Toward More Realistic Forecasting of Dark Energy Constraints from Galaxy Redshift Surveys

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by  Yun Wang

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed11January2013 (MNLATEXstylefilev2.2)
Toward More Realistic Forecasting of Dark Energy Constraints
from Galaxy Redshift Surveys
Yun Wang⋆1, Chia-Hsun Chuang†2, & Christopher M. Hirata3
1HomerL.DodgeDepartmentofPhysics&Astronomy,Univ.ofOklahoma,440WBrooksSt.,Norman,OK73019,U.S.A.
2InstitutodeF´ısicaTeo´rica,(UAM/CSIC),UniversidadAuto´nomadeMadrid,Cantoblanco,E-28049Madrid,Spain
3CaltechM/C350-17,Pasadena,CA91125,U.S.A.
3
1
0 11January2013
2

n
a ABSTRACT
J

0 Galaxyredshiftsurveysarebecomingincreasinglyimportantasadarkenergyprobe.We
1 improvetheforecastingofdarkenergyconstraintsfromgalaxyredshiftsurveysbyusingthe
“dewiggled”galaxypowerspectrum,Pdw(k),intheFishermatrixcalculations.SincePdw(k)
] isagoodfittorealgalaxyclusteringdataovermostofthescalerangeofinterest,ourapproach
O
ismorerealisticcomparedtopreviousworkinforecastingdarkenergyconstraintsfromgalaxy
C redshift surveys. We find that our new approach gives results in excellent agreement when
. comparedtotheresultsfromtheactualdataanalysisoftheclusteringoftheSloanDigitalSky
h
SurveyDR7luminousredgalaxies.Weprovideforecastsofthedarkenergyconstraintsfrom
p
aplausibleStageIVgalaxyredshiftsurvey.
-
o
r Keywords: cosmology:observations,distancescale,large-scalestructureofuniverse
t
s
a
[

2 1 INTRODUCTION 2 METHOD
v
2 One of the most important discoveries in modern cosmology The redshift-space galaxy power spectrum P(k,µ) is a rich
3 is the accelerated expansion of the Universe (Riessetal. 1998; source of cosmological information. It includes the BAO feature
5 Perlmutteretal.1999).Thepowerspectrumor2-pointcorrelation (Blake&Glazebrook2003;Seo&Eisenstein2003),whichhasre-
0 functionmeasuredfromgalaxyredshiftsurveyshasprovidedone ceivedagreatdealofattentionasastandardrulerthatcanbeused
1. of the primary probes of cosmic acceleration, both through the inboththetransversedirection(tomeasuredistances)andtheradial
1 broadbandmeasurementoftheshapeimprintedbymatter-radiation direction (to measure the Hubble rate). However, the full galaxy
2 equality(e.g.Percivaletal.2001;Tegmark2004)andthroughthe power spectrum at largescales isalsosensitive totheunderlying
1 baryon-acousticoscillation(BAO)featureimprintedatrecombina- matterpowerspectrum,tothegrowthofstructureviaredshift-space
: tion(e.g.Eisensteinetal.2005).Galaxyclusteringalsoallowsusto distortions(Kaiser1987),andtostandardrulereffects.Thisaddi-
v
differentiatesmoothdarkenergyandmodifiedgravityasthecause tional information requires some work to extract, since one must
i
X forcosmicaccelerationthroughthesimultaneousmeasurementsof simultaneouslymeasurethecosmologyandthegalaxybiasingpa-
r the cosmic expansion history H(z), and the growth rate of cos- rameters. Nevertheless, the galaxy power spectrum provides the
a miclargescale structure, f (z) (Guzzoetal. 2008; Wang2008a; mostpowerfulconstraintsondarkenergyandgravity.Inthispaper,
g
Blakeetal.2012). wefocusontheanalysisofthefullsetof2-pointgalaxystatistics,
The Fisher matrix approach has generally been used in the anddonotlimitourselvestoonlytheBAOinformation.
forecasts of future galaxy redshift surveys. In this paper, we im-
provetheFishermatrixapproachbymakingitmorerealistic.This
enables its use in cross-checking dark energy and gravity con-
straintsfromcurrent galaxy clusteringdata, aswellasinmaking 2.1 Formalism
theforecastsforfuturegalaxyredshiftsurveysmorerobustandre-
Our Fisher matrix approach is derived from that of
liable.
Seo&Eisenstein (2003), and based on Wang (2006, 2008a,
WepresentourmethodinSection2,ourresultsinSection3,
2010) and Wangetal. (2010). Inthe limitwhere thelength scale
andsummarizeandconcludeinSection4.
corresponding tothesurveyvolumeismuchlargerthanthescale
ofanyfeaturesintheobservedgalaxypowerspectrum P (k),we
g
can assume that the likelihood function for the band powers of a
⋆ E-mail:[email protected] galaxy redshift survey is Gaussian (Feldman,Kaiser,&Peacock
† MultiDarkFellow 1994),withameasurementerrorinlnP(k)thatisproportionalto
(cid:13)c 0000RAS
2 Wang,Chuang, &Hirata
[Veff(k)]−1/2,withtheeffectivevolumeofthesurveydefinedas The biasbetween galaxy and matterdistributions isdenoted
by b(z). The linear RSD parameter β(z) = f (z)/b(z) (Kaiser
2 g
V (k,µ) dr3 n(r)Pg(k,µ) 1987),wherefg(z)isthelineargrowthrate;itisrelatedtothelin-
eff ≡ Z (cid:20)n(r)Pg(k,µ)+1(cid:21) eargrowthfactorG(z)(normalizedsuchthatG(0)=1)asfollows
2
nP (k,µ) dlnG(z)
= g V , (1) f (z)= . (6)
(cid:20)nPg(k,µ)+1(cid:21) survey g dlna
Wehaveassumethatthepeculiarvelocitiesofgalaxiescanbemod-
wherethecomovingnumberdensitynisassumedtoonlydepend
eledwithaprobabilitydistribution
ontheredshift(andconstantineachredshiftslice)forsimplicityin
thelaIsntpoardrteroftothperoepqaugaatitoent.hemeasurementerrorin lnPg(k)into f(v)= σp1√2e−√2|v|/σp, (7)
measurementerrorsfortheparametersp ,weusetheFishermatrix
i whereσ isthepairwisepeculiarvelocitydispersion.TheFourier
(Tegmark1997) p
transformoff(v)is1/[1+k2µ2σ2 /2],thesmallscaleRSDfac-
r,p
F = kmax ∂lnPg(k)∂lnPg(k)V (k) dk3 , (2) torincludedinEq.(4)(Hamilton1998).Notethatσr,p isthedis-
ij Zkmin ∂pi ∂pj eff 2(2π)3 tσan,ctehudsisσpers=ioHnc(ozr)r[eas(pzo)nσding],taondthephysicalvelocitydispersion
p p r,p
where p are the parameters to be estimated from data, and the
i σ
derivativesareevaluatedatparametervaluesofthefiducialmodel. σr,p = H(z)pa(z). (8)
NotethattheFishermatrixF istheinverseofthecovariancema-
ij
trixoftheparametersp ifthep areGaussiandistributed. NotethatwehaveadoptedminimalsmallscaleRSDmodelingin
i i
Weadoptthestandardnotationthatkcanbedecomposedinto this work (see Eq.[7]), since we only consider quasilinear scales
aline-of-sightcomponentk andthetransverseorin-the-plane-of- for a conservative approach. The limitations of Eq.(7) have been
the-skycomponentk .Thekcosineoftheanglebetweenkandthe discussed in detail by Scoccimarro (2004). When smaller scales
lineofsightvectoris⊥denotedbyµ=k /k. (< 20h−1Mpc) areincluded intheanalysis, itwillbecriticalto
k | | u∼seanimprovedRSDmodel,see,e.g.,Chuang&Wang(2012b).
An additional damping factor, e−k2µ2σr2,z, is inserted to ac-
2.2 Themodelforthegalaxypowerspectrum count for redshift uncertainties, with σr,z = (∂r/∂z)σz. Thisis
intendedtoincorporatethetrueredshiftuncertaintyresultingfrom
Atcosmologicaldistances,the“true”galaxypowerspectrumisnot fittingthecentroidofanemissionline(inanemissionlinesurvey),
adirectobservable,sinceonecanmeasureagalaxy’spositiononly butthisfactorcouldalsoabsorbothersmallerrorsintheredshift
inangularandredshiftcoordinatesandnotinitstrue3Dcomoving (e.g. due to the emission line velocity not being exactly equal to
coordinates.Thisisofcoursethebasisforextractionofthe“stan- zerointherestframeofthegalaxy’shosthalo).
dardruler”information,includingtheAlcock&Paczynski(1979) Nonlinear smearing of the BAO feature occurs due to the
effect.Thereforestandardpracticeistoprojectthegalaxiestotheir small-scale(i.e. s 150Mpc)displacementsduringstruc-
BAO
comovingpositionsassumingsomereferencecosmology(orfidu- tureformation.T≪hesedisp∼lacementstakesharp,coherent features
cialcosmology),andthenapowerspectrumorcorrelationfunction inthecorrelationfunctionatlargescales(e.g.theBAO)andsmear
estimatorisapplied.Theobservedgalaxypowerspectrumis then them out; in Fourier space, this corresponds to a damping of the
relatedtothetruegalaxypowerspectrumviaacoordinatetransfor- oscillatorypartofP(k).Thiseffectismodeledbyusingthedewig-
mation:thewavenumberkref inthereferencecosmologyisrelated gledmatterpowerspectrumatredshiftz,givenby
tothewavenumberinthetruecosmologyvia
P (k,z)=G2(z)P knST2 (k,z). (9)
dw 0 dw
D (z) Href(z)
kref = A k and kref = k . (3) HereT2 (k,z)isgivenby
⊥ Dref(z) ⊥ k H(z) k dw
A
Based on Seo&Eisenstein (2003) and Chuang&Wang Td2w(k,z) ≡ T2(k)e−gµk2/(2k∗2)
(2012a),ourmodelforPDgr(ekf()zc)an2Hthe(nz)bewritt1en+asβµ2 2 +Tn2w(k)h1−e−gµk2/(2k∗2)i, (10)
Pg(k⊥ref,kkref) = [(cid:2)DAA(z)]2(cid:3)Href(z)b21(cid:0)+k2µ2σr2,(cid:1)p/2 pwuhreerCeDTM(k()noisbtahreyolninse)atrramnsaftetrerfutnracntisofnergifvuennctbioynE,isTennwst(eki)ni&s Hthue
Pdw(k)ze−k2µ2σr2,z +Pshot, (4) (1998,Eq.29),and
×
where H(z) = a˙/a (with a denoting the cosmic scale factor) is gµ(k,z) G2(z) 1 µ2+µ2[1+fg(z)]2 (11)
≡ { − }
theHubbleparameter,andDA(z) = r(z)/(1+z)istheangular describes the enhanced damping along the line of sight due to
diameterdistanceatz,withthecomovingdistancer(z)givenby theenhancedpower.Thenonlineardampingfactor,e−gµk2/(2k∗2),
r(z)=c|Ωk|−1/2sinn(cid:20)|Ωk|1/2 Z0z Hd(zz′′)(cid:21), (5) wb(2ai0ttih0o7ng)µsuggsriivonewgnNwby-ibtEhoqdc.yo(1ssm1i)mi,cuwltaaimstidoeen,rsti.hveNedolibtneyeatEhriasrteegnsiisnmteceiene,dxSepneasoni,tdy&spWaesrhtwuitree-
where sinn(x) = sin(x),x,sinh(x)for Ω < 0, Ω = 0, and gotohigherredshifts.Hencethefunctiong scaleswiththelinear
k k µ
Ω >0respectively.Inadditiontothegeometricaldistortion,this growthfactorG(z)squared,whichcorrespondstothescaleofthe
k
modelincludesthelineargalaxybiasandredshift-spacedistortion linearregimeincreasingwith1/G(z)athighredshifts.
(RSD), nonlinear smearing of the BAO feature, halo shot noise, Thescalek isrelatedtothepercentageofnonlinearityfrom
∗
small-scalepeculiarvelocities,andredshifterrors. Seo&Eisenstein(2007),p ,via
NL
(cid:13)c 0000RAS,MNRAS000,000–000
Toward MoreRealisticForecastingofDarkEnergyConstraintsfromGalaxyRedshiftSurveys 3
k−1 =8.355h−1Mpc(σ8/0.8)pNL. (12) wherek¯ k/[hMpc−1],andj1(kr)issphericalBesselfunction.
∗ Notethat≡I =I (ω ,ω ,n ,h).Sincek andk scaleasH(z)
Thetruegalaxypowerspectrumshouldhave pNL = 1.Recently 0 0 m b s k ⊥
“reconstruction” algorithms have been proposed (Eisensteinetal. and1/DA(z)respectively,Pgobs(k)inEq.(14)doesnotdependon
2007) and implemented (Padmanabhanetal. 2012) that reverse h.
Eq.(14)isthemodelwewilluseinthispaper.Itsabsorption
some of the flows and move galaxies back closer to their origi-
nal(Lagrangian)positions.Ifsuchanalgorithmisappliedtodata, ofthebiasfactorisanalogoustotheapproachofSong&Percival
thenonlinearitypercentagecanbereduced.BAOreconstructionis (2009), who proposed the use of fg(z)σ8(z) to probe growth
of large scale structure. The difference is that Eq.(14) uses
arapidlydevelopingfield,butisinitsearlystagesandhigh-zred-
shiftsurveysmayhavetodealwithsurveygeometriesthataremore fg(z)σm(z) ≡ fg(z)G(z)P˜01/2,whichdoesnotintroduceanex-
complex and bias-weighted galaxy densities b2n that are smaller plicitdependenceonh(asinthecaseofusingfg(z)σ8(z)).
thanthatofe.g.BOSS.Forthepresentwork,weconsiderarange
ofvaluesforp .Theoptimisticcaseofp =0.5correspondsto
NL NL 2.3 Parametersandassumptions
k 0.24h/Mpc,whereasthemostconservativecaseofp =1
NL
∗ ≃
(no reconstruction) corresponds to k 0.12h/Mpc, assuming In our method, the full set of parameters that describe the
∗ ≃
σ8 =0.8. observed Pg(k) are: lnH(zi), lnDA(zi), ln[fg(zi)σm(zi)],
{
Foranintuitiveunderstanding ofthedewiggled powerspec- lnσ (z ), Pi ; ω , ω , n ,k ,σ /(1 + z) , where i indi-
trumofEq.(9),wecanrewriteitscorrespondingtransferfunction, catesg thie i-tshhotredshmift slbice,sand∗ ωz Ω }h2, and ω
m m b
≡ ≡
Eq.(10),asfollows Ω h2. We marginalize over lnσ (z ),Pi in each red-
b { g i shot}
Td2w(k,z) = Tn2w(k)+ T2(k)−Tn2w(k) e−gµk2/(2k∗2) mshaiftrtixslifcoer, alsnwHe(llzia)s,lnk∗DAan(dzi)σ,zln/([f1g(+zi)zσ),mt(ozio)]b;tωaimn,aωbF,inshser.
≡ Tn2w(k)+T(cid:2)B2AO(k)e−gµk2/((cid:3)2k∗2) (13) TpahriasmfueltlerFsi,s{ihserprmojaetcritxe,doirntaostmhealsletarnsdeatrmdasregtinoaflcizoesdmoovloegrivcaarliopua}s-
wherewehavedefined T2 (k) = T2(k) T2 (k),thediffer-
BAO − nw rameters w0,wa,ΩX,Ωk,ωm,ωb,ns,lnAs .Therearefourdif-
ence between the linear matter transfer functions with and with- { }
ferentwaysofutilizingtheinformationfromP(k)(Wang2012).
outbaryons.Clearly,theexponentialdampingduetononlinearef-
Itisimportanttonotethatwhenevaluatingthederivativesof
fectsisonlyappliedtothetransferfunctionassociatedwithBAO. P (kref)withrespecttotheparametersdescribedabove(required
g
Anguloetal.(2008)havecomparedthespherically-averagedform
tocalculatetheFishermatrix),weshould not extractinformation
ofthismodelwithmeasurementsfromsimulateddata,andfound
from the damping factors due to systematic uncertainties, in or-
thatitworksextremelywellonthelinearandquasilinearscales;the
dertoadheretoaconservativeandrobustapproach.Thesedamp-
assessmentofitsaccuracyispresentlylimitedbytheshotnoiseof
ingfactorsareonlyincludedtorepresentthelossofinformationat
currentlyavailablenumericalsimulations.Infuturework,wewill
smallscalesduetononlineareffects(and,ifapplicable,redshiftun-
testthismodelwithoutsphericalaveragingusingnumericalsimu-
certainties).Wetreatthesedampingfactorsasfollowswhenderiva-
lationstofullyassessit.Wedonotexpectthismodeltocontinue
tivesaretaken.
working well on the smallest scales, where the nonlinear damp- The gµ in the nonlinear damping factor, e−gµk2/(2k∗2), is
ing is coupled with RSD (e.g. Jennings,Baugh,&Pascoli 2011;
fixedatfiducialmodelvalueswhenderivativesaretaken,toavoid
Reid&White2011;Chuang&Wang2012b).
deriving cosmological information from the NL damping itself.
Toavoidthedirectmeasurementoftheunknowngalaxybias
Notethat k is scaled aswe vary T (k) forconsistency, and we
dw
b(z),we rewriteour model forthe measured galaxy power spec-
marginalizeoverk toallowforthesignificantuncertaintyinthe
trumas(Wang2012) ∗
NLdamping.
Pg(kref,kref) Thedampingfactorduetoredshiftuncertainty,e−k2µ2σr2,z,is
⊥ k computed with∂r/∂z fromthe fiducial model, toavoidderiving
Pg(kref,kref)/(h−1Mpc)3 cosmologicalinformationfromthedampingitself.Wemarginalize
≡ ⊥ k
D (z)ref 2H(z) overσz/(1+z)toallowfortheuncertaintyinourknowledgeof
= A σ (z)+f (z)σ (z)µ2 2 redshiftaccuracy.
(cid:2)[D (z)]2H(cid:3) (z)ref g g m
A (cid:2) (cid:3) TheRSDfactorduetosmallscalerandommotionofgalaxies,
k nsT2 (k,z) e−k2µ2σr2,z +P , 1/[1+k2µ2σr2,p/2],isfixedatfiducialmodelvalueswhenderiva-
×(cid:18)Mpc−1(cid:19) dw 1+k2µ2σr2,p/2 shot tivesaretaken.NotethataswevaryH(z),thisRSDfactorremains
unchanged, sincekµ = k H(z),whileσ 1/H(z).The
(14) RSDfactor isincluded hekre∝to represent thesru,ppr∝ession of power
wherewehavedefined due to galaxy peculiar velocities, and not to provide an accurate
σ (z) b(z)G(z)P˜1/2 and σ (z) G(z)P˜1/2. (15) modelingofRSDonallscales.
g ≡ 0 m ≡ 0 Since our model fits real data well on these scales
The dimensionless power spectrum normalization constant P˜0 is (Chuang&Wang 2012a), it represents a step forward in making
justP0inEq.(9)inappropriateunits: Fishermatrixforecasting forgalaxyredshift surveys more realis-
P σ2 tic.
P˜ 0 = 8 . (16)
0 ≡ (Mpc/h)3(Mpc)ns I0hns
ThesecondpartofEq.(16)isrelevant ifσ isusedtonormalize
8 3 RESULTS
thepowerspectrum.Notethat
I0≡Z0∞dk¯k¯2nπs+22 T2(k¯·hMpc−1) (cid:20)3j18(k¯8k¯)(cid:21)2, (17) Wxhe(zw)il≡lpHres(ezn)tsr/ecsulatsnodn xd(z)≡DA(z)/s, (18)
(cid:13)c 0000RAS,MNRAS000,000–000
4 Wang,Chuang, &Hirata
wheres r (z )isthesoundhorizonatthedragepoch,whichis assumptionsarenotneededfortheanalysisofrealormockdata,
s d
≡
thecharacteristicscaleofBAO. sincetheoverallamplitudeismarginalizedover(Chuang&Wang
We assume the fiducial model adopted by the FoMSWG 2012a).
(Albrechtetal. 2009), itself based on the 5-year WilkinsonMi- SincetheFishermatrixresultsdepend onthefiducialmodel
crowaveAnisotropyProbe(WMAP)results(Dunkleyetal.2009): assumed, we give our Fisher matrix forecasts for two differ-
ω Ω h2 = 0.1326, ω Ω h2 = 0.0227, h = 0.719, ent fiducial models in Table 1: the FoMSWG fiducial model
m m b b
Ω =≡0,w= 1.0,n =0.963≡,andσ =0.798. (Albrechtetal. 2009) with with ω Ω h2 = 0.1326, ω
k s 8 m m b
− ≡ ≡
WewillfirstpresentresultsforSDSSDR7LRGs,inorderto Ω h2 = 0.0227, h = 0.719, Ω = 0, w = 1.0, n =
b k s
−
comparewiththeresultsfromactualdataanalysis.Theanalysisof 0.963, andσ = 0.798, andtheEuclidRedBookfiducialmodel
8
currentGCdatarequiretheassumptionofcosmologicalpriors;we (Laureijsetal. 2011) with with ω Ω h2 = 0.1225, ω
m m b
≡ ≡
imposethesamebroadpriorsasChuang&Wang(2012a). Ω h2 = 0.021805, h = 0.7,Ω = 0,w = 0.95,n = 0.963,
b k s
−
Next, we will present results for StageIV+BOSS spectro- and σ = 0.8. These fiducial models lead to σlnx (z) and
8 h
scopic galaxy redshift surveys, and compare these with those σlnx (z)thatdifferby3.1%and6.8%respectively.
d
from the previously widely adopted approach derived from The model used here, Eq.(14), differs somewhat from that
Seo&Eisenstein(2007)and developed indetail inWang(2012). used by Chuang&Wang (2012a). Our new model, Eq.(14),
No priors are used in deriving x (z),x (z),f (z)σ (z)/sα usesaisotropicdewigglingwhereasChuang&Wang(2012a)used
h d g m
{ }
constraints for future surveys, since these provide model- isotropicdewiggling,whichneglectstheadditionaldampingalong
independent constraints on the cosmic expansion history and the thelineof sight due tothe enhanced Lagrangian displacement in
growthrateofcosmiclargescalestructure.Theseallowthedetec- redshiftspace.4 UsingourFishermatrixmethod, wefindthatas-
tion of dark energy evolution, and the differentiation between an sumingisotropicdewigglingleadstoanunder-estmateofσ
lnxh(z)
unknownenergycomponentandmodifiedgravityasthecausesfor andσ of 23 24%and 12 13%respectively.
lnxd(z) ∼ − ∼ −
theobservedcosmicacceleration. Inordertomakeanaccuratecomparison,wehaverepeatedthe
Inordertoderivedarkenergyfigureofmerit(FoM),asdefined analysisoftheSDSSDR7LRGsampleusedbyChuang&Wang
bytheDETF(Albrechtetal.2006),weprojectourFishermatrices (2012a) using Eq.(14) as part of this work. We find that the data
intothestandardsetofdarkenergyandcosmologicalparameters: andmocksgiveσ andσ thatdifferby13%and30%
lnxh(z) lnxd(z)
w ,w ,Ω ,Ω ,ω ,ω ,n ,lnA . To include Planck priors,1 respectively,withthemockresultsagreeingwithourFishermatrix
0 a X k m b s s
{ }
weconvertthePlanckFishermatrixfor44parameters(including forecastsat alevelof 10%orbetter,giventhedependence of the
36parametersthatparametrizethedarkenergyequationofstatein Fishermatrixforecastsontheassumedfiducialmodel.
redshift bins) fromtheFoMSWGintoaPlanckFishermatrixfor OurFishermatrixforecastsforthemeasurement uncertainty
thissetofdarkenergyandcosmologicalparameters. onf (z)σ (z)/s4areinexcellentagreementwiththeresultsfrom
g m
the MCMC analysis of the LasDamas SDSS LRG mocks, while
theresultsfromtheMCMCanalysisofSDSSDR7LRGdatagive
3.1 Comparisonwithanalysisofdata significantlysmalleruncertaintyonf (z)σ (z)/s4.Thisislikely
g m
due to the apparent excess clustering of SDSS DR7 LRGs along
Togaugetheaccuracyofourforecastingmethodologycomparedto
the line of sight (this is apparent from comparing the mock and
thefullanalysisofrealdata,wepresentourforecastsfortheSDSS
data panels of Fig.1 in Chuang&Wang 2012a), which is likely
DR7setof87,000LRGsintheredshiftrange0.16–0.44analyzed
alsoresponsibleforthesmallerthanexpectedmeasurementuncer-
byChuang&Wang(2012a)inTable1,andcomparethemwiththe
taintyofx (z)andx (z)fromthisdatasample.Thisexcesspower
actualmeasurementsperformedaspartofthiswork,usingboththe h d
along the line of sight explains the widely noted excess power
SDSSDR7LRGsandtheSDSSLRGmocksfromLasDamas2.
on large scales for the spherically-averaged galaxy correlation
The scale range analyzed by Chuang&Wang (2012a) is
function(see,e.g.,Cabre&Gaztanaga(2009);Kazinetal.(2010);
r =40-120h−1Mpc,whichcorrespondstothek =2π/rrangeof Chuang,Wang,&Hemantha (2012)). Since the BOSS CMASS
0.0524-0.157hMpc−1. Chuang&Wang (2012a) used flat priors galaxiesdonothavetheexcessclusteringonthesamelargescales
onω andn thathavewidthsof 7σ (withσ given
b s ± WMAP WMAP (Reidetal.2012),theexcesslargescaleclusteringofSDSSDR7
bytheWMAPseven yearresultsfromKomatsuetal.(2011)).In
LRGsmustbeduetosamplevarianceorunknown systematicef-
addition, Chuang&Wang (2012a) imposed flat priors of 0.1 <
fects.
β < 0.6, 0 < σ < 500km/s, and 0.09 < k (z =
p Takingallthefactorsdiscussedaboveintoconsideration,our
0.35)/[hMpc−1] < 0.13. We use Gaussian priors on th∗e same Fishermatrixforecastsareinexcellentagreementwiththe results
parameterswiththesamewidthsforthepriorsasChuang&Wang
from actual data analysis. It is reassuring that our Fisher matrix
(2012a),andwithmeansofσp =250km/s,andk /[hMpc−1]= method gives very similar results compared to actual data analy-
0.11G(z = 0.33) = 0.0939 (the width of k∗/[hMpc−1] is sis, making it a reliable tool for parameter forecasting for future
∗
0.02G(z = 0.33) = 0.01707). Notethatourdefinitionof k is
surveys.
∗
independentofredshift,thusitisdividedbythegrowthfactoratthe
effectiveredshiftofthedatasetusedbyChuang&Wang(2012a).3
Inaddition,weassumearedshiftaccuracyofσln(1+z) = 5 3.2 ForecastsforaStageIVgalaxyredshiftsurvey
10−4,andabiasofb=2.2fortheSDSSLRGs.Theseaddition×al
We perform forecasts for a Stage IV galaxy redshift survey cov-
eringanareaof15,000deg2,usingslitlessgrismspectroscopy to
1 For a general and robust method for including Planck priors, see detect the Hαemission line. A wavelength range of 1.1–2.0 µm,
Mukherjeeetal.(2008). correspondingto 0.7 < z < 2.0,wasassumed. Thedepthofthe
2 URL:http://lss.phy.vanderbilt.edu/lasdamas/
3 Chuang&Wang (2012a) scaled their results from zeff = 0.33 to
zeff =0.35inordertocomparewithpreviousresultsbyothergroups. 4 Thisisequivalenttosettinggµ→G2(z)inEq.(11).
(cid:13)c 0000RAS,MNRAS000,000–000
Toward MoreRealisticForecastingofDarkEnergyConstraintsfromGalaxyRedshiftSurveys 5
Model Method xh(z) xd(z) β fg(z)σm(z)/s4
FoMSWG Fishermatrix 7.31% 4.99% 21.98% 20.69%
EuclidRB Fishermatrix 7.09% 4.67% 22.61% 21.05%
None MCMCanalysisofdata 5.80% 3.74% 14.89% 14.01%
None MCMCanalysisofmocks 6.64% 5.37% 23.72% 22.61%
Table1.OurFishermatrixestimateofthepercentageprecisionofmeasurementofxh(z)≡H(z)s/c,xd(z) ≡DA(z)/s,β,andfg(z)σm(z)/s4atan
effectiveredshiftofz =0.35fromSDSSDR7LRGs,comparedtotheactualmeasurementsusingtheanisotropiccorrelationfunctionperChuang&Wang
(2012a).Eq.(14)isusedinallcases.
surveywascomputedusinginstrumentparameters(throughput,ex- size,butisintherangeof(2.2 3.6) 10−16ergs−1cm−2fora
− ×
posuretime,etc.)similartothoseprovidedfortheEuclidmissionin sourcehalf-lightradiusof0.3arcsec.
Laureijsetal.(2011);itisthusrepresentativeofanext-generation Asapointofcomparison,werantheHirataetal.(2012)code
space-basedgalaxysurvey,thoughitmaynotcorrespondprecisely ontheHubbleSpaceTelescopeWideFieldCamera3(HST/WFC3)
tothefinalEuclidnumbers. G141grism(1.1 < λ < 1.7µm),usingparametersfromtheIn-
All our forecast results are shown for Stage IV plus BOSS. strument Handbook (Dressel 2011). Wefind that foran exposure
The BOSS survey is assumed to cover 10,000 (deg)2, a redshift timeof2700s,the5σ sensitivityshouldbe(4.6 8.2) 10−17
− ×
range of 0.1 < z < 0.7, with a fixed galaxy number density of ergs−1cm−2,withthelower(better)numbersattheredendofthe
n=3 10−4h3Mpc−3,andafixedlinearbiasofb=1.7. bandpass.Thisisingoodagreement( 20percent)withtheme-
× ∼
WediscussthegalaxyyieldsfromaStageIVgalaxyredshift diansensitivityactuallyachievedbyWFC3observations–seee.g.
surveyindetailinSec.3.2.1.Forclusteringanalysis,wealsorequire Figure5ofAteketal.(2010).
thegalaxybias.Weusethegalaxybiasfunctionforemissionline The line flux sensitivity and completeness are only part of
galaxiesgivenbyOrsietal.(2010),whichincreaseswithredshift determining the number of redshifts obtained by a survey – one
reachingb = 1.7atz = 2.Again,wenotethatthisislikelytobe also needs a luminosity function. In the past decade of space-
conservative: therecent biasdetermination of Geachetal. (2012) based redshift survey mission planning, the Hα luminosity func-
forHαemittersisb =2.4+0.1 atz = 2.23.Weassumearedshift tion(HαLF)hasbeenamatterofvigorousdebate:directmeasure-
0.2
accuracyofσz/(1+z)=0−.001,andapeculiarvelocitydispersion ments have suffered from small-number statistics, while indirect
ofσp =290km/s. methods (based on scaling from rest-frame ultraviolet or [O II]
Weconsidertwodifferentcutoffsinscale:kmax =0.2h/Mpc luminosities) have had difficult-to-quantify systematic errors. In-
andkmax =0.2h/Mpc,inordertoincludethequasilinearregime deed,theestimatesusedforspacemissionplanning(e.g.Yanetal.
only in our forecasts. The choice of kmax = 0.2h/Mpc is con- 1999; Hopkinsetal. 2000; Reddyetal. 2008; Jouveletal. 2009;
servative, and represents the lower bound of the scale range in Geachetal.2010;Sobraletal.2012)havespannedafactorof 3
∼
which our model works well in analyzing real data. The choice innumber density, even accounting forthe different cosmologies
ofkmax =0.3h/Mpcismoreoptimistic,butrepresentsafeasible assumed.Fortunately,empiricalmeasuresoftheHαLFacrossthe
goalforthelowerboundofthescalerangeinwhichfuturestudies relevantrangeofredshiftswithlarge-numberstatistics(dozensof
willenablerobustandaccuratemodeling. objectsintherelevantfluxrange,inmultiplefields)arenowavail-
able.
We use the HαLF of Sobraletal. (2012), with conversions
3.2.1 GalaxyyieldsforaStageIVredshiftsurvey describedinHirataetal.(2012) 3Etoensureconsistencywiththe
§
exposuretimecalculatorinputs.Thisisbasedonblindnarrow-band
Galaxy yields were computed using the exposure time calculator
surveys,andupdatesthepreviousestimatebyGeachetal.(2010).
described in Hirataetal. (2012). Two exposures on each field in
ThenewHαLFislowerthanthepreviousestimate;inapproximate
eachgrismbandpasswereassumed.Thezodiacalbackgroundwas
decreasingorderofimportance,themaindifferencesare:
settothatat45◦eclipticlatitudeand90◦awayfromtheSunatthe
meanoftheannualcycle,andweincludeaforegrounddustcolumn
1. Consistenttreatmentofinternal(hostgalaxy)extinctioncorrec-
ofE(B V)=0.05magnitudes;thesevaluesvaryoveranyreal-
− tions,whichareappliedtosomeHαLFresultsandmustbeundone
isticsurveybutarerepresentative.Standardreadnoiseassumptions
topredictredshiftsurveyyields.
forthe2k 2kTeledyneHgCdTedetectorswereused(32channel
× 2. Improvedstatisticsandadditionofdataatnewredshifts.
readout,1frameper1.3s,20electronsrmspercorrelateddouble
3. Redshift-averagingeffectsinsomeofthegrismluminosityfunc-
sample,withanoisefloorof5electronrmsformanyreads).The
tions(thisdoesnotoccurinnarrowbandsurveys).
galaxy survey was assumed to be 70 per cent complete down to
thefluxlimitfora 7σ significancematched-filterdetection.5 The 4. Aperturecorrections.
extinction-correctedHαfluxlimitvarieswithredshiftandgalaxy 5. ConversiontotheWMAP-5/FoMSWGcosmology.
The narrowband surveys do not cleanly separate the Hα 6563 A˚
5 Notethatsomeforecastsintheliteratureuseotherdefinitionsofdetection linefromthe[N II]doublet at6548,6583 A˚,andtheSobraletal.
significance,basedonotherextractionapertures.Thedifferencesareoften (2012)HαLFremovestheestimated[NII]contribution.Ofcourse,
tensofpercentsandoccasionallyaslargeasafactorof2.Thematched-filter inagrismsurveythetwolineswillbeapartialblend,thuswemay
methodgivesthehighestreportedsignificance. beunderestimatingthefinaldetectionsignificanceofthegalaxies.
(cid:13)c 0000RAS,MNRAS000,000–000
6 Wang,Chuang, &Hirata
Forthisreason,weexpectthatouranalysisissomewhatconserva- k nsT2(k)e−12k2Σ2nle−k2µ2σr2;z,p
tive. ×(cid:18)Mpc−1(cid:19)
Table2givesourresultantgalaxyyieldsasafunctionofred-
+P . (22)
shot
shift.
Table4liststheFoMfor(w ,w )forthesamefourcasesas
0 a
listedinTable3.EachlineinTable4anditscorrespondinglinein
3.2.2 Darkenergyfigureofmeritresults Table3assumethesamelevelofnonlinearityandthesamecutoff
k .Theonlydifferencebetweenthetwotablesisthemodelas-
Table 3 shows the dark energy figure-of-merit (FoM) (Wang max
sumed for P(k): Eq. (14) (fromEq. [4]) is assumed for Table3,
2008b),
whileEq.(22)(fromEq.[20])isassumedforTable4.
1
FoM(w ,w ) (19) NotethatthetwoassumedmodelsofP(k)givesimilarFoM
0 a
≡ detCov(w ,w ) forallthecasesthatmarginalizeoverthegrowthinformation,and
0 a
p for the cases that include growth information but assume only a
forthefourdifferentapproachestoutilizingtheinformationfrom
nonlinearity level of 50% (p = 0.5 or k = 0.24h/Mpc).
themeasuredanisotropicgalaxypowerspectrum(Wang2012): NL ∗
When we assume a nonlinearity level of 100% (p = 1 or
(1) x (z),x (z) fromP(k); NL
{ h d } k = 0.12h/Mpc), ournew model (Eq.[14]) givessignificantly
(2){xh(z),xd(z),fg(z)σm(z)/sα}fromP(k); la∗rgerFoMforthecasesthatincludethegrowthinformation.This
(3)P(k),marginalizedoverf (z)σ (z);
g m isbecausetheoldmodelinEq.(22)simplydampsthelinearmat-
(4)P(k)+f (z);P(k)includingf (z)σ (z).
g g m terpowerspectrumexponentially,whilethenewmodelinEq.(14)
It is clear from Table 3 that for a given cutoff k , the
max only damps the BAO oscillations, and retain smaller scale infor-
FoM for (w ,w ) increases as we increase the dewiggling scale
0 a mation via the “no-wiggle” matter power spectrum P (kz) =
kli∗ne(air.eit.y,d(ei.cer.e,afisexetdhekno),nltihneeaFroeMffefcotsr).(wFor,wa fi)xeindclreevaeselsofasnowne- G2(z)P0knsTn2w(k)(withTnw(k)denotingthezerobanrwyon|trans-
∗ 0 a ferfunction fromEisenstein&Hu(1998)).Thisresultsinsignif-
increase the cutoff k . The scaling of f (z)σ (z) with s de-
max g m icantly smaller uncertainties in lnβ(z) (and lnf (z)σ (z)/sα)
pends on the level of nonlinearity assumed: for 50% nonlinear- g m
whenour new model isused, which inturnleads tosignificantly
ity (k = 0.24h/Mpc), α 4, while for 100% nonlinearity
(k =∗0.12h/Mpc),α 5.T≃hisisnotsurprising,sincethescal- largerFoM(w0,wa)whengrowthinformationisincluded.
ing∗of f (z)σ (z) with≃s4 (i.e.,α = 4) originatesfromthe lin- Whilethescalingoffg(z)σm(z)withsdependsonthelevel
g m of nonlinearity assumed in our new model (see discussion in the
ear matterpower spectrum (Wang2012). When nonlinear effects
previoussubsection),f (z)σ (z)scaleswiths4 intheoldmodel
arefullyincluded (and not assumed tobereduced due todensity g m
(Wang2012).WefindthatourFoMresultsarenotsensitivetothe
fieldreconstruction),theappropriatemodelforP(k)(i.e.,Eq.[14])
exactchoiceofα;wehavechosenα=4foralltheFoMtabulated
deviates significantly from the linear power spectrum, leading to
inTables3and4whengrowthinformationisincluded.
modificationofthescalingoff (z)σ (z)withs.
g m
The choice of α does not affect the FoM(w ,w )
0 a
from the x ,x ,f σ /sα case, as the correlations between
h d g m
f (z)σ (z{)/sα and x (z)},x (z) depend on α. Choosing the 4 CONCLUSION
g m h d
αthatminimizesthe{uncertaintiesin}fg(z)σm(z)/sα doesmaxi- Wehaveshownthattheforecastingofdarkenergyconstraintsfrom
mizetheFoM(w0,wa)whenPlanckpriorsareincluded. galaxy redshift surveys can be improved in fidelity by using the
“dewiggled”galaxypowerspectrum,P (k),intheFishermatrix
dw
calculations. Since P (k) is a good fit to real galaxy clustering
3.3 Comparisonwithpreviouswork dw
dataovermostofthescalerangeofinterest,ourapproachis more
Previously,theforecastsofdarkenergyconstraintsfromfullP(k) realisticcomparedtopreviousworkinforecastingdarkenergycon-
assumedthat straintsfromgalaxyredshiftsurveys.
D (z)ref 2H(z) Wetested ourmethodology by comparing our Fishermatrix
Pold(kref,kref) = A forecasts withresultsfrom actual data analysis, and found excel-
g ⊥ k (cid:2)[DA(z)]2H(cid:3)(z)ref lentagreement(seeTable1).OurFishermatrixmethodgivesvery
b2 1+βµ2 2P (kz) similarresultscomparedtoactualdataanalysis,makingitareliable
lin
× | toolforparameterforecastingforfuturesurveys.
e−(cid:0)12k2Σ2nle−k(cid:1)2µ2σr2;z,p+Pshot, (20) Using ournew approach, westudied aStageIV galaxy red-
×
shiftsurvey,incombinationwithBOSS,withoutandwithPlanck
where the linear matter power spectrum P (kz) =
lin | priors.Wefindthatinthisnewapproach,increasingnonlinearef-
G2(z)P knsT2(k) (with T(k) denoting the linear matter
0 fectsfrom50%(bestcase)to100%(mostconservative)hasasig-
transferfunction),and
nificantlyreducedimpactonthedarkenergyfigureofmeritcom-
∂r 2 σ 2 paredtopreviouswork.Thisindicatesthattheerasureofinforma-
σ2 = σ2+ p (21)
r;z,p (cid:16)∂z(cid:17) (cid:20) z (cid:16) c (cid:17) (cid:21) tionby nonlinearsmearing ishavingonly amodest effecton our
abilitytoconstraincosmologyusingthe“fullP(k)”methodinour
Alternatively,wecanwrite
newrealisticapproach.
Pold(kref,kref)
g
⊥ k
≡ Pgold(k⊥ref,kkref)/(h−1Mpc)3 ACKNOWLEDGMENTS
= DA(z)ref 2H(z) σ (z)+f (z)σ (z)µ2 2 Y.W.wassupportedinpartbyDOEgrantDE-FG02-04ER41305.
(cid:2)[D (z)]2H(cid:3)(z)ref g g m C.C. was supported by the Spanish MICINNs Consolider-
A (cid:2) (cid:3)
(cid:13)c 0000RAS,MNRAS000,000–000
Toward MoreRealisticForecastingofDarkEnergyConstraintsfromGalaxyRedshiftSurveys 7
z λ EE50 dV/(dz·dA) [email protected]′′ n dN/dz·dA
(µm) (arcsec) (Mpc3deg−2) (Wm−2) (Mpc−3) (deg−2)
0.700 1.1160 0.2297 5.56334E+06 3.56250E−19 1.70876E−04 9.50640E+02
0.750 1.1489 0.2315 6.06057E+06 3.33879E−19 1.89962E−04 1.15128E+03
0.800 1.1817 0.2335 6.54313E+06 3.15425E−19 2.11061E−04 1.38100E+03
0.850 1.2145 0.2354 7.00894E+06 3.01278E−19 2.23002E−04 1.56301E+03
0.900 1.2474 0.2373 7.45644E+06 2.89615E−19 2.01417E−04 1.50185E+03
0.950 1.2802 0.2393 7.88451E+06 2.79942E−19 1.81900E−04 1.43419E+03
1.000 1.3130 0.2413 8.29240E+06 2.72837E−19 1.62666E−04 1.34889E+03
1.050 1.3458 0.2433 8.67969E+06 2.67418E−19 1.44931E−04 1.25796E+03
1.100 1.3787 0.2454 9.04621E+06 2.63267E−19 1.28950E−04 1.16651E+03
1.150 1.4115 0.2474 9.39200E+06 2.60703E−19 1.13930E−04 1.07003E+03
1.200 1.4443 0.2494 9.71731E+06 2.64718E−19 9.39210E−05 9.12659E+02
1.250 1.4771 0.2696 1.00225E+07 2.80452E−19 6.69603E−05 6.71109E+02
1.300 1.5100 0.2714 1.03081E+07 2.68660E−19 6.78461E−05 6.99362E+02
1.350 1.5428 0.2731 1.05746E+07 2.59173E−19 6.78039E−05 7.16996E+02
1.400 1.5756 0.2749 1.08226E+07 2.51215E−19 6.73077E−05 7.28446E+02
1.450 1.6084 0.2768 1.10529E+07 2.44917E−19 6.61038E−05 7.30641E+02
1.500 1.6413 0.2786 1.12662E+07 2.39935E−19 6.34722E−05 7.15093E+02
1.550 1.6741 0.2804 1.14632E+07 2.35608E−19 6.03252E−05 6.91521E+02
1.600 1.7069 0.2823 1.16446E+07 2.31880E−19 5.73065E−05 6.67313E+02
1.650 1.7397 0.2841 1.18112E+07 2.29092E−19 5.40911E−05 6.38881E+02
1.700 1.7726 0.2860 1.19637E+07 2.26675E−19 5.11258E−05 6.11651E+02
1.750 1.8054 0.2879 1.21027E+07 2.24586E−19 4.83888E−05 5.85637E+02
1.800 1.8382 0.2898 1.22291E+07 2.22811E−19 4.58519E−05 5.60729E+02
1.850 1.8710 0.2917 1.23435E+07 2.21390E−19 4.34464E−05 5.36281E+02
1.900 1.9039 0.2936 1.24465E+07 2.20333E−19 4.11589E−05 5.12286E+02
1.950 1.9367 0.2955 1.25388E+07 2.20230E−19 3.86029E−05 4.84035E+02
2.000 1.9695 0.2975 1.26210E+07 2.20856E−19 3.59559E−05 4.53799E+02
Table2.Galaxyyieldsfora2-exposureStageIVgalaxyredshiftsurveyasdiscussedinSec.3.2.1.Columnsindicate:theredshift;observer-framewavelength
ofHα;thehalf-lightradius(encircledenergy50%,EE50)ofthepoint-spreadfunction;thecosmologicalvolumeelementdV/(dz·dA);thelimitingfluxfor
a0.3arcsechalf-lightradiusgalaxy;thenumbernofobservedsourcesperunitcomovingvolume;andthenumberofsourcesdN/dz·dAperunitredshiftper
unitsolidangle.
kmax k FoM FoMGR (FoM,dγ) FoM FoMGR (FoM,dγ)
(hMpc−1) (hMp∗c−1) {xh,xd} {xh,xd,fgσm/sα} {P(k)} {P(k)+fg}
0.2 0.12 6.56 30.78 (22.80, 0.0514) 14.81 40.48 (24.47, 0.0476)
0.2 0.24 10.05 45.30 (31.07, 0.0470) 23.37 54.59 (35.05, 0.0456)
0.3 0.12 9.83 44.11 (33.98, 0.0437) 19.94 65.32 (35.19, 0.0386)
0.3 0.24 12.73 62.51 (42.41, 0.0394) 29.81 79.57 (46.43, 0.0374)
StageIV+BOSS+Planck
0.2 0.12 58.30 139.24 (80.49, 0.0392) 61.30 171.90 (106.98, 0.0341)
0.2 0.24 92.64 193.61 (119.58, 0.0376) 96.22 238.63 (152.10, 0.0329)
0.3 0.12 85.24 209.98 (110.10, 0.0344) 89.29 240.11 (136.75, 0.0311)
0.3 0.24 119.88 273.20 (152.31, 0.0322) 123.55 315.23 (184.21, 0.0295)
Table3.OurFishermatrixforecastsforStageIV+BOSSgalaxyredshiftsurveysusingournewgalaxypowerspectrummodel,Eq.(14),forthefourcases
discussedinWang(2012).FoMGRdenotedtheFoMassuminggeneralrelativity.Theparameterγisdefinedbyfg(z)=[Ωm(a)]γ.
Ingenio 2010 Programme under grant MultiDark CSD2009- sion Figure of Merit Science Working Group,” preprint,
00064 and grant AYA2010-21231. C.H. was supported by DOE arXiv:0901.0721
DOE.de-sc0006624,andtheDavidandLucilePackardFoundation. AlcockA.,PaczynskiB.1979,Nature281,358
AnguloR.,BaughC.,FrenkC.,LaceyC.2008,MNRAS383,755
AtekH.etal.2010,ApJ723,104
BlakeC.,GlazebrookG.2003,ApJ594,665
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