Measuring the galaxy power spectrum with future redshift surveys

23 Pages·0.45 MB·English

by  Max Tegmark

ToappearinApJ,499(1998) 1
PreprinttypesetusingLATEXstyleemulateapjv.04/03/99
MEASURING THE GALAXY POWER SPECTRUM WITH FUTURE REDSHIFT SURVEYS
Max Tegmarka
aHubbleFellow.
Institute forAdvancedStudy, Princeton,NJ08540; [email protected]
Andrew J. S. Hamilton
JILAandDept. ofAstrophysical,PlanetaryandAtmosphericSciences,
Box440,Univ. ofColorado,Boulder,CO80309, USA;[email protected]
Michael A. Straussa,b
8
9 aAlfredP.SloanFoundation Fellow.
9 bCottrellScholarofResearchCorporation.
1 DepartmentofAstrophysicalSciences,PrincetonUniversity,Princeton,NJ08544;
[email protected]
n Michael S. Vogeley1
a
DepartmentofAstrophysicalSciences,PrincetonUniversity,Princeton,NJ08544;
J
[email protected]

0 and
3
Alexander S. Szalay

2 DepartmentofPhysicsandAstronomy,JohnsHopkinsUniversity,Baltimore,MD21218; [email protected]
v To appear in ApJ, 499 (1998)
0
2
ABSTRACT
0
8 PrecisionmeasurementsofthegalaxypowerspectrumP(k)requireadataanalysispipelinethatisboth
0 fast enough to be computationally feasible and accurate enough to take full advantage of high-quality
7 data. Wepresentarigorousdiscussionofdifferentmethodsofpowerspectrumestimation,withemphasis
9 on the traditional Fourier method, and linear (Karhunen-Lo`eve; KL), and quadratic data compression
/ schemes, showing in what approximations they give the same result. To improve speed, we show how
h
p many of the advantages of KL data compression and power spectrum estimation may be achieved with
- a computationally faster quadratic method. To improve accuracy, we derive analytic expressions for
o
handling the integral constraint, since it is crucial that finite volume effects are accurately corrected
r
t for on scales comparable to the depth of the survey. We also show that for the KL and quadratic
s techniques, multiple constraints can be included via simple matrix operations, thereby rendering the
a
: resultslesssensitiveto galacticextinctionandmis-estimatesofthe radialselectionfunction. We present
v a data analysis pipeline that we argue does justice to the increases in both quality and quantity of
i
X data that upcoming redshift surveys will provide. It uses three analysis techniques in conjunction: a
traditional Fourier approach on small scales, a pixelized quadratic matrix method on large scales and a
r
a pixelized KL eigenmode analysis to probe anisotropic effects such as redshift-space distortions.
1. INTRODUCTION This is especially important since a wide variety of mod-
els currently appear to be at least marginally consistent
Observational data on galaxy clustering are rapidly in-
with the present data (Peacock 1997; White et al. 1996;
creasing in both quantity and quality, which brings new
Vogeley 1997), so smaller error bars on the power spec-
challenges to data analysis. As for quantity, redshifts had
trum will be needed to discriminate between them. How-
been published for a few thousand galaxies 15 years ago.
ever,the increaseddataquantitymakesit arealchallenge
Today the number is 105 (Huchra, private communica-
∼ to perform such an accurate analysis; as we will discuss
tion), and ongoing projects such as the AAT 2dF Survey
at some length, a straightforward application of any such
and the Sloan Digital Sky Survey (hereafter SDSS; see
methodiscomputationallyunfeasiblefordatasetsaslarge
Gunn &Weinberg1995)willraiseitto 106 inafew years.
as those from 2dF and SDSS.
Comprehensive reviews of past redshift surveys are given
Thispaperaddressesboththeaccuracyandspeedissues
by Efstathiou (1994), Vogeley (1995), Strauss & Willick
for measuring the galaxy power spectrum P(k). For the
(1995) and Strauss (1997), the last also including a de-
highest accuracy, we advocate the use of lossless pixelized
tailed description of 2dF and SDSS. As for quality, more
data compressionmethods, both linear (Karhunen-Lo`eve)
accurate and uniform photometric selection criteria (en-
and quadratic. To improve the speed, we present a fast
abledbye.g.thewell-calibrated5-bandphotometryofthe
implementation of a pixelized quadratic power estimation
SDSS) reduce potential systematic errors. This increased
method, and show how it can reproduce Karhunen-Lo`eve
data quality makes it desirable to avoid approximations
results (Vogeley & Szalay 1996;Tegmark,Taylor& Heav-
in the data analysis process and to use methods that can
ens 1997 — hereafter TTH) exactly, without the need to
constraincosmologicalquantitiesasaccuratelyaspossible.
Available in color from h t t p://www.sns.ias.edu/˜max/galpow.html
2
solve eigenvalue problems, although it does not have the density fluctuation field obeys the simple relation
ability to measure redshift-space distortions.
Therestofthepaperisorganizedasfollows. Section2is δ (k)∗δ (k′) =(2π)3δD(k k′)P(k) (1)
r r
a“buyersguide”,wherewelistthevariouspropertiesthat h i −
are important when deciding which data analysis method
for some fubnctionbP which depends only on the magni-
to use. In Section 3, we review the existing methods for
tude of k, not on its direction2 P is known as the power
powerspectrumestimationinacommonframework,make
spectrum. Becauseofequation(1),P containsalltheinfor-
various extensions of them and present the fast quadratic
mation needed to compute any statistical quantities that
data compression and power spectrum technique. Sec-
depend quadratically on δ , for instance the variance or
tion 4 describeshow the variousmethods areinterrelated, r
the correlation function on different scales. Moreover, if
and Section 5 discusses how they can be immunized from
the random field δ is Gaussian (which means that the
various systematic effects, such as errorsin the extinction r
jointprobabilitydistributionofδ atanynumberofpoints
model; many of the technical details are given in Appen- r
is a multivariate Gaussian), then P characterizes δ com-
dicesAandB.Section6discussestheprosandconsofthe r
pletely and contains all the information needed to answer
different techniques. We conclude thatto meetallcriteria
any statistical question whatsoever about δ . Inflationary
on the wish list of Section 2, it is necessary to combine r
theory (e.g., Peebles 1993)and observations of large-scale
three of the principal methods described. The data anal-
structure (e.g., Strauss & Willick 1995) imply that δ is
ysis pipeline that we propose is summarized in Figure 1, r
Gaussianonlargescales,makingP animportantquantity
and the reader may wish to glance at this before delving
in cosmology. The power spectrum estimation problem,
into the details of Section 3, to obtainan overviewof how
which is the topic of this paper, is to estimate P(k) given
everything fits together. We ignore redshift-space distor-
the observed realization of n(r).
tions for mostof this paper, andassume that clustering is
isotropic. In Section 5 and Appendix C, we discuss this
issue further. 2.2. The traditional approach
Let us parameterize the power spectrum P(k) by some
set of parameters θ ,i = 1,2,..., grouped into a vector
i
2. COMPARINGMETHODS:AWISHLIST Θ. These may be either the band powers in a set of nar-
rowbands,orphysicallymotivatedparameterssuchasthe
Since the ultimate goal of the analysis of clustering in
normalization σ , the shape parameter Γ, the primordial
a galaxyredshift surveyis to constraincosmologicalmod- 8
spectral index n, etc. Let us package our data set into a
els, we wanta method that minimizes the statisticalerror
barsoncosmologicalparameters1andisrobustagainstpo- vector x; much of the distinction between different meth-
ods discussed in Section 3 lies in the way this packaging
tential systematic errors. We will now discuss the former
is done. The standard approach to parameter estimation
issue in some detail, and return to the latter in Section 5
is to write down the expression for the probability dis-
and Appendix B.
tribution f(x;Θ). Here we interpret f as a probability
distributionoverx fora fixed Θ. In a Bayesianstatistical
2.1. The problem to be solved
analysis with a uniform prior probability distribution for
The data set from a galaxy redshift survey consists Θ, one reinterprets f as a probability distribution over Θ
of N three-dimensional vectors rα (α = 1,...,N) giving for a given data set x, and to clarify this distinction re-
the measured positions of the galaxies. Following Pee- namesf thelikelihoodfunction. Thefinalresultsareoften
bles (1973, 1980), we model these positions as generated presented as contour plots of this likelihood function, as
by a random Poissonian point process where the galaxy at the bottom of Figure 1.
density is modulated by both selection effects and fluctu- If we take x to be the raw data set, i.e., the measured
ations in the underlying matter distribution. The former coordinates r (α=1,...,N) of the N measured galaxies,
α
are describedby n¯(r), the selection function of the galaxy thenthe likelihoodfunctionf isunfortunatelyhopelessto
surveyunderconsideration,definedastheexpectedgalaxy computenumerically,sinceitinvolvestheN-pointcorrela-
density. Thus n¯(r)dV is the expected (not the observed) tionfunction. EvenintheGaussianapproximationthatf
number ofgalaxiesina volumedV aboutrinthe absence is given by a product over two-point correlationfunctions
of clustering. This function typically falls off at large dis- (e.g., White 1979; Fry 1984), this requires evaluating a
tances, and can exhibit small angular variations due to multivariate polynomial of degree N/2 in the correlations
extinction. It vanishes outside the survey volume. The oftheN(N+1)/2galaxypairs,andtheCPUtimerequired
fluctuations in the underlying matter density are givenby forthisgrowsfasterthanexponentiallywithN (Dodelson,
the field δr(r), which is not to be confused with the Dirac Hui,&Jaffe1997). Thetraditionalapproachhastherefore
delta function δD. This means that the observed galaxy beentotakextobesomethingelse: band-powerestimates
distribution n(r) = αδD(r − rα) is modeled as a 3D of the power spectrum. These are essentially computed
Poisson process with intensity λ(r)=n¯(r)[1+δr(r)]. by multiplying the observed density field by some weight
The density fluctuaPtions δr are modeled as a homoge- function,Fouriertransformingit,takingthesquaredmod-
neousandisotropic(butnotnecessarilyGaussian)random ulusoftheresultandaveragingovershellsink-space(Sec-
field. This implies that the Fourier transform δ of the tion3.3). Inthe(sometimespoor)approximationthatthe
r
1This is not a mere unimportant detail, since doubling the error bars (which an inferiormethod can easily do) is comparable to reducing
thesurveyvolumeandthenumberofgalaxiesprobedbybafactoroffour.
2Note that equation (1) only holds if positions are measured in real space. Redshift-space distortions couple modes of different k; see
AppendixCformoredetails.
2
3
probability distribution for x is a multivariate Gaussian, By computing the Fisher matrix separately from each
its probability distribution is ofthe intermediatedatasetsinFigure1,wecantrackthe
flow of information down the data pipeline and check for
f(x;Θ) C−1/2e−21(x−m)†C−1(x−m), (2) leaks. For instance, if the Fisher matrix computed from
∝| |
therawpositionaldataisidenticaltothatcomputedfrom
where C denotesthedeterminantofC,andthemeanvec- the (much smaller) data set consisting of the band power
tor m| |x andthe covariancematrixC xx† mm† estimates, then the power spectrum estimation method is
≡h i ≡h i−
depend on P(k) and hence on the unknown parameters losslessinthesensethatnoinformationaboutthe param-
Θ. Muchofthe model testing to date hasbeen ratherap- etersofinteresthasbeenlostintheprocessofcompressing
proximate, often little more than a “χ-by-eye” fit of the- the data set from, say, 106 numbers down to 50; cf., the
oretical power spectra to the data, which is tantamount discussion in Section 2.4. We will use this criterion when
to ignoring the correlations between the power estimates comparing different data analysis techniques below.
(the off-diagonal elements of C). This approach clearly
does not utilize all the information present in the data, 2.3.3. The power spectrum Fisher matrix
and can also bias the results.
Whether a method is lossless or not generally depends
Ifwehadinfinitecomputerresources,wewouldimprove
onwhichparametersweare interestedin estimating. For-
the situation by simply performing an exact brute force
tunately, as shown by Tegmark (1997a, hereafter T97),
likelihood analysis on the raw data set. Is there a faster
certain methods can be shown to be lossless for any set
way of obtaining the same result?
of parameters in a large class. An important special case
2.3. The notion of a lossless method arequantitiesthatparameterizethepowerspectrumP(k),
suchasσ andΓ;alltheinformationontheseparametersis
8
Wewillcallamethodforanalyzingadatasetunbeatable
retainedifthepowerspectrumitself(parameterizedbythe
oroptimal ifnoothermethodcanplacetighterconstraints
power in many narrow bands) can be measured with the
oncosmologicalmodels(asparameterizedbyΘ)usingthis
minimalerrorbars. Thismeansthatonecantestwhether
data. In this paper, we are focussed on the power spec-
amethodislosslesssimplybycomputingtheFishermatrix
trum measured in real space, and so we restrict ourselves
for the band powers. This also means that band powers
to measurements of the parameters which determine the
havea specialstatus comparedto other parameters: if we
power spectrum itself.
simply measure P(k) as accurately as possible, this mea-
suredfunctionwillretainalltheinformationaboutallcos-
2.3.1. The Fisher Information Matrix
mologicalparameters. Allthisisofcourseonlytruewithin
This can be made precise using the formalism of the theframeworkofGaussianandisotropicmodels,whichare
Fisher information matrix (see TTH for a comprehensive uniquely characterized by their power spectrum. Specifi-
review of this application), which offers a simple and a cally,thesemethodsdo loseinformationaboutparameters
useful way of measuring how much information each step that affect the data set not only via P(k). Important ex-
in the pipeline of Figure 1 destroys. Given any set of cos- amples to which we will return are redshift-space distor-
mological parameters of interest denoted θi, i = 1,2,..., tions and uncorrected galactic extinction, both of which
their Fisher matrix F gives the smallest error bars with introduce differences between the angular and radialclus-
which the parameters can possibly be measured from a tering patterns. On small scales, nonlinear clustering cre-
given data set. If the probability distribution for the data ates non-Gaussianfluctuations where inaddition to P(k),
set giventhe parameter values is f(x;Θ), then the Fisher higher order moments also contain cosmological informa-
matrix is defined by (Fisher 1935) tion.
∂2lnf
F . (3) 2.4. Data compression, simplicity and speed
ij
≡− ∂θ ∂θ
(cid:28) i j(cid:29) A second and rather obvious criterion for comparing
Crudely speaking, F−1 can be thought of as the best pos- dataanalysismethodsistheirnumericalfeasibility. Forin-
stance, the brute-force likelihood analysis of the raw data
sible covariancematrix for the measurementerrorson the
set (the galaxy positions) described above is lossless, but
parameters. Indeed, the Cram´er-Rao inequality (Ken-
too time-consuming to be numerically feasible when N,
ney & Keeping 1951; Kendall & Stuart 1969) states that
no unbiased method whatsoever can measure the ith pa- the number of galaxies,is large.
When the brute-force method is unfeasible, the general
rameter with error bars (standard deviation) less than
approach is to perform some form of data compression,
1/√F . If the other parameters are not known but are
ii
whereby the data set is reduced to a smaller and simpler
estimated from the data as well, the minimum standard
one which is easier to analyze. If the data compression
deviation rises to (F−1)1/2. This formalism has recently
ii step is lossless, a brute-force analysis on the compressed
been used to assess the accuracy with which cosmological
datasetclearlygivesjustassmallerrorbarsasoneonthe
parameters can be measured from future galaxy surveys
raw data would have done.
(Tegmark 1997b; Goldberg & Strauss 1998; Hu, Eisen-
To facilitate parameter estimation further down the
stein,&Tegmark1997)andcosmicmicrowavebackground
pipeline,itisusefulifthestatisticalpropertiesofthepower
experiments (Jungman et al. 1996; Bond, Efstathiou, &
spectrum estimates are simple and easy to calculate. The
Tegmark 1997;Zaldarriaga,Seljak, & Spergel 1997).
simplestcase(whichoftenoccurswiththepixelizedmeth-
ods described below) is that where the data set is a mul-
2.3.2. Checking for leaks in the pipeline
tivariate Gaussian, described by the likelihood function
3
4
of equation (2). Then the slowest step in the likelihood With the exception of the brute force maximum likeli-
calculation is computing the determinant of the N N hood technique, all of the methods described below com-
×
covariance matrix C, for which the CPU time scales as pute bandpowerestimatesq thatarequadraticfunctions
i
N3, so it is desirable to make the compressed data set as of the density field n, which means that they can be writ-
small as is possible without losing information3. It is also ten as
convenient if the statistical properties of the compressed
n(r)n(r′) E (r ,r )
data set, in particular, its covariance matrix, can be com- q = E (r,r′) d3rd3r′ = i α β
putedanalytically. Wewillseethatthisisthecaseforthe i i n¯(r)n¯(r′) n¯(r )n¯(r )
Z Z α,β α β
Karhunen-Lo`eveandquadraticdatacompressionmethods X
(4)
describe below, but not for the the maximum-likelihood
forsomereal-valuedsymmetricpairweightingfunctionsE
i
method, where it requires numerical computation of the
whicharedesignedtoisolatedifferentrangesofwavenum-
entire likelihood surface. Finally, the simplest covariance
ber k — the methods simply differ in their choices of E .
matrix one can desire is clearly one that is diagonal, i.e., Taking the expectation value of n(r)n(r′)/n¯(r)n¯(r′) proi-
where the errors on the elements of the compressed data duces three terms: “1” from the mean density, δD(r
set are uncorrelated. r′)/n¯(r) fromshotnoise andδ (r)δ (r′) fromdensity fluc−-
r r
tuations. By a derivationanalogousto FKPandT95,one
2.5. The wish list
finds that these three terms give
In summary, the ideal data analysis/data compression
method would d3k
q =W (0)+b + W (k)P(k) , (5)
h ii i i i (2π)3
1. be lossless, at least for the parameters of interest, Z
where W , the three-dimensional window functions, are
2. give easy-to-compute and uncorrelated errors, i
given by
3. be computationally feasible in practice, Wi(k)=Ei(k,k) (6)
and
4. allow one to account for redshift-space distortions b
and systematic effects.
E (k,k′) E (r,r′)e−ik·reik′·r′d3rd3r′ (7)
i i
≡
The first two items can be summarized by saying that we
Z
want the method to retain the cosmological information
is a Foubrier transform of E . We will often find it
content of the original data set, distilled into a set of mu- i
convenient to rewrite the last term of equation (5) as
tuallyexclusive(2)andcollectivelyexhaustive(1)chunks.
∞
W (k)P(k)dk, where the one-dimensional window
In other words, the information chunks should be inde- 0 i
function is the angular average
pendent and together retain all the information from the
R
original data set
W (k)=k2 W (k)dΩ . (8)
3. POWERSPECTRUMESTIMATIONMETHODS—AN i i k
Z
OVERVIEW
The shot noise bias is given by
Inthissection,wereviewthevariousmethodsforpower
spectrum estimation that have been proposed in the lit-
E (r,r)
erature and present various extensions. We start in Sec- b i d3r. (9)
i ≡ n¯(r)
tion 3.1 by developing a formalism that is common to all Z
our approaches. In Section 3.2, we discuss how the data
Alternatively, b can be made to vanish by omitting the
might be discretized; that is, different ways of packaging i
terms with α = β from the double sum in equation (4),
thedataintoaconvenientformx. Wethendiscussvarious
as described in Appendix A. The term W (0) simply
methods of power spectrum estimation: traditional meth- i
probes the mean density of the survey, and as described
ods which take the square of the amplitude of the Fourier
in Section 5 and Appendix B, the functions E should al-
modes (Section 3.3), the method of brute force likelihood i
ways be chosen such that this term vanishes, i.e., so that
(Section 3.4), the linear Karhunen-Lo`eve data compres-
E (r,r′)d3rd3r′ =0,therebyimmunizingthepoweresti-
sionmethod(Section3.5),andanewquadraticdatacom- i
mates to errors in normalization of n¯ (Section 5.1)4. The
pressionmethod(Section3.6). Oursuggestedapproachto R
desirability of choosing windows with this property was
powerspectrumestimationinvolvesacombinationofthese
first explicitly pointed out by Fisher et al. (1993), and
methods in different regimes, as summarized in Figure 1,
this prescription was used also by e.g. Hamilton (1992)
and described in more detail in Section 7. Throughout
and Cole, Fisher & Weinberg (1994). We want to inter-
Section 3, we ignore redshift-space distortions and other
pret q in equation (5) as probing a weighted average of
systematic effects, andassumethat clusteringis isotropic. i
the power spectrum, with the window function giving the
We return to this topic in Section 5.
weights, so E should be normalized so that W (k) inte-
i i
grates to unity (throughout this paper, we will write the
3.1. Density field, shot noise and window functions
volume element in Fourierspace as d3k/(2π)3 ratherthan
3Another problem with large N is the memory requirements for an N ×N covariance matrix; if N = 105, for example, this is currently
beyondtheRAMcapacityofallbuttheverylargestcomputers.
4Infact,thestrongerconstraint Ei(r,r′)d3r′=0forallrshouldbeenforced,asdiscussedinAppendixB.
R
4
5
d3k, since this minimizes the number of 2π-factors else- in general impractical to implement numerically, as it in-
where). Using equation (6) and Parseval’s theorem, this volves a computationally cumbersome infinite series ex-
gives pansion — only in the small-scale limit does it become
simple, as will be described in Section 4.3. To proceed
1= ∞W (k) dk = W (k) d3k = E (r,r)d3r, numerically, it is therefore convenient to discretize the
i (2π)3 i (2π)3 i problem. This reduces it to one similar to that of cosmic
Z0 Z Z
(10) microwave background (CMB) experiments: estimating a
so if we think of E (r,r′) as a matrix with indices r and powerspectrumgivennoisyfluctuationmeasurementsina
i
r′, this normalization condition is simply tr[E ]=1. numberofdiscrete“pixels”. Oncethepixelizationisdone,
i
As described in Hamilton (1997a),the window function the remaining steps are quite analogous to the CMB case
has a simple geometrical interpretation. Let us rewrite (T97), and involve mere matrix operations such as inver-
equation (8) as sion and diagonalization. These operations can often be
furthersimplifiedbyamoresuitablechoiceofpixelization.
∞
W (k)=4πk2 w (r)j (kr)dr, (11) LetusdefinetheoverdensityinN “pixels”x1,...,xN by
i i 0
Z0 n(r)
where the separation weighting is defined by xi ≡ n¯(r) −1 ψi(r)d3r (15)
Z (cid:20) (cid:21)
w (d) E (r,r′)δD(r r′ d)d3rd3r′. (12) for some set of functions ψ . We discuss specific choices
i i i
≡ | − |−
Z Z of ψi in some detail below. One generally strives either
to make these functions fairly localized in real space (in
TheseparationweightingistheaverageofE overallpairs
i
whichcase the pixelization is a generalizedformof counts
of points separated by a fixed distance d, weighted by the
in cells) or fairly localized in Fourier space (in which case
number of such pairs. Equation (11) shows that the only
we will refer to the functions ψ as “modes” and to x as
aspect of E that affects the window function is the sepa- i i
i
expansioncoefficients). Thereadershouldbe warnedthat
rationweighting,sincetwodifferentE thatgivethe same
i
thelattercaseisfarfromwhatwouldnormallybethought
w will produce identical window functions.
i
of as “pixels”, and that we will be using this terminology
An important special case, to which we return in the
nonethelesstostressthatalldiscretizationschemescanbe
next subsection, is that where E is of rank one, i.e., of
i
treated in a mathematically equivalent way.
the separable form
The“ 1”terminequation(15)simplysubtractsoffthe
E (r,r′)=ψ (r)ψ (r′)∗ (13) mean de−nsity. As we discuss in Section 5 and Appendix
i i i
B, one can and should choose the weight functions ψ to
i
for some ψi. In this case, equation (4) can be written as have zero mean, making this term irrelevant (note indeed
qi = xi 2, where xi = ψi(r)n(r)/n¯(r) (see equation (15) our use of this fact immediately preceding equation(14)).
| |
below), and the window function becomes simply This corresponds to requiring that
R
W (k)= ψ (k)2. (14)
i | i | ψi(r)d3r =0. (16)
(Here and throughout, hats denote Fourier transforms; Z
b
ψ(k) e−ik·rψ(r)d3r — see Appendix D.) We will see Letus groupthe pixelsx intoanN-dimensionalvector
i
≡
that both the traditional methods (Section 3.3) and the x. From equations (15) and (16) and a generalization of
R
KbL method (Section 3.5) are of this separable form, while the derivation of equation (5), T95 shows that
the quadratic data compression method that we present
in Section 3.6 is not. We will occasionally refer to the x =0, (17)
h i
formermethodsaslineardatacompression,sincetheybe- xx† =C N+S, (18)
gin by taking linear combinations of the data with weight h i ≡
functions ψi(r); cf., equation (15). The latter method, on where the shot noise covariance matrix is given by
the other hand, is intrinsically quadratic since q is not
i
merely the square of some quantity that is linear in the ψ (r)ψ (r)∗
N = i j d3r (19)
data. This is because its pair weighting E is optimized ij n¯(r)
to provide a minimum variance power estimator, which Z
makes E a quadratic form of rank greater than one. and the signal covariance matrix is
Toavoidconfusion,the readershouldbearinmindthat
when we distinguish between linear and quadratic meth- d3k
ods below,we arereferringto linear versusquadraticdata Sij = ψi(k)ψj(k)∗P(k)(2π)3. (20)
compression. Thepowerestimatorq isofcoursequadratic Z
i
in both cases. How should we choobse oubr ψ ’s to pixelize space? For a
i
pixelizationtobe useful,weclearlywantthe datasetxto
3.2. Pixelization
retainaslargeafractionaspossibleofthecosmologicalin-
Hamilton (1997a)has recently derived the functions E formationfromtheoriginaldataset(thegalaxypositions),
i
that provide the minimum-variance power spectrum esti- while simultaneously simplifying subsequent calculations.
mates for an arbitrary selection function and survey ge- We here list several natural options, most of which have
ometry. Unfortunately, this optimal weighting scheme is appeared in the literature.
5
6
3.2.1. Counts in cells (1992), Fisher et al. (1993), and Park et al. (1994). The
secondchoiceisusedwhendeterminingtheangularpower
Here one partitions the survey volume into N mutually
spectrumofasamplewithoutredshifts,e.g.,theAPMsur-
exclusive and collectively exhaustive volumes V , and de-
i
vey (Baugh & Efstathiou 1994). Then all galaxies by de-
fines ψ (r) = n¯(r) if r V , ψ (r) =0 elsewhere. Thus x
i i i i
∈ fault receive equal weight (moreover,modes can of course
issimply thenumberofgalaxiesobservedinV ,minus the
i
only be computed in the directions perpendicular to the
expectedaverage. Asetofusefulapproximationsforcom-
line of sight). The third choice, advocated by Feldman,
puting N and S for this case is derived in VS96. To keep
Kaiser & Peacock (1994, hereafter FKP), minimizes the
the number of cells from becoming intractably large, one
variance in the limit when k−1 the depth of the survey
might choose the cells to be larger in distant and poorly ≪
andis discussedindetailin Section4.3.1. HereP denotes
sampled regions of space than nearby.
ana priori guessofthe powerinthe bandunder consider-
Withthesesharp-edgedcells,anylinearcombinationsof
ation. The fourth choice (Tegmark 1995, hereafter T95),
pixels will correspond to a weight function that is discon-
gives the narrowest window function for a given variance,
tinuous at cell boundaries. To avoid power leakage prob-
where the constant γ determines the tradeoff.
lems that this can in principle cause, one might use cells
with “fuzzy” boundaries instead, for instance Gaussians
3.2.3. Spherical harmonic modes
as described in T95.
One cangreatlysimplify the computationofthe covari- The spherical wave choice
ance matrix by choosing all cells to have the same shape
and to be spherically symmetric (e.g., spheres or Gaus- ψi(r)=Yℓm(r)jℓ(knr), (27)
sians), sincethe resultingS willdepend onlyonthe sep-
ij
whereY isasphericalharmonicandj asphericalBessel
aration of the pixel centers and this correlation function ℓm b ℓ
function, is well-suited for full-sky redshift surveys, and
can be pre-computed and splined once and for all. For
has the advantage (Fisher, Scharf, & Lahav 1994; Heav-
the volume-limited case (n¯ constant), performing the ap-
ens & Taylor 1995) of greatly simplifying inclusion of the
propriateintegralsforidenticalsphericalcellsofradiusR,
effect of redshift-space distortions in the analysis.
separated by uR, gives
(2−u)2(4+u)n¯V if u<2, 3.2.4. Guessed eigenmodes
Nij = 0 16 if u 2, (21) In Section 3.5.1, we describe a set of smooth functions
(cid:26) ≥ known as continuum signal-to-noise eigenmodes. These
where V =4πR3/3. modes have a number of useful properties. In particu-
lar, there is an integer N such that the first N eigen-
3.2.2. Fourier modes modes retain virtually all the cosmological information
All of the above-mentioned pixels were fairly well- in the survey. If one has a reasonable guess as to the
localizedinrealspace. Tomakepixelsreasonablylocalized shape of these functions a priori, before computing them
exactly(they dependonthesurveygeometry),the firstN
in Fourier space, one can choose modes ψ that are plane
i
waves, tapered by some weight function φ to make them oftheseguessedmodesareobviouslyagoodchoiceforthe
square integrable: pixelizationfunctions ψi, since the amount of information
destroyed by the pixelization process will then be small.
ψ (r)=φ(r)eiki·r (22)
i
3.3. Traditional methods
for some grid points k in Fourier space. Choosing all
i Thetraditionalapproachhasbeentoestimatethepower
modes that are periodic in a box containing the survey
by simply squaring the pixels (q = x 2), choosing the
volume will ensure that this is a complete set, although i | i|
pixelstobeFouriermodesasdescribedabove. Thiscorre-
some high-frequency cutoff is of course necessary to keep
sponds to the pair weighting of equation (13), where ψ is
the number of pixels finite. Four different choices of the i
givenbyequation(22)andφisspecifiedbyequation(23),
volume weighting function φ have appeared in the litera-
(24), (25) or (26). Equation (5) shows that
ture:
φ(r)∝ 01oinustisdideesusurvrevyeyvovloulummee (23) hqii=φ(0)2+ φn¯((rr))2d3r+(2π1)3 |φ(k−ki)|2P(k)(2dπ3k)3,
(cid:26) Z Z (28)
φ(r) n¯(r) (24) b b
∝ the last term of which is simply the true power spec-
n¯(r)
φ(r) (25) trum convolved with the function φ(k)2. The 3D win-
∝ 1+n¯(r)P dow function is thus W (k) = φ(k| k| )2. Using this
i i
γ and Parseval’s theorem (or equ|atiobn−(10)|directly), one
φ(r) eigenfunction of 2 (26)
∝ ∇ − n¯(r) finds that the window function nbormalization constraint
(cid:20) (cid:21) W (k)d3k/(2π)3 =1 corresponds to simply
i
All are to be normalized so that the corresponding win-
dowfunctions integrateto unity– φ(r)d3r =1asshown R
φ(r)2d3r =1. (29)
below. Note that without careful choice of the k ’s, equa-
i
tion (16)will not be satisfiedingeRneralfor these pixeliza- Z
tions. The first choice, which weights all volume elements Thesesimple expressionshavefrequentlybeenusedinthe
in the survey equally, was employed by e.g. Vogeley et al. literature, but an annoying complication has often been
6
7
neglected. As described in Section 5, the fact that the the likelihood function at a dense grid of points in pa-
normalization of n¯ is not known a priori, but is deter- rameter space and rescales it to integrate to unity. These
mined from the observed galaxies (the so-called integral finalresultsareoftenillustratedincontourplots,asatthe
constraintproblem)canbe eliminated by choosingweight bottom of Figure 1.
functions ψ that are orthogonal to the monopole, i.e.,
i
suchthatψ (0)=0. Althoughthisiseasytoarrangewith 3.5. Linear (Karhunen-Lo`eve) method
i
thepixelizedmethodsdescribedbelow,thechoiceofequa- As will be discussed in Section 6, the traditional meth-
tion (22) dboes generally not havethis important property. ods generally destroy information, while the brute force
To obtain a correct answer with the traditional methods, method is lossless but computationally impractical. The
this must be corrected for. As shown in Appendix B, the Karhunen-Lo`eve(KL)method(Karhunen1947)maintains
power estimator the advantage of the brute-force method (indeed, it can
produce an essentially identical answer faster), and has
P˜ |xi|2−bi (30) additional useful features as well, as we detail below. It
i
≡ A was first introduced into large-scale structure analysis by
(cid:20) i (cid:21)
Vogeley & Szalay (see VS96), and it has also been suc-
is unbiased and incorporates the integral constraint cor- cessfully applied to Cosmic Microwave Background data
rection(when n¯ is normalizedso thatxi =0 for ki =0) if (e.g. Bunn 1995; Bond 1995; TTH; Bunn & White 1997;
the normalization factor Ai and the shot noise correction Jaffe, Knox, & Bond 1997). Like any method designed to
bi are given by minimizeerrorbars,theKL-techniquerequiresanapriori
assumptionfor the power spectrum. This is referredto as
2 the fiducial P(k). As will be described in sections 3.5.4,
φ(k ) φ(k )
A = 1+ i a(0) 2Re i a(k ) ,(31) 4.2 and 6.3.5, this is not a problem in practice, since a
i i
 (cid:12) φ(0) (cid:12)  − ( φ(0) ) badfiducialmodeldoesnotbiasthe result. Moreover,the
(cid:12)b (cid:12) b
(cid:12) (cid:12) measurementitselfcanbe usedasthefiducialmodelinan
 (cid:12) (cid:12)2
(cid:12)φb(k )(cid:12) φb(k ) iterative procedure if desired.
b = 1+ i b(0) 2Re i b(k ) ,(32)
i i We start by defining signal-to-noise eigenmodes in
 (cid:12) φ(0) (cid:12)  − ( φ(0) )
(cid:12)b (cid:12) b Section 3.5.1, before generalizing the technique in Sec-
(cid:12) (cid:12)
 (cid:12) (cid:12)  tion 3.5.2.
where the funct(cid:12)iobns a(cid:12)and b are defined bby
3.5.1. Signal-to-noise eigenmodes
a(k) φ(r)2eik·rd3r, (33) Thesignal-to-noiseeigenmodemethodconsistsofdefin-
≡
Z ing a new data vector
φ(r)2
b(k) eik·rd3r. (34)
≡ n¯(r) y Bx, (35)
Z ≡
If the survey is volume limited, then n¯ is independent of whereb,the rowsofthematrixB,arethe N eigenvectors
r, b(k)=a(k)/n¯, and b =A /n¯. of the generalized eigenvalue problem
i i
After computing power estimates q at a large grid of
i
points k , one finally takes some weighted averages of the Sb=λNb, (36)
i
q to obtain power estimates in some bands of (scalar) k
i
(= k). The problem of finding the weights that min- sortedfromhighestto lowesteigenvalueλandnormalized
imiz|e|the variance of the band power estimators is un- so that b†Nb=I. This implies that
fortunately quite a difficult one, and in general involves
y y =δ (1+λ ), (37)
solving a numerically unpleasant quadratic programming i j ij i
h i
problem (T95). For this reason, the customary approach
which means that the transformed data values y have the
hasbeentosimplygiveallq equalweightsinsomespher-
i desirable property of being statistically orthogonal, i.e.,
ical shells in k-space although, as described in VS96, this
uncorrelated. In the approximation that the distribution
is in general far from optimal.
function of x is a multivariate Gaussian, this also implies
thattheyarestatisticallyindependent—thenyismerely
3.4. Brute force likelihood method
avectorofindependentGaussianrandomvariables. More-
Let us parameterize the power spectrum P(k) by some over, since the eigenmodes diagonalize both S and N si-
parameter vector Θ as in Section 2.2. In the approxima- multaneously,equation(36)showsthat the eigenvaluesλ
i
tion that the probability distribution for the pixel vector can be interpreted as a signal-to-noise ratio S/N. Since
x is a multivariate Gaussian, it is given by equation (2) equation (37) shows that the quantity y2 1 on average
with mean m = 0. The maximum likelihood estimator of equals this signal-to-noise ratio, it is a usie−ful band power
Θ, denoted Θml, is simply that Θ-vector that maximizes estimatorwhennormalizedsothatitswindowfunctionin-
the likelihood f(x;Θ). This maximization problem can tegrates to unity. The window function is given by equa-
unfortunately notbe solvedanalyticallywhenthenumber tion (14) with ψ replaced by the continuous KL mode
i
of pixels exceeds one, so Θml is a complicated non-linear defined by
function of x that must be computed by solving the max- N
imization problem numerically. Since one generally wants ψ′(r) B ψ (r), (38)
i ≡ ij j
error bars on the estimate as well, one typically evaluates
j=1
X
7
8
since y = [n(r)/n¯(r) 1]ψ′(r)d3r. For the volume- b†Cb = 1 instead of b†Nb = 1, since this is more con-
i − i
limitedcase,thenoisepowerissimply1/n¯,sothecorrectly venient throughout the rest of the paper. The matrix ele-
normalized aRnd bias-corrected KL band power estimators mentsB arethus afactor(1+λ )1/2 smallerthaninthe
ij i
are simply “signal = noise signal-to-noise”,i.e., previoussubsection. As shownin TTH, solving the eigen-
×
valueequation(40)isusefulforanyparameterθ onwhich
i
y2 1 C depends, eventhose whichdo not affect only the power
q i − ; (39)
i ≡ n¯ spectrum (e.g., redshift-space distortions, cf., Appendix
C), and the signal-to-noise eigenmode method discussed
compare this with equation (30). Since the matrix B is above is just the special case where θ is the power nor-
i
invertible, the final data set y clearly retains all the in- malization. The three properties listed above continue to
formation that was present in x. In summary, the KL hold in the general case, and the “information content”
transformation partitions the information content of the in item 3 above now refers to the information about the
original data set x into N chunks that are parameter θ .
i
The KL method is a very general data analysis tool.
1. mutually exclusive (independent), Note that the eigenmodes continue to be mutually exclu-
sive and collectively exhaustive if we replace C, by any
i
2. collectively exhaustive (jointly retaining all the in-
symmetric matrix M in equation (40). To ensure that
formation), and
theKLmodesgivenarrowwindowfunctionsrangingfrom
small to large k, one can therefore choose M to be the
3. sorted from best to worst in terms of their informa-
signal covariance matrix S that would arise from some
tion content.
monotonically decreasing fiducial power spectrum, for in-
stanceP(k) k−3. Inthatcase,themodesbecomesorted
Typically, most of the KL coefficients y have a signal-to-
i ∝
by the ratio of “signal” in the fiducial power spectrum to
noise ratio λ 1, so that the bulk of the cosmologicalin-
formation is r≪etained in the first N′ coefficients, N′ N, noise;astheformerismonotonic,themodesaresortedby
≪ scale. We discuss this point further in Section 6.3.3.
which is why the KL method is often referredto and used
asdata compression. Onecanthusthrowawayallbutthe
3.5.3. KL modes are asymptotically
first N′ numbers y without any appreciable information
i pixelization-independent
loss, and this compresseddata set will still satisfy proper-
The pixelizations listed in Section 3.2 were all in some
ties 1 and 3 exactly, and 2 to a good approximation.
sense arbitrary, and generally somewhat redundant. The
Bunn (1995) and VS96 have pointed out that the S/N-
coefficients y are useful for power spectrum estimation KL modes can eliminate this arbitrariness, as mentioned
in Section 3.2.4. If the pixelization functions ψ formed a
since as long as the galaxy survey probes only scales i
complete set, spanning the space of all square-integrable
smaller than the peak in the power spectrum (cf., the dis-
cussioninSection6.3.3),thefirstN′ windowfunctionsW functions over the survey volume, the continuum KL
i
modes defined by equation (38) would be some smooth
have the following two properties:
functions ψ′(r) that were independent of the pixelization
i
1. They are narrow in k-space, used to compute them and depended only on P(k), n¯(r)
andthe geometryof the survey. In practice, the functions
2. Asiincreasesfrom1toN′,theyprobeallthescales ψ do not of course form a complete set, since one is lim-
i
accurately measured by the survey, from largest to ited to a finite number of pixels, but the continuum KL
smallest. modesψ′(r)cannonethelessbecomputednumerically. By
i
choosing a pixelization that can resolve all features down
Since they are also uncorrelated, these power spectrum tosomescaleR,equation(38)willaccuratelyapproximate
estimators therefore have all desirable properties that one allcontinuummodeswhosewindowfunctionsW (k)probe
i
may wish for: they distill the cosmological information onlyscalesk−1 R. Ifthenumberofpixelsareincreased
content of the data set into a set of mutually exclusive further to probe≫smaller scales, the first N′ modes remain
and collectively exhaustive chunks, which correspond to stable against this perturbation: the first N′ eigenvalues
the bandpowersin aset ofnarrowbands. Inthe approxi- and eigenvectors would only change by a small amount
mationthatyhasaGaussiandistribution,theprobability (Szalay & Vogeley 1997). The new modes, on the other
distributionsofthepowerestimatesqiaresimple: theyare hand, wouldrepresentthe small scalenoise probed by the
independentχ2 distributionswithonedegreeoffreedom.5 new pixel scale. By using the functions ψ′(r) to pixelize
i
the data, i = 1,...,N, one can thus make sure that all of
3.5.2. General KL modes the cosmologicalsignaldownto the scaleprobedbyψ′ is
N
Let us write C = θ S+N, where the power spectrum retained.
i
normalization parameter θ = 1 in the fiducial model.
i 3.5.4. Using KL modes for trouble spotting
Since S = C, ∂C/∂θ , equation (36) can be rewritten
i i
as (TTH) ≡ Iftheassumedpowerspectrummodeliscorrect,theKL
C, b=λ′Cb, (40) coefficientsyidefinedbyequation(35)willbeindependent
i
Gaussian random variables with zero mean and unit vari-
where λ′ = λ/(1+λ) or, equivalently, λ = λ′/(1 λ′). ance. This offers an efficient way of testing whether the
−
From now on, we will normalize the eigenvectors so that data are inconsistent with this model. The detection of,
5To prevent power spectrum plots frombecoming too cluttered withpoints with large error bars, it is convenient to combine neighboring
bandpowerestimateswithaweightedaverage—thesebroaderbandpowerswillofcoursestillbeuncorrelatedsincealltheqi are.
8
9
say,a6 σoutlier(y >6)wouldprovidestrongevidence k , where
i i+1
− | |
that there is either more variance on the scale probed by
theithmodethanthefiducialpowerspectrumassumed,or 0=k1 <k2 <...<kN′+1 = , (42)
∞
that the probability distribution is strongly non-Gaussian
on that scale. Even if it goes undetected, an incorrect and groupthem into an N′-dimensionalvector p. For the
assumed model does not bias the estimate of the power method to be strictly lossless, these bands should be cho-
spectrum, as discussed below in Section 6.3.5. sentobequitenarrowcomparedtothescalesonwhichthe
power spectrum varies. One first computes a compressed
3.5.5. Using KL modes for linear filtering data vector q whose N′ elements are quadratic functions
of the data set x. These are defined by
TheKLeigenmodeshaveanadditionaluse. Theprocess
of throwing away the eigenmodes with low signal-to-noise 1
ratiossplitsthespaceofallpossibledensityfieldsgiventhe qi z†C,iz, (43)
≡ 2
data into two — one subspace that mostly contains noise,
and one that is dominated by our generalized signal. The where the vector z is given by
two are statistically orthogonal to one another. The ex-
pansion of a given dataset over the signal subspace will z C−1x, (44)
≡
substantially reduce the noise, thus representing a useful
linearfilteringofthe data(cf., Seljak1997). Since itmax- and the matrix C,i is defined by
imizes the signal for a given number of included modes,
the KL transform is sometimes referred to as “optimal d3k
(C ) ψ (k)ψ (k)∗ . (45)
sduabtaspsaecte filtering”. More generally,let us define a filtered ,i ab ≡Zki<|k|<ki+1 a b (2π)3
N That is, C is the derivativebof thbe covariance matrix C
x′ (CB†) w y (41) ,i
i ≡ ij j j (equation (18)) with respect to the normalization of the
j=1 ith band, in the limit of narrow bands. Rewriting this as
X
for some weights w . The generalized eigenvector orthog-
j 1
onality relation gives BCB† =I (cf., VS96, TTH), which qi x†Eix, (46)
≡ 2
implies that C−1 = B†B and CB†B = I. Hence equa-
tion (35) gives CB†y = x, which implies that we recover where
theoriginaldataset(x′ =x)inequation(41)ifwechoose E C−1C, C−1, (47)
i i
theweightsw =1. Anothersimpleexampleistheoptimal ≡
j
subspace filtering mentioned above, which corresponds to we see that the matrix Ei is simply a discrete version of
the choice wj = 1 for j N′, wj = 0 otherwise. Finally, the pair weightingfunction Ei(r,r′) inequation(4). Note
it is easy to show that t≤he choice w = λ′ gives Wiener that it is not separable, i.e., it has rank greater than one.
j j
For the Gaussian case, the Fisher information matrix
filtering, whichis definedbyx′ =SC−1x. Inotherwords,
for p defined by equation (3) reduces to (VS96, TTH)
Wiener filtering becomes diagonal in the KL basis, since
it diagonalizes S and C simultaneously. Indeed, Wiener
1
filtering is but one of many linear filters that are straight- F = tr C−1C C−1C , (48)
ij ,i ,j
2
forward to implement in the KL basis.
(cid:2) (cid:3)
and T97 shows that both the mean and the covariance of
3.6. Quadratic method q are given in terms of F:
In both traditional methods and the pixelized KL tech-
q =Fp, (49)
nique,thepowerspectrumestimatesq aresomequadratic
i h i
functions of the observed density field n(r), i.e., of the qqt q q t =F. (50)
h i−h ih i
form given by equation (4). Hamilton (1997a) adopted
a more ambitious approach, and computed the unbiased This means that F−1q is an optimal estimator of p, since
quadratic power estimators that have minimal variance, its covariance matrix is precisely the inverse of the Fisher
using a series expansion. T97 subsequently showed that matrix. Moreover,as showninT97,compressingthe data
these estimators are unbeatable; their Fisher informa- set x into the coefficients q for some sufficiently narrow
tion matrix is identical to that of the raw data, so no bands is a strictly lossless procedure, retaining all the in-
non-quadratic unbiased estimators can give smaller vari- formation about those cosmological parameters that af-
ance. Moreover, they can be computed without recourse fect galaxy clustering through the power spectrum alone.
tothenumericallycumbersomeseriesexpansionofHamil- Equation (49) shows that, in terms of the band powers,
ton (1997a) (c.f. T97 and Knox, Bond, & Jaffe 1997 for thewindowfunctionsW ofequation(8)forthecoefficients
applications to CMB observations). Here we show how qi are simply proportional to the rows of the ubiquitous
this method can be applied to galaxy surveys, and its re- Fisher matrix.
lation to the KL method. Just as for the KL technique, a The coefficients F−1q tend to be both correlated and
fiducial power spectrum is assumed. noisy, and therefore it is better to work with the trans-
Wewillparameterizethepowerspectrumasapiecewise formed coefficients defined by
constant function with N′ “steps” of height p , which we
i
term the band powers. Thus P(k) = p for k k < p˜ F−1/2q, (51)
i i
≤ ≡
9
10
renormalized so that their window functions integrate to 1997). This is because the quadratic method can be de-
unity. Here F1/2 denotes the symmetric matrix whose rived by expanding the logarithmic likelihood lnf to sec-
squareisF–itisreadilycomputedbydiagonalization. As ond order around the fiducial point and maximizing it.
shown by Tegmark & Hamilton (1997), these coefficients The iteration thus seeks the maximum of lnf by repeat-
are all uncorrelated (multiply equation (50) on both the edly approximating it with a parabola. This is essentially
right and the left by F−1/2 to see this), and moreover the maximum-gradient method for maximization, which
tend to be very well-behaved numerically, with narrow is known to have excellent convergence properties. In the
non-negative window functions (the rows of F1/2) span- one-dimensionalcase,thisisequivalenttofinding thezero
ning the entire range of scales probed by the survey. of (lnf)′ by repeated linear approximations. This is sim-
ply the Newton-Raphson root-finding method, known to
4. RELATIONSBETWEENTHEMETHODS convergeexponentiallyfastandasymptoticallydoublethe
4.1. Relation between KL and quadratic method number of correct decimals in each iteration.
Although the ML-result can be computed by iterating
As we will nowshow, the linear and quadraticpixelized
the quadratic method, it should be stressed that this is
methods arecloselyrelated— the latter is simply a faster
notnecessarilya goodthing to do. If the measuredpower
way of computing the same band power estimates.
spectrumisnoisy,sothatthebandpoweruncertaintiesdo
Letusoptimize ourKLmodestoestimate nottheover-
not satisfy ∆P P, then iteration can produce mislead-
allpowernormalization,butthebandpowerinbandi. As ≪
inglysmallerrorbars,asthefollowingexampleillustrates.
we showed in Section 3.5.5,
Supposesamplevarianceorashotnoisefluctuationmakes
abandpowermeasurementtentimessmallerthanthetrue
C−1 =B†B. (52)
value and this is used as the new fiducial band power P
(prior). Then equation (62) shows that the nominal ∆P
Introducing the diagonal matrix Λ diag λ′ , equa-
≡ { j} from sample variance will be ten times too small as well.
tion (40) implies that
If iteration is nonetheless desired, it is crucial not to let
the priorover-fitthe data. Abetter approachthansimple
BC B† =Λ. (53)
,i iteration is therefore to use as prior a fit to the measured
P(k) with a smooth parameterized fitting function, and
We are partially suppressing the index i here for simplic-
keep adding fitting parameters until the the value of χ2
ity, although the eigenvectors in B and the eigenvalues
per degree of freedom drops below unity (Tegmark 1997a;
in Λ of course depend on which power band i we opti-
Seljak 1997).
mizefor. Equations(35),(52)and(53)allowustorewrite
equation (43) as
4.3. Relation between quadratic and FKP methods: the
2q =x†C−1C C−1x=x†B†BC B†Bx small scale limit
i ,i ,i
Fora traditionalmethodwithavolumeweightingfunc-
=y†Λy= λ′y2, (54)
j j tion φ(r), we define a quantity L implicitly by
j
X
d3k 1
i.e., as a weighted average of the squared KL coefficients φ(k)2 = , (55)
yj,withweightsgivenbytheKLeigenvaluesλ′j. Thusthe ZkL<1| | (2π)3 2
coefficients are weighted by their inverse variance, which
b
meansthatthisistheminimum-variancebandpoweresti- i.e., L−1 istheradiusofasphereinFourierspacecontain-
mator based on the squared KL coefficients, so after sub- inghalfofthek=0windowfunction. Forsimplevolumes
tractingoffshotnoiseandnormalizingcorrectly,thelinear such as a pencil beams, slices or spheres, L is of the order
(KL) and quadratic methods give exactly the same esti- of the width of the survey in its narrowest direction. We
mator for the band power. As described in Section 6.4, willnowshowthatinthesmall-scalelimitwherek−1 L,
≪
the quadratic estimator is is simply faster to compute. A the quadratic method reduces to the FKP method, which
moreintuitivewayofunderstandingwhythetwomethods implies thatthe latter islosslessandhence unbeatable for
give the same result is to note that the quadratic method measuring the power spectrum on the smallest scales, as
wasderivedexplicitlytobetheminimumvarianceestima- was pointed out by Hamilton (1997a).
tor (T97; cf., equation (49)), and that VS96 showed that
the KL approach is guaranteed to minimize the variance 4.3.1. Derivation of the FKP method
as well.
The FKP method (Feldman, Kaiser& Peacock1994)is
the traditionalmethod described in Section 3.3, using the
4.2. Relation between quadratic and ML methods:
volumeweightingofequation(25). Therankonepoweres-
iteration
timates x2 areaveragedwithequalweights6forallmodes
| a|
If the quadratic method is repeated using the output in a spherical shell k = k , so integrating equation (13)
a
| |
(measured) power spectrum as the input (fiducial) power over this shell using equation (22) for the pixels, we see
spectrum, then this iteration will eventually converge to that this corresponds to the pair weighting
thepowerspectrumthatwouldbeobtainedwiththebrute
force maximum-likelihood method (Bond, Jaffe, & Knox E (r,r′)=φ(r)φ(r′)j (k r r′ ), (56)
i 0 i
| − |
6Anintuitivewaytounderstandwhyalldirectionsink-spaceshouldreceiveequalweightwhen∆k≫L−1 istonotethatinthislimit,the
number of coherence volumes that fit into a given solid angle in the shell is independent of its shape (and hence independent of direction in
k-space),beingmerelytheratiooftheshellsubvolumetothe coherence volume.
10