and the length of, the caustic curves of the lenses.
C usps require asym m etric lenses.A sym m etric,substructured clusters are thus m uch m ore e cient in producing
large arcs than sym m etric clusters,provided the individual cluster sublum ps are com pact enough (B artelm ann,
Steinm etz,& W eiss1995).B oth argum entsshow thatthe
in uence ofcosm ology on the structure and the form ation
tim escale ofclustersshould strongly a ecttheir ability to
form large arcs.C lusters that are being assem bled from
com pactsubclustersatredshiftsw here lensesare m ostefcient, 0:2 < z < 0:4, should produce m any m ore arcs
than clustersw hich form atlaterredshiftsfrom sublum ps
w hich are less com pact.
D oesthisline ofreasoning im ply thatdi erentcosm ologicalm odelscan be distinguished by the num berofarcs
thatareexpected in them ? M oreprecisely,doesthe higher
com pactness ofclusters in low -density universes,and the
later form ation tim e ofclusters in high-density universes,
lead to such di erent num bersoflarge arcsthat lim its on
cosm ologicalparam eterscould be obtained from counting
arcs? T his is the question addressed by this paper.
In orderto pursue it,we use galaxy clusterssim ulated
in a variety ofcosm ologicalm odels.T he sim ulations are
described in Sect.2.T he sim ulated clusters are then investigated as to their strong-lensing e ects,follow ing the
prescription in Sect.3. R esults are presented in Sect.4,
and sum m arised in Sect.5.T he paper concludes w ith a
discussion in Sect.6.
W u & M ao (1996) already considered the in uence of
on arc statistics,however w ith spherically sym m etric,
non-evolving clusters.T hey found that this cluster m odel
predicts 2 tim es m ore arcs in a low -density,spatially
at universe ( 0 = 0:3,
= 0:7) than in an Einstein-de
Sitter universe,and that the latter m odelfalls short by a
factorof 4 to explain theobserved arcabundance.U sing
a sim ilar m odel,H am ana & Futam ase (1997)pointed out
thatthe expected num berofobserved arcsincreasesw hen
the evolution ofthe background sources is taken into account.H attori,W atanabe,& Yam ashita (1997) included
observationalselection e ects in a study ofarc statistics
using spherically sym m etric cluster m odels.U sing C D M
cluster m odels at a single redshift in an Einstein-de Sitter universe, van K am pen (1996) studied the in uence
ofthe norm alisation ofthe power spectrum on the num ber ofarcs per cluster and concluded that the norm alisation should be som ew hathigherthan derived from cluster
abundance.
2.C luster sim ulations
the purpose ofstudying cluster m ass pro les in di erent
cosm ologies.D i erent num ericaltechniques were used for
the two sim ulation sets.T hey havein com m on thatw ithin
each set,the sam e random phases are used for the initial
density eld in all cosm ologicalm odels studied, so that
the clusterscan be com pared individually.T he norm alisation ofthe power spectra agrees approxim ately w ith the
norm alisation to the localcluster num ber density.A part
from im proved statistics, the approach using two di erently sim ulated cluster sets allow s us to test w hether different num ericaltechniques yield di erent results for arc
statistics.A lthough the norm alisations for the two sim ulation setsareslightly di erent,the arcstatisticsobtained
from the two sets individually agree wellw ith each other.
T he clusters show density pro les (r) that are well tted by the two-param eterfunction suggested by N avarro,
Frenk,& W hite (1996),
(r)=
s
x (1 + x)2
; x=
r
:
rs
(1)
N avarro etal.(1996)introduced the concentration param eter c = r200=rs, w here r200 is the radius enclosing an
average overdensity of 200 tim es the cosm ic background
density.A t z = 0,we nd
8
< 5 for S/ C D M
c
7 for C D M
:
(2)
:
9 for O C D M
T he di erent values for c re ect the di erent cluster form ation tim es.W hen clustersform earlier,theirconcentration is higher.
2.1.Firstcluster sam ple:G IF sim ulations
2.1.1.T he G IF project
T he G IF project is a joint e ort of astrophysicists from
G erm any and Israel.Its prim ary goalis to study the form ation and evolution ofgalaxiesin a cosm ologicalcontext
using sem i-analyticalgalaxy form ation m odels em bedded
in large high-resolution N -body sim ulations.T his is done
by constructing m ergertreesofparticle haloesfrom darkm atter only sim ulations and placing galaxies into them
using a phenom enological m odelling (for a detailed description of this procedure as well as results cf. K au m ann etal.1997).In orderto achieve both good statistics
and an accurate treatm ent of early epochs, high resolution sim ulations are needed w hich nevertheless contain a
fair sam ple ofthe U niverse,thus accounting correctly for
the in uence oflarge-scale structure on galaxy form ation.
T hose characteristicsalso m ake these sim ulationssuitable
for the present project.
W e use two di erentsetsofclustersim ulations.A rstset
2.1.2.T he sim ulations
of ve clusters is taken from high-resolution cosm ological
sim ulations kindly m ade available by the G IF collabora- T hecodeused fortheG IF sim ulationsiscalled H ydra.Itis
tion,and a second set offour clusters was sim ulated for a paralleladaptiveparticle-particleparticle-m esh (A P 3 M )
M atthias B artelm ann et al.:A rc statistics w ith realistic cluster potentials
code (for details on the code cf.C ouchm an,T hom as,&
Pearce 1995;Pearce & C ouchm an 1997).T he currentversion wasdeveloped aspartofthe V IR G O supercom puting
project and was kindly m ade available by them for the
G IF project.T he sim ulations were started on the C R AY
T 3D at the C om puter C entre ofthe M ax-Planck Society
in G arching (R ZG )on 128 processors.O nce the clustering
strength required an even largeram ountoftotalm em ory,
they were transferred to the T 3D at the Edinburgh ParallelC om puter C entre (EPC C ) and nished on 256 processors.
A set of four sim ulations w ith N = 2563 and w ith
di erent cosm ological param eters was run. A part from
the ducialC old D ark M atter (C D M ) scenario,denoted
SC D M ,w hich has 0 = 1 and h = 0:51,another 0 = 1
and h = 0:5 m odelwas run ( C D M ) w hich has the sam e
shape param eter for the power spectrum , = 0:21, as
the rem aining two m odels. T hose are both m odels w ith
0 = 0:3,the rst one being a at m odelw ith a cosm ologicalconstant( C D M ,
= 0:7,h = 0:7),and the last
m odelbeing an open m odel(O C D M ,
= 0,h = 0:7).
T he case of the C D M m odelis particularly interesting
because it shares the 0 = 1 dynam ics w ith the SC D M
m odel, but has a power spectrum w ith the sam e shape
param eter as the two low - 0 m odels.A value of = 0:21
is usually preferred by analyses of galaxy clustering, cf.
Peacock & D odds (1994).T his is achieved in the C D M
m odel despite 0 = 1 and h = 0:5 by assum ing that a
m assive neutrino (usually taken to be the neutrino)had
existed during the very early evolution of the U niverse.
It m ust have decayed later,thus shifting the epoch w hen
m atterstarted to dom inate overradiation in the U niverse,
and the neutrino m ass and lifetim e are chosen such that
= 0:21.For a detailed description ofsuch a m odelsee
W hite,G elm ini,& Silk (1995).
T he G IF sim ulations adopt the power spectrum
P (k)=
Ak
[1 + [ak + (bk)3=2 + (ck)2 ] ]2=
w ith
a = 6:4
1
b = 3:0
c = 1:7
1
1
h
1
M pc ;
h
h
1
M pc ;
M pc ;
1
3
uctuationson scalesm uch largerthan those pertinentto
the sim ulations.H ence,one has to assum e that there is
no additionalphysics that could alter the resultlike,e.g.,
gravitationalwaves or a slight tilt ofthe initialspectrum
away from the scale-invariant form .T he approach taken
here avoidsthis problem .T he m ass function ofobjects in
the U niverse is very steep at the high-m ass end.In other
words,m assive objects (like clusters ofgalaxies) are not
only rare,but their abundance sensitively depends on the
am plitude of the power spectrum .W hite, Efstathiou,&
Frenk (1993) introduced this way of xing the am plitude
by determ ining 8 ,the square root ofthe variance ofthe
density eld sm oothed over 8h 1 M pc spheres,such that
the observed abundance ofrich clusters is m atched.T hey
used the clusterm assfunction.R ecentstudiesofthe clusterX -ray tem perature function (Eke,C ole,& Frenk 1996;
V iana & Liddle 1996) nd sim ilar results. For the low density G IF sim ulations,the result by Eke et al.(1996)
was taken,
8
=
(0:52
(0:52
0:04)
0:04)
0:46+ 0:10
0
0:52+ 0:13
0
0
0
for
for
0
= 0
+
= 1
(5)
For the 0 = 1 sim ulations, slightly larger values than
suggested by eq.(5) were adopted,according to the earlier resultby W hite et al.(1993).Table 1 sum m arisesthe
m odelparam eters.
T able 1.C osm ologicalparam eters ofthe G IF m odels. 0 and
are the density param eters for m atter and the cosm ologicalconstant,h is the H ubble param eter, 8 is the variance of
the density eld in spheres of 8h 1 M pc,and is the shape
param eter of the pow er spectrum . A lso given are the size of
the cosm ological sim ulation box and the m ass M m ax w ithin
1:5h 1 M pc radius ofthe m ost m assive cluster in 1015 M .
M odel
h
0
8
(3)
SC D M 1
CD M 1
CD M 1
O CD M 1
1.0
1.0
0.3
0.3
0.0
0.0
0.7
0.0
0.5
0.5
0.7
0.7
0.60
0.60
0.90
0.85
0.50
0.21
0.21
0.21
B ox Size M m ax
[M pc=h]
85
0.74
85
0.74
141
0.84
141
0.85
T he param eters show n in Table 1 were chosen not
onl
y
to ful l cosm ological constraints, but also to al= 1:13
low a detailed study ofthe clustering properties at very
(4)
early redshifts. T he m asses of individual particles are
1:0 1010 h 1 M
and 1:4 1010 h 1 M
for the high(B ond & Efstathiou 1984). In order to com pletely x
and low - 0 m odels, respectively.T he gravitationalsoftP (k), the norm alisation, A , has to be chosen. T his can
ening was taken to be 30h 1 kpc.
be done on the basis of m easurem ents of the m icrowave
C lusters are obtained from the sim ulation as folbackground anisotropiesby the C O B E satellite.H owever,
l
ow
s
.H igh-density regions are searched using a standard
this approach su ers from the fact that C O B E m easured
friends-of-friends group nder w ith a linking length of
1
b = 0:05 tim es the m ean interparticle separation.T his seA s usual, the H ubble constant is w ritten as H 0 = 100h
km s 1 M pc 1 .
lectsonly the dense coresofany collapsed object.A round
4
M atthias B artelm ann et al.:A rc statistics w ith realistic cluster potentials
the centres of these, all particles are collected w hich lie
w ithin a sphere ofradius rA = 1:5h 1 M pc,w hich corresponds to A bell’s radius.T hese objects are taken as clusters.For our analysis,the ve m ost m assive clusters are
cutoutofthesim ulation volum es.T hisprocedureneedsto
be expanded ifthe centres oftwo large clusters are closer
together than rA .In this case usually the m ore m assive
clusteristaken,and the otherone isdeleted from the list.
In ourcase,however,thisproblem did notoccur.T he initialdensity elds for the di erent cosm ologies share the
sam e random phases.
2.2.Second cluster sam ple
T able 2.C osm ologicalparam etersofthem odelsforthesecond
cluster sam ple.T he m eaning ofthe sym bols is the sam e as in
Tab.1.
M odel
SC D M 2
CD M 2
O CD M 2
h
0
1.0
0.3
0.3
0.0
0.7
0.0
0.5
0.7
0.7
8
0.60
1.12
1.12
0.50
0.21
0.21
B ox size M m ax
[M pc=h]
144
0.75
201.6
1.67
201.6
1.41
spectrum afterin ation isoftheH arrison-Zel’dovich form .
T he C O B E norm alisation yields a slightly higher 8 than
that derived by Eke et al.(1996).T he box size L ofthe
sim ulation volum e is xed to 288 M pc in physicalcoordinates for each run to achieve the sam e physicalspatial
resolution in allm odels.T hem assresolution in thecentral
sphere is 4:9 1010 h 1 M for the high-density m odel,
and 2:9 1010 h 1 M for the low -density m odels.
Since the sim ulations result in one m assive cluster in
the high-resolution region,it can easily be identi ed by
looking forthe deepestpotentialwellin this region.H ow ever,particular attention has to be paid in order to set
up initialconditions w ith a suitable overdense region in
the centralsphere,representing the seed for the m assive
cluster. T his is done in the follow ing way. First, a presim ulation is perform ed by lling the w hole sim ulation
box w ith particles ofthe sam e m ass as those in the second layer.A t z = 0,the cluster-like objects are identi ed
using a specialgroup- nding algorithm (H ussetal.1997).
T heparticles nally form ing theseobjectsde nethecorresponding overdenseregion atthe initialredshift.T he nal
starting con guration is then centred on one ofthese regions.B y adding sm allscale powerto such a density peak,
the clustering properties of the region can be changed.
H ence,it is possible that the sim ulation nally arrives at
severallow -m assobjects ratherthan at one m assive halo.
T his can be avoided by testing the clustering properties
using the Zel’dovich approxim ation.W hen propagated to
low redshifts w ith this technique,the particlesin the centralsphere m ust form a distinct m atter accum ulation in
the centreratherthan show ing only lam entary structure.
W ith this procedure, four suitable overdense regions
are identi ed in the SC D M 2 m odel.For the C D M 2 and
for the O C D M 2 clusters,the starting con gurations are
centred on the sam e regions as for the SC D M 2 clusters.
T his ispossible since the clustering propertiesare de ned
m ainly by the localrealisation ofthe random eld,w hich
isthe sam e forallthree m odels.H owever,the nalcluster
haloesneed notrepresentthe m ostm assive clustersin the
sim ulation box.
T he second cluster sam ple is sim ulated using a special
m ulti-m asstechnique w hich isexplained in detailin H uss,
Jain,& Steinm etz (1997).In contrast to the G IF sim ulations,this technique gives only one m assive cluster per
run.H owever,it allow s one to study the evolution ofone
individualclusterw ithouttheneed forextensivecom puter
resources.
T he essentialpart ofthe m ulti-m ass technique is the
initialparticle arrangem ent.It consists ofthree spherical
layersem bedded into a cubic volum e,each lled w ith particles of di erent m ass. T he centralsphere encom passes
the leastm assiveparticlesand issurrounded by two shells
of m ore m assive particles.T he rest of the cubic sim ulation volum e is lled up w ith the m ost m assive particles.
T he innersphere m ustinitially be large enough to enclose
allparticles w hich end up in a cluster.T he gravitational
forces on the particles are calculated using a com bined
G R A PE/PM N -body code assum ing periodic boundary
conditions.T he PM partperform sforce calculationsw ith
periodic boundary conditions for allparticles.In the inner three shells,the force is additionally calculated w ith
a PM code using vacuum boundary conditions.T hisforce
issubtracted from the periodically com puted PM force to
obtain the periodic contribution to the force only.T his is
added to thehighly resolved forceprovided by theG R A PE
board for the particles in the inner shells.
T he second cluster sam ple consists of 12 clusters in
total.Four clusters are sim ulated for each ofthree di erent cosm ologies,w hich resem ble the SC D M , C D M ,and
O C D M m odels of the G IF project. T he m odel param eter are sum m arised in Table 2.A llclusters belonging to
one cosm ologicalm odelare partofthe sam e realisation of
the corresponding density eld.In addition,the phasesof
the initialG aussian random eld are identicalin the three
cosm ologicalm odels.
T he initial conditions are calculated using eq.(3) to
m atch the power spectrum of the di erent cosm ologies.
T he norm alisation of P (k) is chosen as determ ined by
W hite et al. (1993) for SC D M 2, and for O C D M 2 and 3.Sim ulations of arcs
C D M 2 itm atchesthe C O B E C M B anisotropy m easurem ents.C ontributionsfrom gravitationalwavesneed notbe O ur m ethod to investigate the arc-form ation statistics of
considered sinceeq.(3)assum esthattheprim ordialpower the num ericalcluster m odels was described in detailby
M atthias B artelm ann et al.:A rc statistics w ith realistic cluster potentials
B artelm ann & W eiss (1994).W e w ill therefore keep the
presentdescription briefand referthereaderto thatpaper
for further inform ation.For generalinform ation on gravitationallensing,see Schneider,Ehlers,& Falco (1992) or
N arayan & B artelm ann (1997),and references therein.
T he num ericalcluster m odels yield the spatialcoordinates and velocities ofdiscrete particles w ith equalm ass.
In order to use them for gravitational lensing, we need
to com pute the surface m ass density distribution ofeach
cluster m odelin each ofthe three independent directions
ofprojection.T hem assdensity is rstdeterm ined by sorting the particles into a three-dim ensionalgrid and subsequently sm oothing w ith a G aussian lter function. T he
grid resolution and the w idth ofthe G aussian are adapted
to the num erical resolution of the N -body codes in order not to lose spatialresolution by the sm oothing ofthe
density eld.T he sm oothed density eld isthen projected
onto the three sides ofthe com putation volum e to obtain
three surface-density elds for each cluster.
T hephysicalsurfacem assdensity eldsarethen scaled
by the critical surface m ass density for lensing, w hich
apart from the cosm ologicalparam eters depends on the
cluster-and source redshifts.W e keep the redshift for all
sources xed at zs = 1,and the clusters are at 0 < zc <
1. T his nally yields three two-dim ensionalconvergence
elds (x;z c) for each cluster m odelat redshift zc.From
(x;zc),allquantitiesdeterm ining thelocallensm apping,
i.e.,the de ection angle and itsspatialderivatives,can be
com puted.W e determ ine the lens properties ofthe clus00
ters on grids w ith an angularresolution of0:
3 in the lens
plane in order to ensure that lensed im ages be properly
resolved.
Sources are then distributed on a regular grid in the
sourceplane.T heresolution ofthissourcegrid can bekept
low close to the eld boundaries because there no large
arcs occur. C lose to the caustics of the clusters, w here
the large arcsare form ed,the source-grid resolution is increased w ith the increasing strength ofthe lens.For our
later purpose ofstatistics,sources are weighted w ith the
inverse resolution of the grid on w hich they are placed.
T he sources are taken to be intrinsically random ly oriented ellipses w ith their axis ratios draw n random ly from
the interval[0:5;1],and their axes determ ined such that
00
theirarea equalsthatofcirclesw ith radius0:
5.A lthough
this choice of source properties appears fairly sim ple, it
should not a ect the arc statistics because these m ainly
re ectthe localpropertiesofthe lens m apping,w hich are
independentofthe particularchoice ofsourcesize orellipticity distribution.W e checked that a change in average
source size did not change the results.
T hesourcesarethen viewed through theclusterlenses.
A ll im ages are classi ed in the way detailed by B artelm ann & W eiss(1994).A m ong otherthings,the classi cation yields for each im age its length L,its w idth W ,and
its curvature radius R .In total,we classify the im ages of
about 1:3 106 sources.
5
K now ing the area covered by the cluster elds, and
having determ ined the frequency of occurrence of im age
properties such as a given length, w idth, and curvature
radius,we can com pute crosssections forthe form ation
ofim ages w ith such properties.T he arc cross section ofa
cluster is de ned as the area in the source plane w ithin
w hich a source has to lie in order to be im aged as an arc.
W e m ostly focus on crosssections for the length-to-w idth
ratio r ofarcs.A partfrom the im age properties,the cross
sections depend on redshift, = (z).
G iven cross sections (z), we com pute the optical
depth forthe form ation oflarge arcs.T he opticaldepth
is the fraction ofthe entire source plane w hich is covered
by cluster cross sections,
Z
nc
=
4 D s2
zs
dz(1 + z)3
0
dV (z)
dz
(z);
(6)
w here nc is the present cluster num ber density,D s is the
angular-diam eterdistance to the source plane,and dV (z)
isthe propervolum eofa sphericalshellofw idth dz about
z.T he factor (1 + z)3 accounts for the cosm ologicalexpansion factor.
4.R esults
4.1.C ross sections
A ccording to the prescription in Sect.3,we rstcalculate
crosssectionsforeach individualm odelcluster,projected
in each ofthe three independent spatialdirections.Interpolating in redshift between the redshifts of the m odel
clusters, this yields cross sections (z) as a function of
redshift.W e then averagethese crosssections(1)overthe
three projection directions and (2) over all m odel clusters w ithin a given cosm ologicalm odel. W e thus obtain
cross sections for the four cosm ologicalm odels.Figure 1
show san exam ple,the crosssectionsforarcsw ith lengthto-w idth ratio r 7:5.
T he averaged cross sections in Fig.1 revealhuge differencesbetween the cosm ologicalm odels.W hile the cross
sectionsforstandard C D M (SC D M ) and C D M are com parable,the m axim um crosssectionsfor C D M and open
C D M (O C D M ) exceed that for SC D M by about half
and one order of m agnitude, respectively, and the redshift range w here (z) > 0 is w ider in O / C D M than
in S/ C D M .C ross sections for other,large values ofthe
length-to-w idth ratio r, or for large arc lengths, show a
qualitatively sim ilar behaviour.
Since the clusters in di erent cosm ologies arise from
initial density perturbations w ith the sam e random
phases, they can also be com pared individually rather
than statistically. O n the w hole, the individual clusters
show the sam e qualitativebehaviourasthe averaged cross
sectionsshow n in Fig.1.R esultsobtained foreach cluster
set individually agree wellw ith each other.
6
M atthias B artelm ann et al.:A rc statistics w ith realistic cluster potentials
F ig. 1. A veraged cross sections for arc length-to-w idth ratio
r 7:5 for clusters in the four di erent cosm ological m odels,
distinguished by line types as indicated.T he gure show s that
clustersin the C D M m odelproduce the few est arcs,and clusters in the open C D M m odelthe m ost.N ote the logarithm ic
scale ofthe ordinate:T he m axim a ofthecrosssectionsdi erby
m ore than an order of m agnitude.T he redshift ranges w here
(z)> 0 are larger in O / C D M than in S/ C D M .
F ig. 2.N orm alised,di erentialopticaldepth as a function of
redshift,forarcsw ith length-to-w idth ratio r 7:5.T he curves
indicate the m ost probable redshift for a cluster form ing arcs
for the four cosm ologicalm odels used.T he bars show the 1redshift range,the dots indicate the m ean arc-cluster redshift.
T he plot show s that there is no signi cant di erence in arc
cluster redshift betw een the four cosm ologicalm odels.
4.3.O pticaldepth
4.2.A rc-cluster redshift
W e now com pare the opticaldepth forform ation ofarcs
w ith given length-to-w idth ratio r in the four cosm ologicalm odels.A sbefore,the opticaldepth iscalculated from
eq.(6).W e do not specify the cluster num ber density nc
yet,but calculate the opticaldepth per unit cluster density,nc 1 .R esults are show n in Fig.3.
Figure 3 con rm s the trends indicated by the cross
sections in Fig.1,but allow s to com pare optical depths
for a w ide range of arc length-to-w idth ratios r. T here
is an intervalat interm ediate r, 5 < r < 10,w here the
optical depths for SC D M and C D M are alm ost equal.
O nly at r > 10 does the opticaldepth in C D M m odels
drop below that ofSC D M m odels.For r > 4,the optical
depth for C D M m odelsisconstantly higherthan thatfor
SC D M m odels by a factor of 10.T he opticaldepth for
the O C D M m odelis highest,exceeding the SC D M value
by up to 2 orders ofm agnitude at large r.
T hehatched region around theSC D M curveillustrates
1- bootstrap errors,w hich we obtained by bootstrapping
the cluster sam ple.T hey give an im pression ofthe uncertainty ofthe opticaldepth due to the lim ited num ber of
clusters in our sam ples.T he uncertainty due to the speT able 3.M ean redshifts and redshift ranges for clusters pro- ci c realisation of the lensed background galaxy sam ple
are sm aller by about a factor of ve.
ducing large arcs in the four di erent cosm ologicalm odels.
In orderto em phasise the results,Fig.4 show sthe opm odel
zc
h(zc zc)2 i1=2
ticaldepths for C D M , C D M ,and O C D M ,divided by
SC D M
0:29
0:09
the opticaldepth for SC D M .
A t w hat redshifts do we expect to nd the m ost clusters that produce large arcs? In other words,clusters at
w hich redshift contribute m ost to the arc opticaldepth?
To answer this question, we com pute the optical depth
from eq.(6) and the di erential optical depth d =dz,
and plotin Fig.2 the norm alised di erentialopticaldepth
1
d =dz asa function ofredshift for the four cosm ologicalm odels.
T he curves in Fig.2 show that the di erentialoptical
depth peaksaround z 0:3 0:4.T he barsinserted in the
gure indicate the 1- redshift range,and the dots show
the average arc-clusterredshift.A lthough there isa slight
tendency that the m ean arc-cluster redshift is sm allest in
theSC D M m odel,largerfor C D M ,and largestfor C D M
and O C D M ,theredshiftvariancesarelargeenough forthe
redshiftrangesin the cosm ologicalm odelsto overlap.T he
gure furtherm ore suggests that the di erences between
cosm ologicalm odelsaredom inated by noise.N um bersare
given in Table 3.
CD M
CD M
O CD M
0:38
0:36
0:39
0:06
0:09
0:12
5.D iscussion
M atthias B artelm ann et al.:A rc statistics w ith realistic cluster potentials
F ig. 3. O ptical depth for arc form ation as a function of arc
length-to-w idth ratio r,forthefourdi erentcosm ologicalm odels.T he opticaldepth for the C D M and the standard C D M
m odels are com parable for interm ediate length-to-w idth ratios
r. For large r, the optical depth is sm allest for the C D M
m odel.T he largest opticaldepth isproduced by clusters in the
open C D M m odel,follow ed by those in the C D M m odel.For
r 10,the opticaldepths for open C D M , C D M ,and standard C D M di er by about an order of m agnitude each. T he
hatched area around the SC D M curve indicates 1- bootstrap
errors.
5.1.Results
W e have used num ericalsim ulations to calculate the opticaldepth forthe form ation oflarge arcsin di erentcosm ologies.T he sim ulated clusters are cut out oflarge cosm ologicalsim ulation volum es,so that the tidale ects of
surrounding m atter are taken into account. T he cosm ological sim ulations were norm alised to the observed localnum ber density ofrich galaxy clusters.Four di erent
cosm ogonic m odels were used.T hese are:standard C D M
(SC D M ),w ith 0 = 1,
= 0,h = 0:5, 8 = 0:6,and
shape param eter = 0:5; a C D M m odel w ith reduced
sm all-scalepower( C D M ),w hich di ersfrom SC D M only
by the shape param eter = 0:21;open C D M (O C D M )
w ith = 0:3,
= 0, h = 0:7, 8 = 0:9 or 1:1, and
= 0:21; and nally a spatially at, low -density C D M
m odel ( C D M ) w ith 0 = 0:3,
= 0:7, h = 0:7,
= 0:21. A ll cosm ological sim u8 = 0:9 or 1:1, and
lations start from density perturbations w ith the sam e
random phases,so that allclusters can be com pared individually in di erent cosm ogonies.For SC D M ,O C D M ,
and C D M ,we sim ulated nine clusters,and ve clusters
for C D M .
T he lensing properties ofthe clusters w ith respect to
large arcs were calculated in a way that has extensively
been described earlier(B artelm ann & W eiss1994;B artelm ann et al. 1995). T he calculations result in averaged
cross sections for the clusters as a function of redshift,
7
F ig. 4. O ptical depths for large arcs, norm alised by the arc
opticaldepth in the standard C D M m odel.T he curvesem phasise the large di erence in opticaldepth for the four m odels.
For the largest r plotted, there are m ore than tw o orders of
m agnitude betw een open C D M and standard C D M clusters,
about one order of m agnitude betw een C D M and standard
C D M ,and clusters in the C D M m odelare less e cient than
the standard C D M m odels for r > 10.
w hich can then be converted to opticaldepths for the
form ation oflarge arcs.
O ur m ain result is that the optical depths for large
arcs,w ith length-to-w idth ratio r 10,di erby ordersof
m agnitude for the di erent cosm ologies.G enerally,clustersin theSC D M and C D M m odelsproducethe sm allest
opticaldepth.Forr 10,the opticaldepthsforthese two
m odelsare com parable,butforlargerr,the opticaldepth
in the C D M m odelfallsbelow thatin the SC D M m odel.
In the C D M m odel, the arc optical depth is larger by
about an order of m agnitude than for SC D M , and the
optical depth is largest in the O C D M m odel, exceeding
the SC D M opticaldepth by about two orders ofm agnitude.W e em phasise that these results are independent of
w hether our cluster sam ples are in any sense com plete or
not, because the sim ulations are designed such that the
clusters can be com pared individually. O ur conclusions
do, however,rest on the assum ption that the sim ulated
clusters are typical for the clusters w ith the largest m ass
in each ofthe cosm ologicalm odels.W e believe thatthisis
guaranteed by thelargesizeofthecosm ologicalsim ulation
volum es from w hich the clusters were taken.
Itisa com bination ofe ectsthatleadsto the largedifference in arc opticaldepth across the cosm ologicalm odels that we have investigated.(i) C lusters form earlier in
low -density than in high-density universes.In SC D M ,norm alised to the cluster abundance,the form ation of such
clusters w hich would in principle be m assive enough for
strong lensing is delayed to such low redshifts that they
failto be e cient lenses for sources at redshifts z s
1.
8
M atthias B artelm ann et al.:A rc statistics w ith realistic cluster potentials
(ii) Forlow -density universes,the propervolum e per unit
redshift is larger than for high-density universes. G iven
the observed num ber density of clusters today,this volum e e ect increases the num ber of clusters between the
sourcesphereand theobserverw hen 0 issm all.(iii)C lusters that form early are m ore concentrated than clusters
thatform late.C lustersin low -density universestherefore
reach higher centralsurface m ass densities than clusters
in high-density universes.(iv)Strong gravitationallensing
is a highly non-linear e ect.T his is because the arc cross
section ofa clustersensitively dependson thelength ofthe
caustic curve and the num ber ofcusp points contained in
it.T he propertiesofthe caustic curve do notonly depend
on the surface m ass density,but also on the tidal eld of
a cluster,w hich is in uenced by the cluster m orphology.
(v) B ecause of (iv), asym m etric clusters are m uch m ore
e cient in producing large arcs than sym m etric clusters
(B artelm ann etal.1995).T he degree ofsubstructure ofa
clusteristhereforevery im portantforarcstatistics.W hile
clustersare in the processofform ation,they are expected
to behighly asym m etric.Ifthishappensatredshiftsw here
lensing ise cientfora given sourcepopulation,the asym m etric cluster m orphology further increases the stronglensing crosssection.M ostclustersin O C D M and C D M
form at z
0:3, exactly w here lensing is m ost e cient
for sourcesat zs 1.C lusters in SC D M and C D M form
later,atz 0:1,w here theirlensing e ciency and thatof
their sublum ps is already suppressed by the lensing geom etry. A lthough clusters in S/ C D M form later than
in O / C D M and should therefore be m ore asym m etric,
the sublum psin O / C D M clustersare m ore com pactand
thus tend to persist for a longer tim e after m erging w ith
the cluster.
because the cross section per cluster should scale approxim ately w ith D e2 (z;zs).T hisquantity is plotted in Fig.5
for 0 = 1 and 0 = 0:3,w ith
= 0.O f course,this
sim ple estim ate com pletely neglects the in uence of the
change in cluster concentration across the cosm ological
m odels,and the non-linearitiesofthestrong lensing e ect.
H owever,it su ces to dem onstrate that large di erences
in the arc cross section are expected between high- and
low -density universes.
F ig. 5. Estim ate for the num ber of e cient lensing clusters
per redshift interval, dN lens=dz, as given in eq.(8). R esults
for 0 = 1 and 0 = 0:3, both w ith
= 0,are plotted,as
indicated.T he gure illustrates that the delayed form ation of
clusters in a high-density universe,com bined w ith the e ects
oflensing e ciency and cosm ic volum e,already account for a
large di erence in the expected num ber ofarcs.
5.2.Illustration
A sim ple Press-Schechter type argum ent illustrates the
in uence ofform ation tim e and cosm icvolum e.A ccording 5.3.In uence of\m issing clusters"
to Press& Schechter(1974),the(com oving)fraction ofthe A llcosm ologicalsim ulation volum es from w hich we have
cosm ic m atter that is contained in clusters is
taken the cluster m odels are of order a few tim es
106 h 3 M pc3. Since the cluster m ass function is very
1
c
F c(z)= erfc p
;
(7) steep, we are therefore likely to m iss the m ost m assive
2
2 R D (z)
clusters. In order to estim ate their in uence on the arc
w ith c 1:686, R thevarianceofthedensity contraston cross sections, we have repeated the arc sim ulations for
clusterscalestoday,and D (z)the (cosm ology-dependent) SC D M w ith surface-m assdensitiesrescaled to highercluslinear grow th factor ofdensity perturbations.T he cluster term ass.LetM 0 and M > M 0 betheoriginaland rescaled
fraction at redshift z, norm alised to the present cluster cluster m asses,respectively.T hen,we take
#
"
fraction, provides an estim ate for the change in cluster
1=3
1=3
M
M
0
0
num berdensity w ith redshift.M ultiplying w ith theproper
(x;zc;M )=
(9)
x;zc;M 0
M0
M
cosm icvolum e4 D 2(z)jd(ct)=dzjdz ofa shellofw idth dz
and thesquared e ectivelensing distanceD e (z;zs)yields
an estim ate forthe num berofe cientlensing clustersper forthe rescaled surface-m assdensity and com pute the arc
crosssection from that.In e ect,wecalculatethearccross
redshift interval,
section (zc;M ) of a cluster of sim ilar structure as the
d(ct)
dN lens
originalone,but w ith higher totalm ass.For each cluster
3
2
2
;(8)
= F c(z) (1+ z) D e (z;zs) 4 D (z)
m odelatredshiftzc,wethen averagethearccrosssections
dz
dz
M atthias B artelm ann et al.:A rc statistics w ith realistic cluster potentials
over m ass,weighted by the Press-Schechter cluster m ass
function n(M ),
R
dM n(M ) (zc;M )
R
:
(10)
h (zc)i=
dM n(M )
Finally,we com pute m ass-averaged opticaldepths h i by
substituting h (zc)i for (zc) in eq.(6).
W efound thath idi ersfrom only by 10-15 percent.
A lthough the arc crosssection isa fairly steep function of
m ass,the increase in is m ore than com pensated by the
steep decrease ofthe cluster m ass function.For exam ple,
quadrupling the m asses of the C D M clusters increases
the averaged cross section for arcs w ith r 10 by about
two orders ofm agnitude,but decreases the cluster m ass
function by aboutthree ordersofm agnitude,alm ostcom pletely cancelling the e ectofthe largerarc crosssection.
For clusters in the other cosm ologicalm odels,the change
in theopticaldepth should besm allerthan thatin C D M .
C lusters w ith higher m asses than those contained in our
sam ple can therefore safely be neglected.
W e have only studied the m ostm assive clusters found
in each cosm ologicalsim ulation.Even then,the leastm assive clustersin each sam ple contribute little ornothing to
the arc opticaldepth.Extending our sam ples by including less m assive clusters would therefore change the arc
opticaldepth negligibly or not at all.
Since we select the m ost m assive clusters at redshift
zero and study their progenitors, it could be that at
higher redshift other clusters in the cosm ological sim ulations would be m ore m assive.In other words,it is not
im m ediately clear that the progenitors ofthe m ost m assive clusters are also the m ost m assive clusters present
at higher redshift.In order to test that,we selected the
C D M m odel,w here clustersform atthe lowestredshifts,
and checked w hether the progenitorsofthe oursam ple of
the ve m ostm assive clustersisidenticalw ith the sam ple
ofthe ve m ost m assive clusters at the redshifts relevant
for lensing.T his turned out to be the case.In the other
m odels,w here clusters form at higher redshifts,possible
m isidenti cations are even less likely than in C D M .
G iven these results,we are con dent that our cluster
sam plesfairly re ectthose clustersthatdom inate the opticaldepth for the form ation oflarge arcs.
9
T he num ber density of such clusters is estim ated to be
nc 2 10 6 h3 M pc 3 (Le Fevre et al.1994).A rc surveys in this sam ple have show n that the num ber ofarcs
w ith r
10 and a lim iting m agnitude of B = 22:5 (or
R = 21:5;these are the arc criteria setup by W u & H am m er 1993) is roughly 0:2 0:3 per cluster (Le Fevre et
al.1994;G ioia & Luppino 1994).
C lusters w ith L X
2 1044 ergs 1 should be fairly
represented by the m assive sim ulated clustersin oursam ples. H aving velocity dispersions > 800km s 1 , the em piricalrelation between velocity dispersion and X -ray lum inosity obtained by Q uintana & M elnick (1982) im plies
X -ray lum inosities in the right range. W e can therefore
assum e that the arc cross sections ofour sim ulated clusters are typicalfor X -ray lum inous clusters in the EM SS
survey.
T hecurvesin Fig.3 givenc 1 .U sing the num berdensity ofbright EM SS clusters given above,
8
< 2:9 10 6 (O C D M )
3:3 10 7 ( C D M )
:
(11)
(r 10)
:
4:4 10 8 (S/ C D M )
Since the w hole sky has 4:1 104 square degrees,the
totalsolid angle in w hich sources at zs 1 are im aged as
large arcs w ith r 10 is
8
< 1:2 10 1 sq.deg. (O C D M )
1:4 10 2 sq.deg. ( C D M )
:
(12)
!
:
1:8 10 3 sq.deg. (S/ C D M )
T hesourcesw hich areim aged asarcsw ith theaboveproperties,r 10 and R
21:5 correspond to sources w ith
R < 23:5 because ofthe m agni cation.Taking the num berdensitiescom piled and m easured by Sm ailetal.,there
are 2 104 such sources per square degree.T he average redshift ofsuch sources is 0:8 1 (e.g.Lilly et al.
1995).Since the average arc-cluster redshift in our m odels is at zc
0:3 0:4,the exact redshift of sources at
z 0:8 1 hasonly very littlein uence;thecriticalsurface
m ass density changes by 10% w hen sources are shifted
from z = 0:8 to z = 1:2.It follow s that the num ber of
such arcson the w hole sky expected from oursim ulations
is
8
< 2400 (O C D M )
6.C om parison w ith observations
280 ( C D M )
:
(13)
N arcs
:
So can we constrain cosm ologicalparam etersthrough arc
36 (S/ C D M )
statistics? For that,we would have to com pare the num ber of observed arcs to that predicted by our m odels. T here are 7500 clusterson the sky w hich m atch the criT his com parison is ham pered by the fact that there is teria ofthe EM SS bright cluster sam ple (Le Fevre et al.
no com plete sam ple ofobserved clustersselected by m ass, 1994).Taking the num ber ofarcsper clusterfound in the
as it should be for a fair com parison.T here is one clus- EM SS clusters,the expected num berofarcson the w hole
ter sam ple,however,w hose de nition com es close to this sky is 1500 2300.D espite the obvious uncertainties
criterion,nam ely the EM SS sam ple ofX -ray bright clus- in this estim ate,the only of our cosm ologicalm odels for
ters, for w hich the X -ray lum inosity in the EM SS en- which the expected num berofarcscom esnearthe observed
ergy band is L X
2 1044 ergs 1 (h = 0:5,q0 = 0:5). num ber is the open C D M m odel.T he othersfailby one or
10
M atthias B artelm ann et al.:A rc statistics w ith realistic cluster potentials
two orders ofm agnitude.T he large di erences in arc opticaldepth between the cosm ologicalm odels investigated
m akesthisresultfairly insensitive to m oderate uncertainties.Itthereforeappearsfairto concludethatarcstatistics
in the fram ework ofcluster-norm alised C D M m odels dem ands that 0 is low ,and that
is sm all.C onversely,
if 0 < 1,clusters have to form earlier than in our m odels.W eestim atew ith thesim plePress-Schechterapproach
sketched abovethatin orderto achievethat, 8 1:2 1:3
would be necessary in the SC D M case.
W e have neglected the potentialin uence ofcD galaxies or cooling ow s on the arc cross sections that could
increase the centralsurface m ass densities ofthe clusters
and thus also theirarc crosssections.M ostprobably,this
in uence is sm all com pared to the huge di erences between the cosm ologicalm odels.N onetheless,we w illstudy
this issue in detailin a further paper because individual
cluster galaxies m ay wella ect arc m orphologies,if not
their totalnum ber.
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