Lensing and Supernovae: Quantifying The Bias on the Dark Energy Equation of State

0.22 MB·English

by  Devdeep Sarkar

DRAFTVERSIONFEBRUARY2,2008
PreprinttypesetusingLATEXstyleemulateapjv.10/09/06
LENSINGANDSUPERNOVAE:QUANTIFYINGTHEBIASONTHEDARKENERGYEQUATIONOFSTATE
DEVDEEPSARKAR1,ALEXANDREAMBLARD1,DANIELE.HOLZ2,3,ANDASANTHACOORAY1
1DepartmentofPhysicsandAstronomy,UniversityofCalifornia,Irvine,CA92617
2TheoreticalDivision,LosAlamosNationalLaboratory,LosAlamos,NM87545
3KavliInstituteforCosmologicalPhysicsandDepartmentofAstronomyandAstrophysics,UniversityofChicago,Chicago,IL60637
DraftversionFebruary2,2008
ABSTRACT
ThegravitationalmagnificationanddemagnificationofTypeIasupernovae(SNe)modifytheirpositionson
8
theHubblediagram,shiftingthedistanceestimatesfromtheunderlyingluminosity-distancerelation.Thiscan
0
introducea systematic uncertaintyin the darkenergyequationof state (EOS) estimated from SNe, although
0
thissystematicisexpectedtoaverageawayforsufficientlylargedatasets. UsingmockSN samplesoverthe
2
redshiftrange0<z≤1.7wequantifythelensingbias. We findthatthebiasonthedarkenergyEOSisless

n thanhalfa percentforlargedatasets(&2,000SNe). However,ifhighlymagnifiedevents(SNedeviatingby
a more than 2.5σ) are systematically removedfrom the analysis, the bias increasesto ∼ 0.8%. Given thatthe
J
EOS parametersmeasured from such a sample have a 1σ uncertainty of 10%, the systematic bias related to

0 lensinginSNdataouttoz∼1.7canbesafelyignoredinfuturecosmologicalmeasurements.
3
Subjectheadings: cosmology: observations—cosmology: theory—supernova—parameterestimation—

gravitationallensing
]
h
p
- 1. INTRODUCTION wide-area(>5deg2) searchesforSNe (inlieu ofsmall-area
o pencil-beamsurveys).
Since the discovery of the accelerating expansion of the
r InadditiontothestatisticalcovarianceofSNdistanceesti-
t universe(Riessetal.1998;Perlmutteretal.1999;Knopetal.
s mates,gravitationallensingalsointroducessystematicuncer-
a 2003; Riessetal. 2004), the quest to understandthe physics
taintiesintheHubblediagrambyintroducinganon-Gaussian
[ responsible for this acceleration has been one of the major
dispersionintheobservedluminositiesofdistantSNe. Since
challenges of cosmology. At present the dominant expla-
2 lensing conserves the total number of photons, this system-
nation entails an additional energy density to the universe
v atic bias averagesaway if sufficiently large numbersof SNe
called dark energy. The physics of dark energy is gener-
3 per redshift bin are observed. In this case the average flux
allydescribedintermsofitsequationofstate(EOS),thera-
4 of the many magnified and demagnified SNe converges on
tio of its pressure to density. In some models this quantity
1
theunlensedvalue(Holz&Linder2005). Nonetheless,even
4 can varywith redshift. While thereexist a variety of probes
with thousandsof SNe in the total sample it is possible that
. to explore the nature of dark energy, one of the most com-
0 the averaging remains insufficient, given that one may need
pellingentailstheuseoftypeIasupernovaetomaptheHub-
1 to bin the Hubblediagramat verysmall redshiftintervalsto
ble diagram, and thereby directly determine the expansion
7 improve sensitivity to the EOS. Furthermore, SNe at higher
historyoftheuniverse.Withincreasingsamplesizes,SNdis-
0 redshifts are more likely to be significantly lensed. If “ob-
tancescanpotentiallyprovidemultipleindependentestimates
: vious” outliersto the Hubble diagramare removedfromthe
v oftheEOSwhenbinnedinredshift(Huterer&Cooray2005;
i Sullivan,Cooray,&Holz2007;Sullivanetal.2007). Several sample,thisintroducesanimportantbiasincosmologicalpa-
X rameterdetermination,andcanleadtosystematicerrorsinthe
present and future SN surveys, such as SNLS (Astieretal.
r determinationofthedarkenergyEOS.
2006)andtheJointDarkEnergyMission(JDEM),areaimed
a
In this paper we quantify the bias introduced in the es-
atconstrainingthevalueofthedarkenergyEOStobetterthan
timation of the dark energy EOS due to weak lensing
10%.
of supernova flux. We consider the effects due to the
Although SNe have been shown to be good standardiz-
non-Gaussian nature of the lensing magnification distri-
able candles, the distance estimate to a given SN is de-
butions (Wang,Holz,&Munshi 2002), performing Monte-
gradedduetogravitationallensingofitsflux(Frieman1997;
CarlosimulationsbycreatingmockdatasetsforfutureJDEM-
Wambsganssetal. 1997; Holz&Wald 1998). The lensing
like surveys. The paper is organizedas follows: In §2.1 we
becomes more prominent as we observe SNe out to higher
discussourparameterizationofthedarkenergyEOS,§2.2dis-
redshift,withtheextradispersioninducedbylensingbecom-
cussesgravitationallensing,and§3isanin-depthdescription
ing comparable to the intrinsic dispersion (of ∼0.1 magni-
ofourmethodology.Wepresentourresultsin§4.
tudes)atz&1.2(Holz&Linder2005).Inadditiontothisdis-
persion,whichleadstoanincreaseintheerrorassociatedwith 2. BACKGROUND
distance estimate to each individual supernova, lensing also
2.1. ParameterEstimationUsingSNeIa
correlatesdistanceestimatesofnearbySNeonthesky,since
the lines-of-sight pass through correlated foreground large- ForaSNwithintrinsicluminosityL,thedistancemodulus
scalestructure(Cooray,Huterer,&Holz2006;Hui&Greene isgivenby
2006). Althoughthiscorrelationerrorcannotbe statistically
L/4πd 2 d
eliminatedbyincreasingthenumberofSNeintheHubbledi- m- M=- 2.5log L =5log L +25,
agram,theerrorscanbecontrolledbyconductingsufficiently 10"L/4π(10pc)2# 10(cid:18)Mpc(cid:19)
(1)
2 LENSINGANDSUPERNOVAE
where, in the framework of FRW cosmologies, the lumi-
nosity distance d is a function of the cosmological pa-
L
rameters and the redshift. We consider the two-parameter,
time-varying dark energy EOS (Chevalier&Polarski 2001;
Linder 2003)adoptedbytheDarkEnergyTaskForce(DETF)
(Albrechtetal. 2006). We take a flat universe describedby
the cosmologicalparametersΘ={h,Ω ,w ,w }, with h the
m 0 a
dimensionlessHubbleconstant,Ω thedimensionlessmatter
m
density,andw andw theparametersdescribingthedarken-
0 a
ergyEOSw(z)=w +w (1- a).Onecanthenexpressd as
0 a L
z cdz′
d (z)=(1+z) , (2)
L H(z′)
Z0
whereH(z)istheHubbleparameteratredshiftz:
H(z)=H0 Ωm(1+z)3+(1- Ωm)(1+z)3(1+w0+wa)e- 3(1w+azz) 1/2.
h i (3)
FIG. 1.—Top panel: The Hubble diagram with a linear redshift scale
2.2. WeakLensingofSupernovaFlux showingthedistancemoduluswithredshiftofasubsetofa2000SNmock
dataset.Bottompanel:Theresidualsofthedatarelativetothefiducialmodel
Light from a distant SN passes through the intervening ofaflatuniversewithΩm=0.3,w0=- 1,wa=0,andh=0.732.
large-scale structure of the universe. This causes a modifi-
cationoftheobservedfluxduetogravitationallensing:
at a given redshift, to the probability distribution for the
Fobs,lensed(z,nˆ)=µ(z,nˆ)Fobs,true, (4) reduced convergence, η, through P(µ)=P(η)/(2|κmin|) (see
Wang,Holz,&Munshi (2002); Eq. (6)), where κ is the
whereµ(z,nˆ)isthelensinginducedmagnificationatredshift min
minimum convergence. The free parameter in normalizing
zinthe directionofthe SN onthe sky,nˆ, andFobs,true is the
P(η), and hence, P(µ), is η , the maximum value for the
fluxthatwouldhavebeenobservedintheabsenceoflensing. max
reducedconvergence.Wedeterminetheuniquevalueofη
Themagnificationµcanbeeithergreaterthan(magnified)or max
which yields both P(µ)dµ= 1 and hµi = µP(µ)dµ= 1
lessthan(demagnified)one,withµ=1correspondingtothe
(to better than 10- 6). Since the mode is not equal to the
(unlensed)pureFRWscenario. R R
mean, the distribution is manifestly non-Gaussian. The
This magnification (or demagnification) of the observed
majority of distant supernovaeare slightly demagnified, and
flux leads to an errorin the distance modulus, which can be
theinferredluminositydistancesareskewedtohighervalues.
expressedas
The SN redshifts, on the other hand, remain unaffected,
Fobs,lensed since gravitational lensing is achromatic. Although lensing
[∆(m- M)] =- 2.5log =- 2.5log (µ),
lensing 10 Fobs,true 10 magnification is insignificant at low redshifts, at redshifts
(cid:18) (cid:19) (5) above one both weak and strong lensing become more
wherewehavewrittenµ(z,nˆ)asµforbrevity. prominent. ForsmallSN samplestheoverallbiasistowards
Evenintheabsenceoflensingthemeasureddistancemod- a larger cosmologicalacceleration (e.g., too large a value of
ulustoaTypeIaSNsuffersanintrinsicerror,sincethesuper- ΩΛ for ΛCDM models). The lensing also addsa systematic
novaearenotperfectstandardcandles. Thiserroristypically biasto estimates of the darkenergyEOS, shiftingto a more
takentobeaGaussiandistributionineitherfluxormagnitude, negativevalueoftheEOS(e.g.,lessthan- 1.0forauniverse
with a redshift-independent standard deviation of σ . On withacosmologicalconstant). Alongwithalargefractionof
int
theotherhand,aslensingintrinsicallydependsontheoptical demagnifiedSNe,lensingproducesasmallnumberofhighly
depth,whichincreaseswithredshift,thescatter(orvariance) magnifiedsources. Ifthishigh-magnificationtailcanbewell
due to lensing also increases with redshift (Holz&Linder sampled, the average of the lensing distribution is expected
2005). The probabilitydistribution function (PDF) for lens- toconvergeonthetruemean,andthebiascanbeeliminated
ingmagnification,P(µ,z),ofabackgroundsourceatredshift (Wang2000;Holz&Linder2005).
z, dependsonboththeunderlyingcosmologyandonthena-
ture of the foregroundstructure responsible for lensing. We 3. METHODOLOGY
makeuseofananalyticformforP(µ)thatwascalibratedwith We generate mock SN samples, and quantify the bias in
numericalsimulations(Wang,Holz,&Munshi2002), which the estimation of the dark energy EOS due to gravitational
isvalidforeventsthatarenotstronglylensed(µislessthana lensing. We pick as our fiducial cosmology a flat universe
few). with Ω =0.3, w =- 1, w =0, and h=0.732with a Gaus-
m 0 a
The generic gravitational lensing PDF peaks sian prior of σ(h)=0.032. We consider a range of possible
at a demagnified value, with a long tail to upcomingsurveys, and Monte-Carlo SN samples of varying
high magnification (Wang,Holz,&Munshi 2002; sizesdistributeduniformlyintheredshiftrange0.1<z<1.7.
Sereno,Piedipalumbo,&Sazhin 2002). However, since If the SN intrinsic errorsare Gaussian in flux, then it makes
thetotalnumberofphotonsisconservedbylensing,alllens- sensetoanalyzethedataviafluxaveraging. Inthiscaseboth
ing distributions preserve the mean: hµi = µP(µ)dµ= 1. the intrinsic errors and the lensing errors will average away
To ensure that this criterion is met (to an accuracy of one for sufficiently large SN samples. However, if the intrinsic
part in a million), we re-normalize the magRnification PDF errorsare Gaussian in magnitude,thena magnitudeanalysis
in a slightly different way than what was originally sug- maybemoreappropriate,althoughthiswillleadtoabiasdue
gested in Wang,Holz,&Munshi (2002). P(µ) is related, tolensing(whichisonlybias-freeforfluxanalysis). Inwhat
SARKARETAL. 3
FIG. 2.—Leftpanel: Histogramsshowingthedistribution(after10,000realizations)ofthevaluesobtainedforw0usingthemagnitudeanalysis(§3.1),after
marginalizingoverwaandh,forthreedifferentsamplesizes. Theun-filledhistogramdepictsthecaseofthe300SNsample,theshadedhistogramshowsthe
distributionforasamplesizeof2000,andthehatchedhistogramshowsthecasefor10,000SNsample.Theverticallinesat-1.0095(dashedanddotted),-1.003
(dashed), and-1.002(solid)showtheaveragew0 valuesforthe300SN,2000SN,and10,000SNcases,respectively. Thewidthofthedistributions isdue
primarilytotheintrinsicnoiseoftheSNe,whiletheshiftedmodeisduetofinitesamplingofthegravitationallensingPDF.Rightpanel: Histogramsshowing
thedistribution(after10,000realizations)ofthevaluesobtainedforw0usingtheflux-averagingtechnique(§3.2),aftermarginalizingoverwaandh,forthree
differentsamplesizes.Theun-filledhistogramdepictsthecaseofthe300SNsample,theshadedhistogramshowsthedistributionforasamplesizeof2000,and
thehatchedhistogramshowsthecasefor10,000SNsample. Theverticallinesat-1.007(dashedanddotted),-1.0025(dashed),and-1.001(solid)showthew0
valuesthatwereobtainedafteraveragingover10,000realizationsforthe300,2000,and10,000samplesizes,respectively.
followsweperformbothanalyses. tooneinflux,butnotinmagnitude.Wethusalsoanalyzethe
mockdata sets in flux. The flux averagingis donesuch that
3.1. MagnitudeAnalysis foreachsupernovawefirstuseEq.(2)tocalculatethevalue
For each SN at a redshift z we draw a value of µ at ran- ofdL(z),andthenevaluatethefiducialfluxusing
dom from the re-normalized lensing PDF, P(µ,z), obtained
L
fromWang,Holz,&Munshi(2002),anduseEq.(5)toevalu- Ffid(z)= , (8)
ate[∆(m- M)] . Tomodeltheintrinsicscatterwedrawa 4π[dLfid(z)]2
lensing
numberatrandomfromaGaussiandistributionofzeromean where we can take any a priori fixed value of L. We take
and standard deviation given by σint =0.1 mag (Astieretal. the lensing PDF, P(µ), and convolveit with a Gaussian dis-
(2006) find a scatter at the level of 0.13 mag). We combine tribution having zero mean and dispersion 0.1 (this corre-
theintrinsicandlensingnoisewiththe“true”underlyingdis- sponds to ∼ 5% error in distance estimates). We then ran-
tancemodulustogettheobserveddistancemodulus: domlydrawfromtheconvolveddistribution,andmultiplythis
(m- M)data(z)=(m- M)fid(z)+[∆(m- M)] +[∆(m- M)] . numberbyFfid(z)togettheobservedfluxforagivensuper-
int lensing nova. WerepeatthisprocessforeachoftheN SNetoobtain
(6)
amockdataset. We thenfollowthefluxaveragingrecipeof
By repeating the above procedure for each of the SNe in a
Wang&Mukherjee (2004) and perform the likelihood anal-
sample, we generate a mock Hubble diagram. We create
ysis to quantify the bias. As before we perform the above
mocksurveyswithdifferentnumbersofSNe: N=300,2000,
& 10,000; and then for each fixed number of SNe we gen- simulationalarge(∼10,000)numberoftimestogetadistri-
butionforthebiasinparameterestimation.
erate at least 10,000 independent mock samples to properly
We further assume that our SNe samples are complete in
samplethedistributions. InFigure1weshowanexampleof
the sense that there is no Malmquist bias or any bias due to
onesuchdatasetwithaSNesamplesizeof2000. Theupper
detectioneffectsovertheredshiftrangeconsidered. Thisisa
panelshows the Hubble diagram and the lower panel shows
reasonableassumptionsinceweconsiderSNeouttoaredshift
theresidualsofthedistancemodulirelativetothatofthefidu-
of1.7andsuchacompletecatalogisexpectedfromthenear-
cialcosmologicalmodel.
infraredimagingcapabilitiesoftheJDEMprogram.
ForeachmockHubblediagramwecomputethelikelihood
ofaparametersetpbyevaluatingtheχ2-statistic:
4. RESULTSANDDISCUSSION
χ2(p)= N (m- M)data(zi)- (m- M)fid(zi) 2, (7) We first presentour results for the magnitudecase, as de-
σ2(z) scribedin§3.1. TheleftpanelofFigure2showshistograms
Xi=1 (cid:2) i (cid:3) of the best-fit values of w0 from the likelihood analysis, af-
whereσ(zi)istheerrorbarforthedistancemodulusoftheith termarginalizingoverwaandh. Theemptyhistogram,which
supernova,takentobe0.1magthroughout.Theprojectedbias peaksat-1.009(markedwithaverticaldot-dashedline),isfor
intheestimationofw0cannowbecomputedbymarginalizing the model with 300 SNe. The shaded histogram, represent-
overwaandh(keepingΩm fixed). ingthe2,000SNcase,peaksat-1.003(verticaldashedline),
whilethehatchedhistogramrepresentingthe10,000SNcase
3.2. Flux-AveragingAnalysis
hasitspeakat-1.002(verticalsolidline). Thesedistributions
Wang(2000)arguesthataveragingtheflux,asopposedto have1σwidthsof0.016,0.006,and0.003,respectively.This
themagnitudes,ofobservedSNenaturallyremovesthelens- scatterisprimarilyduetotheintrinsicuncertaintyassociated
ingbias,sincethemeanofthelensingdistributionsareequal with absolute calibration, and is not dominated by lensing.
4 LENSINGANDSUPERNOVAE
some SNe to be highly magnified, and it is conceivablethat
these “obvious” outliers are subsequently removed from the
analysis. In thiscase the mean of the samplewill be shifted
away from the true underlying Hubble diagram, and a bias
willbeintroducedinthebest-fitparameters. Toquantifythis
effect,weremoveSNewhichdeviatefromtheexpectedmean
luminosity-distancerelation in the Hubble diagram by more
than 25% (corresponding roughly to a 2.5σ outlier). The
SN scatter is a result of the convolution of the intrinsic er-
ror(Gaussianinflux ofwidth0.1)andthelensingPDF, and
the outlier cutoff leads to a removalof ∼ 50 SNe out of the
2,000 SNe. These outliers are preferentially magnified, due
to the strong lensing tail of the magnification distributions.
The demagnification tail is cut off by the empty-beam lens-
inglimit, and thereforeisn’tas prominent. The hatchedhis-
togram in Figure 3 shows the distribution when events with
convolvederrorgreaterthan 2.5σ are removed. The vertical
FIG. 3.—Histogramsshowingthedistributionofthevaluesobtainedfor dot-dashedlineat-1.0075showstheaveragevalueofw0 ob-
w0aftermarginalizingoverwaandhforthe2,000SNcase(usingtheflux- tained in this case, representing a bias in the estimate of w0
averagingtechnique). Theshadedhistogramassumesthatthefullsampleof
roughlythree times larger than when the full 2,000 SNe are
2,000SNeisusedforparameterestimation.Thehatchedhistogramshowsthe
analyzed(shownby shaded histogram). Thisbias is a result
shiftwhenoutliers(SNethatareshiftedaboveorbelowtheHubblediagram
bymorethan25%oneitherside)areremovedfromthesample. Thebiasin of cutting off the high magnification tail of the distribution,
thedistributionisduetotheremovalofhighly-magnifiedlensingeventsfrom andthusshiftingthedatatowardsanetdimmingofobserved
thesample.
SNe,leadingtoamorenegativevalueofw .
0
We also apply a cutoff at 3σ, in addition to the 2.5σ dis-
Without the inclusion of lensing, however, the distributions cussedabove.Thisresultsinaremovalof∼20SNeonaver-
peakatexactly-1,andshownobias. Theshiftedmodegives age,foreach2,000SNsample,andleadstoabiasof∼0.6%.
us a rough idea of the bias to be expected, on average, due Any arbitrary cut on the (non-Gaussian) convolved (lensing
tolensing. Wefindthat68%ofthetimearandomsampleof +intrinsic)sampleleadstoanetbiasinthedistancerelation,
300SNewillhaveanestimatedvalueforw within3%ofits andevenforlargeoutliersandlargeSNsamples,thiscanlead
0
fiducialvalue, andthisdropsto 0.5%whena samplesize of topercent-levelbiasinthebest-fitvaluesforw .
0
10,000SNeisconsidered. Tosummarize,wehavequantifiedtheeffectofweakgrav-
TherightpanelofFigure2showsthesamedistributionsas itationallensing on the estimation of darkenergyEOSfrom
theleftpanel,butthistimeusingtheflux-averagingtechnique type Ia supernova observations. With generated mock sam-
instead of averaging over magnitudes. The empty, shaded, ples of 2,000 SNe distributed uniformly in redshift up to
and hatched histograms peak at -1.007, -1.003, and -1.001, z∼1.7 (as expected in future surveys like JDEM), we have
respectively, showing the mean bias for the 300, 2,000 and shown that the bias in parameter estimation due to lensing
10,000SNcases(markedwithdot-dashed,dashed,andsolid is less than 1% (whichis well within the 1σ uncertaintyex-
vertical lines). With flux-averaging, we expect that 68% of pectedforthesemissions).Analyzingthedatainfluxormag-
the time a random sample of 300 SNe will yield a value of nitudedoesnotalterthisresult. Iflensedsupernovaethatare
w within 2.5% of the fiducial value, and within 0.5% for a highlymagnified(suchthattheconvolvederrorismorethan
0
sampleof10,000SNe. 25%fromtheunderlyingHubblediagram)aresystematically
The 1σ parameteruncertaintyon w rangesfromthe 20% removedfromthesample,wefindthatthebiasincreasesbya
0
level(for300SNe)tolessthan5%(for10,000SNe),dwarf- factorofalmostthree. Thus,solongasallobservedSNeare
ingthebiasduetolensing. Thus, weneednotbeconcerned used in the Hubble diagram, including ones that are highly
about lensing degradation of dark energy parameter estima- magnified,thebiasduetolensingintheestimateofthedark
tion for future JDEM-like surveys. We note, however, that energyEOSwillbesignificantlylessthanthe1σuncertainty.
ourestimatedbiasontheEOSislargerthanthelensingbias Evenforapost-JDEMprogramwith10,000SNe,lensingbias
ofw<0.001quotedinTable7ofWood-Vaseyetal. (2007). canbesafelyignored.
Thisisnotsurprising,giventheiruseofthesimpleGaussian
approximationto lensingfromHolz&Linder(2005), which
islesseffectiveforlowstatistics. Nonetheless,weagreewith
theirconclusionthatlensingisnegligible. Asimilar conclu- We thank Yun Wang for usefuldiscussions. AC acknowl-
sionwasalsoreachedbyMartel&Premadi (2007)whoused edges support from NSF CAREER AST-0645427. DEH
acompilationof230TypeIaSNe (Tonryetal. 2003)in the acknowledges a Richard P. Feynman Fellowship from Los
redshiftrange0<z<1.8toshowthatthelensingerrorsare Alamos National Laboratory. DS, AC, and DEH are par-
smallcomparedtotheintrinsicSNeerrors. tiallysupportedbytheDOEatLANLandUCIrvinethrough
We now discuss the bias which arises if anomalous SNe IGPPGrantAstro-1603-07.AAacknowledgesaMcCueFel-
are removed from the sample. Gravitational lensing causes lowshipatUCIrvine.
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