# The Three-Dimensional Power Spectrum of Galaxies from the Sloan Digital Sky Survey

Max Tegmark1 , Michael R. Blanton1 , Michael A. Strauss1 , Fiona Hoyle1 , David Schlegel1 , Roman Scoccimarro1 , Michael S. Vogeley1 , David H. Weinberg1 , Idit Zehavi1 , Andreas Berlind1 , Tamás Budavari1 , Andrew Connolly1 , Daniel J. Eisenstein1 , Douglas Finkbeiner1 , Joshua A. Frieman1 1 , James E. Gunn1 , Andrew J. S. Hamilton1 , Lam Hui1 , Bhuvnesh Jain1 , David Johnston1 1 , Stephen Kent1 , Huan Lin1 , Reiko Nakajima1 , Robert C. Nichol1 , Jeremiah P. Ostriker1 , Adrian Pope1 , Ryan Scranton1 , Uroš Seljak1 , Ravi K. Sheth1 , Albert Stebbins1 , Alexander S. Szalay1 , István Szapudi1 , Licia Verde1 , Yongzhong Xu1 , James Annis1 , Neta A. Bahcall1 , J. Brinkmann1 , Scott Burles1 , Francisco J. Castander1 , Istvan Csabai1 , Jon Loveday1 , Mamoru Doi1 , Masataka Fukugita1 , J. Richard Gott III1 , Greg Hennessy1 , David W. Hogg1 , Željko Ivezić1 , Gillian R. Knapp1 , Don Q. Lamb1 , Brian C. Lee1 , Robert H. Lupton1 , Timothy A. McKay1 , Peter Kunszt1 , Jeffrey A. Munn1 , Liam O’Connell1 , John Peoples1 , Jeffrey R. Pier1 , Michael Richmond1 , Constance Rockosi1 , Donald P. Schneider1 , Christopher Stoughton1 , Douglas L. Tucker1 , Daniel E. Vanden Berk1 , Brian Yanny1 , Donald G. York1 1 , for the SDSS Collaboration
1affiliationmark:
###### Abstract

We measure the large-scale real-space power spectrum using a sample of 205,443 galaxies from the Sloan Digital Sky Survey, covering 2417 effective square degrees with mean redshift . We employ a matrix-based method using pseudo-Karhunen-Loève eigenmodes, producing uncorrelated minimum-variance measurements in 22 -bands of both the clustering power and its anisotropy due to redshift-space distortions, with narrow and well-behaved window functions in the range . We pay particular attention to modeling, quantifying and correcting for potential systematic errors, nonlinear redshift distortions and the artificial red-tilt caused by luminosity-dependent bias. Our results are robust to omitting angular and radial density fluctuations and are consistent between different parts of the sky. Our final result is a measurement of the real-space matter power spectrum up to an unknown overall multiplicative bias factor. Our calculations suggest that this bias factor is independent of scale to better than a few percent for , thereby making our results useful for precision measurements of cosmological parameters in conjunction with data from other experiments such as the WMAP satellite. The power spectrum is not well-characterized by a single power law, but unambiguously shows curvature. As a simple characterization of the data, our measurements are well fit by a flat scale-invariant adiabatic cosmological model with and for galaxies, when fixing the baryon fraction and the Hubble parameter ; cosmological interpretation is given in a companion paper.

large-scale structure of universe — galaxies: statistics — methods: data analysis

Σ

\submitted

Submitted to the Astrophysical Journal June 18 2003, report received Sept. 21, resubmitted Oct. 27

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Department of Physics, University of Pennsylvania, Philadelphia, PA 19101, USA 2222Center for Cosmology and Particle Physics, Department of Physics, New York University, 4 Washington Place, New York, NY 10003 3333Princeton University Observatory, Princeton, NJ 08544, USA 4444Department of Physics, Drexel University, Philadelphia, PA 19104, USA 5555Department of Astronomy, Ohio State University, Columbus, OH 43210, USA 6666Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, IL 60510, USA 7777Center for Cosmological Physics and Department of Astronomy & Astrophysics, University of Chicago, Chicago, IL 60637, USA 8888Department of Physics and Astronomy, The Johns Hopkins University, 3701 San Martin Drive, Baltimore, MD 21218, USA 9999University of Pittsburgh, Department of Physics and Astronomy, 3941 O’Hara Street, Pittsburgh, PA 15260, USA 10101010Department of Astronomy, University of Arizona, Tucson, AZ 85721, USA 11111111JILA and Dept. of Astrophysical and Planetary Sciences, U. Colorado, Boulder, CO 80309, USA, Andrew.H 12121212Department of Physics, 5000 Forbes Avenue, Carnegie Mellon University, Pittsburgh, PA 15213, USA 13131313Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USA 14141414Apache Point Observatory, 2001 Apache Point Rd, Sunspot, NM 88349-0059, USA 15151515Dept. of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 16161616Institut d’Estudis Espacials de Catalunya/CSIC, Gran Capita 2-4, 08034 Barcelona, Spain 17171717Sussex Astronomy Centre, University of Sussex, Falmer, Brighton BN1 9QJ, UK 18181818Inst. for Cosmic Ray Research, Univ. of Tokyo, Kashiwa 277-8582, Japan 19191919U.S. Naval Observatory, Flagstaff Station, Flagstaff, AZ 86002-1149, USA 20202020Dept. of Physics, Univ. of Michigan, Ann Arbor, MI 48109-1120, USA 21212121Physics Dept., Rochester Inst. of Technology, 1 Lomb Memorial Dr, Rochester, NY 14623, USA 22222222Dept. of Astronomy and Astrophysics, Pennsylvania State University, University Park, PA 16802, USA 23232323Enrico Fermi Institute, University of Chicago, Chicago, IL 60637, USA

## 1 Introduction

The spectacular recent cosmic microwave background (CMB) measurements from the WMAP satellite (Bennett et al. 2003) and other experiments have increased the importance of non-CMB measurements for the endeavor to constrain cosmological models and their free parameters. These non-CMB constraints are crucially needed for breaking CMB degeneracies (Eisenstein et al. 1999; Efstathiou & Bond 1999; Bridle et al. 2003); for instance, WMAP alone is consistent with a closed universe with Hubble parameter and no cosmological constant (Spergel et al. 2003; Verde et al. 2003). Yet they are currently less reliable and precise than the CMB, making them the limiting factor and weakest link in the quest for precision cosmology. Much of the near-term progress in cosmology will therefore be driven by reductions in statistical and systematic uncertainties of non-CMB probes such as Lyman forest and galaxy clustering and motions, gravitational lensing, cluster studies, and supernovae Ia distance determinations. Galaxy redshift surveys can play a key role in breaking degeneracies and providing cross checks (Tegmark 1997a; Goldberg & Strauss 1998; Wang et al. 1999; Eisenstein et al. 1999), but only if systematics can be controlled to high precision. The goal of the present paper is to do just this, using over 200,000 galaxies from the Sloan Digital Sky Survey (SDSS; York et al. 2000) to measure the shape of the real-space matter power spectrum , accurately quantifying and correcting for the effects of light-to-mass bias, redshift space distortions, survey geometry effects and other complications.

The cosmological constraining power of three-dimensional maps of the Universe provided by galaxy redshift surveys has motivated ever more ambitious observational efforts such as the CfA/UZC (Huchra et al. 1990; Falco et al. 1999), LCRS (Shectman et al. 1996), and PSCz (Saunders et al. 2000) surveys, each well in excess of galaxies. The current state of the art is the AAT two degree field galaxy redshift survey (2dFGRS; Colless et al. 2001; Hawkins et al. 2003; Peacock 2003 and references therein). Analysis of the first 147,000 2dFGRS galaxies (Peacock et al. 2001; Percival et al. 2001, 2002; Norberg et al. 2001, 2002; Madgwick et al. 2002) have supported a flat dark-energy dominated cosmology, as have angular clustering analyses of the parent catalogs underlying the 2dFGRS (Efstathiou & Moody 2001) and SDSS (Scranton et al. 2002; Connolly et al. 2002; Tegmark et al. 2002; Szalay et al. 2003; Dodelson et al. 2002). Tantalizing evidence for baryonic wiggles in the galaxy power spectrum is presented by Percival et al. (2001) and Miller et al. (2001a,b, 2002), and cosmological models have been further constrained in conjunction with cosmic microwave background (CMB) data (e.g., Spergel et al. 2003; Verde et al. 2003; Lahav et al. 2002).

The SDSS is the most ambitious galaxy redshift survey to date, whose goal, driven by large-scale structure science, is to measure of order galaxy redshifts. Zehavi et al. (2002) computed the correlation function using about 30,000 galaxies from early SDSS data (Stoughton et al. 2002). In conjunction with the first major SDSS data release in 2003 (hereafter DR1; Abazajian et al. 2003), a series of papers will address various aspects of the 3D clustering of a much larger data set involving over 200,000 galaxies with redshifts. This paper is focused on measuring the power galaxy spectrum on large scales, dealing with complications such as luminosity-dependent bias and redshift distortions only to the extent necessary to recover an undistorted measurement of the real-space matter power spectrum. Zehavi et al. (2003a) measure and model the real space correlation function, mainly on smaller scales, focusing on departures from power-law behavior, and Zehavi et al. (2003b) will study how the correlation function depends on galaxy properties. Pope et al. (2003) measure the parameters which characterize the large-scale power spectrum with a complementary approach involving direct likelihood analysis on Karhunen-Loève eigenmodes, as opposed to the quadratic estimator technique employed in the present paper.

This paper is organized as follows. In Section 2, we describe the SDSS data used and how we model it; the technical details can be found in Appendix A. In Section 3, we describe our methodology and present our basic measurements of both the power spectrum and its redshift-space anisotropy. The details of the formalism for doing this are described in Appendix B. In Section 4 we focus on this anisotropy to model, quantify and correct for the effects of redshift-space distortions, producing an estimate of the real-space galaxy power spectrum and testing our procedure with Monte-Carlo simulations. In Section 5, we model, quantify and correct for the effects of luminosity-dependent biasing, producing an estimate of the real-space matter power spectrum. In Section 6, we test for a variety of systematic errors. In Section 7, we discuss our results. The cosmological interpretation of our measurements is given in a companion paper (Tegmark et al. 2003, hereafter “Paper II”).

## 2 Data and data modeling

The SDSS uses a mosaic CCD camera (Gunn et al. 1998) to image the sky in five photometric bandpasses denoted , , , , 111 The Fukugita et al. (1996) paper actually defines a slightly different system, denoted , , , , , but SDSS magnitudes are now referred to the native filter system of the 2.5m survey telescope, for which the bandpass notation is unprimed. (Fukugita et al. 1996). After astrometric calibration (Pier et al. 2003), photometric data reduction (Lupton et al. 2003, in preparation; see Lupton et al. 2001 and Stoughton et al. 2002 for summaries) and photometric calibration (Hogg et al. 2001; Smith et al. 2002), galaxies are selected for spectroscopic observations using the algorithm described by Strauss et al. (2002). To a good approximation, the main galaxy sample consists of all galaxies with -band apparent Petrosian magnitude ; see Appendix A. Galaxy spectra are also measured for a luminous red galaxy sample (Eisenstein et al. 2001), for which clustering results will be reported in a separate paper. These targets are assigned to spectroscopic plates by an adaptive tiling algorithm (Blanton et al. 2003) and observed with a pair of fiber-fed CCD spectrographs (Uomoto et al., in preparation), after which the spectroscopic data reduction and redshift determination are performed by automated pipelines (Schlegel et al., in preparation; Frieman et al., in preparation). The rms galaxy redshift errors are 30 km/s and hence negligible for the purpose of the present paper.

Our analysis is based on SDSS sample11 (Blanton et al. 2003c), consisting of the 205,443 galaxies observed before July 2002, all of which will be included in the upcoming SDSS Data Release 2. From this basic sample, we produce a set of subsamples as specified in Table 1. The details of how this basic sample was processed, modeled and subdivided are given in Appendix A. The bottom line is that each sample is completely specified by three entities:

1. The galaxy positions (a list of RA, Dec and comoving redshift space distance for each galaxy)

2. The radial selection function , which gives the expected (not observed) number density of galaxies as a function of distance

3. The angular selection function , which gives the completeness as a function of direction in the sky

Our samples are constructed so that their three-dimensional selection function is separable, i.e., simply the product of an angular and a radial part; here and are the comoving radial distance and the unit vector corresponding to the position . The conversion from redshift to comoving distance was made for a flat cosmological model with a cosmological constant — below we will see that our results are insensitive to this assumption.

### 2.1 Angular selection function

The angular selection function is shown in

or the baseline sample, it covers a sky area of 2499 square degrees. The function is defined to be the completeness, i.e., the probability that a galaxy satisfying the sample cuts actually gets assigned a redshift (including the 6% of the total which are determined based on the nearest neighbor redshift as described in Appendix A). Therefore the completeness is a dimensionless number between zero and one. The effective area is square degrees, corresponding to an average completeness of . As detailed in Appendix A.2, we model as a piecewise constant function. We specify this function by giving its value in each of a large number of disjoint spherical polygons, within each of which it takes a constant value. There are 2914 such polygons for the baseline sample, encoding the geometric boundaries of spectroscopic tiles, holes and other relevant entities. Figure 1 shows that the sky coverage naturally separates into three fairly compact regions of comparable size: north of the Galactic plane (in the center of the figure), there is one region on the celestial equator and another at high declination; south of the Galactic plane, there is a set of three stripes near the equator. For the purpose of testing for systematic errors, we define angular subsamples A1, A3 and A4 corresponding to these regions (see Table 1), which have effective areas of 809, 1007 and 600 square degrees, respectively.

Our estimate of the radial selection function for the baseline sample is shown in

ogether with a histogram of the galaxy distances. The full details of the derivation of the radial selection function can be found in Appendix A.4, including both evolution and K-corrections. Our basic sample has magnitude limits at the bright end (since the survey becomes incomplete for bright galaxies with large angular size) and at the faint end (Appendix A.4). Figure 5: Same as Figure 6: u t plotted as comoving number density. The grey (background) histogram shows the full flux-limited sample and the others show the volume-limited subsamples, with lines indicating their predicted constant selection functions.

Figure 8 shows all galaxies within of the equator in a pie diagram, color coded by their absolute magnitude, and illustrates one of the fundamental challenges for our project (and indeed for the analysis of any flux-limited sample): luminous galaxies dominate the sample at large distances and dim ones dominate nearby. A measurement of the power spectrum on very large scales is therefore statistically dominated by luminous galaxies whereas a measurement on small scales is dominated by dim ones (since they have much higher number density). Yet it is well-known that luminous galaxies cluster more than dim ones (e.g., Davis et al. 1988; Hamilton 1988; Norberg et al. 2001; Zehavi et al. 2002; Verde et al. 2003), so when comparing on large and small scales we are in effect comparing apples with oranges, and may mistakenly conclude that the power spectrum is red-tilted (with a spectral index , say) even if the truth is . So far, no magnitude-limited galaxy power spectrum analysis has been corrected for this effect. We do so in Section 5. Although this effect is is not large in an absolute sense, we find that it must nonetheless be included for precision cosmology applications.

The first step is to quantify the luminosity-dependence of bias. For this purpose, we define a series of volume-limited samples as specified in Table 1, constructed by discarding all galaxies that are too faint to be included at the far limit or too bright to be included at the near limit. This gives a radial selection function (

hat is constant within the radial limits and zero elsewhere. The radial limits are set so that galaxies at the far (near) radial limit and the dim (luminous) end of the absolute magnitude range in question have fluxes at the faint (bright) flux limits, respectively. Because the flux range spans exactly three magnitudes, these subsamples overlap spatially only with their nearest neighbor samples, and have a near limit that would be equal to the far limit of the sample that is two notches more luminous if it were not for evolution and K-corrections. This is clear in Figures 9 and 10. These samples have the advantage that the measured clustering is that of a well-defined set of objects whose selection is redshift-independent. Although we have not accounted for our surface brightness limits in defining these samples, very few of even the lowest luminosity galaxies in our sample are affected by the surface-brightness limits of the survey (Blanton et al. 2002c; Strauss et al. 2002).

## 3 Method and basic analysis

We now turn to our basic goal: accurately measuring the shape of the matter power spectrum on large scales using the data described above, i.e., measuring a curve that equals the large-scale matter power spectrum up to an unknown overall multiplicative bias factor that is independent of scale. This involves four basic challenges:

1. Accounting for the complicated survey geometry

2. Correcting for the effect of redshift-space distortions

3. Correcting for bias effects, which as described in Section 5 cause an artificial red-tilt in the power spectrum

4. Checking for potential systematic errors

Before delving into detail, let us summarize each of these challenges and how we will tackle them.

### 3.1 Battle plan

#### 3.1.1 Survey geometry and method of estimating power

It is well-known that since galaxies in a redshift survey probe the underlying density field only in a finite volume, the power spectrum estimated with traditional Fourier methods (e.g., Percival et al. 2001; Feldman, Kaiser & Peacock 1994), is complicated to interpret: it corresponds to a smeared-out version of the true power spectrum, can underestimate power on the largest scales due to the so-called integral constraint (Peacock & Nicholson 1991) and has correlated errors. We therefore measure power spectra with an alternative, matrix-based approach which, although more numerically demanding, has several advantages on the large scales that are the focus of this paper. It facilitates tests for radial and angular systematic errors. If galaxies were faithful tracers of mass, then it would produce unbiased minimum-variance power spectrum measurements with uncorrelated error bars that are smaller than those from traditional Fourier methods. The power smearing is quantified by window functions that are both narrower than with traditional Fourier methods and can be computed without need for approximations or Monte-Carlo simulations. (We do, however, use Monte Carlo simulations in Section 4 to verify that the method and software work as advertised.)

#### 3.1.2 Redshift space distortions

Our basic input data consist of galaxy positions in three-dimensional “redshift space”, where the comoving distance is that which would explain the observed redshift if the galaxy were merely following the Hubble flow of the expanding Universe. The same gravitational forces that cause galaxies to cluster also cause them to move relative to the Hubble flow, and these so-called peculiar velocities make the clustering in redshift space anisotropic (Kaiser 1987; Hamilton 1998). Although this effect can be modeled and accounted for exactly on very large scales on which the clustering is linear (Kaiser 1987), nonlinear corrections cannot be neglected on some of the scales of interest to us (Scoccimarro et al. 2001; Seljak 2001; Cole et al. 1994; Hatton & Cole 1998). Section 4 is devoted to dealing with this complication, going beyond the Kaiser approximation with a three-pronged approach:

1. We precede our power spectrum analysis by a nonlinear “finger-of-God” compression step with a tunable threshold, to quantify the sensitivity of our results to nonlinear galaxy groups and clusters.

2. We measure three power spectra (galaxy-galaxy, galaxy-velocity and velocity-velocity spectra) rather than one, quantifying the clustering anisotropy and allowing the real-space power to be reconstructed beyond linear order.

3. We analyze an extensive set of mock galaxy catalogs to quantify the accuracy of our results and measure how the non-linear correction factor grows toward smaller scales.

Our mock analysis will also allow us to quantify the effects of nonlinear clustering on the error bars and band-power correlations. Step 1 is optional, and we present results both with and without it.

#### 3.1.3 Bias

All we can ever measure with galaxy redshift surveys is galaxy clustering, whereas what we care about for constraining cosmological models is the clustering of the underlying matter distribution. Our ability to do cosmology with the real-space galaxy power spectrum is therefore only as good as our understanding of bias, i.e., the relation of to the matter power spectrum . Pessimists have often argued that since we do not understand galaxy formation at high precision, we cannot understand bias accurately either, and so galaxy surveys will be relegated to a historical footnote, having no role to play in the precision cosmology era. Optimists retort that no matter how complicated the gas-dynamical and radiative processes involved in galaxy formation may be, they have only a finite spatial range (a few Mpc, say), leading to a generic prediction that bias on much larger scales (Mpc, say) should be scale-independent for any particular type of galaxy (Coles 1993; Fry and Gaztañaga 1993; Scherrer & Weinberg 1998; Mann, Peacock & Heavens 1998; Coles, Melott & Munshi 1999; Heavens, Matarrese & Verde 1999; Blanton et al. 2000; Narayanan et al. 2000). This theoretical expectation is supported by visual inspection of the galaxy distribution. Comparing early and late type galaxies in the 2dF galaxy redshift survey (Peacock 2003; Madgewick et al. 2003) shows that whereas the small-scale distribution differs (ellipticals display a more “skeletal” distribution than do cluster-shunning spirals), their large-scale clustering patterns are indistinguishable.

We will devote Section 5 to the bias issue, arguing that both the pessimists and the optimists have turned out to be right: yes, biasing is indeed complicated on small scales (where the galaxy power spectrum will therefore teach us more about galaxy formation than about cosmology) but no, this in no way prevents us from doing precision cosmology with the galaxy power spectrum on very large scales. Our main tool will be analyzing our volume-limited magnitude subsamples, showing that their large-scale power spectra are consistent with all having the same shape and differing merely in amplitude.

Although the argument above for scale-independent bias holds only for a volume-limited subsample, we wish to use our full galaxy sample over a broad range of redshifts, both to expand the range of -scales probed and to reduce shot noise. We will therefore use our measured luminosity-dependence of bias to compute and remove the artificial red tilt in our full magnitude-limited baseline sample.

In future papers, we will constrain galaxy bias empirically using clustering measurements on smaller scales (e.g., Zehavi et al. 2003), which will allow us to calculate the effects of scale-dependent bias on the power spectrum in the non-linear regime, and thus to extend the measurement of the matter power spectrum shape to smaller scales.

#### 3.1.4 Systematic errors

As the old saying goes, the devil you know poses a lesser threat than the devil you don’t. We will therefore devote Section 6 to testing for the sort of effects that are not included in our Monte Carlo simulations. This includes both radial modulations (due to mis-estimates of evolution or -corrections) and angular modulations (due to effects such as uncorrected dust extinction, variable observing conditions, photometric calibration errors and fiber collisions). Our tests use two basic approaches:

1. Analyzing subsets of galaxies: we compare the power spectra from different parts of the sky (subsamples A1-A4 from Table 1) and different distance ranges (subsamples R1-R3) looking for inconsistencies.

2. Analyzing subsets of modes: we look for excess power in purely angular and purely radial modes of the density field, which act like lightning rods for angular and radial modulations such as those mentioned above.

### 3.2 Three Step Power Spectrum Estimation

Our matrix-based power spectrum estimation approach is described in Tegmark et al. (1998). It starts by expanding the galaxy density field in a set of functions known as Pseudo-Karhunen-Loève eigenmodes. This step compresses the data set into a much smaller size (from hundreds of thousands of galaxy coordinates to a few thousand expansion coefficients) while retaining the large-scale cosmological information in which we are interested. It also reduces the power spectrum estimation problem to a mathematical form equivalent to that encountered in CMB analysis, enabling us to take advantage of a powerful set of matrix-based tools that have been fruitfully employed in the CMB field. Our basic analysis in the remainder of this section therefore consists of the following three steps:

1. Finger-of-god compression to remove redshift-space distortions due to virialized structures; Section 3.3

2. Pseudo-Karhunen-Loève eigenmode expansion; § 3.4

3. Power spectrum estimation using quadratic estimators; § 3.6.

As mentioned, the third step measures not one but three power spectra, encoding clustering anisotropy that contains information about redshift space distortions. In this section, we merely present the basic measurement of these three curves, which involves no assumptions about linearity, the nature of biasing or anything else. We then return to modeling and interpreting these curves in terms of real-space power in Section 4 and to bias modeling in Section 5.

### 3.3 Step 1: Finger-of-god compression

Since our analysis method is motivated by (although not limited to) the linear Kaiser approximation for redshift space distortions, it is crucial that we are able to empirically quantify our sensitivity to the so-called finger-of-god (FOG) effect whereby radial velocities in virialized clusters make them appear elongated along the line of sight. We therefore start our analysis by compressing (isotropizing) FOGs, as illustrated in

he FOG compression involves a tunable threshold density, and in Section 4 we will study how the final results change as we gradually change this threshold to include lesser or greater numbers of FOGs.

We use a standard friends-of-friends algorithm, in which two galaxies are considered friends, therefore in the same cluster, if the density windowed through an ellipse 10 times longer in the radial than transverse directions, centered on the pair, exceeds a certain overdensity threshold. To avoid linking well-separated galaxies in deep, sparsely sampled parts of the survey, we impose the additional constraint that friends should be closer than in the transverse direction. The two conditions are combined into the following single criterion: two galaxies separated by in the radial direction and by in the transverse direction are considered friends if

 [(r∥/10)2+r2⊥]1/2≤[43π¯n(1+δc)+r−3⊥max]−1/3 (1)

where is the 3D selection function at the position of the pair, and is an overdensity threshold. Note that represents not the overdensity of the pair as seen in redshift space, but rather the overdensity of the pair after their radial separation has been reduced by a factor of 10. Thus is intended to approximate the threshold overdensity of a cluster in real space. We have chosen somewhat larger than the virial diameter of typical clusters to be conservative, minimizing the risk of missing FOGs — for our baseline threshold , our results are essentially unaffected by this choice of . Having identified a cluster by friends-of-friends in this fashion, we measure the dispersion of galaxy positions about the center of the cluster in both radial and transverse directions. If the one-dimensional radial dispersion exceeds the transverse dispersion, then the cluster is deemed a FOG, and the FOG is then compressed radially so that the cluster becomes round, that is, the transverse dispersion equals the radial dispersion. We perform the entire analysis five times, using density cutoffs , 200, 100, 50 and 25, respectively; in our analyses below, we will explore the sensitivity of our results to this cutoff. The infinite threshold corresponds to no compression at all.

Figure 11 illustrates FOG compression with threshold density , and unless explicitly stated otherwise, all results presented in this paper are for this threshold density. We make this choice to be on the safe side: Bryan & Norman (1998) estimate that the overdensity of a cluster at virialization is about 337 in a CDM model, rising further as the Universe expands and the background density continues to drop.

### 3.4 Step 2: Pseudo-KL pixelization

The raw data consist of three-dimensional vectors , , giving the measured positions of each galaxy in redshift space, with the number of galaxies given in Table 1 for each sample. As in Tegmark et al. (1998), we expand the observed three-dimensional density field in a basis of noise-orthonormal functions , ,

 xi≡∫Vn(r)¯n(r)ψi(r)d3r (2)

and work with the -dimensional data vector of expansion coefficients instead of the numbers . Here, is the three-dimensional selection function described in Section 2, i.e., is the expected (not the observed) number of galaxies in a volume about in the absence of clustering, and the integration is carried out over the volume of the sample where the selection function is nonzero. We will frequently refer to the functions as “modes”. As we will see below, these modes play a role quite analogous to pixels in CMB maps, with the variance of depending linearly on the power spectrum that we wish to measure.

Galaxies are (from a cosmological perspective) delta-functions in space, so the integral in equation (2) reduces to a discrete sum over galaxies. We do not rebin the galaxies spatially, which would artificially degrade the resolution. It is convenient to isolate the mean density into a single mode , with amplitude

 x1=∫n(r)d3r=Ngal, (3)

and to arrange for all other modes to have zero mean

 ⟨xi⟩=∫ψi(r)d3r=0(i≠1). (4)

The covariance matrix of the vector of amplitudes is a sum of noise and signal terms

 ⟨ΔxΔxt⟩=C≡N+S. (5)

where the shot noise covariance matrix is given by

 Nij=∫Vψi(r)ψj(r)¯n(r)d3r (6)

and the signal covariance matrix is

 Sij=∫ˆψi(k)ˆψj(k)∗Pgg(k)d3k(2π)3 (7)

in the absence of redshift-space distortions (which will be included in Section 3.5). Here hats denote Fourier transforms and the star denotes complex conjugation. is the (real-space) galaxy power spectrum, which for a random field of density fluctuations is defined by . We take the functions to have units of inverse volume, so , , and are all dimensionless.

Our method requires computing the signal covariance matrix , both to calculate power spectrum error bars and to find the power spectrum estimator that minimizes them. Equation (7) shows that this requires assuming a power spectrum . For this spectrum, which we refer to as our prior, we use a simple two-parameter fit as described in Section B.4.1, whose parameters are determined from our measurements by iterating the entire analysis.

As our functions , we use the pseudo-Karhunen-Loève (PKL) eigenmodes defined in Hamilton, Tegmark & Padmanabhan (2000; hereafter “HTP00”) and Tegmark, Hamilton & Xu (2002; hereafter “THX02”). The construction of these PKL modes explicitly uses the three-dimensional selection function , but is model-independent since it does not depend on the power spectrum.

To provide an intuitive feel for the nature of these modes, a sample is plotted in

d

e use these modes because they have the following desirable properties:

1. They form a complete set of basis functions probing successively smaller scales, so that a finite number of them (we use the first 4000, for the reasons given in Section B.4.2) allow essentially all information about the density field on large scales to be distilled into the vector .

2. They are orthonormal with respect to the shot noise, i.e., such that equation (6) gives , the identity matrix. The construction of the modes thus depends explicitly on the survey geometry as specified by , and in regions of space where .

3. They allow the covariance matrix to be fairly rapidly computed.

4. They are the product of an angular and a radial part, i.e., take the separable form , which accelerates numerical computations and helps isolate radial and angular systematic problems.

5. A range of potential sources of systematic problems are isolated into special modes that are orthogonal to all other modes. This means that we can test for the presence of such problems by looking for excess power in these modes, and immunize against their effects by discarding these modes.

We have four types of such special modes:

1. The very first mode is the mean density, . The mean mode is used in determining the maximum likelihood normalization of the selection function, but is then discarded from the analysis, since it is impossible to measure the fluctuation of the mean mode. The idea of solving the so-called integral constraint problem by making all modes orthogonal to the mean (Eq. 4) goes back to Fisher et al. (1993).

2. Modes 2-8 are associated with the motion of the Local Group through the Cosmic Microwave Background at 622 km/s towards (B1950 FK4) RA = , Dec = (Lineweaver et al. 1996; Courteau & van den Bergh 1999). To first order, these modes are the only modes affected by mis-estimates of the motion of the Local Group.

3. Purely radial modes (for example mode 468 in

re to first order the only ones affected by mis-estimates of the radial selection function .

4. Purely angular modes (for example mode 859 in

re to first order the only ones affected by mis-estimates of the angular selection function , as may result from inadequate corrections for, e.g., extinction, the variable magnitude limit, the variable magnitude completeness or photometric zero-point offsets.

The computation of the modes in practice is described in detail in THX02 and in even more detail in Hamilton & Tegmark (2003).

The pixelized data vector is shown in

his data compression step has thus distilled the large-scale information about the galaxy density field from galaxy coordinates into 4000 PKL-coefficients. The order of these coefficients is one of decreasing scale (increasing ) as is shown in Figure 15. If there were no cosmological density fluctuations in the survey, merely Poisson fluctuations, the PKL-coefficients would be uncorrelated with unit variance (since ), so about of them would be expected to lie within the blue/dark grey band. Figure 16 shows that the fluctuations are considerably larger than Poisson, especially for the largest-scale modes (to the left), demonstrating the obvious fact that cosmological density fluctuations are present, as expected.

### 3.5 What we wish to measure: three power spectra, not one

Following HTP00 and THX02, we will measure three separate power spectra, whose ratios encode information about clustering anisotropy due to redshift space distortions. Let us now give their definition and some intuition for how to interpret them.

On large scales where redshift space distortions can be treated in the linear approximation (Kaiser 1987), the signal covariance matrix in equation (5) can be generalized from equation (7) and written in the form

 S=∫∞0[Pgg(k)Pgg(k)+Pgv(k)Pgv(k)+Pvv(k)Pvv(k)]k2dk(2π)3, (8)

where , and are three power spectra defined in real space (as opposed to redshift space) and , and are known dimensionless matrix-valued functions. We will refer to these three power spectra as the galaxy-galaxy power, the galaxy-velocity power and the velocity-velocity power, respectively, or , and for short. Specifically, is the real-space galaxy power spectrum, is the velocity power spectrum and is the cross-power between galaxies and velocity. More rigorously, ‘velocity’ here refers to minus the velocity divergence , which in linear theory is related to the mass (not galaxy) overdensity by , where denotes the comoving gradient in velocity units. Here is the dimensionless growth rate for linear density perturbations (see Hamilton 2001). The three matrix-valued functions are determined directly from the modes , i.e., by geometry alone:

 (Pgg)ij = ∫ˆψi(k)ˆψj(k)∗dΩk, (9) (Pgv)ij = ∫[ˆψi(k)ˆζj(k)∗+ˆζi(k)ˆψj(k)∗]dΩk, (10) (Pvv)ij = ∫ˆζi(k)ˆζj(k)∗dΩk, (11)

in which the velocity mode is related to the position mode by (Fisher, Scharf & Lahav 1994; Heavens & Taylor 1995; Hamilton 1998 eq. 8.13)

 ζi=M†ψi (12)

where is the Hermitian conjugate of the velocity part of the linear redshift distortion operator. In the small-angle, or distant observer, approximation, the operator takes the familiar Kaiser (1987) form, a diagonal operator in Fourier space

 M≈μ2k (13)

where is the cosine of the angle between the wavevector and the line of sight . Here however we do not assume the small-angle approximation, but rather take into account the full radial nature of redshift distortions. Radial redshift distortions destroy translation invariance, so that the radial redshift distortion operator is no longer diagonal in Fourier space, as it is in the small-angle approximation; indeed, the radial redshift distortion operator takes a rather complicated form in Fourier space (Hamilton 1998, eq. 4.37). The radial redshift distortion operator takes a simpler form in real space, where , expressed in the frame of the Local Group, can be written as the integro-differential operator (Hamilton 1998 eq. 4.46)

 M=[∂2∂r2+α(r)ˆrr.(∂∂r−∂∂r∣∣∣r=0)]∇−2 (14)

with the logarithmic derivative of times the selection function ,

 α(r)≡∂ln[r2¯n(r)]∂lnr. (15)

The term inside parentheses in eq. (14), which subtracts from the first term its value at the position of the Local Group, is the term that arises from the motion of the Local Group. The Hermitian conjugate which enters equation (12) for the velocity mode can be written (Hamilton 1998 eq. 4.50)

 M†=∇−2r−2∂∂r(∂∂r−α(r)r)r2−ˆrr2.∂∂r∣∣∣r=0∇−2α(r)r (16)

in which the last term is again the term arising from the motion of the Local Group.

Although the definition of these three power spectra assumes that redshift distortions conform to the linear Kaiser model, they measure useful information from the data even if the linear model fails. In the small-angle (distant observer) approximation, they reduce to simple linear combinations of the monopole, quadrupole and hexadecapole power spectra in redshift space (Cole, Fisher & Weinberg 1994; Hamilton 1998):

 (17)

Whereas the vector on the right hand side is closer to the measurements (and also more familiar in the literature), the vector on the left hand side is closer to the physics of linear redshift distortions. Indeed, inverting equation (17),

 (18)

we see that we can use this last equation as an improved definition of monopole, quadrupole and hexadecapole, remaining valid even in the regime where the small-angle approximation fails. For the reader more used to thinking in terms of the multipole formalism, the bottom line is that our main measurement is basically the monopole power minus half the quadrupole power plus three eights of the hexadecapole power, as per equation (17).

Because redshift distortions displace galaxies only along the line of sight, the transverse, or angular, power spectrum is completely unaffected by redshift distortions, a point emphasized by Hamilton & Tegmark (2002). In the small-angle approximation, the galaxy-galaxy power spectrum equals the redshift space power spectrum in the transverse direction,

 Pgg(k)=Ps(μk=0)=∑\l even(−1)\l/2(\l−1)!!\l!!Ps\l(k), (19)

which is true in all circumstances, linear or nonlinear, regardless of the character of redshift distortions. The coefficients of the expansion (19) are the values of Legendre polynomials in the transverse direction . The first few terms of the expansion (19) are

 Pgg(k)=Ps0(k)−12Ps2(k)+38Ps4(k)−1548Ps6(k)+…, (20)

which shows that our linear estimate of is effectively the expansion (19) truncated at the harmonic, as predicted by linear theory (Kaiser 1987). We expect on general grounds that the linear estimate of will underestimate the true galaxy-galaxy power spectrum at nonlinear scales (Hamilton & Tegmark 2002), although this underestimate should be mitigated by FOG compression.

In Section 4, we will demonstrate with Monte Carlo simulations that faithfully recovers the true real-space galaxy power spectrum on large scales, and we will quantify what constitutes “large”, finding accurate recovery on substantially smaller scales that those where the Kaiser approximation is valid.

A wide range of approximations for and have been introduced in the literature. Using our notation, the Kaiser (1987) approximation becomes simply

 Pgv(k) = β(k)r(k)Pgg(k), (21) Pvv(k) = β(k)2Pgg(k), (22)

where , is the bias factor, is the dimensionless correlation coefficient between the galaxy and matter density (Dekel & Lahav 1999; Pen 1998; Tegmark & Peebles 1998) and was defined above. Since both and can in principle depend on scale, we have two unknown functions and that can in principle be determined uniquely from the two measured ratios and in the Kaiser approximation. Further popular approximations in the literature are that both and are constant, and most workers also assume despite some observational (Tegmark & Bromley 1999; Blanton 2000) and simulational (Blanton et al. 1999; Cen & Ostriker 2000; Somerville et al. 2001) evidence that may be of the order of 0.9 for some galaxies222 Although is normally referred to as stochastic bias, this does of course not imply any randomness in the galaxy formation process, merely that additional factors besides the present-day dark matter density may be important (gas temperature, say). The evidence for thus far comes from scales smaller than those that are the focus of this paper. More details on the relationship between our three power spectra and the stochastic bias formalism are given in Section 3.4 of THX02. .

We will not make any of these approximations in our basic data analysis, simply reporting measurements of , and from the SDSS data. In Section 4 we use the approximation that anisotropies are negligible, assessing its accuracy with Monte Carlo simulations and tunable finger-of-god compression, but the reader wishing to avoid approximations can simply fit better simulations directly to our three measured curves. Specifically, using an axis of a periodic cube as the line-of-sight direction, so that the distant observer approximation holds perfectly, one can compute the monopole, quadrupole, and hexadecapole components of the redshift-space power spectrum and transform them to , , and via equation (17).

### 3.6 Step 3: Power spectrum estimation

All our information about the SDSS density field is encoded in the 4000-dimensional vector plotted in

nd equation (8) shows that the covariance matrix of depends linearly on the three power spectra that we want to measure. We wish to invert equations (5) and (8) to estimate the power spectra from the data vector. This problem is mathematically equivalent to that of measuring the power spectrum from a CMB map, and can be solved optimally with so-called quadratic estimators (Tegmark 1997b; Bond et al. 2000). We describe our analysis method in full detail in Appendix B. However, since it is important for the interpretation, let us briefly review here how the measurements are computed from the input data . Figure 25: Decorrelated and disentangled measurements of the galaxy-galaxy power spectrum (top), the galaxy-velocity power spectrum (middle) and the velocity-velocity power spectrum (bottom) for the baseline galaxy sample. Each point is plotted at the k-value that is the median of its window function, and the horizontal bars range from the 20th to the 80th percentile of the window function. The values of Pgg(k) are given in Table 3. From top to bottom, the three curves shows our prior for Pgg(k), Pgv(k) and Pvv(k). Note that most of the information in the survey is in the galaxy-galaxy spectrum. Band-power measurements with very low information content have been binned into fewer (still uncorrelated) bands. All these measurements are for our baseline FOG compression threshold of 200. Unlike Figure 26: t hese points are uncorrelated (affecting all three panels) and the leakage between gg, gv and vv has been removed (affecting mostly the lower panels), giving much larger (and easier to interpret) error bars.

We parameterize our three power spectra by their amplitudes in 97 separate logarithmically spaced -bands as detailed in Appendix B, so our goal is to measure band powers , . Quadratic estimators are simply quadratic functions of the data vector , and the most general unbiased case can be written as

 ˆpi≡xtQix−tr[NQi], (23)

for some symmetric -dimensional matrices ; the second term merely subtracts off the expected contribution from the shot noise.

The basic idea behind quadratic estimators is that each matrix can be chosen to effectively Fourier transform the density field, square the Fourier modes in the power spectrum band and average the results together, thereby probing the power spectrum on that scale. Grouping the parameters and the estimators into vectors denoted and , the expectation value and covariance of the measurements is given by

 ⟨ˆp⟩ = Wp, (24) cov(ˆp) ≡ ⟨ˆpˆpt⟩−⟨ˆp⟩⟨ˆp⟩t=\SS, (25)

where the matrices and can be computed from the -matrices and the geometry alone via equations (B3) and (B4) in Appendix B.

As detailed in Appendix B, there are several attractive choices of -matrices, each giving different desirable properties to the matrices and . Figure 24 shows the power spectrum measurements for the baseline galaxy sample using the choice of that gives the smallest error bars, and

ows them using a better choice described below.

Although

oks impressive, it fails to convey two important complications. The first is that the error bars are strongly correlated between neighboring bands, i.e., the covariance matrix is far from diagonal. The second complication involves the matrix , known as the window matrix. The -matrices are normalized so that each row of the window matrix sums to unity. Equation (24) shows that this normalization enables us to interpret each band power measurement as a weighted average of the true power spectrum , the elements of the row of giving the weights (the so-called window function). In short, the window functions connect our measurements to the underlying power spectrum parameters . The windows are plotted in

nd we see that they are complicated in two different ways, making

rd to interpret:

1. Smearing: They have a non-zero width , so that our estimate of the power on some scale is really the weighted average of the power over a range of scales around . In other words,
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