THE ASTROPHYSICAL JOURNAL, 546 :2È19, 2001 January 12001. The American Astronomical Society. All rights reserved. Printed in U.S.A.(

CORRELATIONS IN THE SPATIAL POWER SPECTRA INFERRED FROM ANGULAR CLUSTERING:METHODS AND APPLICATION TO THE AUTOMATED PLATE MEASURING SURVEY

DANIEL J. EISENSTEIN1,2,3 AND MATIAS ZALDARRIAGA1,3Received 1999 December 7 ; accepted 2000 July 31

ABSTRACTWe reconsider the inference of spatial power spectra from angular clustering data and show how to

include correlations in both the angular correlation function and the spatial power spectrum. Inclusionof the full covariance matrices loosens the constraints on large-scale structure inferred from the Automa-ted Plate Measuring (APM) survey by over a factor of 2. We present a new inversion technique based onsingular-value decomposition that allows one to propagate the covariance matrix on the angular corre-lation function through to that of the spatial power spectrum and to reconstruct smooth power spectrawithout underestimating the errors. Within a parameter space of the cold dark matter (CDM) shape !and the amplitude we Ðnd that the angular correlations in the APM survey constrain ! to be 0.19Èp8,0.37 at 68% conÐdence when Ðtted to scales larger than k \ 0.2 h Mpc~1. A downturn in power atk \ 0.04 h Mpc~1 is signiÐcant at only 1 p. These results are optimistic, since we include only Gaussianstatistical errors and neglect any boundary e†ects.Subject headings : cosmology : theory È large-scale structure of universe È methods : statistical

1. INTRODUCTION

Even without distance measurements, the large-scaleclustering of galaxies can be measured through its projec-tion on the celestial sphere. The angular correlation func-tion and power spectrum provide useful statistics toquantify this clustering ; however, in order to compare theresults to theoretical models or between di†erent surveys, itis necessary to account for the projection along the line ofsight (Limber 1953 ; Peebles 1973 ; Groth & Peebles 1977).One approach to this is to deproject the angular statistic tothe full spatial power spectrum by assuming the latter to beisotropic in wavenumber. This inversion, however, requiressome form of smoothing, which in turn complicates thepropagation of errors. In particular, the correlations of dif-ferent scales in the angular correlation function and thespatial power spectrum are never negligible and must behandled correctly.

In this paper, we present improvements to two aspects ofthe deprojection problem. First, we calculate the covariancematrix of the angular correlation function, and the spatialpower spectrum derived from it, under the approximationof wide sky coverage and Gaussian statistics. The formercondition means that we neglect boundary e†ects ; the lattercondition means that we neglect contributions from thethree- and four-point functions. These are reasonableapproximations for the large-angle clustering signal inwide-Ðeld sky surveys such as the Automated Plate Mea-suring (APM) galaxy survey (Maddox et al. 1990), thePalomar Digital Sky Survey (DPOSS; Djorgovski et al.1998), and the Sloan Digital Sky Survey (SDSS).4 Weinclude both sample variance and shot noise contributions,although the latter is negligible on large angular scales inthese surveys. While we cannot estimate the e†ects of sys-

1 Institute for Advanced Study, Olden Lane, Princeton, NJ 08540.2 Enrico Fermi Institute, 5640 South Ellis Avenue, Chicago, IL 60637.3 Hubble Fellow.4 The Sloan Digital Sky Survey is available at : http ://www.sdss.org/.

tematic errors, the statistical covariances should provide alower limit on the uncertainties.

Second, we present a new, simple inversion technique,based on singular-value decomposition (SVD). We use SVDto identify those excursions in the power spectrum thatwould have minimal e†ects on the angular clusteringobservables. We then restrict these directions from havingunphysical and numerically intractable e†ects on the inver-sion. The errors on the observed angular correlations can beeasily propagated to the power spectrum, including thenontrivial correlations between di†erent bins. The best-Ðtpower spectra and covariance matrix converge as thebinning in angle and wavenumber is reÐned.

We then apply both of these improvements to theproblem of inferring the spatial power spectrum from theangular clustering of the APM galaxy survey (Maddox,Efstathiou, & Sutherland 1996). Assuming only Gaussianstatistical errors, we reconstruct the binned band powersand their covariance matrix. We Ðnd large anticorrelatederrors. To give a sense of what these results imply for themeasurement of the power spectrum on large scales, weÐt scale-invariant CDM models to the results at k \ 0.2 hMpc~1. We focus exclusively on large scales because this iswhere the details of the shape of the power spectrum drawunambiguous distinctions between cosmological models.On small scales, scale-dependent bias and nonlinear evolu-tion may obscure the di†erences between cosmologies.

The constraints on spatial clustering from the angularclustering of the APM galaxies have been studied pre-viously by Baugh & Efstathiou (1993, hereafter BE93 ;1994), & Baugh (1998), and Dodelson &Gaztan8 aga

(1999, hereafter DG99). While we agree withGaztan8 agathese analyses as to the best-Ðt spatial power spectrum, wedisagree substantially as to the size of the uncertainties. Ourlarge-scale constraints are over a factor of 2 looser than theolder values. We Ðnd that the underestimation of the errorbars in the previous calculations can be traced to the neglectof the o†-diagonal terms of the covariance matrices of eitherthe angular correlation function or the spatial power spec-

2

APM SPATIAL POWER SPECTRA 3

trum. The recovery of the actual uncertainty after includinga smoothing operation in the power spectrum estimatoralso plays a nontrivial role. We will discuss these issues indetail. In the end, however, we can show that the discrep-ancies are not due to di†erences in the inversion technique,since the previous constraints substantially outperform asimple mode-counting limit on the errors of the APMangular power spectrum.

The structure of this paper is as follows. In ° 2, we presentthe deÐnitions for clustering statistics and the relationsbetween them. In ° 3, we show how to calculate the covari-ance matrix for the angular correlation function. Section 4describes how to construct the spatial power spectrumusing SVD. We then apply these methods to the APMangular clustering in ° 5, recovering the correlated band-powers in ° 5.1 and Ðtting them to CDM models in ° 5.2. In° 5.3, we consider the e†ects of non-Gaussianity and esti-mate that they are likely to be small. In ° 5.4, we demon-strate that the constraints obtained are close to the bestpossible errors available to an angular clustering surveywith the selection function and sky coverage of APM. Wecompare our work to previous analyses in ° 5.5. We con-clude in ° 6.

2. DEFINITIONS AND RELATIONS

Following the usual notation, we take the angular posi-tions of the galaxies to deÐne a continuous fractional over-density Ðeld d(x), where x is a position on the sky.We take aÑat-sky approximation and deÐne the Fourier modes of thisdensity Ðeld as for all angular wave-d

K\ / d2xd(x)e~iK Õ x

vectors K. If the random process underlying the density Ðeldis translationally invariant, then ensemble averages of theproduct of two of these Fourier modes is given by the powerspectrum,

SdK

dK{* T \ (2n)2dD(2)(K [ K@)P2(K) , (1)

where is the two-dimensional Dirac delta function. ThedD(2)power spectrum is the sum of the true power spectrumP2and a shot-noise term equal to the inverse of the numberdensity of sources on the sky. We will assume that isP2isotropic. The angular correlation function is deÐned as

w(h)4 Sd(x)d(x ] h)Tx\P d2K

(2n)2 eiK Õ hP2(K)

\P K dK

2nJ0(Kh)P2(K) , (2)

where is the Bessel function.J0(x)Relating these angular correlations to their parent three-

dimensional correlations requires one to include the survey-dependent projection along the line of sight. We adopt theLimber approximation to project the spatial clustering(Limber 1953 ; Groth & Peebles 1977 ; Phillips et al. 1978).This is valid for modes with wavelengths smaller than thesurvey depth and any radial scale over which the popu-lation of galaxies substantially evolves.

The projection is characterized by the redshift distribu-tion dN/dz of the galaxies in the survey. The total number ofgalaxies per unit solid angle is denoted N. The cosmologyand the evolution of clustering a†ect the projection,although for analysis of APM, the di†erences can be scaledout easily. Since we are interested in large scales, we assume

that the power spectrum can be separated into a function ofredshift z and a function of comoving spatial wavenumber k,

P(k, t) \ P(k)(1] z)a

, (3)

where P(k) denotes the present-day spatial power spectrum(BE93). The function of time is a convolution of the growthof perturbations in the mass, the time evolution of bias, andthe e†ects of luminosity-dependent bias between nearby,faint galaxies and distant, bright ones. Following the nota-tion of BE93 and BE94, the angular power spectrum is

P2(K) \ 1KP

dk P(k) fAK

kB

, (4)

where the kernel is

f (ra) \A 1N

dNdz

dzdr

a

B2 F(ra)

(1] z)a. (5)

Here, is the comoving angular diameter distancera\K/k

(or proper-motion distance) to a redshift z. One has thesimple relation

cH0

dzdr

a\ E(z) \ [)

m(1] z)3 ])

K(1] z)2] )"]1@2 ,

(6)

where is the density in nonrelativistic matter, is the)m

)"cosmological constant, and Curvature)K

\ 1 [ )m

[ )".also enters through the volume correction

F(ra) \ J1 ] (H0 r

a/c)2)

K. (7)

For the APM redshift distribution and a CDM power spec-trum, we Ðnd excellent agreement between the approx-imation of equation (4) and a full-sky numerical integration :5% at K \ 10, and improving rapidly at larger wavenum-bers.

Combining equations (2) and (4), we can write theangular correlation function as

w(h) \P0

=kP(k)g(kh)dk , (8)

g(kh) \ 12nP

draJ0(khr

a)

F(ra)

(1] z)aA 1N

dNdz

dzdr

a

B2. (9)

3. COVARIANCE

We are interested in the estimation of the angular corre-lation function on large angular scales in wide-Ðeld surveys.In these surveys, the density of galaxies is large enough thatincluding only shot noiseÈthe sparse sampling of thedensity Ðeld by the galaxiesÈwould severely underestimatethe errors. Instead, the errors are dominated by ““ samplevariance,ÏÏ the uncertainty due to the Ðnite number ofpatches of the desired angular scale available within thesurvey. If the angular extent of the survey is large comparedto the correlation scales and compared to the angular pro-jection of any clustering scales, then corrections from theboundaries of the survey will be small. In this limit, thee†ects of sample variance on the angular correlation func-tion can be easily calculated. Bernstein (1994) presents amore detailed treatment of a speciÐc statistical estimator ;

4 EISENSTEIN & ZALDARRIAGA Vol. 546

however, the approximate form below is considerably easierto use.

Imagine that our survey has a selection window W (x) onthe sky, with W \ 1 in covered regions and W \ 0 else-where. Let h be a vector separation on the sky. Then theestimator of w(h) is simply

wü (h)\ 1A(h)

Pd2x W (x)

]P

d2x@ W (x@)d(x)d(x@)dD(2)(x [ x@[ h) (10)

\P d2K

(2n)2d2K1(2n)2 d

KdK1

* eiK1 Õ hh(K [ K1, h) , (11)

where

A(h)\P

d2xW (x)W (x [ h) (12)

and

h(K, h)\ 1A(h)

Pd2x eiK Õ xW (x)W (x [ h) . (13)

Using equations (1) and (2), one Ðnds that Swü (h)T \w(h).The covariance of this set of estimators can be written

Cw(h, h@)4 S[wü (h)[ w(h)][wü (h@)[ w(h)]T (14)

\P d2K

(2n)2d2K1(2n)2 eiK1 Õ hh(K [ K1, h)

]P d2K@

(2n)2d2K1@(2n)2 deiK1@ Õ h@h(K@[ K1@ , h@)

] [SdK

dK1

* dK { dK1@

p T [ SdK

dK1

*TSdK { dK1@

p T] .

(15)

The expectation of four dÏs involves a Gaussian term as wellas the four-point function :

SdK

dK1

* dK { dK1{*

T [ SdK

dK1

*TSdK { dK1{*

T \(2n)2dD(2)(K ] K @)P2(K)(2n)2dD(2)(K1] K1@)P2(K1)

] (2n)2dD(2)(K [ K1@)P2(K)(2n)2dD(2)(K1[ K @)P2(K1)] (2n)2dD(2)(K [ K1 ] K @[ K1)T4(K, K1, K @, K1@) . (16)

The four-point function primarily includes the four-T4point function of the density, but nonzero shot noise alsointroduces terms involving the two- and three-point func-tions (Hamilton 2000). We can simplify the Gaussianportion of toC

w(h, h@)

Cw(h, h@)\

P d2K(2n)2

d2K1(2n)2 P2(K )P2(K1)

] h(K [ K1, h)h*(K [ K1, h@)] (eiK1 Õ h`iK Õ h{ ] eiK1 Õ h~iK1 Õ h{) . (17)

We drop the non-Gaussian terms from equation (14). Ifone assumes Gaussian initial perturbations, then non-Gaussianity is small (albeit nonzero) on large angular andspatial scales, and one can use the Gaussian approximationto fair accuracy. On smaller scales, we expect that non-Gaussianity will contribute considerably more variance dueto the correlations between modes (Meiksin & White 1999 ;Scoccimarro, Zaldarriaga, & Hui 1999). It is important tonote that angular clustering statistics tend to have smallernon-Gaussian terms than a simple mapping of the spatial

nonlinear scale would suggest. This is because one is pro-jecting many nonlinear regions along the line of sight ; thecentral limit theorem then drives the sum of the Ñuctuationstoward Gaussianity. We note that although this is comfort-ing for the calculation of the angular correlations, it is notclear that the inference of spatial clustering from theangular statistics retains this advantage.

For the particular case of APM, calculations with thehierarchical Ansatz (° 5.3) suggest that non-Gaussian termsbecome equal to the Gaussian terms at whichK Z 100,indicates that our smallest scales are not safely in theGaussian regime. Unfortunately, the Ansatz is not reliableenough to give a useful calculation of the four-point termsin equation (14) (Scoccimarro et al. 1999). As we describe in° 5.3, a simple attempt to include non-Gaussianity degradedour results by D10%. We therefore regard non-Gaussianityas a caveat to our results but not a catastrophic error.

We are interested in the case in which the sky coverage ofthe survey is large, compared both to the angle h and to anyangular correlation length. Here, the function h(K, h)becomes sharply peaked around K \ 0. To leading order, itcan be treated as a Dirac delta function. The coefficient is

P d2K(2n)2 h(K, h)h*(K, h@) \ / d2xW 2(x)W (x [ h)W (x [ h@)

A(h)A(h@)

B1

A), (18)

where is simply the area of the survey. E†ects fromA)boundaries or from features in the power spectrum will besuppressed by additional powers of For wide-angleA).surveys, we can approximate the correlations in w(h) as

Cw(h, h@) \ 1

A)

P d2K(2n)2 P22(K)[eiK Õ (h`h{)] eiK Õ (h~h{)] . (19)

Since we are neglecting all boundary e†ects, we can averageh over angle, i.e., w(h) \ (1/2n)/ d/w(h), to yield

Cw(h, h@) 4 S[wü (h) [ w(h)][wü (h@) [ w(h@)]T

\ 1nA)

P0

=dK KP22(K)J0(Kh)J0(Kh@) . (20)

This is the Gaussian contribution to the covariance of theangular correlation function in the limit of a wide-Ðeldsurvey. Our neglect of the boundary terms is equivalent tothe approximation that working on a fraction of the skyfskysimply increases the variance on the angular power spectraby (Scott, Srednicki, & White 1994 ; Bond, Ja†e, &f sky~1Knox 1998).

It is important to note that as the area of a surveyincreases, the covariance in equation (20) does not approacha limit in which errors on di†erent angular scales are sta-tistically independent. This means that analysis of w(h) inthe sample-variance limit must not neglect the correlationof the error bars on w(h). This runs contrary to the proper-ties of in which di†ering scales do become independentP2,in the large-data-set limit of a Gaussian process. We showbelow that the inclusion of these correlations substantiallyweakens the published constraints on the large-scale powerspectrum from the APM survey.

Our estimate of does not include systematic errors,Cwthe e†ects of non-Gaussian statistics, or aliasing from the

survey boundary. It would be very surprising, however, ifthese complications were to reduce the uncertainty on infer-

No. 1, 2001 APM SPATIAL POWER SPECTRA 5

ring P(k). In this sense, we consider equation (20) as a lowerbound on the errors.

4. SVD INVERSION

We wish to estimate P(k) from observations of w(h). Inpractice, we are given estimates of w in bins centered onNhangles (j\ 1, . . . , We denote the estimates ash

jNh).and place them in a vector w. These measurements havew

ja covariance matrixNh ] Nh Cw.

We then wish to estimate P(k) in bins centered atNk

kj( j\ 1, . . . , The values in these bins are denoted andN

k). P

jformed into a vector P. The integral transform of equation(8) can then be cast as a matrix, yielding

w \ GP , (21)

where G is a matrix.Nh ] NkIn detail, one should calculate the elements of G taking

account of the averaging in the bins of k and h. The methodof averaging can be chosen, but if one takes the estimates w

jto be averages of w(h) according to the weight hdh and treatsP(k) as constant within a bin in k, then the integrals over kand h can be done analytically using properties of the Besselfunction. We Ðnd, however, that for reasonably narrow binsthe approximate treatment of using only the central valuesof h and k produces nearly the same answer as the exactintegration.

With this notation, the best-Ðt power spectrum is simply5and the covariance matrix on this inversion isPŒ \ G~1w,

It is important to note that this covari-CP~1\ GTC

w~1G.

ance matrix is not diagonal, even in the large survey volumelimit. This di†ers from the behavior of estimates of P3(k)from a redshift survey, where individual bins approachindependence in the large-volume limit.

In practice, the matrix G is nearly singular, as one wouldguess from its origin as a projection from three dimensionsto two. By singular, we mean that the matrix has a nonzeronull space, i.e., that there are vectors P that are annihilatedby G. Such null directions cannot be constrained from theangular data. Often, the near singularities come about fromhaving too Ðne a binning in k or from extending the domainin k to values that have negligible impact on the range ofangular scales being measured. Left untreated, these direc-tions introduce wild excursions in P(k) in order to compen-sate for tiny variations in w(h). The resulting covariancematrix has enormous, but highly anticorrelated, errors.C

PSingular-value decomposition o†ers a useful way to treatthis singularity. Motivated by the Gaussian distribution, wemeasure the di†erence of the measured w from their truevalues by the statisticw

ms2\ (w

m[ w)TC

w~1(w

m[ w) . (22)

is related to the true power spectrum by Wewm

wm

\GPm.

will rescale the basis set for by dividing by a set ofPmreference values intended to be deÐned by a ÐducialPnorm,

power spectrum evaluated at the appropriate wave-Pnorm(k)numbers. This produces P@ by dividing each element of P bythe corresponding element of We also deÐne w@\Pnorm.

and here, is constructedCw~1@2 w G@\ C

w~1@2 GPnorm ; C

w~1@2

by taking the inverse square root of the eigenvalues of thepositive-deÐnite matrix. We then haveC

ws2\ oG@P

m@ [ w@ o2 . (23)

5 If G were square ; if not, the formula is exactly asPΠ\CPGTC

w~1 w,

one would get from SVD.

Finding the vector (or subspace) that minimizes s2, andPm@

thereby maximizes the likelihood in a Gaussian treatment,is a prime application of SVD, and the technique allows oneto treat the nearly singular directions in P-space explicitly.Note that we can immediately see that the covariancematrix of the will simply be (G@TG@)~1. We now drop theP

m@

m subscript and refer to the reconstructed power spectrumas P@.

We deÐne the SVD of the G@ matrix by G@\ UW V T (fora review, see Press et al. 1992, °° 2.6 and 15.6), where W is asquare, diagonal matrix of the singular values (SV), V is a

orthogonal matrix, and U is a column-Nk] N

kNh ] N

korthonormal matrix. Singular values close to zero corre-spond to columns in V that contain P@ directions that havealmost no e†ect on w@ and therefore are not well-constrained. To Ðnd the best-Ðt power spectrum, we useP@\ V W ~1UTw@. The covariance matrix, of P@ is simplyC

P,

V W ~2V T, which is the diagonalization of the covariancematrix.

Mathematically, the fact that the W matrix is diagonalmeans that the data w@ are coupled to the power spectrumestimate P@ through distinct modes, in which the match-N

king columns of the U and V matrix specify a matching set ofw@ and P@ excursions. We denote the jth SV as and the jthW

jcolumn of U and V as and respectively. We referUj

Vj,

to the set and as the jth SV mode. In detail,Wj, U

j, V

jeach mode enters the best-Ðt power spectrum as Pi\

Comparing this to the formula forPnorm,iVjiW

j~1 U

jT w@. C

Pshows that is the number of standard deviations byUjT w@

which the jth mode is demanded by w@. Indeed, s2 can berewritten as

s2\ o w@ o2[ ;j

oUjT w@ o2 , (24)

so the value of is the amount by which the inclusion(UjT w@)2

of a mode in P@ will decrease s2. Since the columns of V areunit-normalized, the quantity is a measure ofW

j~1U

jT w@

the size of the contribution that this mode makes to thepower spectrum in units of Pnorm.

Of course, such an analysis is only useful if it converges asthe binning of k and h becomes Ðner. In our APM example(° 5), we Ðnd that this is the case : the columns in U and Vcorresponding to large singular values change very little aswe alter the binning in k or h. The large scale asW

jN

k~1@2,

because if one simply reÐnes the binning in k, the elementsof G scale as due to the smaller range dk in the deÐningN

k~1

integral, while the elements of scale as because it isVj

Nk~1@2

an unit-normalized vector. The primary e†ect of adding orremoving bins is to change the number of tiny singularvalues. The modes with large SV show broad tilts andcurves in P(k) ; the modes with small SV show rapid com-pensating oscillations as well as excursions at very large orsmall k that the w(h) data do not constrain. The ability toidentify the excursions in P(k) space that are well con-strained, in a manner that converges as the binning becomesÐner, is the strength of the SVD method. It should be notedthat the kernel and its SVD decomposition depend on thesurvey geometry, the covariance matrix, and the ÐducialC

wscaling but not on the observed data w itself.Pnorm,Left untreated, the small singular values will have large

inverses and therefore produce large excursions in P@. Suchexcursions are unphysical and can even make numeri-C

Pcally intractable. In the usual spirit of SVD, we wish toadjust the treatment of these singular values. This is compli-

6 EISENSTEIN & ZALDARRIAGA Vol. 546

cated by the fact that SVD relies on the concepts of orthog-onality and normalization and thereby implies a geometricstructure that our P- and w-spaces do not actually have. Tosort the singular values and declare some of them to be““ small ÏÏ requires that we have some sense of comparing wat di†erent values of h or P at di†erent values of k. On thew@-space side, this choice is easy : the absorption of intoC

wG@ and w@ means that the unit-normalization of Ñuctuationsin w@ have the correct role in the s2 statistic. However, forP@-space, the choice is more arbitrary. The Ðducial powerspectrum determines how Ñuctuations in power onPnormdi†erent scales but of equal statistical signiÐcance are to beweighted in the singular values. By choosing to bePnormclose to the observed spectrum, we are opting that equalfractional excursions on di†erent scales receive equalweight. Had we instead chosen to be a constant, aPnorm100% oscillation at the peak of the power spectrum(PB 104 h~3 Mpc3 at k B 0.05 h Mpc~1) would have beensuppressed relative to the same fractional Ñuctuation atsmaller scales (say, PB 102 h~3 Mpc3 at k B 1 h Mpc~1). Itis important to remember that is irrelevant if one isPnormusing all the singular values unmodiÐed. It enters only whenwe place a threshold on the singular values (as describedbelow). When small singular values are altered or elimi-nated, determines how di†erent scales are to be com-Pnormpared in the application of a smoothness condition. Whilethe arbitrary choice of means that an SVD treatmentPnormof the inversion is not unique, we feel that our is aPnormwell-motivated choice : each singular value represents thesquare root of the s2 contribution for a given fractionalexcursion around the best-Ðt power spectrum.

We next describe how we alter the small In detail, weWj.

incorporate two di†erent SV thresholds, one for the con-struction of and another for the construction of P@. SmallC

PSV indicate poorly constrained directions. We would liketo reÑect this, but not to the extent that the matrixC

Pbecomes numerically intractable. Physically, these small Wjare highly oscillatory, and our prior from both theory and

previous observations is that enormous (much greater thanunity) Ñuctuations do not exist in the power spectrum.Hence, when constructing we increase all SV to aC

P,

minimum level of We cannot recommend a choice ofSVC.

for arbitrary applications, because all the will scaleSVC

Wjwith the normalization of However, in our work,Pnorm.

where has an amplitude similar to the actual powerPnormspectrum, a choice of between 0.1 and 1.0 will allowSVCorder unity Ñuctuations in the power spectrum. This is

larger than any oscillations ever seen, but not so large as tomake overly singular.C

PWithout correcting the small the best-Ðt power spec-Wj,

trum P@ becomes wildly Ñuctuating. If we have modiÐed thesmall in then these Ñuctuations will appear to beW

jC

P,

highly signiÐcant. Hence, at a minimum one must use thesame in P@ as those used in so as to keep the Ñuctua-W

jC

P,

tions and the covariance on the same scale. However, sincethe small-SV modes have already been granted an errorbudget larger than what is likely observable, there is noreason to include them in the best-Ðt P@ at all. The di†erencebetween the best-Ðt power spectrum and any reasonablysmooth model power spectrum will be insigniÐcant withrespect to the covariance Hence, we generally onlyC

P.

include in P@ the modes with the largest SV. Essentially, thisis a threshold on for inclusion in the nominal best Ðt. InW

jpractice, when comparing between spectra calculated with

di†erent (as we occasionally do in next section), it isCwbetter to keep a Ðxed number of SV modes than a Ðxed W

jthreshold, because the will change even while the SVWjspectrum and the structure of the U and V matrices remain

fairly constant.The modiÐcation of W ~1 when calculating P@ causes P@

to be a biased estimator of the power spectrum. This bias isstatistically signiÐcant at a level of (1 [ W

j/W

j,used) oUjT w@ o ,

where is the value of actually used in constructingWj,used W

jP@ (O if the mode has been dropped). One can thereby judgethe statistical signiÐcance of the bias imposed by altering a

and decide whether an excursion of the amplitudeWjimplied by the original is physically reasonable. OfW

jcourse, one would only wish to drop modes that are sta-tistically irrelevant or physically unreasonable. It is impor-tant to remember that the small can be stronglyW

jperturbed by small changes in Since one cannot hope toCw.

control all systematic errors in one does not want toCw,

place any weight on the SV modes with small Wj.

The bias from increasing or omitting modes in P@ pullsWjthe amplitude of the altered modes toward zero. Usually

this pulls the power toward zero, but we can alter thisbehavior by subtracting from the w(h) data andGPbiasadding to the reconstructed power spectrum. In otherPbiaswords, we alter equation (23) to

s2 \ oG@(Pm@ [ Pbias) [ (w@[ G@Pbias) o2 , (25)

and reconstruct The bias then pulls towardPm@ [ Pbias.Pbias.Strictly speaking, our use of the s2 statistic is appropriate

only if the likelihood function of w(h) is Gaussian, which it isnot. However, ultimately, one is simply solving w \ GP.The SVD manipulations replace G with a modiÐed matrixthat allows Ñuctuations in P much larger than physicallyexpected, but not so large that they cause numerical prob-lems. This new kernel allows a linear mapping from w-spaceto P-space, including the transference of a non-Gaussianlikelihood of w to one for P. The SVD alterations wouldonly introduce problems if the likelihood surface is so com-plicated that the covariance matrix gives a very mislead-C

wing representation. This pitfall does not occur in similarcosmic microwave background (CMB) problems (Bond etal. 1998). For now, we assume that the P likelihood isGaussian in our APM example. We motivate this by notingthat on our featured scales of k B 0.2 h Mpc~1, APM ismeasuring many hundreds of angular modes, which we arecondensing down to a few bandpowers. One would there-fore expect that the well-measured quantities have fairlyGaussian distributions. We intend to study these issuesmore carefully in a future work.

All the above techniques apply equally well to theproblem of inferring the spatial power spectrum from theangular power spectrum, and it is trivial to alter the equa-tions.

5. ANALYSIS OF THE APM SURVEY

5.1. Reconstructing the Power SpectrumOne of the most inÑuential uses of angular clustering in

the last decade has been its application to the APM survey(Maddox et al. 1990). However, a full treatment of thecovariance matrix on power spectra inferred from APMclustering has not yet been presented, and so we choose thisas our example. We take the data on the APM angularcorrelation function (Maddox et al. 1996) as presented in

No. 1, 2001 APM SPATIAL POWER SPECTRA 7

the binned results of DG99. This includes 40 half-degreebins from to 20¡. Tests show that including data from0¡.5smaller angular scales does not a†ect our results on scaleslarger than k \ 0.2 h Mpc~1. We also Ðnd that scales above10¡ are insigniÐcant for the reconstruction.

We discard the quoted errors on the DG99 w(h) data andinstead use the covariance matrix from ° 3. For we useP2,

P2(K)\ 456002 ] 10~4 K \ 202 ] 10~4(K/20)~1.35 K [ 20 ,

(26)

which is a reasonable Ðt to the results of Baugh & Efsta-thiou (1994). We assume a survey area of 1.31 sr. We add ashot-noise term of based on a numberPshot\ 1/n6 \ 10~6,density of 1 galaxy per square pixel (Baugh & Efsta-n6 3@.5thiou 1994). The shot noise has little impact on the results.We use this to calculate according to equation (20).P2 C

wNote that the errors on the power spectrum will scalelinearly with the overall amplitude of P2.To calculate the projection kernel, we use the redshiftdistribution dN/dz of BE93 ; we have not included theuncertainties in this distribution in our error analysis. A10% underestimate of the true redshifts would shift the esti-mated power spectrum to 10% larger wavenumber and30% smaller amplitude. We also assume )

m\ 1, )" \ 0,

and a \ 0. The results do depend on these choices, butnearly all the behavior can be scaled out in two easy parts.First, the fact that the average galaxy in the survey is atzB 0.11 means that specifying a time evolution of thepower spectrum will cause a shift in the amplitude of(a D 0)the z\ 0 power spectrum. In practice, one can consider thereconstructed power spectrum to be appropriate toz\ 0.11 ; in other words, choosing di†erent time depen-dences for the power spectrum leaves the power at z\ 0.11essentially constant. Second, the cosmology enters throughthe volume available at higher redshift. Models with morevolume per unit redshift (lower have more modesdz/dr

a)

and therefore su†er more dilution in angular clusteringwhen projected. This e†ect scales roughly as E(z\ 0.11)~2.Despite the low median redshift, this is not a small e†ect in" models : using an model yields a)

m\ 0.3, )" \ 0.7 P3(k)

20% higher than our Ðducial model. The e†ect is muchsmaller in open models. In detail, or a change in thea D 0cosmology will also shift the average depth of the surveyslightly, causing the reconstructed power spectrum to movein wavenumber. We Ðnd this e†ect to be less than 5%,which is certainly within the errors.

We use a logarithmic binning in k-space. Our Ðducial setuses three bins per octave, ranging from k \ 0.0125 to 0.8 hMpc~1. There is also a large-scale bin of k \ 0.0125 hMpc~1, for a total of 19 bins. Wavenumbers greater than0.8 h Mpc~1 are not constrained by data at andh [ 0¡.5would simply be degenerate with our last bin. We also triedcoarser and Ðner binnings, using two and four bins peroctave to get 13 and 25 bins, respectively. These threechoices are shown in Table 1 under the names k13, k19, andk25. We Ðnd equivalent results with nonlogarithmic binningschemes.

We set to bePnorm

Pnorm(k)\ 1.5] 104 h~3 Mpc3[1 ] (k/0.05 h Mpc~1)2]0.65 . (27)

This is a rough Ðt to the observed power spectrum untilk B 0.05 h Mpc~1 and has constant power on the largest

TABLE 1

k-SPACE BINS

k13 k19 k25

0.000È0.0125 0.000È0.0125 0.000È0.01250.0125È0.018 0.0125È0.016 0.0125È0.0150.018È0.025 0.016È0.020 0.015È0.0180.025È0.035 0.020È0.025 0.018È0.0210.035È0.050 0.025È0.032 0.021È0.0250.050È0.071 0.031È0.040 0.025È0.0300.071È0.100 0.040È0.050 0.030È0.0350.100È0.141 0.050È0.063 0.035È0.0420.141È0.200 0.063È0.079 0.042È0.0500.200È0.283 0.079È0.100 0.050È0.0590.283È0.400 0.100È0.126 0.059È0.0710.400È0.566 0.126È0.159 0.071È0.0840.566È0.800 0.159È0.200 0.084È0.100

0.200È0.252 0.100È0.1190.252È0.317 0.119È0.1410.317È0.400 0.141È0.1680.400È0.504 0.168È0.2000.504È0.635 0.200È0.2380.635È0.800 0.238È0.283

0.283È0.3360.336È0.4000.400È0.4760.476È0.5660.566È0.6730.673È0.800

NOTE.ÈThree di†erent choices of binning. Allunits are h Mpc~1.

scales. Recall that enters only in the treatment ofPnormsmall singular values and serves to set an upper bound onthe allowed size of Ñuctuations in the power spectrum rela-tive to the best Ðt. The constant power on large scales waschosen so as not to prejudice the results on scales where wehave little information. It is important to choose to bePnormcontinuous because the prior against rapid oscillations actson the ratio of the Ðtted power spectrum to Discon-Pnorm.tinuities in would impose discontinuities in the ÐttedPnormpower spectrum.

With and we can construct the kernel G@ andCw

Pnorm,Ðnd its SV decomposition. Table 2 shows the spectrum ofsingular values for each of these three choices of binning.When one corrects by to account for the defaultN

k1@2

scaling in the large singular values are very stable as theWj,

binning is reÐned. Increasing the number of bins onlyincreases the number of very small singular values. Thisdemonstrates that 19 bins is a Ðne enough grid to character-ize the power spectrum; increasing to 25 bins would onlyadd degrees of freedom that are unconstrained by theangular data. We could have put the factor of into theN

k1@2

deÐnition of so as to make the large stable againstPnorm Wjchanges in binning, but since we hereafter work only with

we opted against the extra complication.Nk\ 19,As described in ° 4, the matching columns and mapU

jVjÑuctuations in w@ to those in P@ with an amplitude equal to

the inverse of the singular value W . Figure 1 displays Ðvepairs of columns from the SVD. One sees that the large W

jare associated with small angular scales in w@ and withsmooth, large k excursions in P@. As the decrease, theW

joscillations become wilder and move to larger angularscales. The and vectors for large remain veryU

jVj

Wjsimilar as one changes from k19 to k25 binning.

8 EISENSTEIN & ZALDARRIAGA Vol. 546

TABLE 2

SINGULAR VALUES, SCALED TO 19 BINS

Wk13 ] (13/19)1@2 W

k19 Wk25 ] (25/19)1@2

17.1 17.1 17.19.2 9.3 9.35.2 5.3 5.33.1 3.2 3.22.0 2.0 2.11.4 1.5 1.51.0 1.1 1.20.80 0.88 0.930.53 0.59 0.620.30 0.37 0.390.12 0.22 0.230.014 0.12 0.132.9] 10~4 0.066 0.073

0.038 0.0400.018 0.0203.8] 10~3 9.6] 10~31.9] 10~4 4.1] 10~33.3] 10~6 1.7] 10~33.7] 10~8 6.6] 10~4

3.0] 10~42.0] 10~51.3] 10~61.5] 10~72.7] 10~84.2] 10~9

NOTE.ÈWe scale the by to remove theWj

(Nk/19)1@2

predicted scaling that occurs when one reÐnes the binningin wavenumber. The choice of wavenumber bins is givenin Table 1.

In Table 3, we look at the overlap of these SV modes withthe APM w@. The quantity is the number of standardU

jT w@

deviations by which the jth mode is demanded by w@. Divid-ing that by yields the amplitude of the e†ect on theW

jpower spectrum (in units of Modes withPnorm). oUjT w@ o[ 1

are not statistically signiÐcant, while modes with

TABLE 3

OVERLAP OF SINGULAR VALUES WITH APM DATA

Wj

UjT w@ W

j~1U

jT w@ W

j,eff~1 UjT w@

17.1 . . . . . . . . . . . . . 21.2 1.2 1.29.3 . . . . . . . . . . . . . . . 7.8 0.84 0.845.3 . . . . . . . . . . . . . . . 9.0 1.7 1.73.2 . . . . . . . . . . . . . . . 4.4 1.4 1.42.0 . . . . . . . . . . . . . . . [4.1 [2.0 [2.01.5 . . . . . . . . . . . . . . . 2.3 1.6 1.61.1 . . . . . . . . . . . . . . . [1.1 [1.0 [1.00.88 . . . . . . . . . . . . . [0.56 [0.64 [0.640.59 . . . . . . . . . . . . . [1.5 [2.6 [2.60.37 . . . . . . . . . . . . . [0.79 [2.1 [1.60.22 . . . . . . . . . . . . . [0.96 [4.4 [1.90.12 . . . . . . . . . . . . . [1.5 [12.4 [3.10.066 . . . . . . . . . . . . [0.30 [4.5 [0.600.038 . . . . . . . . . . . . [0.56 [14.9 [1.10.018 . . . . . . . . . . . . 0.082 4.5 0.163.8] 10~3 . . . . . . 1.2 3.3] 102 2.51.9] 10~4 . . . . . . [0.31 [1.6] 103 [0.613.3] 10~6 . . . . . . 0.36 1.1] 105 0.723.7] 10~8 . . . . . . [1.6 [4.2] 107 [3.1

NOTE.ÈThe singular values are listed as shows theWj. U

jT w@

dot product between the jth column of the U matrix and the datavector w@. is the value of the jth SV rounded up to aW

j,effminimum value of SVC\ 0.5.

put enormous Ñuctuations in the poweroWj~1U

jT w@ o? 1

spectrum that are probably unphysical. One sees that theÐrst six modes are clearly demanded, while the remainderare of marginal signiÐcance. We include eight modes in ourquoted results, because this seems to be the transitionbetween a smooth and an oscillatory reconstruction.However, we also perform Ðts to CDM models with allmodes included in P@. In this regard, we also quote

in Table 3, where is rounded up toWj,eff~1 U

jT w@ W

j,eff WjThis is to remind the reader that the value ofSV

C\ 0.5. W

jused in is the one used in P@.CPIn Figure 2, we show how the reconstructed power spec-

trum develops as we add more SV modes. The largest Wjcontribute mostly small-scale power. With six or eight

modes, a fairly smooth shape appears that matches theexpected form of P(k). Adding smaller modes quickly makesthe spectrum more oscillatory, even with the artiÐcialincrease in the As Table 3 shows, these oscillations areW

j.

not statistically signiÐcant.Figure 2 also shows the reconstructed power spectrum if

one chooses Recall that was chosen toPbias \ Pnorm. Pnormhave a large amount of power on large scales. This meansthat any bias in P@ due to the alteration of SV will pull thespectrum toward high P rather than P\ 0. This allows usto determine how many SVs must be included to avoid biasin the large-scale power spectrum. One sees that with eightmodes, the two power spectra are indistinguishable atk [ 0.02 h Mpc~1. This demonstrates that the smoothportion of the power spectrum is being reconstructed in anunbiased way by the SVD method. In other words, the Ðrsteight modes are all that is needed to describe a smoothpower spectrum like Note that this does not meanPnorm.that features in the resulting power spectrum are sta-tistically signiÐcant ; that depends on the covariance matrixC

P.We present the best-Ðt P(k) using the largest eight SVs in

Table 4. The reduced forms of and usingCP

CP~1, SV

C\ 0.5,

are shown in Table 5, with the diagonal elements in Table 4.The reduced form of shows the correlation coefficientsC

Pbetween the k bins. Neighboring bins are anticorrelated,with correlation coefficients ranging from [0.95 on smallscales to [0.5 on moderate scales to [0.1 on the largestscales. While could be used to calculate the change in s2C

Pfor particular excursions around P, two signiÐcant Ðgures isnot enough to do so correctly. Instead, one should use thequoted For smooth excursions around the best-ÐtC

P~1.

P(k), great accuracy in is not required. However,CP~1

because the oscillatory excursions are more singular,readers should contact the authors for more signiÐcantÐgures if they wish to manipulate these kinds of Ñuctua-tions.

The w(h) using this best-Ðt power spectrum di†ers fromthe input w(h) by s2\ 46. There are 40 bins in h and 19 binsin k, so the naive number of degrees of freedom is 21.However, we have included only eight SV modes in con-structing P(k), so in this sense there are 32 degrees offreedom. If we include all 19 modes, s2 drops to 41. Thisdecrease is small because most of these modes have hadtheir adjusted before use in P@. This causes the modes toW

jbe smaller in amplitude than would suggest. As weCwreduce s2 slowly drops as additional modes reach theirSV

C,

““ natural ÏÏ scale.With 32 degrees of freedom, a s2 of 46 is 5% likely, and

hence the Ðt is only marginal. This may indicate that our

No. 1, 2001 APM SPATIAL POWER SPECTRA 9

FIG. 1.ÈSelected pairs of columns from the U and V matrices. The U column (left) indicates the overlap with the data, while the V columnw@4Cw~1@2 w

(right) indicates the impact on P (as normalized by The singular value of each pair is also given. The noise in the last U vector is the result of the linearPnorm).binning in h.

errors are underestimated. However, we Ðnd that changingthe power on small scales makes a large di†erence to s2, butalmost no di†erence to the reconstruction on large scales orthe Ðts to CDM models. For example, if we calculate C

wusing the two-dimensional projection of a !\ 0.25 CDMmodel with and the nonlinear corrections ofp8\ 0.89Peacock & Dodds (1996), s2 drops to 22. Removing thenonlinear corrections increases s2 to 131. These two modelsbracket the observed on small scales. Neither change toP2a†ects large-scale model Ðts at all. We therefore con-C

wclude that it is our small-scale errors, not our large-scaleones, that are slightly underestimated.

If we spline the P(k) quoted in & Baugh (1998)Gaztan8 agaonto our wavenumber binning and compare this powerspectrum to ours using our covariance matrix, we Ðnd thatat k \ 0.2 h Mpc~1, the two power spectra di†er by a s2 of0.858. Clearly, the two reconstructions agree to well withinthe errors, as they should, since they are derived from thesame survey data, whereas the errors reÑect the fact that theactual data are only one realization of the possible APM-like surveys. The small discrepancies are due to di†erencesin the use of smoothing and to the di†erent weighting of thedata by our use of the covariance matrix in the inversion.

5.2. Constraints on P(k) at L arge Scales

Having reconstructed the power spectrum and its covari-ance matrix, we wish to consider how the results constrainthe large-scale power spectrum. Large scales are importantbecause the spectral signatures that would identify particu-lar cosmologies are strongest there. Moreover, nonlinearevolution erases any residual features on small scales(Peebles 1980, °° 27È28), at which point the potentialproblem of scale-dependent bias might further obscure thelink to cosmology (e.g., Mann, Peacock, & Heavens 1998).While the small-scale power spectrum is certainly impor-tant, it is the large-scale power that can be most cleanlylinked to cosmological parameters.

We begin by discussing two of the important phenom-enological results that have been associated with the APMpower spectrum. First, does the power reach a maximum atk B 0.04 h Mpc~1 and drop at the larger scales (Gaztan8 aga& Baugh 1998)? One can see from the comparison of thetwo curves in Figure 2 that any downturn in the powerspectrum at k \ 0.04 h Mpc~1 is only contributed by modes7 and higher. With the Ðrst six modes, the situation atk \ 0.04 h Mpc~1 is completely prior-dominated. Unfor-

10 EISENSTEIN & ZALDARRIAGA Vol. 546

FIG. 2.ÈEvolution of P(k) as we include smaller singular values. Solid line : Results with Dashed line : Results with The fact thesePbias \ 0. Pbias\ Pnorm .two are identical with eight SVs shows that the resulting power spectrum is not being biased by the SVD reconstruction.

tunately, modes 7 and 8 have and [0.56UjT w@\ [1.1

(Table 3), respectively, so they improve the s2 of the Ðt tothe w(h) data by only 1.52. Modes 9 and higher produceoscillations in P that are inconsistent with small-scale data.Hence, we conclude that this suggestion of a downturn inP(k) at k \ 0.04 h Mpc~1 is not statistically signiÐcant.

Another way to quote this signiÐcance is to look at howwell the covariance matrix constrains a constant power Ñuc-tuation in the Ðrst six k bins (k \ 0.04 h Mpc~1). Contract-ing this submatrix of with the vector of six 1Ïs givesC

P~1

3.4] 10~8, which means that the 1 p limit on such anexcursion is 4500 h~3 Mpc3. Using a submatrix of C

P~1

corresponds to assuming perfect information about smallerscales ; in other words, this Ñuctuation leaves the best-Ðtpower at k [ 0.04 h Mpc~1 unchanged. Allowing the

smaller scales to vary within their errors increases p to 5200h~3 Mpc3. Using a !\ 0.25, CDM model forp8\ 0.9 P2only increases these errors. Comparing to the best-Ðt P, thehypothesis that P(k) \ 15,000 h~3 Mpc3 on all scales belowk \ 0.04 h Mpc~1 can only be rejected at 1.25 or 1.21 p,using the assumptions of perfect and imperfect small-scaleinformation, respectively. The 2 p upper bound on thepower at k \ 0.04 h Mpc~1 is roughly 2 ] 104 h~3 Mpc3.Again, we conclude that the downturn at k \ 0.04 h Mpc~1is not signiÐcant.

While the data do not require a downturn, they areinconsistent with an unbroken extrapolation of the small-scale power-law spectrum. The spectrum at k [ 0.05 hMpc~1 is well Ðtted by a power law, but if one uses the samespectrum at larger scales as well, the total s2 is very large,

No. 1, 2001 APM SPATIAL POWER SPECTRA 11

TABLE 4

RECONSTRUCTED APM POWER SPECTRUM

k Range P(k) (CP)jj1@2 (C

P~1)

jj~1@2

0.0000È0.0125 . . . . . . 6088 22557 180910.012È0.016 . . . . . . . . 3802 27469 247060.016È0.020 . . . . . . . . 6127 26254 223380.020È0.025 . . . . . . . . 9354 24806 198040.025È0.032 . . . . . . . . 12891 22724 168360.031È0.040 . . . . . . . . 15175 19909 136070.040È0.050 . . . . . . . . 14625 17426 108740.050È0.063 . . . . . . . . 11458 14855 83240.063È0.079 . . . . . . . . 7724 11813 56430.079È0.100 . . . . . . . . 5544 9155 34740.100È0.126 . . . . . . . . 5077 6948 20610.126È0.159 . . . . . . . . 4331 5151 12100.159È0.200 . . . . . . . . 2394 3827 7100.200È0.252 . . . . . . . . 978 2731 4170.252È0.317 . . . . . . . . 936 2053 2470.317È0.400 . . . . . . . . 776 1438 1470.400È0.504 . . . . . . . . 206 1055 90.60.504È0.635 . . . . . . . . 252 855 61.00.635È0.800 . . . . . . . . 401 422 55.2

NOTE.ÈThis is the best-Ðt power spectrum using eightSV to construct P@ and in The Ðt to theSV

C\ 0.5 C

w.

observed (eq. [26]) was used to calculate The unitsP2 Cw.

for wavenumber are h Mpc~1 and for power areh~3 Mpc3. Also shown are the diagonal elements of C

Pand converted to give a standard deviation on P(k).CP~1,

These are not very useful without the correlations inTable 5. However, they do give the uncertainty on a singlek bin when marginalizing and not marginalizing, respec-tively, over all others. In detail, this is the power atz\ 0.11 in an cosmology. The power spectrum)

m\ 1

(and errors) would increase by about 20% in a "\ 0.7cosmology due to extra dilution of the angular clusteringcaused by the additional volume at higher z. The correc-tions in an open cosmology are D3%. Errors in the red-shift distribution of APM galaxies have not been includedhere : a 10% shift in the mean redshift would shift P(k) by10% in scale and 30% in amplitude.

ruling out the model at high conÐdence. This remains true ifone uses the power-law spectrum to generate and henceC

wthe errors on P(k).The shape of the APM power spectrum at k B 0.1 h

Mpc~1 scales has been noted for a sharp inÑection thatdoes not Ðt simple CDM models & Baugh(Gaztan8 aga1998 ; Gawiser & Silk 1998). Using the covariance matrix inTable 5, we Ðnd that the BE93 power spectrum and a!\ 0.25, CDM model di†er by only 1.2 p atp8\ 0.89k \ 0.2 h Mpc~1 even if smaller scales are held Ðxed. Alter-natively, Table 4 shows that large (D50%) Ñuctuations inthe power at a single bin at k B 0.1 h Mpc~1 are permitted.We therefore Ðnd that the shape of the BE93 power spec-trum at these scales is not statistically di†erent from that ofthe CDM model.

Unfortunately, the large anticorrelated errors in thespatial power spectrum makes it difficult to visualize theconstraints at large scales. We therefore Ðt a set of theoreti-cal power spectra to the power spectrum and study theresulting constraints on the parameter space. For this weconsider a very restricted set, namely, a scale-invariantCDM model speciÐed by ! and an amplitude Wep8.include nonlinear evolution according to the formulae ofPeacock & Dodds (1996), but one should note that thismeans that we have assumed that the galaxies are unbiasedwith respect to the mass. We include only wavenumbers less

than We use h Mpc~1 in most cases. This isk \ kc. k

c\ 0.2

roughly the transition point between the linear and nonlin-ear regimes, which is where the problems of scale-dependentbias could appear and where our Gaussian assumption incomputing the sample variance will begin to be overly opti-mistic. We have marginalized over the smaller scales whencomputing s2 ; however, we get similar constraints on largescales if we hold the small-scale power spectrum equal to apower law PP k~1.3 with unknown amplitude.

In Figure 3, we show the constraints on these CDMmodels. We use and include eight modes in calcu-SV

C\ 0.5

lating P. All modes are used in calculating The contoursCP.

are drawn at *s2\ 2.30 and 5.41, which are the values for a68% and 95% conÐdence region in a Gaussian ellipse.6 Theconstraints are rather loose. The most likely model has!\ 0.26 and If we marginalize over ! has ap8\ 0.92. p8,range of 0.19È0.37 (68%) and 0.15È0.58 (95%). The strongskewness of the constraint region toward higher ! is anartifact of using ! as a parameter. Increasing ! removeslarge-scale power, but since the error bars are not changingwith !, eventually a small change in power maps to a largechange in !. Because of this skew tendency, we are notgenerally concerned about modest changes in the upperlimit on ! range, since they correspond to small changes inthe actual power.

If we use all 19 SV modes in constructing P, the s2 for thebest-Ðt CDM model is 6. This is based on 11 degrees of

6 In detail, the actual integral of the probability would di†er from this,but we neglect this e†ect, because it would not alter the basic point andwould make the results depend on oneÏs choice of metric in parameterspace.

FIG. 3.ÈConstraints on a two-parameter family of CDM powerspectra when Ðtted to the best-Ðt power spectrum of Table 4. Nonlineartheoretical power spectra are used (Peacock & Dodds 1996). P@ and areC

wcalculated using the observed values (eq. [26]). Values of less thanP2 Wjhave been increased to 0.5 in constructing and only the ÐrstSV

C\ 0.5 C

w,

eight SV modes have been included in P@. All SV modes are used in TheCw.

di†erence between this reconstruction and the model is then used to Ðnds2. The 68% and 95% contours (*s2\ 2.30 and 5.41) are shown. Onlywavenumbers k \ 0.2 h Mpc~1 are used ; we marginalize over the uncer-tainty at larger wavenumbers.

TA

BL

E5

RED

UC

ED

CO

VA

RIA

NC

EM

AT

RIX

AN

DIN

VER

SE

CO

VA

RIA

NC

EM

AT

RIX

FO

RP

(k)

1.00

[0.

24[

0.27

[0.

25[

0.15

00.

100.

080.

01[

0.05

[0.

020.

030.

01[

0.02

00.

01[

0.01

00.

011.

00[

0.10

[0.

09[

0.07

[0.

030.

010.

020.

01[

0.01

[0.

010

00

00

00

01.

001.

00[

0.13

[0.

11[

0.07

[0.

020.

010.

020

[0.

010

0.01

00

00

00

0.39

1.00

1.00

[0.

16[

0.14

[0.

08[

0.02

0.03

0.02

[0.

01[

0.02

0.01

0.01

[0.

010

0[

0.01

00.

440.

291.

001.

00[

0.24

[0.

19[

0.08

0.03

0.06

0[

0.03

0.01

0.02

[0.

010

0.01

00

0.43

0.30

0.38

1.00

1.00

[0.

32[

0.19

[0.

020.

070.

03[

0.04

[0.

010.

030

[0.

020.

010.

01[

0.01

0.33

0.26

0.35

0.46

1.00

1.00

[0.

32[

0.16

0.01

0.07

0[

0.04

0.01

0.02

[0.

020

0.02

[0.

020.

180.

170.

270.

410.

561.

001.

00[

0.37

[0.

160.

060.

08[

0.04

[0.

030.

030

[0.

020.

01[

0.01

0.05

0.09

0.18

0.33

0.50

0.65

1.00

1.00

[0.

45[

0.13

0.11

0.06

[0.

07[

0.02

0.05

[0.

01[

0.03

0.04

[0.

010.

040.

100.

210.

360.

530.

681.

001.

00[

0.51

[0.

080.

160.

01[

0.08

0.05

0.02

[0.

040.

04[

0.02

0.01

0.04

0.09

0.19

0.32

0.50

0.72

1.00

1.00

[0.

56[

0.04

0.17

[0.

02[

0.08

0.04

0.03

[0.

05[

0.01

00.

010.

030.

070.

140.

290.

530.

791.

001.

00[

0.62

0.04

0.18

[0.

11[

0.03

0.08

[0.

080

00

0.01

0.02

0.05

0.12

0.30

0.57

0.83

1.00

1.00

[0.

650.

060.

18[

0.08

[0.

060.

110

00

00.

010.

020.

040.

130.

320.

600.

861.

001.

00[

0.68

0.20

0.11

[0.

140.

110

00

00

0.01

0.02

0.05

0.15

0.34

0.61

0.87

1.00

1.00

[0.

760.

170.

19[

0.27

00

00

00

0.01

0.02

0.06

0.15

0.34

0.61

0.88

1.00

1.00

[0.

660.

170.

040

00

00

00

0.01

0.02

0.06

0.16

0.34

0.61

0.88

1.00

1.00

[0.

810.

620

00

00

00

00.

010.

020.

070.

160.

340.

620.

881.

001.

00[

0.95

00

00

00

00

00.

010.

020.

070.

160.

350.

620.

891.

001.

000

00

00

00

00

00

0.02

0.06

0.17

0.36

0.64

0.90

1.00

00

00

00

00

0[

0.01

[0.

01[

0.01

0.01

0.05

0.17

0.41

0.72

0.94

1.00

NO

TE.È

The

uppe

rtr

iang

lesh

ows

afte

rw

eha

vedi

vide

dea

chro

wan

dco

lum

nby

the

squa

rero

otof

itsre

spec

tive

diag

onal

elem

ent.

The

low

ertr

iang

lesh

ows

with

itsdi

agon

aldi

vide

dou

t.B

oth

CP

CP~1

mat

rice

sar

esy

mm

etric,

ofco

urse

.The

squa

rero

otof

the

diag

onal

ofis

give

nin

the

third

colu

mn

ofT

able

4;t

hein

vers

eof

the

squa

rero

otof

the

diag

onal

ofis

the

last

colu

mn

inth

atch

art.

Ple

ase

CP

CP~1

note

that

wel

l-co

nstr

aine

ddi

rect

ions

have

smal

leig

enva

lues

inan

dla

rge

eige

nval

uesin

With

only

two

sign

iÐca

ntÐg

ures

,the

smal

leig

enva

lues

will

bein

accu

rate

.is

quot

edhe

reon

lyto

show

CP

CP~1

.C

Pth

eco

rrel

atio

nco

effic

ient

s.If

one

wishe

sto

Ðtsm

ooth

pow

ersp

ectr

umm

odel

sus

ing

the

abov

em

atrice

s,on

em

ustus

eT

oca

lcul

ate

the

chan

gein

s2of

apa

rtic

ular

devi

atio

nin

P(k

),di

vide

each

CP~1

.el

emen

tof

the

vect

orof

*PÏs

byth

eco

rres

pond

ing

num

ber

inth

ela

stco

lum

nof

Tab

le4

and

then

cont

ract

asy

mm

etrize

dve

rsio

nof

the

low

ertr

iang

leof

the

mat

rix

abov

eby

the

vect

orof

frac

tion

alva

riat

ions

(i.e.,v

TMv)

.Not

eth

atth

ese

nse

ofth

eo†

-dia

gona

lte

rms

ofis

tope

naliz

eno

nosc

illat

ory

devi

atio

nsin

P(k

)m

ore

than

the

sum

ofth

esign

iÐca

nce

inea

chk

wou

ldsu

gges

t.C

onve

rsel

y,C

P~1os

cilla

tory

Ñuct

uation

sin

pow

erar

em

orepe

rmitte

dth

anth

esu

mof

sign

iÐca

nces

wou

ldsu

gges

t.W

eus

ehe

re.R

eade

rssh

ould

cont

actt

heau

thor

sif

mor

esign

iÐca

ntÐg

ures

arene

eded

.SV

C\

0.5

APM SPATIAL POWER SPECTRA 13

freedom, as calculated from 13 k bins and two parameters.Alternatively, one can think of this as 19 k bins and eightparameters : the two CDM parameters at k \ 0.2 h Mpc~1and six bins of band power at k [ 0.2 h Mpc~1 that havebeen marginalized over. The s2\ 6 on 11 degrees offreedom is small but not statistically abnormal. We wouldtherefore say that the CDM model is an acceptable Ðt to thedata.

If one uses only eight SV modes to construct P, the best-Ðt CDM model has s2\ 0.5. One might imagine that witheight modes and eight parameters, one has zero degrees offreedom. However, the other 11 modes have not beenremoved from the s2 ; they have simply had their amplitudein P set to zero. If all the models had zero overlap with theomitted modes, then we would indeed lose 1 degree offreedom per frozen mode. However, the overlap is smallÈbecause the omitted modes are wiggly, while the models aresmoothÈbut nonzero. Hence, we do not Ðnd the small s2 tobe surprising, but it is difficult to say this quantitatively.

One might worry that allowing the power at k [ 0.2 hMpc~1 to vary within its errors could cause great uncer-tainty on large scales, because we have not included anyangular data on scales below One way to address this is0¡.5.to force the small-scale power spectrum to a smooth form.Holding the power at k [ 0.36 h Mpc~1 equal to a k~1.3power law of unknown amplitude has only a small e†ect onthe allowed region for !. Extending this power law tok \ 0.2 h Mpc~1 causes the conÐdence intervals on ! to be0.19È0.33 (68%) and 0.155È0.50 (95%). This is a minorimprovement for such a strong prior. As a second test, weattempt to include our knowledge of the small-scale powerspectrum directly in the inversion by replacing the DG99w(h) at h ¹ 2¡ with a Ðnely sampled representation of theBE93 Ðtting form to w(h) that extends to This yields0¡.07.conÐdence regions on ! of 0.185È0.36 (68%) and 0.145È0.57(95%). Hence, we conclude that the small scales are wellenough constrained by angular data at that theirh [ 0¡.5uncertainties do not a†ect the reconstruction of the powerspectrum at k \ 0.2 h Mpc~1.

In Figure 4, we vary some of the above assumptions. Thetop row of the Ðgure shows the results as we vary TheSV

C.

left panel shows as in Figure 3. In the right panel,SVC\ 0.5,

we use This gives the ill-constrained directionsSVC\ 0.1.

in the power spectrum Ðt 5 times more freedom. Indeed,their amplitudes will commonly exceed unity, which isunphysical for a positive-deÐnite quantity such as the powerspectrum. The constraints on CDM parameters are slightlyworse but not considerably so. Reducing even moreSV

Cmakes little di†erence. Increasing above 1.0 begins toSVCshrink the allowed region and move the best-Ðt point to

higher !. This is because the modes with contributeWjZ 2

little large-scale power ; if all the modes with smaller SV aresuppressed by setting then the result becomesW

j\SV

C,

biased toward zero power on large scales.The middle row of Figure 4 shows the results when all SV

are included in calculating the best-Ðt power spectrum.Generally the di†erences are small, showing that thesesmaller SV have little e†ect on Ðts to CDM models. Larger! are slightly less favored, but the constraints are still verybroad. One should remember that since the small haveW

jbeen increased to before being added to P, adding suchSVCmodes is not more ““ correct ÏÏ in the sense of yielding an

unbiased estimator or returning the best-Ðt (andnonpositive) power spectrum.

The bottom row of Figure 4 restricts the Ðt to even largerscales, h Mpc~1. The constraints are considerablyk

c\ 0.1

worse : in particular, no interesting upper bound can be seton !. The best-Ðt ! is also higher.

With h Mpc~1, the tilt of the constraint region inkc\ 0.2

the plane is in the sense of a tight constraint on the!-p8rms Ñuctuations on a larger scale. That is, if we were to plotthe constraint on the plane, the region would be!-p24roughly perpendicular to the axes. This is not surprising,because is dominated by k B 0.2 h Mpc~1, and the Ðtp8should focus on larger scales.

Our Ðt to the observed approaches a constant asP2K ] 0, which means that it does not approach scale-invariance on the largest scales. One might worry(P2P K)that this causes an overestimate of the errors on large scales.We can address this by using a CDM power spectrum whencalculating We take a model with !\ 0.25 andC

w. p8\

0.89 and project the nonlinear to While the CDMP3 P2. P2does eventually go to zero at large scales, it actually exceedsour Ðt to observations at K \ 10. Using the CDM toP2calculate we Ðnd constraints in the plane that areC

w, !-p8a very close match to those in Figure 4. In detail, the best-Ðt

! and the conÐdence intervals shift by only 0.01, which is farwithin the errors.

When Ðtting to cosmological models, one can include thefact that the sample-variance portion of the covariancematrix depends on the model itself. For example, one mightworry that large ! models would predict smaller samplevariance and hence be less favored than Figure 3 wouldsuggest. We therefore repeat our Ðts to CDM models, usingthe model at each point to generate the covariance matrixand the best-Ðt power spectrum. We then calculate s2 asbefore. We Ðnd that the conÐdence regions are essentiallyunchanged and that the best-Ðt ! moves by less that 0.01.

5.3. Higher Order Terms in the CovarianceIn our treatment so far, we have only included the Gauss-

ian terms in the covariance matrix We want toCw(h, h@).

estimate the size of the non-Gaussian terms and determinewhether their inclusion could substantially change ourresults. To do this, we use the hierarchical Ansatz for thehigher order moments of the density Ðeld. The four-pointfunction is assumed to be

T4(K1, K2, K3, K4) \ ra[P2(K1)P2(K2)P2(K13)] cyc.]

]rb[P2(K1)P2(K2)P2(K3) ] cyc.] , (28)

where and are constants describing the hierarchicalra

rbamplitudes for the two di†erent topologies of diagrams con-

tributing to the four-point function. With the same set ofapproximations we used to obtain equation (20), the fullcovariance of w(h) is

Cw(h, h@) \ 1

A)

CP0

= dK K2n

2P22(K)J0(Kh)J0(Kh@)

]P0

= dK K2n

P0

= dK@K@2n

] T14(K, K@)J0(Kh)J0(K@h@)D

,

T14(K, K@) \P d2K

aA

r

P d2Kb

Ar@

T4(Ka, [K

a, K

b, [K

b) . (29)

14 EISENSTEIN & ZALDARRIAGA Vol. 546

FIG. 4.ÈSame as Fig. 3, but with changes to parameters of the reconstruction and Ðt. L eft : less than are increased to 0.5. Right : less thanWj

SVC\ 0.5 W

jare increased to 0.1. Top : Only the eight modes with the largest are used in constructing the power spectrum. All modes are used in constructingSVC\ 0.1 W

jthe covariance matrix. Only wavenumbers k \ 0.2 h Mpc~1 are used ; we marginalize over the uncertainty at larger wavenumbers. Middle : Same as top, butall 19 SV are used in constructing the power spectrum. Bottom : Same as top, but only k \ 0.1 h Mpc~1 is used.

The last integral is an angular average of the four-pointfunction over rings in K-space of area centered around KA

rand K@.In order to estimate the size of this contribution, we need

to have an estimate of the hierarchical amplitudes andra

rb.

Szapudy & Szalay (1997) estimated these amplitudes forAPM by measuring two di†erent conÐgurations of the four-point function. They obtained and Ther

a\ 1.15 r

b\ 5.3.

diagonal terms are determined mainly by the com-T14(K, K)bination In Scoccimarro et al.R\ 4(2r

a] r

b)\ 30.4.

(1999), it was shown that the hierarchical Ansatz is not aparticularly good approximation for the conÐgurations of

the four-point function relevant for the variance of thepower spectrum (or the two-point function), i.e., those con-Ðgurations in which two pairs of K add up to zero. Theamplitudes of the important conÐgurations were roughly afactor of 5 smaller than one would naively expect. There-fore, a smaller value of the hierarchical coefficients shouldbe used for the variance calculation. The spatial statisticsmeasured in particle-mesh N-body simulations imply afterprojection that the hierarchical coefficients for APM shouldsatisfy Figure 5 shows the ratio of T /P34(2r

a] r

b) B 12.

along the diagonal for the di†erent choices of andra

rb.

In reality, the four-point function has other contributions

No. 1, 2001 APM SPATIAL POWER SPECTRA 15

FIG. 5.ÈT (K, K) in the hierarchical model for di†erent choices/P23(K)of and The hierarchical amplitudes measured from APM (Szapudy &r

arb.

Szalay 1997) are expected to be larger than the amplitudes relevant for theconÐgurations that determine the variance of the power spectrum.The curves for are each normalized to the T /P3 values (R\r

a\^r

bcalculated when the spatial quantities obtained in4[2ra] r

b]\ 12)

N-body simulations were projected to the angular quantities using theAPM selection function.

due to shot noise, but in the case of APM they are subdomi-nant for the scales of interest. The full four-point functioncan be written as

T1 4full(K, K@)\ 1n6 3] 2

n6 3 [P2(K)] P2(K@)]

]B1 (K, K@)n6

] T14(K, K@) , (30)

where is the averaged bispectrum over the shells. TheB1scaling of the three- and four-point functions with thepower spectrum means that each of the additional termscoming from shot noise are down by a factor of P2(K)n6 ,which is smaller than 1 for the measured APM powerspectra up to K B 1000.

In Figure 6, we show the correlation coefficients for thepower spectra for the di†erent choices of and Ther

arb.

hierarchical model for the four-point function does notguarantee that the correlation coefficient stays smaller thanunity, illustrating that this model cannot correctly describethe correlations induced by gravity. Only the case r

a\[r

bmakes the coefficients stay smaller than 1, but Scoccimarroet al. (1999) show that the shape of the correlation coeffi-cients in the simulations are not particularly well Ðtted bythis choice. The hierarchical model does give a good esti-mate of the order of magnitude of the correlations, butcannot account for their shape ; this can also be seen in theresults of Meiksin & White (1999). In summary, our calcu-lation of the non-Gaussian e†ects should be taken as anorder-of-magnitude estimate.

Figure 7 shows the ratio of the Gaussian to the non-Gaussian terms in the covariance of the angular powerspectrum. We conclude that the two contributions are

FIG. 6.ÈCross-correlation coefficients between di†erent K shells of theangular power spectra for the choices of and listed in Fig. 5. Eachr

arbcurve shows the cross-correlation coefficient between one K shell (the one

with and all the rest.rij\ 1)

about equal at K \ 100, corresponding approximately to1¡. On the 10¡ scale, we expect the inclusion of the four-point function in to alter the error bars by less thanC

w10%.A quick estimate of the e†ect of four-point terms on the

errors of the correlation function can be obtained using asimple approximation. The four-point function scales as P23,so we approximate as where weT4(K, K@) RP1 2P2(K)P2(K@),have introduced a mean power With this simpliÐcation,P1 2.the non-Gaussian term in is InC

w(h, h@) A)~1RP1 2w(h)w(h@).

other words, we simply add an overall random Ñuctuation

FIG. 7.ÈRatio of the non-Gaussian terms to the Gaussian terms in thediagonal elements of the covariance matrix of the angular power spectra.We see that the non-Gaussian terms are subdominant for K \ 100.

16 EISENSTEIN & ZALDARRIAGA Vol. 546

in the amplitude of the correlation function, wü (h) \(1] v)w(h), with an rms amplitude of

Sv2T1@2 \ 0.05AR12

P1 22 ] 10~4

1 srA)

B1@2. (31)

This model reÑects the tendency of modes to becomeextremely correlated in the nonlinear regime, such that theshape of the power spectrum or correlation functionbecomes far better determined than the amplitude.

Taking we add this additional corre-(RP1 2)1@2 \ 0.05,lation to and repeat the calculation of the power spec-C

wtrum. The errors on ! increase by about 7%. If we doublethe amplitude of the e†ects to the errors on(RP1 2)1@2\ 0.1,! increase by about 30%. We expect that this amplitude isan overestimation of the non-Gaussian corrections. Thebest-Ðt power spectra for these two cases yield Ðts to theobserved w(h) with s2\ 42 and 34, respectively, on 32degrees of freedom. Weakening the o†-diagonal terms of thenon-Gaussian portion of so as to step away from totalC

w,

correlation between di†erent h, causes a less severe degrada-tion in the constraints on !. We therefore conclude thatnon-Gaussianity should have a relatively mild e†ect on ouranalysis of APM.

5.4. L ower Bound on L arge-Scale Constraints fromAngular Data

One can set a lower bound on the errors of an angularsurvey by working directly from the angular power spec-trum. For a Gaussian random Ðeld with angular powerspectrum the covariance matrix of the angular powerP2(K),spectrum measured over the full sky is diagonal, with avariance equal to for a bin of width *K. We2P22(K)/2K*Kmake the optimistic assumption that a survey of sky cover-age of sr will retain this variance with a scaling ofA) 4n/A).We can use the transformation in equation (4) to convertthe inverse covariance matrix of the angular power spec-trum into that of the spatial power spectrum. The elementrelating two bins in spatial wavenumber is

CP~1(k, k@)\ dk dk@

P dKK

A)4n

1P22(K)

fAK

kB

fAKk@B

, (32)

where is the survey projection kernel, and the bins havef (ra)

width dk and dk@. To compare two spatial power spectrathat di†er by *P(k), we integrate to Ðnd s2 :C

P~1

s2\P

dkP

dk@ *P(k)*P(k@)CP~1(k, k@) . (33)

DeÐning

*P2(K)\ 1KP

dk fAK

kB*P(k) (34)

as the angular power spectrum corresponding to the projec-tion of the di†erence of the spatial power spectra, we Ðnd

s2\ A)4nP

dK KC*P2(K)

P2(K)D2

. (35)

We now apply this limit to the CDM parameter space inthe case of APM. We assume that the true clustering isgiven by a !\ 0.25, model with nonlinear evolu-p8\ 0.89tion (Peacock & Dodds 1996). We then consider how wellone can constrain an excursion from this model on largescales. We therefore set *P(k) to be zero on scales k [ k

c

and equal to the di†erence between two CDM models (themodel to be tested and the !\ 0.25 model) on larger scales.This corresponds to the limit in which the small scales areconsidered to be perfectly known and not allowed to varywithin their errors. Of course, it also assumes that thisperfect knowledge on small scales says nothing to dis-tinguish CDM models, but we are interested here in thecosmological information available in the large-scale clus-tering. The integral in K is extended from 1 to 1000.

Figure 8 shows the constraints in the plane avail-!-p8able at scales of k \ 0.1, \0.2, and \0.3 h Mpc~1 using thesky coverage and redshift distribution of the APM survey.The ranges of allowed ! for the k \ 0.2 h Mpc~1 case are0.19È0.35 (68%) and 0.15È0.56 (95%) ; the best Ðt is !\ 0.25by construction. This is very similar to the limit assigned tothe Ðt to the power spectrum reconstructed from the actualdata if we use the same CDM model to generate WeC

w.

conclude that a survey with the sky coverage and selectionfunction of APM has too much sample variance to placestrong constraints on the shape of the power spectrum onscales greater than k \ 0.1 h Mpc~1.

While one cannot prove it rigorously, we do not see howone could in practice achieve errors smaller than the limitsimplied by equation (35) and shown in Figure 8. The rele-vant assumptions of Gaussianity, freedom from boundarye†ects, inÐnitesimal bins in angle and wavenumber, totalangular coverage, and perfect information at small scalesare all optimistic. The only subtlety is that equation (35) is a

FIG. 8.ÈConstraints on CDM parameters within the APM survey ifone adopts the optimistic assumptions of eq. (35). A !\ 0.25, p8\ 0.89model is used to calculate the sample variance, and the s2 is calculated forthe di†erence between this model and the grid of other models. We usenonlinear spatial power spectra in all cases. We compare the CDM modelsonly at scales larger than 0.1 (dotted line), 0.2 (solid line), and 0.3 h Mpc~1(dashed line) ; smaller scales are assumed to be known perfectly but tocontain no extra cosmological information. This is the optimistic assump-tion for the extraction of the large-scale power spectrum; allowing thesmall scales to vary within their errors would worsen the constraints. Weview the regions as lower limits on the uncertainty on the large-scale powerspectrum from APM, save for the minor adjustments that would occurwith a likelihood analysis on the actual data.

No. 1, 2001 APM SPATIAL POWER SPECTRA 17

statement about the s2 di†erence between two models,whereas for the actual survey one is concerned with thelikelihood function for model Ðts to the data. This can causesmall di†erences if the likelihood function is non-Gaussian ;in this case, the tendency would be to shift the allowedregion toward larger power, i.e., smaller ! and larger p8.

5.5. Comparison to Previous WorkDespite the limit described in the last section, previous

analyses have found substantially smaller error bars on thelarge-scale power spectrum. In this section, we describe howneglect of correlations and improper use of smoothing haveled to these underestimates.

We would like to compare the covariance matrix derivedfrom theory in ° 3 to that used in previous analyses of thepower spectra inferred from APM angular clustering. Gen-erally, the errors for large-scale correlations have been esti-mated as the deviation between four subsamples of theAPM survey (Maddox et al. 1996 ; Baugh & Efstathiou1993). This procedure is at best marginal for estimating eventhe diagonal elements of the covariance matrix, but it iscompletely inadequate for estimating the full covariancematrix. Indeed, one could only generate four nonzero eigen-values ! Therefore, the covariance matrices of either theangular correlations or the spatial power spectra have beenassumed to be diagonal. Neither of these approximations iscorrect or, as we will see, particularly good.

We begin with the angular correlation function. Withoutthe correlations between bins, it is very easy to overestimatethe power of the data set by using too Ðne a binning in h.Neighboring bins that are highly correlated will show thesame dispersion between the subsamples, but this is countedas two independent measurements rather than one. Theerrors on any Ðt will improve by The visual cue thatJ2.this is occurring is when the subsamples show coherentÑuctuations around the mean rather than rapid bin-to-binscatter. This is clearly occurring in Figure 27 of Maddox etal. (1996).

We can compare our calculation of to the quotedCwobservational errors by setting all our o†-diagonal terms to

zero. We then substitute this new and recalculate limitsCwon ! and amplitude in the manner described in the previous

section. We do the same for a diagonal that uses theCwerrors on w(h) based on the dispersion between four sub-

samples of APM (Maddox et al. 1996 ; Dodelson &1999). As shown in Figure 9, these two choicesGaztan8 aga

give constraint regions that are quite similar to one another.The fact that these two treatments give similar results isevidence that sample variance in the Gaussian limit doesexplain most of the observed scatter in w(h) on large angularscales in APM and further justiÐes the approximations thatunderlie our estimation of C

w.

Importantly, both diagonal treatments give constraintson CDM parameters that are a factor of 2 tighter than thosefound when using the theoretical covariance matrix with itso†-diagonal terms. For example, comparing the 68% semi-range on !, we Ðnd 0.09 in the full case and 0.043 inC

weither of the diagonal counterparts. The best-Ðt ! in thediagonal cases are around 0.3, somewhat higher than in theanalysis with nonzero correlations in w(h) and suggesting abias in the reconstruction.

It should also be noted that when either of these diagonalcovariance matrices are used, the s2 for the w(h) of thebest-Ðt power spectra is less than 3 for 32 degrees offreedom. This is another indication that these matrices donot properly describe the error properties of the data.

DG99 reconstruct the power spectrum based on a diago-nal covariance matrix. However, they obtain limits usingk \ 0.124 h Mpc~1 that are tighter than what we show fork \ 0.2 h Mpc~1 in Figure 9. We believe that this is causedby the way in which their smoothing prior enters the calcu-lation of the covariance matrix, namely, that the quotedcovariance matrix is for the smoothed estimator of thepower, not the actual power itself. On scales at which theconstraints from the data are poor, the smoothing prior willchoose a value of the power based on an extrapolation fromthe wavenumbers with stronger measures of the power. Thevariance between samples of this extrapolated power will befar smaller than the true uncertainty in the power. We thinkthat this underestimate of the errors at large scales is

FIG. 9.ÈConstraints when the covariance matrix is assumed to be diagonal. L eft : Observed APM error bars (Maddox et al. 1996 ; Dodelson &Cw1999) formed into a diagonal covariance matrix. Right : Covariance matrix from ° 3, but with the o†-diagonal terms set to zero. In each case, weGaztan8 aga

use consider only the largest seven SV when constructing P@, and use only wavenumbers k \ 0.2 h Mpc~1 in the Ðt. These constraints should beSVC\ 0.5,

compared to those of Fig. 3.

18 EISENSTEIN & ZALDARRIAGA Vol. 546

responsible for the discrepancy between the SVD treatmentand the method of DG99. Indeed, DG99 found that theerrors on cosmological parameters increase as they relaxedthe smoothing prior.

Baugh & Efstathiou (1993, 1994) did not use the covari-ance on instead, they estimated errors on P(k) directlyC

w;

by using the variance of the power spectra of the four sub-samples, having inverted each separately. Again, four sub-samples was too few to estimate the o†-diagonal terms of

It is clear, however, that these terms are important,CP.

since Figure 9 of BE93 reveals that the P(k) from the foursubsamples do show obvious correlations in their di†er-ences from the mean. The tests on simulations in Gaztan8 aga& Baugh (1998) also neglect the correlations between di†er-ent bins in the reconstructed power spectra.

Unfortunately, we cannot simply use the diagonal termsof our matrix to compare to the BE93 results, becauseC

Pthe inversion procedure of BE93 includes an implicitsmoothing prescription. In order to compare to their esti-mate of the errors on the smoothed power spectrum, wewould need to project our matrix onto their allowedC

Pbasis, removing the variance in any disallowed directions inP-space. Without this step, the large variances we includedfor the ill-constrained directions will give enormousvariance to individual k bins when the detailed correlationsbetween bins are discarded.

One must be especially wary of estimating error barsfrom the variance between subsamples when a smoothingprior has been applied to a wavenumber or angle where thedata is not constraining. In the present context, the iterativeinversion method (Lucy 1974) employed by BE93 containeda smoothing step that pushed P(k) to a particular functionalform. When such a method is used on large scales (e.g.,k \ 0.02 h Mpc~1) where the power is not well-constrained,then all subsamples will tend to reconstruct a power spec-trum value on large scales that is simply an extrapolation ofthe smaller scale result. The dispersion between the sub-samples will not grow with scale as fast as they would in theabsence of smoothing, causing the resulting error bars to besigniÐcantly underestimated on large scales. We suspectthat this e†ect is a signiÐcant contribution to why Baugh &Efstathiou (1993) Ðnd near-constant power and small errorsat k \ 0.05 h Mpc~1.

6. CONCLUSION

Both the angular correlation function and the spatialpower spectrum inferred from angular clustering haveimportant correlations between di†erent bins of angle andwavenumber even if the Ñuctuations are Gaussian. Previousanalyses of the deprojection of angular clustering haveneglected these e†ects. In this paper, we have shown how toinclude sample variance in the covariance matrix for theangular correlation function w(h) under a Gaussian, wide-Ðeld approximation. We have then described how one caninvert w(h) to Ðnd the spatial power spectrum usingsingular-value decomposition (SVD) in such a way as toretain the full covariance matrix. The method allows one tohandle the near-singularity of the projection kernel withoutnumerical difficulty and can yield a smoothed version of thedeprojected power spectrum without sacriÐcing the covari-ances of unsmoothed spectrum.

Using the large-angle galaxy correlations of the APMsurvey as an example, we have shown that correlationsbetween di†erent bins in h and in k are critical for quoting

accurate statistical limits on the power spectrum and modelÐts thereto. With the sample variance properly included, weÐnd that APM does not detect a downturn in P(k) atk \ 0.04 h Mpc~1 ; the signiÐcance is only 1 p. Fitting non-linearly extrapolated, scale-invariant CDM power spectrato the power spectrum at large scales (k \ 0.2 h Mpc~1), weÐnd that APM constrains the CDM parameter ! to be0.19È0.37 (68%). We have investigated a wide range of alter-ations to the method in the hopes of shrinking this range,but have found nothing that makes a signiÐcant di†erence.Indeed, in ° 5.4, we showed that the above constraintsalready approach the best available to a survey with the skycoverage and selection function of APM. Extending theCDM Ðts to smaller scales would improve the constraints,but this depends entirely on the modeling of galaxy bias andnonlinear gravitational evolution. Moreover, such a Ðtwould not validate this particular set of CDM models,because many other models would look similar in the non-linear regime. To conÐrm a model from galaxy clustering,one would like to see the characteristic features of the modeldirectly rather than attempt to leverage a measurement ofthe slope in the nonlinear regime onto a cosmologicalparameter space.

We have made a number of approximations in ouranalysis. In our treatment of large scales, we have ignoredthe ability of boundaries to alias power from one scale toanother and used LimberÏs equation even for modes withwavelengths similar to the scale of the survey. On smallscales, we have ignored three- and four-point contributionsto the covariance matrix of the angular statistics. In general,we have treated the likelihood of the correlation function asa Gaussian and ignored the Ðne details of how cosmologyor evolution of clustering might enter. We have argued thatthe above approximations are likely to be reasonably accu-rate for an analysis of the large-angle clustering of APM.We also have not questioned the redshift distribution func-tion that has been used in past APM analyses nor includedany systematic errors. Conservatively, therefore, one canregard our results as the optimistic limits, because it is veryunlikely that the breakdown of any of the above assump-tions would substantially improve the constraints.

Surveys such as DPOSS and SDSS will be substantiallydeeper and wider than the APM survey. We repeat theanalysis of ° 5.4 for parameters suggestive of SDSS, namely,3.1 sr of sky coverage to a median redshift of 0.35. Thisyields an error on ! of 0.017 (1 p) about a Ðducial model of!\ 0.25. Remember that this is an optimistic limit on theerror, that we have assumed perfect knowledge of the selec-tion function, and that we have only allowed one otherparameter, the amplitude, to vary. With analogous assump-tions, the limit on ! from the SDSS redshift survey of brightred galaxies is roughly 0.007. The redshift survey would bemore strongly preferred if one were interested in narrowerfeatures in the power spectrum (Meiksin, White, & Peacock1999), such as would be needed to separate e†ects in a largerparameter space (Eisenstein, Hu, & Tegmark 1999). Withregard to systematic errors, the angular survey su†ers fromits dependence on purely tangential modes, while the red-shift survey must contend with redshift-space distortions.

If one could use color information to select a clean, high-redshift sample of galaxies, the prospects for interpretingangular correlations improve. For example, using onlythose galaxies with z[ 0.45 in the above SDSS exampledrops the limiting errors on ! to 0.010. This occurs because

No. 1, 2001 APM SPATIAL POWER SPECTRA 19

the obscuring e†ects of smaller scale clustering from lowerredshift galaxies have been removed ; moreover, the projec-tion from three dimensions to two becomes signiÐcantlysharper. This performance is comparable to that of the red-shift survey and would allow measurement of the large-scalepower spectrum in a range of redshifts disjoint from thespectroscopic survey, thereby allowing one to study theevolution of large-scale clustering.

The results of this paper, and particularly the limits set in° 5.4, provide a cautionary note for the interpretation oflarge angular surveys. Even at the depths of the SDSSimaging, the constraints on large-scale clustering fromangular correlations alone are not strong. The inclusion ofphotometric redshifts to separate the sample into multiple

(or even continuous) radial shells could provide a signiÐcantimprovement to this state of a†airs.

We thank Scott Dodelson, Enrique LloydGaztan8 aga,Knox, Jon Loveday, Roman Scoccimarro, Istvan Szapudi,Max Tegmark, and Idit Zehavi for helpful discussions.Support for this work was provided by NASA throughHubble Fellowship grants HF-01118.01-99A (D. J. E.) andHF-01116.01.98A (M. Z.) from the Space Telescope ScienceInstitute, which is operated by the Association of Uni-versities for Research in Astronomy, Inc., under NASA con-tract NAS5-26555. D. J. E. was additionally supported byFrank and Peggy Taplin Membership at the IAS.

REFERENCESBaugh, C. M., & Efstathiou, G. 1993, MNRAS, 265, 145 (BE93)ÈÈÈ. 1994, MNRAS, 267, 323Bernstein, G. M. 1994, ApJ, 424, 569Bond, J. R., Ja†e, A. H., & Knox, L. 1998, Phys. Rev. D, 57, 2117Djorgovski, S. G., Gal, R. R., Odewahn, S. C., de Carvalho, R. R., Brunner,

R., Longo, G., & Scaramella, R. 1998, in Wide Field Surveys in Cosmol-ogy, ed. S. Colombi & Y. Mellier, in press (preprint astro-ph/9809187)

Dodelson, S., & E. 1999, preprint (astro-ph/9706018) (DG99)Gaztan8 aga,Eisenstein, D. J., Hu, W., & Tegmark, M. 1999, ApJ, 518, 2Gawiser, E., & Silk, J. 1998, Science, 280, 1405

E., & Baugh, C. M. 1998, MNRAS, 294, 229Gaztan8 aga,Groth, E. J., & Peebles, P. J. E. 1977, ApJ, 217, 385Hamilton, A. J. S. 2000, MNRAS, 312, 285Limber, D. N. 1953, ApJ, 117, 134Lucy, L. B. 1974, AJ, 79, 745Maddox, S. J., Efstathiou, G., & Sutherland, W. J. 1996, MNRAS, 283,

1227

Maddox, S. J., Sutherland, W. J., Efstathiou, G., & Loveday, J. 1990,MNRAS, 242, 43P

Mann, R. G., Peacock, J. A., & Heavens, A. F. 1998, MNRAS, 293, 209Meiksin, A., & White, M. 1999, MNRAS, 308, 1179Meiksin, A., White, M., & Peacock, J. A. 1999, MNRAS, 304, 851Peacock, J. A., & Dodds, S. J. 1996, MNRAS, 280, L19Peebles, P. J. E. 1973, ApJ, 185, 413ÈÈÈ. 1980, The Large-Scale Structure of the Universe (Princeton :

Princeton Univ. Press)Phillips, S., Fong, R., Ellis, R. S., Fall, S. M., & MacGillivray, H. T. 1978,

MNRAS, 182, 673Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992,

Numerical Recipes (2d Ed. Cambridge : Cambridge Univ. Press)Scoccimarro, R., Zaldarriaga, M., & Hui, L. 1999, ApJ, 527, 1Scott, D., Srednicki, M., & White, M. 1994, ApJ, 421, L5Szapudi, I., & Szalay, A. 1997, ApJ, 481, L1