Simulating Weak Lensing by Clusters and Large-Scale Structure

8 Pages·0.36 MB·English

by  Bhuvnesh Jain

Proceedings:EvolutionofLargeScaleStructure–Garching,August1998
SIMULATING WEAK LENSING BY CLUSTERS AND
LARGE-SCALE STRUCTURE
BhuvneshJain1,UrosSeljak2 andSimonWhite2
1Dept. of Physics and Astronomy, Johns Hopkins University,Baltimore MD, USA
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2Max-Planck-Institut fu¨r Astrophysik, D–85740 Garching
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n ABSTRACT. Selected results on estimating cosmological parameters from
a simulatedweaklensingdatawithnoisearepresented.Numericalsimulations
J ofraytracingthrough N-bodysimulationshave beenusedtogenerate shear
andconvergencemapsduetolensingbylarge-scalestructure.Noiseduetothe
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intrinsicellipticitiesofafinitenumberofgalaxiesisadded.Inthiscontribution
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wepresentourmainresultsonestimationofthepowerspectrumanddensity

parameter Ω from weak lensing data on several degree sized fields. We also
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show that there are striking morphological differences in the weak lensing
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mapsofclustersofgalaxiesformedinmodelswithdifferentvalues ofΩ.
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1 1 Introduction cal work includes Wambsganss, Cen
0 & Ostriker (1998), Premadi, Martel
9 Weaklensingbylarge-scalestruc- & Matzner (1998), van Waerbeke,
9 ture (LSS) shears the images of dis- Bernardeau&Mellier(1998),Bartel-
/ tant galaxies. The first calculations mann et al (1998) and Couchman,
h
of weak lensing by LSS (Blandford Barber&Thomas(1998)
p
et al. 1991; Miralda-Escude 1991; The dark matter distribution ob-
-
o Kaiser 1992), based on the pioneer- tained from N-body simulations of
r ing work of Gunn (1967), showed different models of structure forma-
st that lensing would induce coherent tionisprojectedonto2-dimensional
a ellipticities of order 1% over regions planes lying between the observer
: of order one degree on the sky. Re- andsourcegalaxies.Typicallyweuse
v centlyseveralauthorshaveextended galaxies at z ∼ 1 with ∼ 20−30
Xi this work to probe semi-analytically planes. We propagate ∼ 106 light
thepossibilityofmeasuringthemass rays through these planes by com-
r
a power spectrum and cosmological puting the deflections due to the
parameters from the second and matter at every plane. Fast Fourier
third moments of the induced ellip- Transformsareusedtocomputegra-
ticityorconvergence(Bernardeauet dients of the potential that provide
al.1997;Kaiser1998;Stebbins1996; the shear tensor at each plane. The
JainandSeljak1997;Schneideretal. outcomeofthesimulationisamapof
1997). theshearandconvergenceonsquare
Analytical calculations suggest patches of side length 1−5◦. Sev-
that nonlinear evolution of the den- eral realizations for each model are
sity perturbations that provide the neededtocomputereliablestatistics
lensing effect can significantly al- onscalesrangingfrom1′ to1◦.
ter the predicted signal. It is ex- In the weak lensing regime, the
pected to enhance the power spec- magnificationandinducedellipticity
trum on small scales and makes are given by linear combinations of
the probabilitydistributionfunction the Jacobian matrixof the mapping
(pdf) of the ellipticity and conver- from the source to the image plane.
gence non-Gaussian. We have car- TheJacobianmatrixisdefinedby
ried out numerical simulations of
rlaatyiontradciantga tthorocuogmhpNut-ebotdhyesifmululy- Φij ≡ ∂∂δθθji (1)
nonlinear moments and pdf. Details
of the method and results are pre- where δθi is the i−th component of
sented in a separate (Jain, Seljak & the perturbation due to lensing of
White1999);herewesummarizethe the angular position on the source
method and present some highlights plane, and θj is the j−th compo-
of the results. Other recent numeri- nent of the position on the image
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Jainetal.
Figure1. Thedimensionlesspowerspectrumofκ.Forthecosmologicalmodel
indicated by Ωm and Γ in the panel, the power spectrum from ray tracing
shownbythesolidcurvesiscomparedwiththelinear(long-dashed)andnon-
linear analytical (short-dashed) predictions. The angular wavenumber l is
given in inverse radians – the smallest l plotted corresponds to modes with
wavelength of order L≃3◦, where L isthe side-lengthof the field.
plane.Theconvergence isdefined as 2 Results of Simulations
κ = −(Φ11+Φ22)/2, while the two
components of the shear are γ1 = The N-body simulations used for
−(Φ11−Φ22)/2andγ2=−Φ12.The the ray tracing are taken from four
convergence κ can be reconstructed different cosmological simulations,
from the measured shear γ1, γ2, up theparametersofwhicharesumma-
to a constant which depends on the rizedinTable1.TheN-bodysimula-
mean density in the survey area. If tions use an adaptive P3M method
the survey is sufficiently large and with2563 particles,andwerecarried
there is little power on scales larger out using codes kindly made avail-
thanthesurvey,thiserrorcanbene- able by the Virgo consortium (e.g.
glected. Jenkins et al. 1997). Coupled with
raytracingona20482gridthesesim-
ulationsprovideuswithasmallscale
resolution down to ∼<0.5′, well into
the nonlinear regime for weak lens-
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Numericalsimulationsofweaklensing
Table 1. Summary of the parameters used for the N-body simulations. h is
the Hubble constant inunitsof 100kms−1Mpc−1, Γisthe shape parameter
of the power spectrum, and the other parameters have their conventional
meaning.
Model Ωm ΩΛ h σ8 Γ
SCDM 1.0 0.0 0.5 0.60 0.50
τCDM 1.0 0.0 0.5 0.60 0.21
ΛCDM 0.3 0.7 0.7 0.90 0.21
OCDM 0.3 0.0 0.7 0.85 0.21
ing. found that S3 was sensitive to the
The power spectrum of κ mea- tails of the probability distribution
sured from the simulations is com- function(pdf)andthereforerequired
paredwiththeanalyticalpredictions largesamplesizes.Instead,wefitthe
of Jain & Seljak (1997) in figure pdf to an Edgeworth expansion and
1. We will focus in this contribu- estimatedS3asaparameter.There-
tion on the extraction of parame- sult is shown in figure 4 and shows
ters from simulated noisy data. Fig- that cosmological models with dif-
ure 2 shows the reconstructed con- ferent values of Ωm can be distin-
vergence field from noisy ellipticity guishedatahighlevelofsignificance
dataonasingle3◦field.Thegalaxies (theerrorbarsare1-σ).Thedashed
were randomly distributed and as- curves show the predictions of per-
signedintrinsicellipticiteswitheach turbationtheory.
component drawn randomly from a Finally, figure 5 shows the con-
Gaussian with rms=0.4. The recon- vergence field centered on typical
structedκiscomparedwiththefield richclustersintheEinstein-deSitter
withoutnoise,aswellaswithapure model, τCDM, and the open CDM
noise field, to show the significance model.Thefieldsare10’arcminutes
of reconstructed features. The field onasideandarechosenbecausecur-
is smoothed on the scale at which rentobservationaldataisalreadyca-
the noise and signal should be com- pableofdetectingsignalintheouter
parable. The figure shows that with parts of clusters. Comparison of the
observational parameters feasibleon EdSandopenmodelsshowsthatthe
largetelescopes,afieldafewdegrees convergence fields are morphologi-
onasidecanallowonetoreconstruct cally different: clusters in the open
thelarge-scalefeaturesdominatedby modelappearmorecompact,regular
groups and clusters of galaxies. Sta- andisolated.TheclustersintheEdS
tisticallyonecandomuchbetter. model are not fully relaxed and ap-
Figure 3 shows the power spec- pearlinkedtothesurroundinglarge-
trum of κ measured from simulated scale structure through filaments or
noisy data. On scales larger than moreirregularstructures. Thesedif-
10’ the signal dominates the noise. ferences can be quantified by using
The scales on which the power can topologicalmeasuresandbymeasur-
be measured are dominated by the ingmomentssuchasthequadrupole
density power spectrum on scales of or higher moments. In figure 6 we
about 1−10 h−1Mpc at z ∼ 0.3. simply show the mean profile of the
Thusweaklensingsurveysafewde- convergence around clusters, which
grees on a side will be sensitive to also shows differences between the
the dark matter power spectrum on twomodels.Thesourcegalaxieshave
thesescales. beentakentobeatz=1andthe10
Figure4showsoneattemptates- richestclustersinafield3degreeson
timating the density parameter Ωm a side have been used to obtain the
from simulated data. The simplest meanprofiles.
way to estimate Ωm is to measure
the skewness S3 of the convergence
(Bernardea et al. 1997; Jain & Sel-
jak 1997; Schneider et al. 1998). We
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Jainetal.
Figure2. The reconstructed convergence field from noisy ellipticitydata on
asingle3◦ field. Theupperleftpanel showsthefieldwithout noiseorsparse
sampling. The upper right panel shows the reconstructed field using 2×105
galaxies per square degree at a mean z =1. The lower left panel used 1/4th
asmanygalaxies.Thelowerrightpanelshowsamapofthe“curl”fieldgen-
erated from the same ellipticitydata as in the lower left panel. Fluctuations
in this field are solely due to noise, therefore comparison of the two panels
shows the significance of the reconstructed features inthe lower left panel.
3 Conclusion different cosmologies. Further work
is needed to quantify the differences
Wehaveshownresultsonestimat- and explore the effects of noise and
ingthepowerspectrumandΩmfrom ofvaryingtheredshiftdistributionof
simulated, noisy weak lensing data. sourcegalaxies.
Withseveralfieldsadegreeonaside,
the dark matter power spectrum on
scales of 1 − 10 h−1Mpc can be Acknowledgments
probed and Ωm estimated to within
about 0.1-0.2. We have also shown We are grateful to Matthias
preliminary results on the morpho- Bartelmann and Peter Schneider for
logicaldifferencesbetweenclustersin helpfuldiscussions.Itisapleasureto
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Numericalsimulationsofweaklensing
Figure3. Convergencepowerspectrumestimatedfromsimulatednoisydata.
The shear field on a single field 3◦ on a side is sampled by randomly dis-
tributedgalaxieswithintrinsicellipticitesassignedasinfigure 2.The power
spectrum of the reconstructed κ from the ellipticitydata isshown with error
bars obtained from 10 independent realizations of the noisydata.
thank Anthony Banday for his pa- Premadi, P., Martel, H., Matzner,
tient support in the writing of this R.,1998,ApJ,493,10
contribution.
Schneider, P., Ehlers, J., & Falco,
E.E., 1992, Gravitational Lensing
(SpringerVerlag,Berlin)
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Jainetal.
Figure 4. The density parameter Ωm estimated from the pdf of simulated
noisy data. The skewness parameter S3 is estimated by minimizing the χ2
with respect to an Edgeworth expansion of the pdf. The four curves with
decreasing peak heights are for the open, cosmological constant, Einstein-de
Sittermodels,allwithΓ=0.2CDMpowerspectra,andanEinstein-deSitter
model with Γ=0.5 CDM power spectrum.
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Numericalsimulationsofweaklensing
Figure5. Clusters in open and Einstein-de Sitter cosmologies. The conver-
gence in fields 10’ on a side centered on a rich cluster isshown for the EdS
model in the upper panels and for the open model in the lower panels. The
values of the convergence range from over10% in the centertobelow 1% in
the outermost regions. Figure 6 givesthe mean profiles of theconvergence.
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Jainetal.
Figure6. TheprofileoftheconvergenceinclustersinopenandEdScosmolo-
gies.The dashed curve shows the average convergence profile measured from
10 clusters in the EdS model while the solid curve shows the corresponding
profile for the open model.