The Angular Power Spectrum of EDSGC Galaxies

11 Pages·0.27 MB·English

by  Dragan Huterer

SubmittedtoApJ
PreprinttypesetusingLATEXstyleemulateapjv.16/07/00
THE ANGULAR POWER SPECTRUM OF EDSGC GALAXIES
Dragan Huterer
DepartmentofPhysics,UniversityofChicago,Chicago,Illinois60637, USA,email:
[email protected]
Lloyd Knox1
Department ofAstronomyandAstrophysics,UniversityofChicago, Chicago,Illinois60637, USA,email:
knox@flight.uchicago.edu
and
Robert C. Nichol
1 DepartmentofPhysics,CarnegieMellonUniversity,Pittsburgh,Pennsylvania15213,USA,email:
0 [email protected]
Submitted to ApJ
0
2
ABSTRACT

n We determine the angular power spectrum C of the Edinburgh/Durham Southern Galaxy Catalog
l
a
(EDSGC) and use this statistic to constrain cosmological parameters. Our methods for determining
J
Cl and the parameters that affect it are based on those developed for the analysis of cosmic microwave
0
backgroundmaps. Weexpectthemtobeusefulforfuturesurveys. Assumingflatcolddarkmattermodels
3
with a cosmological constant (constrained by COBE/DMR and local cluster abundances) and a scale–

independent bias b, we find acceptable fits to the EDSGC angular power spectrum with 1.11<b<2.35
2
and 0.2 < Ω < 0.55 at 95% confidence. These results are not significantly affected by the “integral
v m
9 constraint” or extinction by interstellar dust, but may be by our assumption of Gaussianity.
6 Subject headings: cosmology: theory – cosmology: observation – cosmology: galaxies
0
1 1. introduction to obtain the three–dimensional power spectrum P(k).
1
This requires inverting Limber’s equation (Limber 1953).
0 Over the next decade, the quantity and quality of
Baugh&Efstathiou(1993,1994)andGaztan˜aga&Baugh
0 galaxy survey data will improve greatly because of a
/ (1998)usedLucy’s algorithm(Lucy1974)to dothe inver-
h variety of new survey projects underway including the sion,while Dodelson&Gaztan˜aga(2000)usedaBayesian
p Sloan Digital Sky Survey (SDSS; see York et al. 2000).
prior constraining the smoothness of the power spectrum.
- However, most of the galaxies in such surveys will not
o Eisenstein&Zaldarriaga(2000)usedatechniquebasedon
havespectroscopically–determinedredshifts;therefore,the
r singular-valuedecompositiontogetP(k)fromw(θ). They
t studyoftheirangularcorrelationswillbehighlyprofitable
s pointoutthat,oncethecorrelationsintheinvertedpower
for our understanding of the large–scale structure of the
a spectraareincluded,theuncertaintiesoncosmologicalpa-
: Universe.
v rameters from the APM are significantly weakened.
The primary purpose of this paper is to consider an
i Our analysis is a three-step process, similar to what is
X analysis approach which is likely to be useful for deriv-
done with CMB data sets (Tegmark 1997; Bond, Jaffe &
r ing cosmological constraints from these larger surveys. Knox1998;Bond,Jaffe&Knox2000). Thefirststepisthe
a In particular, we use methods that have become stan-
constructionofa pixelizedmapof galaxycounts,together
dard in the analysis of cosmic microwave background
with its noise properties. The second step is the deter-
(CMB) anisotropy maps, such as those from Boomerang
mination of the angular power spectrum C of the map
(de Bernardisetal.2000;Langeet al.2000)andMaxima- l
using likelihood analysis, together with window functions
I (Hanany et al. 2000, Balbi et al. 2000).
and a covariance matrix. In the final step, we compare
Estimation of the two–point angular correlation func-
our observationally–determinedC to the C predicted for
l l
tion w(θ) from galaxy surveys without redshfit informa-
a given set of parameters in order to get constraints on
tion has a long history. Early work (Peebles & Hauser
those parameters. We assume that the errors in C are
l
1974; Groth & Peebles 1977) found the angular correla-
log–normally distributed.
tion function to vary as w(θ) = θ1−γ with γ = 1.77 and
The angular power spectrum C is a useful intermedi-
a break at scales larger than 9h−1 Mpc. The advent of l
∼ ate step on this road from galaxy catalog to parameter
automatedsurveys,suchastheAPMgalaxysurvey(Mad-
constraints. Estimates of the angular power spectrum,
doxet.al.1990)andEdinburgh/DurhamSouthernGalaxy
together with a description of the uncertainties, can be
Catalogue(EDSGC;Collins,Nichol&Lumsden1992)en-
viewed as a form of data compression. One has converted
abledamuchmoreaccuratedeterminationofw(θ),aseach
the 1millionEDSGCgalaxies(forexample)intoahand-
surveycontainedangularpositionsforoveramilliongalax- ∼
ful of power spectrum constraints, together with window
ies.
functions and covariance matrices. Thus if one wishes to
One way to compare the measured angular correla-
make other assumptions about bias and cosmological pa-
tion functionwith theoreticalpredictionsis to invertw(θ)
rameters than we have done here and determine the re-
1 sultingconstraints,onecandosowithouthavingtoreturn
CurrentlyatInstitutd’AstrophysiquedeParis to the cumbersome galaxy catalog.
1
2
We use C instead of its historically–preferredLegendre useful for other current and future data sets—even those
l
transformw(θ) forseveralreasons. First,theerrormatrix with large numbers of measured redshifts. For example,
structure is muchsimpler: δClδCl′ is band-diagonaland theSloanDigitalSkySurvey(SDSS)willspectroscopically
h i
becomesdiagonalinthelimitoffull–skycoverage,whereas determine the redshifts of a million galaxies, but there
δw(θ)δw(θ′) is muchmorecomplicatedanddoesnotbe- will be about one hundred times as many galaxies in the
h i
comediagonaleveninthefull–skylimit. Second,therela- photometric data, without spectroscopic redshifts. One
tion between C and the correspondingthree–dimensional can generalize the methods presented here to analyze sets
l
statistic P(k)issimplerthanthatbetweenw(θ) andP(k) of maps produced from galaxies in different photometric
(oritsFouriertransformξ(r)(Baugh&Efstathiou1994)). redshift slices.
We use likelihood analysis to determine C because the In 2, we review likelihood analysis and the use of the
l
§
likelihood is a fundamental statistical quantity. The like- quadratic estimator to iteratively find the maximum of
lihood is the probability of the data given C , which by the likelihood function. In 3 we describe our calculation
l
§
Bayes’ theorem is proportional to the probability of C of P(k) and its projection to C . In 4 we show how to
l l
§
given the data. Another advantage of likelihood analysis compare the calculated C to the measured C in order to
l l
is that, as explained below, it allows for straightforward determine parameters. In 5 we apply ourmethods to the
§
control of systematic errors (due to, e.g., masking) via EDSGC, and discuss some possible sources of systematic
modifications of the noise matrix. error in 6. This is followed by a discussion of our results
§
Only on sufficiently large scales do we expect the like- in 7 and a brief conclusion in 8. An appendix outlines
§ §
lihood function to be a Gaussian that depends only on the derivation of the projection of P(k) to Cl.
C and not any higher-order correlations. We therefore
l 2. the likelihood function and quadratic
restrict our analysis to l values less than some critical
estimation
value. On small scales the likelihood function becomes
much more complicated and its form harder to predict Thelikelihoodisafundamentalstatisticalquantity: the
a priori. Mode-mode coupling due to nonlinear evolu- probability of the data given some theory. According to
tion leads to departures of the Cl covariance matrix from Bayes’ theorem, the probability of the parameters of the
band-diagonal. Therefore some of the advantages of like- assumedtheoryisproportionaltothelikelihoodtimesany
lihood analysis and the angular power spectrum are lost prior probability distribution we care to give the param-
on smallerscales where other techniques may be superior. eters. Thus, determining the location of the likelihood
The Gaussianity assumption is perhaps the weakestpoint maximum and understanding the behavior of the likeli-
of the approach outlined here. Below, we briefly discuss hood function in that neighborhood (i.e., understanding
how the analysis can be improved in this regard with fu- the uncertainties) is of great interest.
ture data sets. Despiteitsfundamentalimportance,anexactlikelihood
The EDSGC, with over a million galaxies and cover- analysis is not always possible. Two things can stand in
ing over 1000 sq. degrees, offers us an excellent test bed our way: insufficient computer resources for evaluation of
for applying our algorithms (Nichol, Collins & Lumsden thelikelihoodfunction(operationcountscalesasN3 and
pix
2000). We convert this catalog into a pixelized map and memory use scales as N2 ), and, even worse, the absence
determine its angular power spectrum together with win- pix
of an analytic expression for the likelihood function.
dow functions and covariance matrix. As an illustrative
Inthispaperweassumethatthepixelizedmapofgalaxy
application of the angular power spectrum, we constrain
counts is a Gaussian random field—an assumption which
a scale–independent bias parameter b and the cosmologi-
provides us with the analytic expression for the likeli-
calconstantdensityparameterΩ inaCOBE–normalized
Λ hoodfunction. FormodelswithGaussianinitialconditions
ΛCDMmodelwithzeromeanspatialcurvature. Ourcon-
(which are the only models we consider here), we expect
straints on the bias are improved by including constraints
thistobeagoodapproximationonsufficientlylargescales.
ontheamplitudeofthepowerspectrumderivedfromnum-
Since we restrict ourselves to studying large-scalefluctua-
ber densities of low–redshift massive clusters of galaxies
tions, we can use large pixels, thereby reducing N and
(Viana & Liddle 1999, hereafter VL99; also see Pierpaoli, pix
ensuring that the likelihood analysis is tractable. We also
Scott & White 2000). These number densities are sensi-
check the Gaussianity assumption with histograms of the
tive to the amplitude of the matter power spectrum cal-
pixeldistribution. Onthe largescalesofinterest here and
culated in linear perturbation theory, near the range of
for a given three-dimensional length scale, Gaussianity is
length scales probed by the EDSGC.
a better approximationfora galaxycountsurveythanfor
The angular power spectrum of the APM catalog
a redshift survey, due in part to the redshift-space dis-
was previously estimated by Baugh & Efstathiou (1994)
tortions which affect the latter (Hivon et al. 1995). The
though not via likelihood analysis. Very recently, Efs-
projection from 3 dimensions to 2 dimensions also tends
tathiou&Moody(2000)haveappliedthesametechniques
to decrease non-Gaussianity.
we do here to estimating C for the APM survey. Their
l Where likelihood analysis is possible, it naturally han-
approach differs from ours in how they constrain cosmo-
dlestheproblemsofotherestimators(suchasedgeeffects).
logical parameters. Instead of projecting the theoretical
Likelihood analysis also provides a convenient framework
three–dimensionalpower spectra P(k) into angularpower
for taking into account various sources of systematic er-
spectra, they transform their C constraints into (highly
l ror, such as spatially varying reddening and the “integral
correlated) constraints on P(k) and then compare to the-
constraint” discussed in 6.
oretical P(k). §
To begin our likelihood analysis, we assume that the
We expect the analysis methods presented here to be
data are simply the angular position of each galaxy
HUTERER ET AL. 3
observed—though it is possible to generalize the follow- this iterative procedure converges to well within the size
inganalysisanduseeithermagnitudeinformationorcolor of the error bars quite rapidly.
redshifts. We pixelize the sky and count the number of The small sky coverage prevents us from determining
galaxiesineachpixelGi. Thenwecalculatethefractional each multipole moment individually; thus we determine
deviation of that number from the ensemble average: thepowerspectruminbandsofl instead,callthem“band
G G¯Ω powers”,and denote them by B where
∆ i− i (1) C
i ≡ G¯Ω
i l(l+1)C
l
where G¯ is the ensemble average number of galaxies per Cl ≡ 2π = χB(l)CB (8)
unit solid angle and Ω is the pixel solid angle. We do XB
i
not actually know the ensemble mean. In practice, we andχ is unity forl (B)<l <l (B) wherel (B) and
B(l) < > <
approximateitwiththesurveyaverageG˜. Wediscussthis l (B) delimit band B.
>
approximationin 6anddemonstratethatithasnegligible Although we view equation (6) as a means of finding
§
impact on our results. themaximumofthelikelihoodfunction,onecanalsotreat
We model the fractional deviation in each pixel from +δ (withnoiteration)asanestimatorinitsownright
l l
C C
the meanashavingacontributionfrom“signal”andfrom (Tegmark 1997; Bond, Jaffe & Knox 1998). It is referred
“noise”, so that toasaquadraticestimatorsinceitisaquadraticfunction
∆i =si+ni. (2) ofthe data. One canview equation(6)as a weightedsum
over (∆∆T C), with the weights chosen to optimally
The covariancematrix, Cij, for the fractionaldeviationin change so−that C is closer to ∆∆T in an averagesense.
each pixel from the mean is given by: Cl
Various sources of systematic error can be taken into
account by including extra terms in the modeling of the
C ∆ ∆ =S +N (3)
ij ≡h i ji ij ij data(equation(2))andworkingouttheeffectonthedata
where S s s and N n n are the signal and covariance matrix, C. Below we see specific examples as
ij i j ij i j
≡ h i ≡ h i
noisecovariancematrices. Roughlyspeaking,signalispart we take into account the “integral constraint” and pixel
ofthe datathatisdue tomassfluctuationsalongthe line- masking. The reader may also wish to see the appendix
of-sight (see the appendix), and noise are those fluctua- of Bond, Jaffe & Knox (1998), Tegmark et al. 1998 and
tions due to anything else. Knox et al. (1998) for more general discussions.
ThesignalcovariancematrixS dependsontheparam-
ij
eters of interest (the angular power spectrum Cl) via: 3. calculation of cl
S =w(θ )= 2l+1C P (cosθ )e−l2σb2 (4) We needtobe abletocalculateCl foragiventheory,in
ij ij Xl 4π l l ij ocardlceurlatotioconmispaartehirteew-sitthepCplreoscteimssa.tSetdepfroomnetihsetodactaal.cuTlahties
whereθij istheangulardistancebetweenpixelsiandjand the matter power spectrum P(k) in linear perturbation
we have assumed a Gaussian smoothing of the pixelized theory. Step two is to then use some biasing prescription
galaxy map with fwhm = √8ln2σb. In practice, we do toconvertthistothegalaxynumbercountpowerspectrum
not estimate eachCl individually but binned Cls with bin PG(k). StepthreeistoprojectthisP(k)toCl. Wefurther
widths greater than π/θ where θ is a typical angular discuss these steps in the following subsections.
∼
dimension for the survey.
The noise contribution to the fluctuations n is due to 3.1. The 3D matter power spectrum, P(k)
the fact that two regions of space with the same mass
We take the primordial matter power spectrum to be a
density can have different number of galaxies. We model
power-law with power-spectral index n and amplitude δ2
thisadditionalsourceoffluctuationsasaGaussianrandom H
process with variance equal to 1/G¯, so that attheHubbleradius. Wewritethematterpowerspectrum
today P (k) (calculated using linear perturbation theory)
0
Nij ninj =1/G¯δij. (5) as a product of the primordial spectrum and a transfer
≡h i
More sophisticated modeling of the noise is not necessary function T(k):
btoectahuesenoaitseailsl ml vuaclhuessmoafllienrtethreasnt,tthheesavamrpialencvearinianCcle.due P0(k)≡ k32Pπ02(k) =δH2 (k/H0)3+nT2(k) (9)
To find the maximum of the likelihood function, we it-
where H = 100hkmsec−1Mpc−1 is the Hubble param-
eratively apply the following equation: 0
eter today. The transfer function, T(k), goes to unity
1
δCl = 2Fl−l′1Tr (∆∆T −C)(C−1∂C/∂Cl′C−1) , (6) caetslsaersgferosmcaalelstesriinncgetchaeusspaelicttyrupmreavtenlatsrgmeisccraolpesh.yAsitcahligphreor-
(cid:2) (cid:3)
where F is the Fisher matrix given by k itdependsonh,Ωmh,andΩbh2. Tocalculatethetrans-
fer function we use the semi–analytic approximation of
1 ∂C ∂C Eisenstein & Hu (2000). It is also available as an out-
Fll′ = 2Tr(cid:18)C−1∂ lC−1∂ l′(cid:19), (7) putfromthepublicallyavailableCMBfastBoltzmanncode
C C (Seljak & Zaldarriaga1996).
andforlaterconvenienceweareusing l(l+1)C/(2π)
l l Our power spectrum is now parametrized by 5 param-
C ≡
instead of Cl. That is, start with an initial guess of l, eters: n, δ , Ω h, h and Ω h2. In the following analy-
C H m b
update this to +δ , and repeat. We have found that
l l sis, we eliminate two of these parameters by simply fixing
C C
4
h=0.7 and Ω h2 =0.019. The dependence of our results σ constraints forces n to change as well. For Ω =
b 8 m
on variations in h can be derived analytically, which we 0.15,0.3,0.35,0.4,1, n = 1.55,1.00,0.91,0.84,0.47respec-
do in 7. Measurements of deuterium abundances in the tively. Onecanunderstandthisbyconsideringthesimpler
§
Ly-α forest, combined with the dependence of primordial caseofδ andσ heldconstantwithoutΩ andndepen-
H 8 m
abundances on the baryon density, lead to the constraint dence. Then the only effect of changing Ω h is to change
m
Ω h2 =0.019 0.002 at 95% confidence (Burles & Tytler the transfer function. For fixed δ , increasing Ω h in
b H m
±
1998; Burles, Nollett & Turner 2000). this case leads to increased power on small scales. One
Of the remaining parameters, two more, δH and n, can thereforeneeds todecreasethe tiltinorderto keepσ8 un-
befixedbyinsistingonagreementwithboththeamplitude changed. Now,thefactthatourtwoamplitudeconstraints
ofCMBanisotropyonlargeangularscalesasmeasuredby dodependonΩm alsohasaneffectonhownchangeswith
COBE/DMR, and the number density of massive clusters changing Ωm. However, this is a subdominant effect be-
at low redshifts. The COBE constraint can be expressed cause these dependences are quite similar.
with the fitting formula
δ =1.94 10−5Ω−0.785−0.05lnΩm
H × m ×
exp 0.95(n 1) 0.170(n 1)2 , (10)
h− − − − i
whichisvalidfortheflatΛCDMmodelsweareconsidering
(Bunn & White 1995).
The cluster abundance constraint can be expressed as
a constraint on σ which is the rms fluctuation of mass
8
in spheres of radius r = 8h−1Mpc, calculated in linear
theory:
dk
σ2 = (3j (kr)/(kr))2 (k) (11)
8 Z k 1 P0
wherej (x)=(xcos(x) sin(x))/x2. VL99findthemost
1
likely value of σ to be σ− =0.56Ω−0.47.
8 8 m
The reason for the choice of the scale of 8h−1Mpc is
that a sphere of this size has a mass of about 1015M , Fig. 1.—MatterpowerspectraandCl derivatives. Frombottom
⊙ totopatlowlaretheCOBEandclusterconsistentpredictionsfor
whichisthemassofalargegalaxycluster. MostoftheΩm Ωm = 0.15,0.3,1, all with b = 1 (dashed lines show linear theory
dependenceofσ8 comesfromthefactthatthepre-collapse predictions). The l=20 and l=80 curves show k∂Cl/∂Pk for for
lengthscalecorrespondingtoagivenmassdependsonthe thesetwomultipolemoments (arbitrarynormalization).
matter density. Thus, in a low density Universe the pre-
collapse scale is larger, and since there is less fluctuation
power on larger scales, the σ normalization has to be
8 3.2. The biasing prescription
higher for fixed cluster abundance.
The shift in pre-collapselength scale with changing Ω Although biasing in general is stochastic, non–linear,
m
is very slow, scaling as Ω1/3. Thus, although the param- redshift and scale dependent, we adopt the simplest pos-
m
sible model here in which the galaxynumber density fluc-
eters that govern the shape of the power spectrum affect
tuations are directly proportional to the matter density
the normalization, their influence is quite small. For ex-
fluctuations. Thenwecanwriteb δ /δ,whereδ =δρ/ρ¯
ample, the scale shift for changing Ωm by a factor of 3 is ≡ G
31/3 =1.44,and overthis rangeanuncertaintyinn of0.2 is the matter density contrast, δG is the galaxy number
density contrast, and b is the bias factor.
translates into an uncertainty in power of 8%.
With this description P (k) = b2P(k), where P(k) is
Of course there are uncertainties in both the constraint G
thematterpowerspectrum. Notethatabovewehaveonly
from COBE and the constraint from cluster abundances.
calculatedthelineartheorymatterpowerspectrum. Non-
More significant of the two is the uncertainty in clus-
linearcorrectionsareimportantovertheEDSGC rangeof
ter abundance constraint. Consequently, we extend our
grid of models to cover a range of values of σc where length scales, and we must incorporate these effects. We
σ =σcΩ−0.47. VL99findthattheprobabilityofσ8c islog- derive P(k) from the linear theory power spectra P0(k)
no8rmal8lymdistributed with a maximum at σc = 0.856 and by useofa fitting formula(Peacock& Dodds 1996)which
8 provides a good fit to the results of n-body calculations.
a variance of lnσc of 0.25ln2(1+0.20Ω0.2log10Ωm). The
8 m The resulting power spectra are shown by the solid lines
COBE uncertainty is only 7%. We ignore this source of
in Fig. 1.
uncertaintyanddonotexpectitto affectourresultssince
Wehaveassumedthatthegalaxynumberdensityfluctu-
such a small departure from the nominal large-scale nor-
ationsarecompletelydeterminedbythelocaldensitycon-
malizationcanbeeasilymimicked,overtherangeofscales
trast. Thenumberdensityofgalaxiesmustalsohavesome
probed by EDSGC, by a very small change in the tilt n.
non–localdependence onthe density contrast. Morecom-
In Fig. 1 we plot (k) (dashed lines) for several
P0 plicatedmodelingoftherelationship,or“biasingschemes”
models that satisfy the COBE/DMR and VL99 con-
(e.g., Cen & Ostriker 1992; Mann, Peacock & Heavens
straints. Changing Ω h and also satisfying the δ and
m H 1998, Dekel & Lahav 1999) are beyond the scope of this
HUTERER ET AL. 5
paper. In the applications that follow we assume the bias ourapproximationtothereforebeoverestimatesofC ,but
l
tobeindependentoftimeorscale,althoughourformalism since we do not use data that reachvery far into the non-
allows inclusion of both of these possibilities. linearregime,wedonotexpectthiserrortobesignificant.
From analytic theory (e.g., Seljak 2000) we expect the
biastobescale–independentonscalesthatarelargerthan 4. extraction of parameters
any collapsed dark matter halos. Numerical simulations
To find the maximum-likelihood power spectrum, we
show this to be the case as well (see Blanton et al. 2000;
haveiterativelyappliedthebinnedversionofequation(6).
Nayayanan, Berlind & Weinberg 2000) on scales larger
Althoughequation(6)isusedasaniterativemeansoffind-
than 10h−1Mpc. Moreover,recentobservations by Miller,
ingthemaximumofthe likelihood,itisalsoconvenientto
Nichol & Batuski (2001) show that a scale–independent,
write it as the equivalent equation for , instead of the
linear, biasing model works well when scaling cluster & CB
correction δ :
galaxy data over the range of 200 to 40h−1 Mpc. Our CB
results are determined mostly by information from these
large scales. Since we find acceptable fits to the data us- = 1 F−1 Tr ∆∆T N C−1 ∂C C−1 , (15)
ing our constant bias model, we have no evidence for a CB 2XB′ BB′ (cid:20)(cid:0) − (cid:1) ∂CB′ (cid:21)
scale–dependent bias.
where the right–hand side is evaluated at the previous it-
eration value of , RHS, and = RHS +δ is the
3.3. The projection to 2D CB CB CB CB CB
updated power spectrum.
Asdescribedintheappendix,Cl canbecalculatedfrom We have shownhow to calculate l fromthe theoretical
P0(k) and the selection function as parameters. We now needtocalculaCte whatCB weexpect
for this . One can show that the expectation value for
C =4π (k)f (k)2dk/k (12) Cl
l 0 l ,giventhatthedataarerealizedfromapowerspectrum
Z P CB
, is
where Cl
f (k) 1 dzdrj(kr)r2g¯(z)D(z)bT (k,z), (13) hCBi= FB−B1′ Fll′Cl
l ≡ G¯ Z dz l nl Xl XB′ l′X∈B′
r is the comoving distance along our past light cone, g¯(z) WB
= l (16)
isthemeancomovingnumberdensityofobservablegalax- l Cl
Xl
ies, D(z) is the growth of perturbations in linear theory
where the Fisher matriceson the right-handside are eval-
relative to z = 0 and T (k,z) is the correction factor for
nl uatedat RHS,andthe lastline servesto define the band-
non-linear evolution (Peacock & Dodds 1996). CB
power window function WB. Note that the sum over l′
Equations (12) and (13) are valid for all angular scales. l
is only from l (B′) to l (B′). This equation reduces to
It becomes time–consuming to evaluate the Bessel func- < >
tion on smaller angular scales. Although we always used equation(8)ofKnox(1999)inthe limit ofdiagonalFBB′.
Itisthisexpectationvaluethatshouldbecomparedtothe
equations(12)and(13),thereadershouldknowthatthere
measured .
is a much more rapid approximation which works well at B
C
l >30: AsshownbyBond,Jaffe&Knox(2000),theprobability
∼ 1 dr distribution of Cl is well–approximated by an offset log-
C = dz P(k =l/r,z)[g¯(z)D(z)bT (k =l/r,z)]2. normal form. In the sample–variance limit, which applies
l G¯2 Z dz nl for our analysis of EDSGC, this reduces to a log–normal
(14) distribution. Thereforewetaketheuncertaintyineach
B
In order to calculate Cl, we need to know g¯(z). Since to be log–normally distributed and evaluate the followiCng
r2g¯(z)dr/dz = dG¯/dz (Baugh & Efstathiou 1993; 1994; χ2:
our appendix) it is sufficient to know dG¯/dz, whose mea-
surement is described in 5.
§ χ2 (Ω ,b,σc)= (17)
To give an idea of how C depends on (k) we plot EDSGC m 8
l
k∂Cl/∂P(k)inFig.1forl=20andl =80. PThis quantity lnCB −lnCBt CBFBB′CB′ lnCB′ −lnCBt′
is the contribution to Cl from each logarithmic interval BXB′(cid:0) (cid:1) (cid:0) (cid:1)
in k. Note that it is the breadth of these derivatives that WB
explainsthecorrelationsthatappearinanyattempttore- t l (Ω ,b,σc) (18)
CB ≡ l Cl m 8
constructP(k)fromangularcorrelationdata. Thederiva- Xl
tives have some dependence on cosmology; those plotted
where σ =σcΩ−0.47.
are for the Ω =0.3 case. 8 8 m
m Ourtotalχ2 =χ2 +χ2 includesthe contribution
The angular power spectrum is sensitive not only to EDSGC VL
from the cluster abundance constraint which is also log-
the power spectrum today, but to the power spectrum in
normal:
the past as well. In linear theory, the evolution of the χ2 =(lnσc ln0.56)2/σ2 (19)
power spectrum is separable in k and z; one can write VL 8−
P(k,z) = P(k,0)D2(z) where D(z) is the growth factor where σ = 1ln 1+0.32Ω0.24log10Ωm (Viana and Liddle
well-described by the fitting formula of Carroll, Press & 2 m
(cid:16) (cid:17)
Turner(1992). Wealsoassumethatthisrelationholdsfor 1996,hereafterVL96). Note that here andthroughoutwe
the non-linear power spectra. In truth, non-linear evolu- have adopted the more conservative uncertainty in VL96,
tion is more rapid at higher k than at lower k. We expect as opposed to the VL99 uncertainty.
6
5. application to the edsgc 80000
The Edinburgh/Durham Southern Galaxy Catalogue
(EDSGC) is a sample of nearly 1.5 million galaxies cover-
ing over 1000 deg2 centered on the South Galactic Pole. 60000
ThereaderisreferredtoNichol,Collins&Lumsden(2000)
forafulldescriptionoftheconstructionofthisgalaxycat- z
alogue as well as a review of the science derived from this d
/ 40000
G
survey. For the EDSGC data, the reader is referred to
d
www.edsgc.org.
Fortheanalysisdiscussedinthispaper,weconsideronly
the contiguous region of the EDSGC defined in Nichol, 20000
Collins & Lumsden (2000) and Collins, Nichol & Lums-
den (1992) (right ascensions 23 < α < 3 hours, through
zero hours, and declinations 42◦ < δ < 23◦). We also 0
restrict the analysis to the m−agnitude ra−nge 10 < b < 0 0.1 0.2 0.3 0.4 0.5
J z
19.4. The faint end of this range is nearly one magni-
tude brighter than the completeness limit of the EDSGC
(see Nichol, Collins & Lumsden 2000) but corresponds to Fig. 2.— The selection function for the EDSGC; i.e., the mean
numberofgalaxiespersteradianperredshiftinterval.
the limiting magnitude of the ESO Slice Project(ESP) of
Vettolani et al. (1998) which was originally based on the
EDSGC. The ESP survey is 85% complete to this limit-
ing magnitude (bj = 19.4) and consists of 3342 galaxies of galaxies in the EDSGC, the resulting shot noise is still
with redshift determination. This allows us to compute lessthanthefluctuationpower,evenatthesmallestscales
the selection function of the whole EDSGC survey which we consider.
is shownin Fig.2. The datashowninthis figurehas been We binned the map into 5700 pixels with extent 0.5◦
corrected for the 15% incompleteness in galaxies brighter in declination and 0.5◦ in right ascension (RA). The pix-
than b = 19.4 with no measured redshifts as well as the
j els are slightly rectangular with varying solid angles; the
mean stellar contamination of 12% found by Zucca et al.
RA widths correspond to angular distances ranging from
(1997) in the EDSGC. These corrections are not strong 0.46◦ at δ = 23◦ to 0.37◦ at δ = 42◦. This pixeliza-
functions of magnitude; therefore, we apply them as con- − −
tionis fine enoughso asnotto affect ourinterpretationof
stant values across the whole magnitude range of the sur-
the large–scale fluctuations; it causes a 4% suppression
vey. ∼
of the fluctuation power at l = 80. We have varied the
Asmentionedabove,weneedtocorrectourpowerspec- pixelization scale to test this and find that with 1◦ 1◦
trum estimates for stellar contamination in the EDSGC pixels the estimated s change by less than half an e×rror
l
map. Ifthestarsareuncorrelated(whichweassume)then bar for l<80. C
their presence will suppress the fluctuation power as we
We alsotook into accountthe “drillholes”,locations in
now explain. Let T be the total count in pixel i, consist-
i the map which were obstructed (e.g. by bright stars). In
ing of galaxies and stars: Ti =Gi+Si (for simplicity, we the case of 0.5◦ 0.5◦ pixelization about 75 pixels were
consider equal-area pixels). Let α = 0.12 be the fraction ×
corrupted by drill holes. Those pixels were assigned large
of the total that are stars, so that G¯ = (1 α)T¯. Then,
diagonal values in the noise matrix (e.g., Bond, Jaffe &
defining∆G =(G G¯)/G¯ and∆S =(S S¯−)/S¯,wehave
i i− i i− Knox 1998), and thus had negligible weight in the subse-
T T¯ quent analysis. The 0.5◦ 0.5◦ pixelized map is shown in
i
∆ − (20) ×
i≡ T¯ Fig. 3.
=(1 α)∆G+α∆S. (21) In Fig. 4 we plot the estimated angular power spec-
− i i trum from the EDSGC data. Also shown in Fig. 4 are
∆G is what we are after: density contrast in the absence
i our predicted s. For each of these, we can calculate the
of stellar contamination. The second term amounts to a Cl
expected values of by summing over the window func-
small additional source of noise. Since, as mentioned in CB
tions,showninthebottompanelforthesixlowestlbands.
2, the noise is completely unimportant on the scales of
§ Thejaggednessresultsfromourpracticeofcalculatingthe
interest, we neglect this term. Therefore,
Fisher matrix notfor everyl, but for fine bins of l labeled
h∆Gi ∆Gj i=(1−α)−2h∆i∆ji. (22) by b. We then assume Fll′ =Fbb′/(δl(b)δl(b′)).
We have accordingly corrected all our estimates and We apply equation (17) with the sum restricted to the
B
their error bars upwards by (1 α)−2 C1.29. six at lowest l. First we keep σc fixed to the preferred
By selection function we me−an dG¯/≈dz where G¯ is the valuCeBof 0.56 (VL99) resulting in a8χ2 whose contours are
mean number of EDSGC galaxies per steradian. The shownas the dashed lines in Fig. 5. The minimum of this
smooth curve in Fig. 2 was chosen to fit the histogram, χ2 is 8.1 for 6 2 = 4 degrees of freedom at Ωm = 0.35
and is given by andb=1.3,wh−eren=0.91. Thisisanacceptableχ2;the
dG¯ z 3 probability of a larger χ2 is 9%. Moving towards higher
=4 105exp( (z/0.06)3/2) . (23) Ω decreasestheVL99preferredvalueofσ ,andthusthe
dz × − (cid:16)0.1(cid:17) prmeferredvalue of b increases. Increasing Ω8 also changes
m
Restrictingourselvestob <19.4leavesaround200,000
J the transfer function, requiring a decrease in n in order
galaxies. Althoughthis isonly 15%ofthe totalnumber
∼
HUTERER ET AL. 7
Fig. 3.—MapoftheEDSGCthatweusedinouranalysis(23<ra<3hoursand−42<dec<−23hours,bJ <19.4). Fiveofthelargest
masksareindicatedwithsquares.
to agree with both COBE/DMR and cluster abundances.
This change in the shape of the angular power spectrum
leadstoanincreaseinχ2 . MovingtowardslowerΩ
EDSGC m
generatesabluertilttotheC shapeintwodifferentways.
l
ItleadstohighernforconsistencywithCOBE/DMRand
clusterabundancesanditalsoincreasestheimportanceof
non–linear corrections. These combined effects lead to a
rapidly increasing χ2 for Ω <0.2.
EDSGC m
The uncertaintiesonσc fromcluster abundances(as we
8
interpret them) are significantly larger than the EDSGC
constraintsonbforfixedσc. Ifwetakethemintoaccount,
8
we must include additional prior information in order to
obtain an interesting constraint on the bias. Since (at
fixed Ω ) changing σc changes n, prior constraints on n
m 8
will help to constrain σc. Therefore we work with the
8
total χ2 = χ2 +χ2 +χ2. From a combined anal-
EDSGC VL n
ysis of Boomerang-98, Maxima-I and COBE/DMR data,
Jaffe et al. (2000) find n = 1 0.1; hence we adopt
χ2 =(n 1)2/0.12. We marginali±ze the likelihood, which
isnpropor−tional to e−χ2/2, over σc.
8
Marginalizing over the amplitude constraint from clus-
ter abundances, we find 1.07 < b < 2.33 at the best-fit
value of Ω = 0.35, and 1.11 < b < 2.35 after marginal-
m
izing over Ω (both ranges 95% confidence). These con-
m
straints correspond to the solid and dashed contours re-
spectively in Fig. 5. Figure 6 shows the likelihood of
bias, when marginalized either over σ (solid line) or Ω
8 m
(dashed line). Marginalizing over the bias leads to weak
Fig. 4.— Angular power spectra estimated from the data and
constraints on Ω , unless one insists on allowing only
predicted for various models. From bottom to top at low l arethe m
COBE and cluster consistent predictions for Ωm =0.15,0.3,1 and smalldeparturesfromscale-invariance. With the assump-
b=1(dashedlinesshowlineartheorypredictions). Thelowerpanel tionthattheprimordialpowerspectralindexisn=1 0.1,
showsthewindowfunctions forthefirstsixbands. wefind0.2<Ω <0.55at95%confidence. Furtherm±ore,
m
itisinterestingthatnotonlydo“concordance”–typemod-
els, with scale–independent biases provide the best fits to
the EDSGC data, but they also provide acceptable fits.
6. systematic errors
Inthissectionwediscussthreesourcesofsystematicer-
ror: spatially–varying extinction by interstellar dust, de-
viation of the survey mean from the ensemble mean and
deviationfromGaussianity. Abovewehaveassumedtheir
impact on the data to be negligible. In the following we
use maps with three different pixelizations: BIGPIX (1.5
8
deg 1.5 deg pixels, a total of N = 650 of them), MED-
×
PIX (1.0 deg 1.0 deg, N = 1425) and FINEPIX (0.5
×
deg 0.5 deg, N = 5700). Note that FINEPIX was ul-
×
timately used to obtain the cosmological parameter con-
straints. Coarserpixelizations,however,areeasiertowork
with due to a much smaller number of pixels (in particu-
lar, N N matrices have to be repeatedly invertedin the
×
quadratic estimator).
6.1. Interstellar Dust
The first possible source of systematic error, interstel-
lar dust, we can dispense with quickly due to the work
of Nichol & Collins (1993) and, more recently, Efstathiou
& Moody (2000). The former investigated the effects of
interstellar dust (using HI and IRAS maps as tracers of
the dust) on the observed angular correlation function
of EDSGC galaxies (see Collins et al. 1992) and found
no significant effect on the angular correlations of these
galaxies to b = 19.5. We note that Nichol & Collins
j
(1993)alsoinvestigatedplate–to–platephotometric errors
and concluded they were also unlikely to severely effect
Fig. 5.—Contoursofconstantχ2 intheΩm vs. biasplane. The the angular correlations of EDSGC galaxies. Efstathiou
dashedlinesareforσ8 chosentobeatVianaandLiddlemaximum- & Moody (2000)used the latest dust maps from Schlegel,
likelihood value. The solid line is the result of marginalizing over
σ8, with the VL99 prior and a prior in n of 1±0.1. The contour Finkbeiner & Davis (1998)to make extinction corrections
levelsshowtheminimumaswellas2.3and6.17abovetheminimum, to the APM catalog and found that for galactic latitudes
correspondingto68%and95.4%confidencelevelsifthedistribution of b >20◦, the corrections have no significant impact on
wereGaussian. | |
the angular power spectrum. Since all the EDSGC sur-
vey area resides at galactic latitudes of b > 20◦ and has
| |
been thoroughly checked for extinction–induced correla-
tions, we conclude that spatially–varying dust extinction
has not significantly affected our power–spectrum deter-
minations either.
6.2. Integral Constraint
We are interested in the statistical properties of devi-
ations from the mean surface density of galaxies. This
effort is complicated by our uncertain knowledge of the
mean. Ourbest estimate ofthe ensemble meanis the sur-
vey mean. But assuming that the survey mean is equal
to the ensemble mean leads to artificially suppressed esti-
mates of the fluctuation power on the largestscales of the
survey. Thisassumptionisoftenreferredtoas“neglecting
the integralconstraint”(for discussions,see,e.g., Peacock
& Nicholson (1991); Collins, Nichol & Lumsden (1992)).
Let G¯ be the ensemble average number of galaxies in a
pixel. Let us denote the survey averageas
1
G˜ = G . (24)
i
n
pix Xi
Sincewedonotknowtheensembleaverage,inpracticewe
use the survey average to create the contrast map:
G G˜ 1
∆˜ = i− = (∆ ǫ) (25)
Fig. 6.— Left panel: likelihood of b marginalized over σ8 (with i G˜ 1+ǫ i−
n = 1±0.1 prior) at Ωm = 0.35 (solid line), and additionally
marginalizedoverΩm (dashedline). Rightpanel: likelihoodofΩm where
with no priors (dotted line), n prior (dashed line), n and our VL Gi G¯
priors(solidline). ∆i ≡ G−¯ (26)
is the contrast map made with the ensemble averageand
G˜ G¯
ǫ − (27)
≡ G˜
HUTERER ET AL. 9
is the fractional difference between the two averages (for
simplicity of notationwe are assuming equal–areapixels).
Our likelihood function should not have the covariance
matrix for ∆ , but instead for ∆˜ . These are related by
i i
∆˜ ∆˜ = ∆ ∆ ǫ(∆ +∆ ) + ǫ2 (28)
i j i j i j
h i h i−h i h i
plus higher order terms2. The extra terms of the above
equation are easily calculated with the following expres- )
1
sions: −
s
e
e
r
g
1 e
ǫ∆ = ∆ ∆ d
h ii N h i ji (
pix Xj 0)
(
1 S
ǫ2 = ∆ ∆ (29) /
h i N2 h i ji Ω
pix Xij /
)
θ
Each correction term typically contributes 10-20% to S(
the corresponding terms of the covariance matrix (they πθ
do not cancel, since there are two linear correction terms; 2
see equation (28)). The main contribution comes from
the lowest multipoles, corresponding to largest angles θ.
θ (degrees)
Indeed, the correction terms come almost entirely from
our lowest multipole bin. Dropping this bin (or using a Fig. 7.—Theareaunderthecurveisapproximatelyequaltothe
ΛCDM C ) reduces the correction terms to 2% or less. integralconstraintcorrectiontermsofequation(28)inunitsofS(0);
l
seeequation(30). TheassumedmodelisΩM =0.3withVL99and
The amplitude of the correction terms can be under- COBE/DMR normalization. (The oscillations are due to the fact
stoodfromtheweaknessofthesignalcorrelationsonscales thatonlythecontributionsfrommultipolemomentsatl<180were
approaching the smaller survey dimension of 19◦. In that included.)
case, we can write:
1
∆ ∆ 2π S(θ)θdθ/Ω. (30)
i j
N h i≈ Z
pix Xj
where Ω is the area of the survey, and S(θ) is the signal
covariance, given by the RHS of equation (4) (we have
neglected pixel noise). We plot the integrand in Fig. 7 in
units of S(0).
Fortunately, even though the correction terms are not
entirely negligible, their inclusion makes the estimated
l
C
change very little. This is shown in Fig. 8. The most
0.1
significantchangeisa 20%broadeningoftheerrorbarof
with correction terms
∼
the lowestmultipole. Including this effect has a negligible without
consequence on our cosmologicalparameter constraints. 0.08
6.3. Gaussianity π)
2 0.06
On large enough scales, we expect the maps to be C/(l
Gaussian–distributed. Figure 9 shows histograms of the )
1
data for the three pixelizations we examined. The his- + 0.04
tograms are overplotted with the Gaussians with zero l(l
mean and variance equal to the pixel variance. One can
0.02
seetheimprovedconsistencywithGaussianityasthepixel
size increases. MEDPIX
We applied a Kolmogorov–Smirnov test (e.g., Press et 0
0 50 100 150
al. 1992)to check for consistency of the above histograms l
with their corresponding zero–mean Gaussians. We find center
probabilitiesthat theseGaussiansarethe parentdistribu-
tionsof<10−10,0.001,and4.5%forFINEPIX,MEDPIX strFaiign.t8c.o—rreTcthioenC.l SdhetoewrmniinsedthweiMthEaDnPdIwXitchaosuet, tahnedinatbesgcrisaslaceono-f
andBIGPIX respectively,indicating thatGaussianityis a pointswereslightlyoffsetforeasierviewing.
better approximation on large scales than it is on small
scales, as expected. We also determined the skewness of
the maps in units of the variance to the 1.5 power, and
2
An exact expression to all orders is given by equation (20) of
Gaztan˜aga &Hui(1999).
10
Note though that Bayesian approaches may still be vi-
able,ifitcanbeshownthatnon–Gaussiananalyticexpres-
sionsfor the likelihoodprovide anadequate descriptionof
the statistical properties of the data. See Magueijo, Hob-
son & Lasenby (2000) and Contaldi et al. (2000).
To get our constraints on cosmological parameters we
fixed the Hubble constant at 70 km/sec/Mpc, or h=0.7.
∆
/d We now explain how our bias results and Ωm results scale
N for different values of the Hubble constant.
d
Thetransferfunctiondependsonthesizeofthehorizon
atmatter–radiationequalityλ whichis proportionalto
EQ
1/(Ω h2), or, in convenient distance units of h−1Mpc,
m
1/(Ω h). The latter quantity is the relevant one since
m
all distances come from redshifts and the application of
Hubble’s law (in this case the redshifts taken for our se-
lection function) with the result that distances are only
known in units of h−1Mpc. Thus there is a degeneracy
∆ betweenmodels with the same value ofΩ h anddifferent
m
Fig. 9.— Histograms of the data, overplotted with Gaussians values of h.
centered at zero with variances equal to the pixel variances, for This degeneracyisbrokenbythe Ω dependence ofthe
maps made with three different pixel sizes. From top to bottom m
COBE-normalizationof δ and the cluster normalization
theyare: BIGPIX,MEDPIX,FINEPIX. H
of σ . Increasing h at fixed values of Ω h means Ω
8 m m
decreases, raising both δ and σ . Ignoring non–linear
H 8
effects, this can be mimicked by an increase in the bias
find the same trend of decreasing non–Gaussianity with and only a very slight reddening of the tilt (since δ has
H
scale: 1.21, 0.85, 0.79. risenonlyslightlymorethanσ andthereisalongbaseline
8
The trend with increasing angular scale and the weak- to exploit).
ness of the 2σ discrepancy for the BIGPIX map are The end result is that our constraints on b are actually
∼
reassuring for our analysis that considered only moments constraintsonb(h/0.7)−0.5,andourconstraintsonΩ (at
m
l < 80. Note that a spherical harmonic with l = 80 has leastwhenmarginalizedoverbias)areactuallyconstraints
3 BIGPIX pixels in a wavelength. However, a normalized on Ω (h/0.7).
m
skewness near unity is worrisome—and this skewness is
not decreasing rapidly with increasing angular scale. We 8. conclusions
discusspossiblewaysofdealingwiththisnon–Gaussianity
in the next section. We have presented a general formalism to analyze
galaxy surveys without redshift information. We pixelize
7. discussion thegalaxycountsonthesky,andthen,usingthequadratic
estimatoralgorithm,extracttheangularpowerspectrum–
Wereducedoursensitivitytothenon–Gaussianityofthe
aprocedurealreadyinuseinCMBdataanalysis. Justlike
databyrestrictingourcosmologicalparameteranalysisto
intheCMBcase,oneeffectivelyconvertscomplexinforma-
l < 80. However, the map may still be significantly non–
tioncontainedintheexperiment(inthis case,locationsof
Gaussian even on these large scales. Future analyses of
severalhundredthousandgalaxies)intoahandfulofnum-
more powerful data sets that result in smaller statistical
bers – the angular power spectrum. One can then use the
errors will have to quantify the effects of the Gaussianity
angular power spectrum for all subsequent analyses.
assumption, which we have not done here.
We apply this method to the EDSGC survey. We com-
The non–Gaussianity may force us towards a Monte–
pute the angular power spectrum of EDSGC, and com-
Carlo approach. An analysis procedure similar to the one
bineitwithCOBE/DMRandclusterconstraintstoobtain
utilized here mayhaveto be repeatedmanytimes onsim-
constraints on cosmological parameters. Assuming flat
ulateddata—wherethesimulationsincludethenon–linear
ΛCDM models with constant bias between galaxies and
evolutionthatpresumablyisthesourceoftheGaussianity.
dark matter, we get 1.11<b<2.35 and 0.2<Ω <0.55
The distribution of the recovered parameters can then be m
at 95% confidence.
used to correct biases and characterize uncertainties.
One advantage of our formalism is that it does not re-
Monte–Carloapproachesmaybenecessaryforotherrea-
quire galaxy redshifts, but only their positions in the sky.
sons as well. Recently Szapudi et al. (2000) have tested
Thisshouldmakeitusefulforsurveyswithverylargenum-
a quadratic estimatorfor C with a simpler (sub–optimal)
l ber of galaxies, only a fraction of which will have redshift
weighting scheme that only requires on the order of N2
information. For example, the ongoing SDSS is expected
operations (orN√N operationsusing the new algorithms
to collect about one million galaxies with redshift infor-
of Moore et al. 2001) instead of N3. A drawback is that
mation,butalsoastaggeringonehundredmilliongalaxies
evaluation of analytic expressions for the uncertainties re-
with photometric information only. Using the techniques
quires on the order of N4 operations. Fortunately, the
presented in this paper, one will be able to convert that
estimationof C is rapidenoughto permita Monte–Carlo
l information into the angular power spectrum, which can
determinationofthe uncertaintiesinareasonableamount
then be used for various further analyses.
of time.