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12-11-2006 Cosmological Constraints from the SDSS Luminous Red Galaxies Max Tegmark Massachusetts nI stitute of Technology
Daniel J. Eisenstein University of Arizona
Michael Strauss Princeton University Observatory
David H. Weinberg Ohio State University
Michael R. Blanton New York University
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Recommended Citation M. Tegmark et al. (SDSS Collaboration), Phys. Rev. D 74, 123507 (2006) https://doi.org/10.1103/PhysRevD.74.123507
This Article is brought to you for free and open access by RIT Scholar Works. It has been accepted for inclusion in Articles by an authorized administrator of RIT Scholar Works. For more information, please contact [email protected]. Authors Max Tegmark, Daniel J. Eisenstein, Michael Strauss, David H. Weinberg, Michael R. Blanton, Joshua A. Frieman, Masataka Fukugita, James E. Gunn, Andrew J. S. Hamilton, Gillian R. Knapp, Robert C. Nichol, Jeremiah P. Ostriker, Nikhil Padmanabhan, Will J. Percival, David J. Schlegel, Donald P. Schneider, Roman Scoccimarro, Uroš Seljak, Hee-Jong Seo, Molly Swanson, Alexander S. Szalay, Michael S. Vogeley, Jaiyul Yoo, Idit Zehavi, Kevork Abazajian, Scott .F Anderson, James Annis, Neta A. Bahcall, Bruce Bassett, Andreas Berlind, John Brinkman, Tamás Budavari, Francisco Castander, Andrew Connolly, Istvan Csabai, Mamoru Doi, Douglas P. Finkbeiner, Bruce Gillespie, Karl Glazebrook, Gregory S. Hennessy, David W. Hogg, Željko Ivezić, Bhuvnesh Jain, David Johnston, Stephen Kent, Donald Q. Lamb, Brian C. Lee, Huan Lin, Jon Loveday, Robert H. Lupton, Jeffrey A. Munn, Kaike Pan, Changbom Park, John Peoples, Jeffrey R. Pier, Adrian Pope, Michael Richmond, Constance Rockosi, Ryan Scranton, Ravi K. Sheth, Albert Stebbins, Christopher Stoughton, István Szapudi, Douglas L. Tucker, Daniel E. Vanden Berk, Brian Yanny, and Donald G. York
This article is available at RIT Scholar Works: http://scholarworks.rit.edu/article/1184 hoSaeUiest,Clmu,O 31,USA; 43210, OH Columbus, University, State Ohio arXiv:astro-ph/0608632v2 30 Oct USA; 85721, AZ Tucson, Arizona, of 2006University ahntnP. e ok Y103 USA; 10003, NY York, New Pl., Washington 4 USA; Frieman a Tegmark Max Richmond Glazebrook .Bahcall A. USA; eryA Munn A. Jeffrey 1 eeihP Ostriker P. Jeremiah USA; 19 taaCsir 1 41 ret,Italy; Trieste, 34014 11, Costiera Strada et fPyis ascuet nttt fTcnlg,C Technology, of Institute Massachusetts Physics, of Dept. hcg,I 03,USA; 60637, IL Chicago, enyvnaSaeUiest,Uiest ak A16802, PA Park, University University, State Pennsylvania et fAtooy nv fWsigo,Bx318,Seattl 351580, Box Washington, of Univ. Astronomy, of Dept. .Vogeley S. tpe Kent Stephen oa Scoccimarro Roman nrwConnolly Andrew A114 USA; 19104, PA USA; aai,I 01,USA; 60510, IL Batavia, Istv´an Szapudi 71SnMri rv,Blioe D228 USA; 21218, MD Baltimore, Drive, Martin San 3701 29 20Es aionaBv. aaea A915 USA; 91125, CA Pasadena, Blvd., California East 1200 6 hb,2788,Japan; 277-8582, Chiba, 8-05 Japan; 181-0015, , 12 7 e rplinLbrtr,40 a rv r,Psdn CA, Pasadena Dr., Grove Oak 4800 Laboratory, Propulsion Jet Z80182,USA; 86001-8521, AZ Korea; nv fCiao hcg,I 03,USA; 60637, IL Chicago, Chicago, of Univ. aaaaFukugita Masataka , arneBree ainlLbrtr,Bree,C 94720 CA Berkeley, Laboratory, National Berkeley Lawrence ole,C 00,USA; 80309, CO Boulder, amr rgtnB19J UK; 9QJ, BN1 Brighton Falmer, H44106-7215; OH itbrh A120 USA; 15260, PA Pittsburgh, 35 3 Spain; 36 15 A02138; MA ot Africa; South rc Bassett Bruce , osac Rockosi Constance , Dtd umte oPy.Rv .Ags 220,rvsdOc revised 2006, 22 August D. Rev. Phys. to Submitted (Dated: omlgclCntansfo h DSLmnu e Galaxi Red Luminous SDSS the from Constraints Cosmological 0 E,Uie Kingdom; United 2EG, P01 aeTw,SuhAfrica; South Town, Cape nttt o srnm,Uiest fHwi,28,Woodl 2680, Hawaii, of University Astronomy, for Institute rgr .Hennessy S. Gregory , 1 USA; 16 ailJ Eisenstein J. Daniel , 35 aylYoo Jaiyul , hsc et,RcetrIs.o ehooy obMemori Lomb 1 Technology, of Inst. Rochester Dept., Physics 27 7 24 oadQ Lamb Q. Donald , ak Pan Kaike , 23 nvriyo itbrh eateto hsc n Astron and Physics of Department Pittsburgh, of University 36 ntttdEtdsEpcasd aauy/SC apsUA Campus Catalunya/CSIC, de Espacials d’Estudis Institut 24 17 3 oga .Tucker L. Douglas , ihlPadmanabhan Nikhil , 27 et fAtooy aeWsenRsreUiest,10900 University, Reserve Western Case Astronomy, of Dept. svnCsabai Istvan , 5 26 ..NvlOsraoy lgtffSain 09 .NvlOb Naval W. 10391 Station, Flagstaff Observatory, Naval U.S. Uroˇs Seljak , 22 18 avr-mtsna etrfrAtohsc,6 adnSt Garden 60 Astrophysics, for Center Harvard-Smithsonian 20 pcePitOsraoy 01Aah on d uso,NM Sunspot, Rd, Point Apache 2001 Observatory, Point Apache 32 hoeia iiin o lmsNtoa aoaoy Los Laboratory, National Alamos Los Division, Theoretical 4 , aa n. laatn A94588; CA Pleasanton, Inc., Gatan 21 8 dtZehavi Idit , 28 ae .Gunn E. James , 8 9 nra Berlind Andreas , 22 nt o omcRyRsac,Ui.o oy,515 Kashiw 5-1-5, Tokyo, of Univ. Research, Ray Cosmic for Inst. eateto hsc,Uiest fPnslai,Philad Pennsylvania, of University Physics, of Department 6 IAadDp.o srpyia n lntr cecs Uni Sciences, Planetary and Astrophysical of Dept. and JILA ynScranton Ryan , hnbmPark Changbom , 10 27 2 ihe .Strauss A. Michael , nt fCsooy&Gaiain nv fPrsot,Port Portsmouth, of Univ. Gravitation, & Cosmology of Inst. 6 ai .Hogg W. David , 25 14 , 31 15 21 nt fAtooy nv fTko sw -11 iaa To Mitaka, 2-21-1, Osawa Tokyo, of Univ. Astronomy, of Inst. 3 , 11 rneo nvriyOsraoy rneo,N 84,US 08544, NJ Princeton, Observatory, University Princeton 11 ra .Lee C. Brian , ple ahmtc et,Ui.o aeTw,Cp Town, Cape Town, Cape of Univ. Dept., Mathematics Applied aouDoi Mamoru , 5 6 15 et fPyis rneo nvriy rneo,N 0854 NJ Princeton, University, Princeton Physics, of Dept. 34 17 etrfrCsooyadPril hsc,Dp.o Physics of Dept. Physics, Particle and Cosmology for Center etrfrCsooia hsc n eateto Astronom of Department and Physics Cosmological for Center e-ogSeo Hee-Jong , 7 eateto hsc n srnm,TeJhsHpisUniv Hopkins Johns The Astronomy, and Physics of Department ailE adnBerk Vanden E. Daniel , eateto srnm,SolNtoa nvriy 151-7 University, National Seoul Astronomy, of Department eokAbazajian Kevork , 11 3 ilJ Percival J. Will , nrwJ .Hamilton S. J. Andrew , 5 24 o Brinkmann Jon , 34 aiK Sheth K. Ravi , 7 em ainlAclrtrLbrtr,PO o 500, Box P.O. Laboratory, Accelerator National Fermi onPeoples John , 5 25 32 , , ejoIvezi´cZeljko oga .Finkbeiner P. Douglas , ˇ 3 12 2 ai .Weinberg H. David , 16 ol Swanson Molly , mrde A019 USA; 02139, MA ambridge, unLin Huan , et fPyis rxlUiest,Philadelphia, University, Drexel Physics, of Dept. USA; 31 ,W 98195; WA e, nioFriIsiue nvriyo Chicago, of University Institute, Fermi Enrico 18 33 10 ct .Anderson F. Scott , usxAtooyCnr,Uiest fSussex, of University Centre, Astronomy Sussex 14 USA; , ai .Schlegel J. David , 28 22 7 nentoa etrfrTertclPhysics, Theoretical for Center International eryR Pier R. Jeffrey , 24 letStebbins Albert , Tam´as Budavari , 19 10,USA; 91109, w rv,Hnll,H 62,USA; 96822, HI Honolulu, Drive, awn 7 ra Yanny Brian , , o Loveday Jon , 3 oe 0 cetdOtbr26) October accepted 10, tober hvehJain Bhuvnesh , 9 ila .Knapp R. Gillian , 13 1 et fAtooyadAstrophysics, and Astronomy of Dept. lxne .Szalay S. Alexander , 20 m,34 ’aaStreet, O’Hara 3941 omy, lD,Rcetr Y14623, NY Rochester, Dr, al ot fia srnmclObservatory, Astronomical African South 4 ihe .Blanton R. Michael , ,004Barcelona, 08034 B, 3 , 26 30 et S6 Cambridge, MS46, reet, uldAeu,Cleveland, Avenue, Euclid rc Gillespie Bruce , aionaIs.o Technology, of Inst. California 27 7 lms M87545, NM Alamos, 12 19 33 .R. Flagstaff, Rd., s. oadG York G. Donald , 2 7 dinPope Adrian , 15 eateto Astronomy, of Department hitpe Stoughton Christopher , oadP Schneider P. Donald , oetH Lupton H. Robert , ae Annis James , lha A19104, PA elphia, rnic Castander Francisco , 28 88349-0059, ai Johnston David , .o Colorado, of v. nh,Kashiwa, anoha, 3 oetC Nichol C. Robert , smouth, A; 15 es 4, Michael , e okUniversity, York New , 4 kyo, 15 22 5 Astrophysics, & y et fAstronomy, of Dept. 42, 7 ohaA. Joshua , Michael , Neta , Karl , 6 , 31 ersity, 3 29 , 13 , 23 30 , 7 10 , , , , 2
We measure the large-scale real-space power spectrum P (k) using luminous red galaxies (LRGs) in the Sloan Digital Sky Survey (SDSS) and use this measurement to sharpen constraints on cos- mological parameters from the Wilkinson Microwave Anisotropy Probe (WMAP). We employ a matrix-based power spectrum estimation method using Pseudo-Karhunen-Lo`eve eigenmodes, pro- ducing uncorrelated minimum-variance measurements in 20 k-bands of both the clustering power and its anisotropy due to redshift-space distortions, with narrow and well-behaved window functions in the range 0.01 h/Mpc
I. INTRODUCTION on large scales using the SDSS galaxy redshift survey in a way that is maximally useful for cosmological param- eter estimation, and to explore the resulting constraints The dramatic recent progress by the Wilkinson Mi- on cosmological models. The emphasis of our cosmo- crowave Anisotropy Probe (WMAP) and other experi- logical analysis will be on elucidating the links between ments [1–4] measuring the cosmic microwave background cosmological parameters and observable features of the (CMB) has made non-CMB experiments even more im- WMAP and SDSS power spectra, and on how these two portant in the quest to constrain cosmological models and data sets alone provide tight and robust constraints on their free parameters. These non-CMB constraints are many parameters that complement more aggressive but crucially needed for breaking CMB degeneracies [5, 6]; more systematics-prone analyses of multiple data sets. for instance, WMAP alone is consistent with a closed universe with Hubble parameter h = 0.3 and no cosmo- logical constant [7]. As long as the non-CMB constraints are less reliable and precise than the CMB, they will be the limiting factor and weakest link in the precision cos- mology endeavor. Much of the near-term progress in cos- In a parallel paper, Percival et al. [43] present a power mology will therefore be driven by reductions in statisti- spectrum analysis of the Main Galaxy and LRG samples cal and systematic uncertainties of non-CMB probes of from the SDSS DR5 data set [44], which is a superset of the cosmic expansion history (e.g., SN Ia) and the matter the data used here. There are a number of differences power spectrum (e.g., Lyman α Forest, galaxy clustering in the analysis methods. Percival et al. use an FFT- and motions, gravitational lensing, cluster studies and 21 based method to estimate the angle-averaged (monopole) cm tomography). redshift-space galaxy power spectrum. We use a Pseudo- The cosmological constraining power of three- Karhunen-Lo`eve method [45, 46] (see further discussion dimensional maps of the Universe provided by galaxy and references below) to estimate the real space (as op- redshift surveys has motivated ever more ambitious ob- posed to redshift space) galaxy power spectrum, using servational efforts such as the CfA/UZC [8, 9], LCRS finger-of-god compression and linear theory to remove [10], PSCz [11], DEEP [12], 2dFGRS [13] and SDSS [14] redshift-space distortion effects. In addition, the many projects, resulting in progressively more accurate mea- technical decisions that go into these analyses, regarding surements of the galaxy power spectrum P (k) [15–30]. completeness corrections, angular masks, K-corrections Constraints on cosmological models from these data sets and so forth, were made independently for the two pa- have been most robust when the galaxy clustering could pers, and they present different tests for systematic un- be measured on large scales where one has confidence in certainties. Despite these many differences of detail, our the modeling of nonlinear clustering and biasing (e.g., conclusions agree to the extent that they overlap (as dis- [7, 31–42]). cussed in Section III F and Appendix A 1), a reassuring Our goal in this paper therefore is to measure P (k) indication of the robustness of the results. 3
A. Relation between different samples B. Relation between different methods
The amount of information in a galaxy redshift survey In the recent literature, two-point galaxy clustering about the galaxy power spectrum Pg(k) and cosmological has been quantified using a variety of estimators of both parameters depends not on the number of galaxies per se, power spectra and correlation functions. The most re- but on the effective volume of the survey, defined by [47] cent power spectrum measurements for both the 2dFGRS as [26, 29] and the SDSS [30, 38, 43] have all interpolated the galaxy density field onto a cubic grid and measured n¯(r)P (k) 2 V (k) g d3r, (1) P (k) using a Fast Fourier Transform (FFT). eff ≡ 1+¯n(r)P (k) Z g Appendix A 1 shows that as long as discretization er- wheren ¯(r) is the expected number density of galaxies in rors from the FFT gridding are negligible, this procedure the survey in the absence of clustering, and the FKP ap- is mathematically equivalent to measuring the correlation proximation of [19] has been used. The power spectrum function with a weighted version of the standard “DD- −1/2 2DR+RR” method [52, 53], multiplying by “RR” and error bars scale approximately as ∆Pg(k) Veff (k) , ∝ then Fourier transforming. Thus the only advantage of which for a fixed power Pg is minimized if a fixed to- −1 the FFT approach is numerical speedup, and comparing tal number of galaxies are spaced with densityn ¯ Pg [48]. The SDSS Luminous Red Galaxy (LRG) sample∼ the results with recent correlation function analyses such was designed [49, 50] to contain such “Goldilocks” galax- as [36, 54–56] will provide useful consistency checks. ies with a just-right number density for probing the power Another approach, pioneered by [45], has been to con- around the baryon wiggle scale k (0.05 0.1)h/Mpc. struct “lossless” estimators of the power spectrum with For comparison, the SDSS main∼ galaxy sample− [50] is the smallest error bars that are possible based on infor- much denser and is dominated by sample variance on mation theory [23, 24, 27, 28, 34, 45, 46, 57, 58]. We these scales, whereas the SDSS quasar sample [51] is will travel this complementary route in the present pa- much sparser and is dominated by Poisson shot noise. per, following the matrix-based Pseudo Karhunen-Lo`eve As shown in [36], the effective volume of the LRG sam- (PKL) eigenmode method described in [28], as it has the ple is about six times larger than that of the SDSS main following advantages: galaxies even though the number of LRGs is an order of 1. It produces power spectrum measurements with magnitude lower, and the LRG volume is over ten times uncorrelated error bars. larger than that of the 2dFGRS. These scalings are con- 2 firmed by our results below, which show that (∆Pg/Pg) 2. It produces narrow and well-behaved window func- on large scales is about six times smaller for the SDSS tions. LRGs than for the main sample galaxies. This gain re- sults both from sampling a larger volume, and from the 3. It is lossless in the information theory sense. fact that the LRG are more strongly clustered (biased) 4. It treats redshift distortions without the small- than are ordinary galaxies; Pg for LRGs is about 3 times larger than for the main galaxy sample. angle approximation. We will therefore focus our analysis on the SDSS LRG 5. It readily incorporates the so-called integral con- sample. Although we also measure the SDSS main sam- straint [16, 59], which can otherwise artificially sup- ple power spectrum, it adds very little in terms of sta- press large-scale power. tistical constraining power; increasing the effective vol- ume by 15% cuts the error bar ∆P by only about 6. It allows testing for systematics that produce excess 1/2 (1+0.15) 1 7%. This tiny improvement is eas- power in angular or radial modes. ily outweighed− by∼ the gain in simplicity from analyzing LRGs alone, where (as we will see) complications such as These properties make the results of the PKL-method redshift-dependence of clustering properties are substan- very easy to interpret and use. The main disadvantage tially smaller. is that the PKL-method is numerically painful to im- A complementary approach implemented by [41, 42] plement and execute; our PKL analysis described below is to measure the angular clustering of SDSS LRGs with required about a terabyte of disk space for matrix stor- photometric redshifts, compensating for the loss of radial age and about a year of CPU time, which contributed to information with an order of magnitude more galaxies the long gestation period of this paper. extending out to higher redshift. We will see that this The rest of this paper is organized as follows. We de- gives comparable or slightly smaller error bars on very scribe our galaxy samples and our modeling of them in large scales k < 0.02, but slightly larger error bars on Section II and measure their power spectra in Section III. the smaller scales∼ that dominate our cosmological con- We explore what this does and does not reveal about cos- straints; this is because the number of modes down to a mological parameters in Section IV. We summarize our given scale k grows as k3 for our three-dimensional spec- conclusions and place them in context in Section V. Fur- troscopic analysis, whereas they grow only as k2 for a ther details about analysis techniques are given in Ap- 2-dimensional angular analysis. pendix A. 4
Our analysis is based on 58, 360 LRGs and 285, 804 main galaxies (the “safe13” cut) from the 390, 288 galax- ies in the 4th SDSS data release (“DR4”) [74], processed via the SDSS data repository at New York University [75]. The details of how these samples were processed and modeled are given in Appendix A of [28] and in [36]. The bottom line is that each sample is completely speci- fied by three entities:
1. The galaxy positions (RA, Dec and comoving red- shift space distance r for each galaxy),
2. The radial selection functionn ¯(r), which gives the expected number density of galaxies as a function of distance,
3. The angular selection functionn ¯(r), which gives the completeness as a function of direction in the sky, specified in a set of spherical polygonsb [76].
Our samples are constructed so that their three- dimensional selection function is separable, i.e., simply the productn ¯(r)=¯n(r)¯n(r) of an angular and a radial FIG. 1: The redshift distribution of the luminous red galaxies used part; here r r and r r/r are the comoving radial dis- ≡ | | ≡ is shown as a histogram and compared with the expected distribu- tance and the unit vectorb corresponding to the position tion in the absence of clustering, ln 10× n¯(r)r3dΩ (solid curve) in r. The effective sky areab covered is Ω n¯(r)dΩ 4259 comoving coordinates assuming a flat ΩΛ = 0.75 cosmology. The ≡ ≈ R square degrees, and the typical completenessn ¯(r) exceeds bottom panel shows the ratio of observed and expected distribu- R tions. The four vertical lines delimit the NEAR, MID and FAR 90%. The radial selection functionn ¯(r) forb the LRGs is samples. the one constructed and described in detail inb [36, 56], based on integrating an empirical model of the luminosity function and color distribution of the LRGs against the II. GALAXY DATA luminosity-color selection boundaries of the sample. Fig- ure 1 shows that it agrees well with the observed galaxy The SDSS [14, 60] uses a mosaic CCD camera [61] on distribution. The conversion from redshift z to comoving a dedicated telescope [62] to image the sky in five pho- distance was made for a flat ΛCDM cosmological model tometric bandpasses denoted u, g, r, i and z [63]. Af- with Ωm = 0.25. If a different cosmological model is ter astrometric calibration [64], photometric data reduc- used for this conversion, then our measured dimensionless 3 < tion [65, 66] and photometric calibration [67–70], galax- power spectrum k P (k) is dilated very slightly (by 1% ∼ ies are selected for spectroscopic observations [50]. To a for models consistent with our measurements) along the good approximation, the main galaxy sample consists of k-axis; we include this dilation effect in our cosmological all galaxies with r-band apparent Petrosian magnitude parameter analysis as described in Appendix A 4. r< 17.77 after correction for reddening as per [71]; there For systematics testing and numerical purposes, we are about 90 such galaxies per square degree, with a me- also analyze a variety of sub-volumes in the LRG sam- dian redshift of 0.1 and a tail out to z 0.25. Galaxy ple. We split the sample into three radial slices, labeled spectra are also measured for the LRG sample∼ [49], tar- NEAR (0.155 1000 500 0 -500 -1000 -1000 -500 0 500 1000 FIG. 2: The distribution of the 6,476 LRGs (black) and 32,417 main galaxies (green/grey) that are within 1.25◦ of the Equatorial plane. The solid circles indicate the boundaries of our NEAR, MID and FAR subsamples. The “safe13” main galaxy sample analyzed here and in [28] is more local, extending out only to 600h−1 Mpc (dashed circle). III. POWER SPECTRUM MEASUREMENTS called fingers-of-god (FOGs), virialized galaxy clusters that appear elongated along the line-of-sight in redshift space; we do this with several different thresholds and We measure the power spectrum of our various samples return to how this affects the results in Section IV F 2. using the PKL method described in [28]. We follow the The LRGs are not just brightest cluster galaxies; about procedure of [28] exactly, with some additional numeri- 20% of them appear to reside in a dark matter halo with cal improvements described in Appendix A, so we merely one or more other LRG’s. The second step is to expand summarize the process very briefly here. The first step the three-dimensional galaxy density field in N three- is to adjust the galaxy redshifts slightly to compress so- 6 FIG. 3: The angular distribution of our LRGs is shown in Hammer-Aitoff projection in celestial coordinates, with the seven colors/greys indicating the seven angular subsamples that we analyze. dimensional functions termed PKL-eigenmodes, whose Table 1 – The real-space galaxy power spectrum Pg(k) in units variance and covariance retain essentially all the informa- −1 3 of (h Mpc) measured from the LRG sample. The errors on Pg tion about the k < 0.2h/Mpc power spectrum from the are 1σ, uncorrelated between bands. The k-column gives the galaxy catalog. We use N = 42,000 modes for the LRG median of the window function and its 20th and 80th percentiles; sample and 4000 modes for the main sample, reflecting the exact window functions from their very different effective volumes. The third step is http://space.mit.edu/home/tegmark/sdss.html (see Figure 5) estimating the power spectrum from quadratic combina- should be used for any quantitative analysis. Nonlinear modeling is definitely required if the six measurements on the smallest tions of these PKL mode coefficients by a matrix-based scales (below the line) are used for model fitting. These error bars process analogous to the standard procedure for mea- do not include an overall calibration uncertainty of 3% (1σ) suring CMB power spectra from pixelized CMB maps. related to redshift space distortions (see Appendix A 3). The second and third steps are mathematically straight- forward but, as mentioned, numerically demanding for large N. k [h/Mpc] Power Pg +0.005 0.012−0.004 124884 ± 18775 +0.003 0.015−0.002 118814 ± 29400 +0.004 0.018−0.002 134291 ± 21638 A. Basic results +0.004 0.021−0.003 58644 ± 16647 +0.004 0.024−0.003 105253 ± 12736 The measured real-space power spectra are shown in +0.005 0.028−0.003 77699 ± 9666 Figure 4 for the LRG and MAIN samples and are listed +0.005 0.032−0.003 57870 ± 7264 in Table 1. When interpreting them, two points should +0.006 0.037−0.004 56516 ± 5466 be borne in mind: +0.008 0.043−0.006 50125 ± 3991 0.049+0.008 45076 ± 2956 1. The data points (a.k.a. band power measurements) −0.007 0.057+0.009 39339 ± 2214 probe a weighted average of the true power spec- −0.007 0.065+0.010 39609 ± 1679 trum P (k) defined by the window functions shown −0.008 0.075+0.011 31566 ± 1284 in Figure 5. Each point is plotted at the median −0.009 0.087+0.012 24837 ± 991 k-value of its window with a horizontal bar ranging −0.011 th th +0.013 ± from the 20 to the 80 percentile. 0.100−0.012 21390 778 +0.013 0.115−0.014 17507 ± 629 0.133+0.012 15421 ± 516 2. The errors on the points, indicated by the vertical −0.015 +0.012 ± bars, are uncorrelated, even though the horizon- 0.153−0.017 12399 430 +0.013 ± tal bars overlap. Other power spectrum estimation 0.177−0.018 11237 382 +0.015 methods (see Appendix A 1) effectively produce a 0.203−0.022 9345 ± 384 smoothed version of what we are plotting, with er- ror bars that are smaller but highly correlated. 7 FIG. 4: Measured power spectra for the full LRG and main galaxy samples. Errors are uncorrelated and full window functions are shown in Figure 5. The solid curves correspond to the linear theory ΛCDM fits to WMAP3 alone from Table 5 of [7], normalized to galaxy bias b = 1.9 (top) and b = 1.1 (bottom) relative to the z = 0 matter power. The dashed curves include the nonlinear correction of [29] for A = 1.4, with Qnl = 30 for the LRGs and Qnl = 4.6 for the main galaxies; see equation (4). The onset of nonlinear corrections is clearly > visible for k ∼ 0.09h/Mpc (vertical line). Our Fourier convention is such that the dimensionless B. Clustering evolution power ∆2 of [77] is given by ∆2(k)=4π(k/2π)3P (k). Before using these measurements to constrain cosmo- The standard theoretical expectation is for matter logical models, one faces important issues regarding their clustering to grow over time and for bias (the rela- interpretation, related to evolution, nonlinearities and tive clustering of galaxies and matter) to decrease over systematics. time [78–80] for a given class of galaxies. Bias is also 8 FIG. 5: The window functions corresponding to the LRG band powers in Figure 4 are plotted, normalized to have unit peak height. Each window function typically peaks at the scale k that the cor- responding band power estimator was designed to probe. FIG. 6: Same as Figure 4, but showing the NEAR (circles), MID luminosity-dependent, which would be expected to af- (squares) and FAR (triangles) LRG subsamples. On linear scales, they are all well fit by the WMAP3 model with the same clustering fect the FAR sample but not the MID and NEAR sam- amplitude, and there is no sign of clustering evolution. ples (which are effectively volume limited with a z- independent mix of galaxy luminosities [49]). Since the galaxy clustering amplitude is the product of these two in the matter power spectrum is approximately canceled factors, matter clustering and bias, it could therefore in by a drop in the bias factor to within our measurement principle either increase or decrease across the redshift uncertainty. For a flat Ωm = 0.25 ΛCDM model, the range 0.155 anisotropies due to redshift space distortions. Here “ve- locity” refers to the negative of the peculiar velocity di- vergence. Specifically, Pgg(k) and Pvv(k) are the power spectra of the galaxy density and velocity fields, respec- tively, whereas Pgv(k) is the cross-power between galaxies and velocity, all defined in real space rather than redshift space. In linear perturbation theory, these three power spec- tra are related by [82] Pgv(k) = βrgvPgg(k), (2) 2 Pvv(k) = β Pgg(k), (3) where β f/b, b is the bias factor, rgv is the dimen- sionless correlation≡ coefficient between the galaxy and 0.6 matter density fields [79, 83, 84], and f Ωm is the dimensionless linear growth rate for linear≈ density fluc- tuations. (When computing f below, we use the more accurate approximation of [85].) The LRG power spectrum P (k) tabulated and plotted above is a minimum-variance estimator of Pgg(k) that linearly combines the Pgg(k), Pgv(k) and Pvv(k) estima- tors as described in [28] and Appendix A 3, effectively marginalizing over the redshift space distortion param- FIG. 7: Same as Figure 4, but multiplied by k and plotted with eters β and rgv. As shown in Appendix A 3, this lin- a linear vertical axis to more clearly illustrate departures from a ear combination is roughly proportional to the angle- simple power law. averaged (monopole) redshift-space galaxy power spec- trum, so for the purposes of the nonlinear modeling in the next section, the reader may think of our mea- sured P (k) as essentially a rescaled version of the red- shift space power spectrum. However, unlike the redshift space power spectrum measured with the FKP and FFT methods (Appendix A 1), our measured P (k) is unbiased on large scales. This is because linear redshift distortions are treated exactly, without resorting to the small-angle approximation, and account is taken of the fact that the anisotropic survey geometry can skew the relative abun- dance of galaxy pairs around a single point as a function of angle to the line of sight. The information about anisotropic clustering that is discarded in our estimation of P (k) allows us to mea- sure β and perform a powerful consistency test. Figure 8 shows the joint constraints on β and rgv from fitting equa- tions (2) and (3) to the 0.01h/Mpc k 0.09h/Mpc LRG data, using the best fit WMAP3≤ model≤ from Fig- ure 4 for Pgg(k) and marginalizing over its amplitude. The data are seen to favor r 1 in good agree- gv ≈ ment with prior work [86, 87]. Assuming rgv = 1 (that galaxy density linearly traces matter density on these large scales) gives the measurement β = 0.309 0.035 (1σ). This measurement is rather robust to changing± the FOG compression threshold by a notch (Section IV F 2) FIG. 8: Constraints on the redshift space distortion parameters or slightly altering the maximum k-band included, both β and rgv. The contours show the 1, 2 and 3σ constraints from the observed LRG clustering anisotropy, with the circular dot in- of which affect the central value by of order 0.01. As a dicating the best fit values. The diamond shows the completely cross-check, we can compute β = f(Ωm, ΩΛ)/b at the me- independent β-estimate inferred from our analysis of the WMAP3 dian survey redshift based on our multi-parameter anal- and LRG power spectra (it puts no constraints on rgv, but has ysis presented in Section IV, which for our vanilla class been plotted at rgv = 1). of models gives β = 0.280 0.014 (marked with a di- ± 10 matter power spectrum on small scales. 2. Nonlinear evolution washes out baryon wiggles on small scales. 3. The power spectrum of the dark matter halos in which the galaxies reside differs from that of the underlying matter power spectrum in both ampli- tude and shape, causing bias. 4. Multiple galaxies can share the same dark matter halo, enhancing small-scale bias. We fit these complications using a model involving the three “nuisance parameters” (b,Qnl, k∗) as illustrated in Figure 9. Following [29, 88], we model our measured galaxy power spectrum as 1+ Q k2 P (k)= P (k)b2 nl , (4) g dewiggled 1+1.4k where the first factor on the right hand side accounts for the non-linear suppression of baryon wiggles and the last factor accounts for a combination of the non-linear change of the global matter power spectrum shape and scale-dependent bias of the galaxies relative to the dark FIG. 9: Power spectrum modeling. The best-fit WMAP3 model from Table 5 of [7] is shown with a linear bias b = 1.89 (dotted matter. For Pdewiggled(k) we adopt the prescription [88] curve), after applying the nonlinear bias correction with Q = 31 (the more wiggly solid curve), and after also applying the wiggle Pdewiggled(k)= W (k)P (k)+[1 W (k)]Pnowiggle(k), (5) suppression of [88] (the less wiggly solid curve), which has no effect − on very large scales and asymptotes to the “no wiggle” spectrum 2 where W (k) e−(k/k∗) /2 and P (k) denotes the of [89] (dashed curve) on very small scales. The data points are ≡ nowiggle the LRG measurements from Figure 7. “no wiggle” power spectrum defined in [89] and illus- trated in Figure 9. In other words, Pdewiggled(k) is simply a weighted average of the linear power spectrum and the amond in Figure 8)1. That these two β-measurements wiggle-free version thereof. Since the k-dependent weight agree within 1σ is highly non-trivial, since the second β- W (k) transitions from 1 for k k∗ to 0 for k k∗, equa- ≪ ≫ measurement makes no use whatsoever of redshift space tion (5) retains wiggles on large scales and gradually fades distortions, but rather extracts b from the ratio of LRG them out beginning around k = k∗. Inspired by [88], power to CMB power, and determines Ω from CMB we define the wiggle suppression scale k∗ 1/σ, where m ≡ and LRG power spectrum shapes. σ σ2/3σ1/3(A /0.6841)1/2 and σ and σ are given by ≡ ⊥ k s ⊥ k equations (12) and (13) in [88] based on fits to cosmolog- ical N-body simulations. The expression in parenthesis D. Nonlinear modeling is an amplitude scaling factor that equals unity for the best fit WMAP3 normalization As = 0.6841 of [7]. Es- Above we saw that our k< 0.09h/Mpc measurements sentially, σ is the characteristic peculiar-velocity-induced of the LRG power spectrum were well fit by the linear displacement of galaxies that causes the wiggle suppres- theory matter power spectrum predicted by WMAP3. In sion; [88] define it for a fixed power spectrum normal- contrast, Figures 4, 6 and 7 show clear departures from ization, and it scales linearly with fluctuation amplitude, 1/2 the linear theory prediction on smaller scales. There are i.e., As . For the cosmological parameter range al- ∝ several reasons for this that have been extensively studied lowed by WMAP3, we find that k∗ 0.1h/Mpc, with a in the literature: rather rather weak dependence on cosmological∼ parame- ters (mainly Ω and A ). 1. Nonlinear evolution alters the broad shape of the m s The simulations and analytic modeling described by [29] suggest that the Qnl-prescription given by equa- tion (4) accurately captures the scale-dependent bias of 1 galaxy populations on the scales that we are interested Here β = f(Ωm, ΩΛ)/b is computed with Ωm,ΩΛ and b evaluated at the median redshift z = 0.35, when b = 2.25 ± 0.08, taking in, though they examined samples less strongly biased into account linear growth of matter clustering between then and than the LRGs considered here. To verify the applica- now. bility of this prescription for LRGs in combination with 11 an arguably more realistic way, yet giving results nicely consistent with Figure 10, with a best-fit value Qnl 24. (We will see in Section IV F that FOG-compression≈ can readily account for these slight differences in Qnl-value.) A caveat to both of these simulation tests is that they were performed in real space, and our procedure for mea- suring Pg(k) reconstructs the real space power spectrum exactly only in the linear regime [28]. Thus, these re- sults should be viewed as encouraging but preliminary, and more work is needed to establish the validity of the nonlinear modeling beyond k > 0.1h/Mpc; for up-to-date discussions and a variety of ideas∼ for paths forward, see, e.g., [92–95]. In addition to this simulation-based theoretical evi- dence that our nonlinear modeling method is accurate, we have encouraging empirical evidence: Figure 9 shows an excellent fit to our measurements. Fitting the best- fit WMAP3 model from [32] to our first 20 data points (which extend out to k =0.2h/Mpc) by varying (b,Qnl) gives χ2 = 19.2 for 20 2 = 18 degrees of freedom, where the expected 1σ range− is χ2 = 18 (2 18)1/2 = 18 6, so the fit is excellent. Moreover, Figures± × 7 and 9 show± that that main outliers are on large and highly linear scales, not on the smaller scales where our nonlinear modeling FIG. 10: The points in the bottom panel show the ratio of the real- has an effect. space power spectrum from 51 averaged n-body simulations (see The signature of baryons is clearly seen in the mea- text) to the linear power spectrum dewiggled with k∗ = 0.1h/Mpc. sured power spectrum. If we repeat this fit with baryons Here LRGs were operationally defined as halos with mass exceeding 2 12 8 × 10 M⊙, corresponding to at least ten simulation particles. replaced by dark matter, χ increases by 8.8, correspond- The solid curve shows the prediction from equation (4) with b = ing to a baryon detection at 3.0σ (99.7% significance). < 2.02, Qnl = 27, seen to be an excellent fit for k ∼ 0.4h/Mpc. Much of this signature lies in the acoustic oscillations: if The top panel shows the ratio of the simulation result to this fit. we instead repeat the fit with k∗ = 0, corresponding to Although the simulation specifications and the LRG identification 2 prescription can clearly be improved, they constitute the first and fully removing the wiggles, χ increases by an amount only that we tried, and were in no way adjusted to try to fit our corresponding to a detection of wiggles at 2.3σ (98% Qnl = 30 ± 4 measurement from Table 2. This agreement suggests significance). The data are not yet sensitive enough to that our use of equations (4) and (5) to model nonlinearities is distinguish between the wiggled and dewiggled spectra; reasonable and that our measured Qnl-value is plausible. dewiggling reduces χ2 by merely 0.04. In summary, the fact that LRGs tend to live in high- mass dark matter halos is a double-edged sword: it helps our dewiggling model, we reanalyze the 51 n-body sim- by giving high bias b 2 and luminous galaxies observ- ulations described in [90], each of which uses a 512h−1 able at great distance,∼ but it also gives a stronger non- 3 Mpc box with 256 particles and WMAP1 parameters. linear correction (higher Qnl) that becomes important Figure 10 compares these simulation results with our on larger scales than for typical galaxies. Although Fig- nonlinear modeling prediction defined by equations (4) ure 10 suggests that our nonlinear modeling is highly and (5) for b =2.02, Qnl = 27.0, showing excellent agree- accurate out to k = 0.4h/Mpc, we retain only measure- ment (at the 1% level) for k < 0.4h/Mpc. Choosing ments with k < 0.2h/Mpc for our cosmological parame- a k∗ very different from 0.1h/∼Mpc causes 5% wiggles ter analysis to∼ be conservative, and plan further work to appear in the residuals because of a over- or under- to test the validity of various nonlinear modeling ap- suppression of the baryon oscillations. These simulations proaches. In Section IV F 2, we will see that our data with are likely to be underresolved and the LRG halo prescrip- 0.09h/Mpc< k < 0.2h/Mpc, where nonlinear effects are tion used (one LRG for each halo above a threshold mass clearly visible, allow∼ us to constrain the nuisance param- of 8 1012M ) is clearly overly simplistic, so the true eter Q without significantly improving our constraints × ⊙ nl value of Qnl that best describes LRGs could be somewhat on cosmological parameters. In other words, the cosmo- different. Nonetheless, this test provides encouraging ev- logical constraints that we will report below are quite idence that equation (4) is accurate in combination with insensitive to our nonlinear modeling and come mainly equation (5) and that our Q = 30 4 measurement from from the linear power spectrum at k< 0.09h/Mpc. More ± Table 2 is plausible. Further corroboration is provided by sophisticated treatments of galaxy bias in which Qnl is the results in [41] using the Millennium Simulation [91]. effectively computed from theoretical models constrained Here LRG type galaxies were simulated and selected in by small scale clustering may eventually obviate the need 12 1. Analysis of subsets of galaxies To test for effects that would be expected to vary across the sky (depending on, say, reddening, seasonally variable photometric calibration errors, or observing conditions such as seeing and sky brightness), we repeat our entire analysis for the seven different angular subsets of the sky shown in Figure 3 in search of inconsistencies. To search for potential zero-point offsets and other systematic ef- fects associated with the southern Galactic stripes, they are defined as one of these seven angular subsets (see Fig- ure 3). To test for effects that depend on redshift, we use the measurements for our three redshift slices, plotted in Figure 6. To test the null hypothesis that all these subsamples are consistent with having the same power spectrum, we fit them all to our WMAP+LRG best-fit vanilla model described in Section IV, including our nonlinear cor- rection (this P (k) curve is quite similar to the best-fit WMAP3 model plotted above in, e.g., Figure 4). We include the 20 band-powers with k < 0.2 in our fit, so if the null hypothesis is correct, we∼ expect a mean χ2 of 20 with a standard deviation of √2 20 6.3. Our seven angular subsamples give a mean ×χ2 ≈ 22.6 and 2 2 1/2 h i ≈ FIG. 11: Same as Figure 7, but showing the effect of discarding a scatter (χ 20) 6.9. Our three radial sub- special modes on the large-scale power. The circles with associated samples giveh χ−2 18i .6 and≈ (χ2 20)2 1/2 2.4. All error bars correspond to our measured power spectrum using all of the ten χ2-valuesh i ≈ are statisticallyh − consistenti ≈ with the 4000 full-sample PKL modes. The other points show the effect of removing the 332 purely angular modes (crosses), the 18 purely null hypothesis at the 95% level. We also repeated the radial modes (triangles), and all special modes combined (squares), cosmological parameter analysis reported below with the including seven associated with the motion of the local group as southern stripes omitted, finding no significant change in described in [28]. Any systematic errors adding power to these the measured parameter values. In other words, all our special modes would cause the black circles to lie systematically angular and radial subsamples are consistent with hav- above the other points. These special modes are seen to have less impact at larger k because they are outnumbered: the number of ing the same power spectrum, so these tests reveal no radial, angular, and generic modes below a given k-value scales as evidence for systematic errors causing radial or angular k, k2 and k3, respectively. power spectrum variations. 2. Analysis of subsets of modes to marginalize over this nuisance parameter, increasing the leverage of our measurements for constraining the Because of their angular or radial nature, all poten- linear power spectrum shape [93]. tial systematic errors discussed above create excess power mainly in the radial and angular modes. As mentioned above, one of the advantages of the PKL method is that it allows these modes to be excluded from the analysis, in analogy to the way potentially contaminated pixels in a CMB map can be excluded from a CMB power spec- E. Robustness to systematic errors trum analysis. To quantify any such excess, we therefore repeat our full-sample analysis with radial and/or angu- lar modes deleted. The results of this test are shown in Let us now consider potential systematic errors in the Figure 11 and are very encouraging; the differences are LRG data that could affect our results. Examples of such tiny. Any systematic errors adding power to these special effects include radial modulations (due to mis-estimates modes would cause the black circles to lie systematically of the radial selection function) and angular modulations above the other points, but no such trend is seen, so there (due to effects such as uncorrected dust extinction, vari- is no indication of excess radial or angular power in the able observing conditions, photometric calibration errors data. and fiber collisions) of the density field. As long as such The slight shifts seen in the power on the largest scales effects are uncorrelated with the cosmic density field, are expected, since a non-negligible fraction of the infor- they will tend to add rather than subtract power. mation has been discarded on those scales. Figure 11 13 shows that removing the special modes results in a no- IV. COSMOLOGICAL PARAMETERS ticeable error bar increase on the largest scales and es- sentially no change on smaller scales. This can be read- Let us now explore the cosmological implications of ily understood geometrically. If we count the number of our measurements by combining them with those from modes that probe mainly scales k < k∗, then the num- WMAP. As there has recently been extensive work on ber of purely radial, purely angular and arbitrary modes constraining cosmological parameters by combining mul- 2 3 will grow as k∗, k∗ and k∗, respectively. This means tiple cosmological data sets involving CMB, galaxy clus- that “special” modes (radial and angular) will make up tering, Lyman α Forest, gravitational lensing, supernovae a larger fraction of the total pool on large scales (at Ia and other probes (see, in particular, [7, 39]), we will fo- small k), and that the purely radial ones will be outnum- cus more narrowly on what can be learned from WMAP bered by the purely angular ones. Conversely, the first and the LRGs alone. This is interesting for two reasons: 12 modes are all special ones: the monopole, the seven modes related to local-group motion, one radial mode 1. Less is more, in the sense that our results hinge and three angular modes. This means that almost all in- on fewer assumptions about data quality and mod- formation on the very largest scales is lost when discard- eling. The WMAP and LRG power spectra suf- ing special modes. Figure 11 illustrates this with the left- fice to break all major degeneracies within a broad most point labeled “generic” both having large error bars class of models, yet they are also two remarkably and being shifted to the right, where more information clean measurements, probing gravitational cluster- remains — yet it is consistent, lying about 1.3σ above ing only on very large scales where complicated an imaginary line between the two leftmost black points. nonlinear physics is unlikely to cause problems. We also repeated the cosmological parameter analysis re- 2. Since the LRG power spectrum is likely to be in- ported below with the special modes omitted, finding no cluded (together with WMAP and other data sets) significant change in the measured parameter values. in future parameter analyses by other groups, it is important to elucidate what information it contains about cosmological parameters. We will therefore place particular emphasis on clarifying the links be- tween cosmological parameters and observable fea- F. Other tests tures of both the LRG and WMAP power spectra, notably the LRG matter-radiation equality scale, the LRG acoustic scale, the CMB acoustic scale, We have found no evidence for systematic errors af- unpolarized CMB peak height ratios and large-scale flicting our power spectrum, suggesting that such effects, CMB polarization. if present, are substantially smaller than our statistical We then compare our constraints with those from other errors. For additional bounds on potential systematic cosmological probes in Section VC. We also compare errors in the SDSS LRG sample, see [43]. our results with the analysis of [36] below, which had A direct comparison of our P (k)-measurement and the narrower focus of measuring the LRG acoustic scale; that of [43] is complicated because these are not mea- the correlation function analysis in that paper comple- surements of the same function. [43] measures the angle- ments our present analysis, since the acoustic oscillations averaged redshift-space galaxy power spectrum, whereas in P (k) correspond to a readily measured single localized our PKL-method attempts to recover the real space feature in real space [36, 96]. galaxy power spectrum, using finger-of-god (FOG) com- We work within the context of the arguably simplest pression and linear theory to remove redshift-space dis- inflationary scenario that fits our data. This is a hot tortion effects [28]. The galaxy selection is also different, Big Bang cosmology with primordial fluctuations that with [43] mixing main sample galaxies in with the LRGs. are adiabatic (i.e., we do not include isocurvature modes) Both of these differences are expected to affect the non- and Gaussian, with negligible generation of fluctuations linear corrections. In addition, the quantity P (k) plotted by cosmic strings, textures or domain walls. We assume in [43] has correlated points with broader window func- the standard model of particle physics with three active tions than our uncorrelated points, and the angular cov- neutrino species, very slightly heated during the era of erage of the sample used in [43] is about 20% larger. To electron/positron annihilation [97]. Within this frame- make a direct but approximate comparison with [43], we work, we parameterize our cosmological model in terms perform our own FKP analysis, both with and without of 12 parameters that are nowadays rather standard, aug- FOG-compression, and as described in Appendix A 1, we mented with the two nuisance parameters b and Qnl from obtain good agreement with [43] on linear scales for the equation (4): case of no defogging. p (Ωtot, ΩΛ,ωb,ωc,ων ,w,As,r,ns,nt,α,τ,b,Qnl). We further investigate the robustness of our results to ≡ (6) systematic errors in Section IV F below, this time focus- Table 2 defines these 14 parameters and another ing on their potential impact on cosmological parameters. 45 that can be derived from them; in essence, 14 (Ωtot, ΩΛ,ωb,ωc,ων , w) define the cosmic matter bud- ΩΛ = 0.763, ωb = 0.0223, ωc = 0.105, As = 0.685, get, (As,ns, α, r, nt) specify the seed fluctuations and ns = 0.954, τ = 0.0842, b = 1.90, Qnl = 31.0. As cus- (τ,b,Qnl) are nuisance parameters. We will frequently tomary, the 2σ contours in the numerous two-parameter use the term “vanilla” to refer to the minimal model figures below are drawn where the likelihood has dropped space parametrized by (ΩΛ,ωb,ωc, As,ns,τ,b,Qnl), set- to 0.0455 of its maximum value, which corresponds to 2 ting ων = α = r = nt = 0, Ωtot = 1 and w = 1; this ∆χ 6.18 and 95.45% 95% enclosed probability for is the smallest subset of our parameters that provides− a a two-dimensional≈ Gaussian≈ distribution. good fit to our data. Since current nt-constraints are too We will spend most of the remainder of this paper di- weak to be interesting, we make the slow-roll assumption gesting this information one step at a time, focusing on nt = r/8 throughout this paper rather than treat nt as what WMAP and SDSS do and don’t tell us about the − a free parameter. underlying physics, and on how robust the constraints All our parameter constraints were computed using the are to assumptions about physics and data sets. The now standard Monte Carlo Markov Chain (MCMC) ap- one-dimensional constraints in the tables and Figure 12 proach [98–104] as implemented in [33] 2. fail to reveal important information hidden in param- eter correlations and degeneracies, so we will study the joint constraints on key 2-parameter pairs. We will begin A. Basic results with the vanilla 6-parameter space of models, then intro- duce additional parameters (starting in Section IV B) to quantify both how accurately we can measure them and Our constraints on individual cosmological parameters to what extent they weaken the constraints on the other are given in Tables 2 and 3 and illustrated in Figure 12, parameters. both for WMAP alone and when including our SDSS First, however, some of the parameters in Table 2 de- LRG information. Table 2 and Figure 12 take the Oc- serve comment. The additional parameters below the cam’s razor approach of marginalizing only over “vanilla” double line in Table 2 are all determined by those above parameters (ΩΛ,ωb,ωc, As,ns,τ,b,Qnl), whereas Table 3 the double line by simple functional relationships, and shows how key results depend on assumptions about the fall into several groups. non-vanilla parameters (Ω ,ω , w, r, α) introduced one tot ν Together with the usual suspects under the heading at a time. In other words, Table 2 and Figure 12 use the “other popular parameters”, we have included alterna- vanilla assumptions by default; for example, models with tive fluctuation amplitude parameters: to facilitate com- ω = 0 are used only for the constraints on ω and other ν ν parison with other work, we quote the seed fluctuation neutrino6 parameters (Ω , ξ , f and M ). ν ν ν ν amplitudes not only at the scale k =0.05/Mpc employed The parameter measurements and error bars quoted by CMBfast [113], CAMB [114] and CosmoMC [102] (de- in the tables correspond to the median and the cen- noted A and r), but also at the scale k = 0.002/Mpc tral 68% of the probability distributions, indicated by s employed by the WMAP team in [7] (denoted A.002 and three vertical lines in Figure 12. When a distribution r ). peaks near zero, we instead quote an upper limit at .002 The “cosmic history parameters” specify when our uni- the 95th percentile. Note that the tabulated median verse became matter-dominated, recombined, reionized, values are near but not identical to those of the maxi- started accelerating (¨a> 0), and produced us. mum likelihood model. Our best fit vanilla model has Those labeled “fundamental parameters” are intrinsic properties of our universe that are independent of our observing epoch tnow. (In contrast, most other parame- ters would have different numerical values if we were to 2 To mitigate numerically deleterious degeneracies, the in- dependent MCMC variables are chosen to be the param- measure them, say, 10 Gyr from now. For example, tnow eters (Θs, ΩΛ,ωb,ωd,fν ,w,Apeak,ns, α, r, nt,Aτ ,b,Qnl) from would be about 24 Gyr, zeq and ΩΛ would be larger, and i.e. Table 2, where ωd ≡ ωc + ων , , (Ωtot,ωc,ων ,As, τ) are re- h, Ωm and ωm would all be smaller. Such parameters −2τ placed by (Θs,ωd,fν ,Apeak,e ) as in [33, 105]. When impos- are therefore not properties of our universe, but merely ing a flatness prior Ωtot = 1, we retained Θs as a free parameter alternative time variables.) and dropped ΩΛ. The WMAP3 log-likelihoods are computed with the software provided by the WMAP team or taken from The Q-parameter (not to be confused with Qnl!) is WMAP team chains on the LAMBDA archive (including all un- the primordial density fluctuation amplitude 10−5. polarized and polarized information) and fit by a multivariate The curvature parameter κ is the curvature that∼ the 4th order polynomial [106] for more rapid MCMC-runs involving galaxies. The SDSS likelihood uses the LRG sample alone and Universe would have had at the Planck time if there is computed with the software available at http://space.mit. was no inflationary epoch, and its small numerical value edu/home/tegmark/sdss/ and described in Appendix A 4, em- 10−61 constitutes the flatness problem that infla- ∼ ploying only the measurements with k ≤ 0.2h/Mpc unless oth- tion solves. (ξ, ξb, ξc, ξν ) are the fundamental parame- 6 erwise specified. Our WMAP3+SDSS chains have 3 × 10 steps ters corresponding to the cosmologically popular quar- each and are thinned by a factor of 10. To be conservative, we do not use our SDSS measurement of the redshift space distor- tet (Ωm, Ωb, Ωc, Ων ), giving the densities per CMB pho- tion parameter β, nor do we use any other information (“priors”) ton. The current densities are ρi = ρhωi, where i = whatsoever unless explicitly stated. m,b,c,ν and ρh denotes the constant reference density 15 Table 2: Cosmological parameters measured from WMAP and SDSS LRG data with the Occam’s razor approach described in the text: the constraint on each quantity is marginalized over all other parameters in the vanilla set (ωb,ωc, ΩΛ, As,ns,τ,b,Qnl). Error bars are 1σ. Parameter Value Meaning Definition Matter budget parameters: +0.010 Ω 1.003 Total density/critical density Ω = Ω + Ω = 1 − Ω tot −0.009 tot m Λ k +0.017 Ω 0.761 Dark energy density parameter Ω ≈ h−2ρ (1.88 × 10−26kg/m3) Λ −0.018 Λ Λ +0.0007 ω 0.0222 Baryon density ω = Ω h2 ≈ ρ /(1.88 × 10−26kg/m3) b −0.0007 b b b +0.0041 ω 0.1050 Cold dark matter density ω = Ω h2 ≈ ρ /(1.88 × 10−26kg/m3) c −0.0040 c c c 2 −26 3 ων < 0.010 (95%) Massive neutrino density ων = Ων h ≈ ρν /(1.88 × 10 kg/m ) +0.087 w −0.941 Dark energy equation of state p /ρ (approximated as constant) −0.101 Λ Λ Seed fluctuation parameters: +0.045 A 0.690 Scalar fluctuation amplitude Primordial scalar power at k = 0.05/Mpc s −0.044 r < 0.30 (95%) Tensor-to-scalar ratio Tensor-to-scalar power ratio at k = 0.05/Mpc +0.016 n 0.953 Scalar spectral index Primordial spectral index at k = 0.05/Mpc s −0.016 +0.0096 n + 1 0.9861 Tensor spectral index n = −r/8 assumed t −0.0142 t +0.027 α −0.040 Running of spectral index α = dn /d ln k (approximated as constant) −0.027 s Nuisance parameters: +0.028 τ 0.087 Reionization optical depth −0.030 +0.074 b 1.896 Galaxy bias factor b = [P (k)/P (k)]1/2 on large scales, where P (k) refers to today. −0.069 galaxy +4.4 Q 30.3 Nonlinear correction parameter [29] P (k) = P (k)b2 (1 + Q k2)/(1 + 1.7k) nl −4.1 g dewiggled nl Other popular parameters (determined by those above): +0.019 h 0.730 Hubble parameter h = (ω + ω + ω )/(Ω − Ω ) −0.019 b c ν tot Λ +0.018 q Ω 0.239 Matter density/critical density Ω = Ω − Ω m −0.017 m tot Λ +0.0019 Ω 0.0416 Baryon density/critical density Ω = ω /h2 b −0.0018 b b +0.016 Ω 0.197 CDM density/critical density Ω = ω /h2 c −0.015 c c 2 Ων < 0.024 (95%) Neutrino density/critical density Ων = ων /h +0.0095 Ω −0.0030 Spatial curvature Ω = 1 − Ω k −0.0102 k tot +0.0044 ω 0.1272 Matter density ω = ω + ω + ω = Ω h2 m −0.0043 m b c ν m fν < 0.090 (95%) Dark matter neutrino fraction fν = ρν /ρd At < 0.21 (95%) Tensor fluctuation amplitude At = rAs Mν < 0.94 (95%) eV Sum of neutrino masses Mν ≈ (94.4 eV) × ων [107] +0.042 A 0.801 WMAP3 normalization parameter A scaled to k = 0.002/Mpc: A = 251−ns A if α = 0 .002 −0.043 s .002 s r.002 < 0.33 (95%) Tensor-to-scalar ratio (WMAP3) Tensor-to-scalar power ratio at k = 0.002/Mpc +0.035 2 σ 0.756 Density fluctuation amplitude σ = {4π ∞[ 3 (sin x − x cos x)]2P (k) k dk }1/2, x ≡ k × 8h−1Mpc 8 −0.035 8 0 x3 (2π)3 +0.024 R σ Ω0.6 0.320 Velocity fluctuation amplitude 8 m −0.023 Cosmic history parameters: +105 z 3057 Matter-radiation Equality redshift z ≈ 24074ω − 1 eq −102 eq m +0.93 z 1090.25 Recombination redshift z (ω ,ω ) given by eq. (18) of [108] rec −0.91 rec m b +2.2 1/3 z 11.1 Reionization redshift (abrupt) z ≈ 92(0.03hτ/ω )2/3Ω (assuming abrupt reionization; [109]) ion −2.7 ion b m +0.059 z 0.855 Acceleration redshift z = [(−3w − 1)Ω /Ω ]−1/3w − 1 if w < −1/3 acc −0.059 acc Λ m +0.0045 t 0.0634 Myr Matter-radiation Equality time t ≈(9.778 Gyr)×h−1 ∞ [H /H(z)(1 + z)]dz [107] eq −0.0041 eq zeq 0 +0.0040 t 0.3856 Myr Recombination time t ≈(9.778 Gyr)×h−1R ∞ [H /H(z)(1 + z)]dz [107] rec −0.0040 req zrec 0 +0.20 t 0.43 Gyr Reionization time t ≈(9.778 Gyr)×h−1 R ∞ [H /H(z)(1 + z)]dz [107] ion −0.10 ion zion 0 +0.25 t 6.74 Gyr Acceleration time t ≈(9.778 Gyr)×h−1 R ∞ [H /H(z)(1 + z)]dz [107] acc −0.24 acc zacc 0 +0.15 t 13.76 Gyr Age of Universe now t ≈(9.778 Gyr)×h−1R ∞[H /H(z)(1 + z)]dz [107] now −0.15 now 0 0 Fundamental parameters (independent of observing epoch): R +0.051 1/2 Q 1.945 ×10−5 Primordial fluctuation amplitude Q = δ ≈ A × 59.2384µK/T −0.053 h .002 CMB +3.7 κ 1.3 ×10−61 Dimensionless spatial curvature [110] κ = (¯hc/k T a)2k −4.3 B CMB +0.11 ρ 1.48 ×10−123ρ Dark energy density ρ ≈ h2Ω × (1.88 × 10−26kg/m3) Λ −0.11 Pl Λ Λ +1.2 ρ 6.6 ×10−123 ρ Halo formation density ρ = 18π2Q3ξ4 halo −1.0 Pl halo +0.11 ξ 3.26 eV Matter mass per photon ξ = ρ /n −0.11 m γ +0.018 ξ 0.569 eV Baryon mass per photon ξ = ρ /n b −0.018 b b γ +0.11 ξ 2.69 eV CDM mass per photon ξ = ρ /n c −0.10 c c γ ξν < 0.26 (95%) eV Neutrino mass per photon ξν = ρν /nγ +0.20 η 6.06 ×10−10 Baryon/photon ratio η = n /n = ξ /m −0.19 b g b p +135 A 2077 Expansion during matter domination (1 + z )(Ω /Ω )1/3 [111] Λ −125 eq m Λ +0.024 σ∗ 0.561 ×10−3 Seed amplitude on galaxy scale Like σ but on galactic (M = 1012M ) scale early on gal −0.023 8 ⊙ CMB phenomenology parameters: +0.013 A 0.579 Amplitude on CMB peak scales A = A e−2τ peak −0.013 peak s +0.012 A 0.595 Amplitude at pivot point A scaled to k = 0.028/Mpc: A = 0.56ns−1A if α = 0 pivot −0.011 peak pivot peak +0.37 H 4.88 1st CMB peak ratio H (Ω , Ω ,ω ,ω ,w,n , τ) given by [112] 1 −0.34 1 tot Λ b m s +0.0051 12ω0.52 H 0.4543 2nd to 1st CMB peak ratio H = (0.925ω0.18 2.4ns−1)/[1 + (ω /0.0164) m )]0.2 [112] 2 −0.0051 2 m b +0.0088 H 0.4226 3rd to 1st CMB peak ratio H = 2.17[1 + (ω /0.044)2 ]−1ω0.593.6ns−1/[1+ 1.63(1 − ω /0.071)ω ] 3 −0.0086 3 b m b m +0.17 1/2 z 1/2 d (z ) 14.30 Gpc Comoving angular diameter distance to CMB d (z ) = c sinh Ω rec [H /H(z)]dz /Ω [107] A rec −0.17 A rec H k 0 0 k 0 +0.0014 r (z ) 0.1486 Gpc Comoving sound horizon scale r (ω ,ω ) given by eq. (22)R of [108] s rec −0.0014 s m b +0.0009 r 0.0672 Gpc Comoving acoustic damping scale r (ω ,ω ) given by eq. (26) of [108] damp −0.0008 damp m b +0.0020 Θ 0.5918 CMB acoustic angular scale fit (degrees) Θ (Ω , Ω ,w,ω ,ω ) given by [112] s −0.0020 s tot Λ b m +1.0 ℓ 302.2 CMB acoustic angular scale ℓ = πd (z )/r (z ) A −1.0 A A rec s rec 16 Table 3: How key cosmological parameter constraints depend on data used and on assumptions about other parameters. The columns compare different theoretical priors indicated by numbers in italics. wc denotes dark energy that can cluster as in [7]. Rows labeled “+SDSS” combine WMAP3 and SDSS LRG data. Data Vanilla Vanilla+Ωtot Vanilla+r Vanilla+α Vanilla+ων Vanilla+w Vanilla+wc 1 +0.064 1 1 1 1 1 Ωtot WMAP 1.054−0.046 1 +0.010 1 1 1 1 1 +SDSS 1.003−0.009 +0.032 +0.14 +0.038 +0.051 +0.082 +0.071 +0.064 ΩΛ WMAP 0.761−0.037 0.60−0.17 0.805−0.042 0.708−0.060 0.651−0.086 0.704−0.100 0.879−0.168 +0.017 +0.020 +0.018 +0.020 +0.024 +0.019 +0.020 +SDSS 0.761−0.018 0.757−0.021 0.771−0.019 0.750−0.022 0.731−0.030 0.757−0.020 0.762−0.021 +0.037 +0.23 +0.042 +0.060 +0.086 +0.10 +0.17 Ωm WMAP 0.239−0.032 0.46−0.19 0.195−0.038 0.292−0.051 0.349−0.082 0.30−0.07 0.12−0.06 +0.018 +0.028 +0.019 +0.022 +0.030 +0.020 +0.021 +SDSS 0.239−0.017 0.246−0.025 0.229−0.018 0.250−0.020 0.269−0.024 0.243−0.019 0.238−0.020 +0.0082 +0.0082 +0.0096 +0.010 +0.011 +0.0083 +0.0082 ωm WMAP 0.1272−0.0080 0.1277−0.0079 0.1194−0.0092 0.135−0.009 0.139−0.011 0.1274−0.0082 0.1269−0.0080 +0.0044 +0.0066 +0.0043 +0.0045 +0.0048 +0.0063 +0.0075 +SDSS 0.1272−0.0043 0.1260−0.0064 0.1268−0.0042 0.1271−0.0044 0.1301−0.0044 0.1248−0.0059 0.1264−0.0079 +0.0007 +0.0008 +0.0011 +0.0010 +0.0009 +0.0007 +0.0008 ωb WMAP 0.0222−0.0007 0.0218−0.0008 0.0233−0.0010 0.0210−0.0010 0.0215−0.0009 0.0221−0.0007 0.0222−0.0007 +0.0007 +0.0007 +0.0009 +0.0010 +0.0008 +0.0007 +0.0008 +SDSS 0.0222−0.0007 0.0222−0.0007 0.0229−0.0008 0.0213−0.0010 0.0221−0.0008 0.0223−0.0007 0.0224−0.0007 ων WMAP 0 0 0 0 < 0.024 (95%) 0 0 +SDSS 0 0 0 0 < 0.010 (95%) 0 0 Mν WMAP 0 0 0 0 < 2.2 (95%) 0 0 +SDSS 0 0 0 0 < 0.94 (95%) 0 0 1 1 1 1 1 +0.23 +0.88 w WMAP − − − − − −0.82−0.19 −1.69−0.85 1 1 1 1 1 +0.087 +0.17 +SDSS − − − − − −0.941−0.101 −1.00−0.19 +0.050 +0.051 +0.064 +0.056 +0.085 +0.054 +0.066 σ8 WMAP 0.758−0.051 0.732−0.046 0.706−0.072 0.776−0.053 0.597−0.075 0.736−0.052 0.747−0.066 +0.035 +0.046 +0.036 +0.036 +0.056 +0.048 +0.057 +SDSS 0.756−0.035 0.747−0.044 0.751−0.036 0.739−0.035 0.673−0.061 0.733−0.043 0.745−0.056 r.002 WMAP 0 0 < 0.65 (95%) 0 0 0 0 +SDSS 0 0 < 0.33 (95%) 0 0 0 0 +0.017 +0.017 +0.032 +0.047 +0.022 +0.017 +0.019 ns WMAP 0.954−0.016 0.943−0.016 0.982−0.026 0.871−0.046 0.928−0.024 0.945−0.016 0.947−0.017 +0.016 +0.017 +0.022 +0.041 +0.017 +0.016 +0.018 +SDSS 0.953−0.016 0.952−0.016 0.967−0.020 0.895−0.042 0.945−0.017 0.950−0.016 0.953−0.017 0 0 0 +0.031 0 0 0 α WMAP −0.056−0.031 0 0 0 +0.027 0 0 0 +SDSS −0.040−0.027 +0.033 +0.15 +0.058 +0.044 +0.065 +0.085 +0.46 h WMAP 0.730−0.031 0.53−0.10 0.782−0.047 0.679−0.040 0.630−0.044 0.657−0.086 1.03−0.37 +0.019 +0.047 +0.022 +0.022 +0.025 +0.031 +0.037 +SDSS 0.730−0.019 0.716−0.043 0.744−0.021 0.713−0.022 0.695−0.028 0.716−0.029 0.727−0.034 +0.16 +1.5 +0.21 +0.20 +0.24 +0.34 +0.49 tnow WMAP 13.75−0.16 16.0−1.8 13.53−0.25 13.98−0.20 14.31−0.33 13.96−0.28 13.44−0.27 +0.15 +0.59 +0.17 +0.19 +0.22 +0.18 +0.26 +SDSS 13.76−0.15 13.93−0.58 13.65−0.18 13.90−0.19 13.98−0.20 13.80−0.17 13.77−0.24 +0.029 +0.029 +0.031 +0.031 +0.029 +0.030 +0.030 τ WMAP 0.090−0.029 0.083−0.029 0.091−0.032 0.101−0.031 0.082−0.030 0.087−0.031 0.087−0.030 +0.028 +0.029 +0.029 +0.032 +0.028 +0.030 +0.030 +SDSS 0.087−0.030 0.088−0.031 0.085−0.031 0.101−0.032 0.087−0.029 0.090−0.031 0.089−0.032 b WMAP +0.074 +0.092 +0.078 +0.081 +0.11 +0.076 +0.10 +SDSS 1.896−0.069 1.911−0.086 1.919−0.072 1.853−0.077 2.03−0.10 1.897−0.072 1.92−0.08 Qnl WMAP +4.4 +4.6 +4.5 +6.1 +6.9 +4.7 +5.0 +SDSS 30.3−4.1 30.0−4.2 30.9−4.1 34.7−5.4 34.9−5.3 31.0−4.3 31.0−4.4 ∆χ2 WMAP 0.0 −2.0 0.0 −3.6 −1.0 −1.0 0.0 +SDSS 0.0 0.0 −0.5 −2.4 −0.5 −0.9 −0.3 2 3(H/h)2/8πG = 3(100kms−1Mpc−1) /8πG 1.87882 [115]3. 10−26kg/m3, so the conversion between the conventional≈ × The tiny value 10−123 of the vacuum density ρ in ∼ Λ and fundamental density parameters is ξi ρi/nγ Planck units where c = G =h ¯ = 1 constitutes the well- 25.646 eV (T /2.726K)ω in units where≡c = 1. The≈ known cosmological constant problem, and the tiny yet × cmb i parameter ξm is of the same order as the temperature at matter-radiation equality temperature, kT 0.22ξ eq ≈ 3 The matter-radiation equality temperature is given by −1 30ζ(3) 7 4 4/3 kTeq = 1+ Nν ξ ≈ 0.2195ξ, (7) π4 8 11 " # where ζ(3) ≈ 1.202, and the effective number of neutrino species in the standard model is Nν ≈ 3.022 [97] when taking into ac- count the effect of electron-positron annihilation on the relic neu- trino energy density. 17 FIG. 12: Constraints on key individual cosmological quantities using WMAP1 (yellow/light grey distributions), WMAP3 (narrower orange/grey distributions) and including SDSS LRG information (red/dark grey distributions). If the orange/grey is completely hidden behind the red/dark grey, the LRGs thus add no information. Each distribution shown has been marginalized over all other quantities in the “vanilla” class of models parametrized by (ΩΛ,ωb,ωc,As,ns,τ,b,Qnl). The parameter measurements and error bars quoted in the tables correspond to the median and the central 68% of the distributions, indicated by three vertical lines for the WMAP3+SDSS case above. When the distribution peaks near zero (like for r), we instead quote an upper limit at the 95th percentile (single short vertical 2 line). The horizontal dashed lines indicate e−x /2 for x = 1 and 2, respectively, so if the distribution were Gaussian, its intersections with these lines would correspond to 1σ and 2σ limits, respectively. 3 4 3 4 similar value of the parameter combination Q ξ explains would form if ρΛ = 0 [115], so if ρΛ Q ξ , dark energy the origin of attempts to explain this value anthropically freezes fluctuation growth before then≫ and no nonlinear [116–123]: Q3ξ4 is roughly the density of the universe structures ever form. at the time when the first nonlinear dark matter halos ∗ The parameters (AΛ, σgal) are useful for anthropic 18 buffs, since they directly determine the density fluctua- ment breaks the severe vanilla degeneracy in the WMAP1 tion history on galaxy scales through equation (5) in [111] data [32, 33] (see Figure 13) and causes the dramatic ∗ (where σgal is denoted σM (0)). Roughly, fluctuations tightening of the constraints on (ωb,ωc, ΩΛ, As,ns) seen ∗ grow from the initial level σgal by a factor AΛ. Marginal- in the figures; essentially, with τ well constrained, the ra- +240 tio of large scale power to the acoustic peaks determines izing over the neutrino fraction gives AΛ = 2279−182, ∗ +0.024 −3 ns, and the relative heights of the acoustic peaks then σgal =0.538−0.022 10 . The group labeled× “CMB phenomenology parameters” determine ωb and ωc without residual uncertainty due contains parameters that correspond rather closely to to ns. Indeed, [127] has shown that discarding all the the quantities most accurately measured by the CMB, WMAP3 polarization data (both TE and EE) and re- placing it with a Gaussian prior τ =0.09 0.03 recovers such as heights and locations of power spectrum peaks. ± Many are seen to be measured at the percent level or parameter constraints essentially identical to those from better. These parameters are useful for both numeri- the full WMAP3 data set. In Section IV F 1, we will re- cal and intuition-building purposes [105, 106, 112, 124– turn to the issue of what happens if this τ-measurement 126]. Whereas CMB constraints suffer from severe de- is compromised by polarized foreground contamination. generacies involving physical parameters further up in The second important change from WMAP1 to WMAP3 is that the central values of some parameters the table (involving, e.g., Ωtot and ΩΛ as discussed be- low), these phenomenological parameters are all con- have shifted noticeably [7]. Improved modeling of noise strained with small and fairly uncorrelated measure- correlations and polarized foregrounds have lowered the ment errors. By transforming the multidimensional low-ℓ TE power and thus eliminated the WMAP1 ev- WMAP3 log-likelihood function into the space spanned idence for τ 0.17. Since the fluctuation amplitude scales as eτ times∼ the CMB peak amplitude, this τ drop by (H2,ωm,fν , ΩΛ, w, Θs, Apivot, H3, α, r, nt, Aτ ,b,Qnl), it becomes better approximated by our quartic polyno- of 0.08 would push σ8 down by about 8%. In addi- mial fit described in Footnote 2 and [106]: for example, tion, better measurements around the 3rd peak and a the rms error is a negligible ∆ ln 0.03 for the vanilla change in analysis procedure (marginalizing over the SZ- case. Roughly speaking, this transformationL≈ replaces the contribution) have lowered ωm by about 13%, causing fluctuation growth to start later (zeq decreases) and end curvature parameter Ωtot by the characteristic peak scale earlier (zacc increases), reducing σ8 by another 8%. These Θs, the baryon fraction by the ratio H2 of the first two effects combine to lower σ8 by about 21% when also tak- peak heights, the spectral index ns by the ratio H3 of the third to first peak heights, and the overall peak am- ing into account the slight lowering of ns. plitude Apeak by the amplitude Apivot at the pivot scale where it is uncorrelated with n . Aside from this nu- s 2. What SDSS LRGs add merical utility, these parameters also help demystify the “black box” aspect of CMB parameter constraints, elu- cidating their origin in terms of features in the data and A key reason that non-CMB datasets such as the in the physics [112]. 2dFGRS and the SDSS improved WMAP1 constraints so dramatically was that they helped break the vanilla banana degeneracy seen in Figure 13, so the fact B. Vanilla parameters that WMAP3 now mitigates this internally with its E- polarization measurement of τ clearly reduces the value Figure 12 compares the constraints on key parameters added by other datasets. However, Table 3 shows that from the 1-year WMAP data (“WMAP1”), the 3-year our LRG measurements nonetheless give substantial im- WMAP data (“WMAP3”) and WMAP3 combined with provements, cutting error bars on Ωm, ωm and h by about our SDSS LRG measurements (“WMAP+LRG”). We in- a factor of two for vanilla models and by up to almost an clude the WMAP1 case because it constitutes a well- order of magnitude when curvature, tensors, neutrinos or tested baseline and illustrates both the dramatic progress w are allowed. in the field and what the key new WMAP3 information The physics underlying these improvements is illus- is, particularly from E-polarization. trated in Figure 14. The cosmological information in the CMB splits naturally into two parts, one “vertical” and one “horizontal”, corresponding to the vertical and 1. What WMAP3 adds horizontal positions of the power spectrum peaks. The first thing to note is the dramatic improvement from WMAP1 to WMAP3 emphasized in [7]. (Plotted WMAP1 constraints are from [33].) As shown in [127], this stems almost entirely from the new measurement of the low-ℓ E power spectrum, which detects the reion- ization signature at about 3σ and determines the corre- sponding optical depth τ = 0.09 0.03. This measure- ± 19 FIG. 13: 95% constraints in the (Ωm,h) plane. For 6-parameter FIG. 14: Illustration of the physics underlying the previous figure. “vanilla” models, the shaded red/grey region is ruled out by Using only WMAP CMB peak height ratios constrains (ωm,ωb,ns) WMAP1 and the shaded orange/grey region by WMAP3; the main independently of As, τ, curvature and late-time dark energy prop- 2 source of the dramatic improvement is the measurement of E- erties. This excludes all but the white band ωm ≡ h Ωm = polarization breaking the degeneracy involving τ. Adding SDSS 0.127 ± 0.017 (2σ). If we assume Ωtot = 1 and vanilla dark en- LRG information further constrains the parameters to the white ergy, we can supplement this with independent “standard ruler” region marked “Allowed”. The horizontal hatched band is re- information from either WMAP CMB (thin yellow/light grey el- quired by the HST key project [137]. The dotted line shows the fit lipse) giving Ωm = 0.239 ± 0.034 (1σ), or SDSS galaxies (thicker −0.32 h = 0.72(Ωm/0.25) , explaining the origin of the percent-level blue/grey ellipse) giving Ωm = 0.239±0.027 (1σ). These two rulers 0.32 constraint h(Ωm/0.25) = 0.719 ± 0.008 (1σ). are not only beautifully consistent, but also complementary, with the joint constraints (small ellipse marked “allowed”) being tighter than those from using either separately, giving Ωm = 0.238 ± 0.017 (1σ). The plotted 2-dimensional constraints are all 2σ. The three black curves correspond to constant “horizontal” observables: con- stant angular scales for the acoustic peaks in the CMB power (dot- ∝ −0.3 ted, h ∼ Ωm ), for the acoustic peaks in the galaxy power (solid, ∝ 0.37 h ∼ Ωm ) and for the turnover in the galaxy power spectrum ∝ −0.93 (dashed, h ∼ Ωm ). This illustrates why the galaxy acoustic scale is even more helpful than that of the CMB for measuring Ωm: although it is currently less accurately measured, its degeneracy di- rection is more perpendicular to the CMB peak ratio measurement 2 of h Ωm. 20 By vertical information, we mean the relative heights of the acoustic peaks, which depend only on the physical matter densities (ωm,ωb,ων ) and the scalar primordial power spectrum shape (ns, α). They are independent of curvature and dark energy, since ΩΛ(z) Ωk(z) 0 at z > 103. They are independent of h, since≈ the physics≈ at those∼ early times depended only on the expansion rate as a function of temperature back then, which is simply ξ1/2T 3/2 times a known numerical constant, where ξ is given by ωm and the current CMB temperature (see Ta- ble 3 in [115]). They are also conveniently independent of τ and r, which change the power spectrum shape only at ℓ 102. By≪ horizontal (a.k.a. “standard ruler”) information, we mean the acoustic angular scale ℓA πdA(zrec)/rs(zrec) defined in Table 2. The ℓ-values of CMB≡ power spectrum peaks and troughs are all equal to ℓA times constants de- pending on (ωm,ωb), so changing ℓA by some factor by al- tering (Ωk, ΩΛ, w) simply shifts the CMB peaks horizon- tally by that factor and alters the late integrated Sachs 2 Wolfe effect at ℓ 10 . Although this single number ℓA is now measured≪ to great precision ( 0.3%), it depends on multiple parameters, and it is popular∼ to break this degeneracy with assumptions rather than measurements. The sound horizon at recombination r (z ) in the de- s rec FIG. 15: 95% constraints in the (Ωm, ΩΛ) plane. The large shaded nominator depends only weakly on (ωm,ωb), which are regions are ruled out by WMAP1 (red/dark grey) and WMAP3 (or- well constrained from the vertical information, and Ta- ange/grey) when spatial curvature is added to the 6 vanilla param- ble 2 shows that it is now known to about 1%. In con- eters, illustrating the well-known geometric degeneracy between trast, the comoving angular diameter distance to recom- models that all give the same acoustic peak locations in the CMB power spectrum. The yellow/light grey region is ruled out when bination dA(zrec) depends sensitively on both the spa- adding SDSS LRG information, breaking the degeneracy mainly tial curvature Ωk and the cosmic expansion history H(z), by measuring the acoustic peak locations in the galaxy power spec- which in turn depends on the history of the dark energy trum. Models on the diagonal dotted line are flat, those below are open and those above are closed. Here the yellow banana has been density: > cut off from below by an h ∼ 0.4 prior in the CosmoMC software. H(z) 2 3 4 1/2 = X(z)ΩΛ +(1+ z) Ωk +(1+ z) Ωm +(1+ z) Ωr) . H0 (8) ment, we can now add two independent pieces of infor- Here X(z) is defined as the dark energy density relative to mation if we are willing to make the vanilla assumptions its present value [128], with vanilla dark energy (a cosmo- that curvature vanishes and dark energy is a cosmological logical constant) corresponding to X(z) = 1. The most constant: If we add the WMAP horizontal information, common (although physically unmotivated) parametriza- the allowed region shrinks to the thin ellipse hugging the tion of this function in the literature has been a simple −0.3 3(1+w) h Ωm line of constant ℓA (dotted). If we instead power law X(z)= (1+z) , although it has also been ∝ 0.93 add∼ the LRG information (which constrains hΩm via constrained with a variety of other parametric and non- −0.37 the P (k) turnover scale and hΩm via the acoustic os- parametric approaches (see [129] and references therein). cillation scale5), the allowed region shrinks to the thick The parameter Ωr refers to the radiation contribution from photons and massless neutrinos, which is given by h2Ω 0.0000416(T /2.726K)4 and makes a negligible r ≈ cmb contribution at low redshift. 5 The origin of these scalings can be understood as follows. −1 Using the vertical WMAP information alone gives a The matter-radiation equality horizon scale req ∝ ωm . The 2 tight constraint on ω h Ω , corresponding to the sound horizon scales as r (z ) ∝ ω−0.25 with a weak depen- m ≡ m s eq m white band in Figure 14, independent of assumptions dence on ωb that is negligible in this context [108]. For the 4 LRG mean redshift z = 0.35, the power law fit dA(z, Ωm) = about curvature or dark energy. To this robust measure- −0.065 −1 −1 −0.065 0.3253(Ωm/0.25) cH0 ∝ h Ωm is quite good within our range of interest, accurate to within about 0.1% for 0.2 < Ωm < 0.3. For z = 1100, the power law fit dA(z, Ωm) ≈ −0.4 −1 −1 −0.4 3.4374(Ωm/0.25) cH0 ∝ h Ωm retains 0.1% accuracy 4 To obtain this ωm-constraint, we marginalized over ℓA by for for 0.19 < Ωm < 0.35. The P (k) turnover angle ∝ 2 −1 −1 −0.065 marginalizing over either Ωk or w; Table 3 shows that these two req/dA(0.35) ∝ (h Ωm) /h Ωm is therefore constant −0.93 approaches give essentially identical answers. for h ∝ Ωm , the P (k) acoustic angle ∝ rs/dA(0.35) ∝ 21 FIG. 16: 95% constraints in the (Ωtot,h) plane for 7-parameter FIG. 17: 95% constraints in the (Ωtot, tnow) plane for 7-parameter curved models. The shaded red/dark grey region was ruled out curved models. The shaded red/dark grey region is ruled out by WMAP1 alone, and WMAP3 tightened these constraints (or- by WMAP1 alone, and WMAP3 tightened these constraints (or- ange/grey region), illustrating that CMB fluctuations alone do not ange/grey region), illustrating that CMB fluctuations do not si- simultaneously show space to be flat and measure the Hubble pa- multaneously show space to be flat and measure the age of the rameter. The yellow/light grey region is ruled out when adding Universe. The yellow/light grey region is ruled out when adding SDSS LRG information. Here the yellow banana has been artifi- SDSS LRG information. The age limit tnow > 12 Gyr shown is > cially cut off for h ∼ 0.4 by a hardwired prior in the CosmoMC the 95% lower limit from white dwarf ages by [132]; for a review of software. recent age determinations, see [7]. ellipse. Let us now spice up the vanilla model space by in- These two independent pieces of horizontal informa- cluding spatial curvature Ωk and a constant dark energy tion are seen to be not only beautifully consistent, but equation of state w as free parameters, both to constrain also complementary: the joint constraints are signifi- them and to quantify how other constraints get weakened cantly tighter than those from using either separately. when dropping these vanilla assumptions. When going beyond vanilla models below, the thin CMB- only ellipse is of course no longer relevant, making the LRG constraints even more valuable. 1. LRGs as a standard ruler at z = 0.35 C. Spacetime geometry Before constraining specific spacetime geometry pa- rameters, let us review the relevant physics to intuitively To zeroth order (ignoring perturbations), the space- understand what CMB and LRGs do and do not teach time geometry is simply the Friedmann-Robertson- us about geometry. As discussed in the previous sec- tion, current CMB data accurately measure only a single Walker metric determined by the curvature Ωk and the cosmic expansion history H(z). The vanilla assumptions number that is sensitive to the spacetime geometry infor- mation in Ωk and H(z). This number is the peak angular imply the special case of no curvature (Ωk = 0) and constant dark energy (H(z) given by equation (8) with scale ℓA, and it in turn depends on the four independent X(z) = 1). parameters (Ωm, Ωk,w,h). (ΩΛ is of course not indepen- dent, fixed by the identity Ω =1 Ω Ω .) Since the Λ − k − m sound horizon size rs is now accurately known indepen- dently of spacetime geometry from CMB peak ratios, the 2 −0.25 −1 −0.065 0.37 CMB ℓA-measurement provides a precise determination (h Ωm) /h Ωm is constant for h ∝ Ωm , and the 2 −0.25 −1 −0.4 Cℓ acoustic angle ∝ rs/dA(zrec) ∝ (h Ωm) /h Ωm is of the comoving angular diameter distance to the last −0.3 constant for h ∝ Ωm . scattering surface, dA(zrec), thus allowing one function 22 of (Ωm, Ωk,w,h) to be accurately measured. dark energy ΩΛ, the Hubble parameter h, and the age of As emphasized in [36, 128, 130, 131], measuring the the universe tnow; without further information or priors, acoustic angular scale at low redshift in galaxy cluster- one cannot simultaneously demonstrate spatial flatness ing similarly constrains a second independent combina- and accurately measure ΩΛ, h or tnow, since the CMB tion of (Ωm, Ωk,w,h), and measuring dA(z) at multiple accurately constrains only the single combination ℓA. In- redshifts with future redshift surveys and current and deed, the WMAP3 degeneracy banana extends towards future SN Ia data can break all degeneracies and allow even larger Ωtot than these figures indicate; the plotted robust recovery of both Ωk and the dark energy history banana has been artificially truncated by a hardwired X(z). For the galaxy approach, the point is that leav- lower limit on h in the CosmoMC software used to com- ing the early universe physics (ωb, ωm, ns, etc.) fixed, pute this particular MCMC. changing the spacetime geometry merely scales the hori- Including our LRG information is seen to reduce the zontal axis of the angular power spectrum of galaxies at curvature uncertainty by about a factor of five, providing a given redshift z as dA(z). More generally, as described a striking vindication of the standard inflationary pre- in detail in [36], the main effect of changing the space- diction Ωtot = 1. The physical reason for this LRG im- time geometry is to shift our measured three-dimensional provement is obvious from the thick ellipse in Figure 14: power spectrum horizontally by rescaling the k-axis. The WMAP vertical peak height information combined with k-scale for angular modes dilates as the comoving an- LRG standard ruler information on dV (0.35) measures gular diameter distance dA(z) to the mean survey red- Ωm rather independently of curvature. shift z 0.35, whereas that for radial modes dilates as Yet even with WMAP+LRG information, the figures ≈ d(dA)/dz = c/H(z) for the flat case. For small varia- show that a strong degeneracy persists between curvature tions around our best fit model, the change in H(0.35) and h, and curvature and tnow, leaving the measurement is about half that of the angular diameter distance. To uncertainty on h comparable with that from the HST model this, [36] treats the net dilation as the cube root key project [137]. If we add the additional assumption of the product of the radial dilation times the square of that space is exactly flat, then uncertainties shrink by the transverse dilation, defining the distance parameter factors around 4 and 10 for h and tnow, respectively, still in beautiful agreement with other measurements. 1/3 2 cz In conclusion, within the class of almost flat models, dV (z) dA(z) . (9) ≡ H(z) the WMAP-only constraints on h, tnow, ΩΛ and Ωtot re- main weak, and including our LRG measurements pro- Using only the vertical WMAP peak height information vides a huge improvement in precision. as a prior on (ωb,ωd,ns), our LRG power spectrum gives the measurement d (0.35) = 1.300 0.088 Gpc, which V ± agrees well with the value measured in [36] using the LRG 3. Dark energy correlation function. It is this geometric LRG informa- tion that explains most of the degeneracy breaking seen Although we now know its present density fairly accu- in the Figures 15, 16, 17 and 18 below. rately, we still know precious little else about dark energy, As more LRG data become available and strengthen and much interest is focused on understanding its nature. the baryon bump detection from a few σ to > 5σ, this Assuming flat space1, Table 3 and Figure 18 show our measurement should become even more robust, not re- constraints on constant w for two cases: assuming that quiring any ωm-prior from WMAP peak heights. dark energy is homogeneous (does not cluster) and that it allows spatial perturbations (does cluster) as modeled in [7]. We see that adding w as a free parameter does not 2. Spatial curvature significantly improve χ2 for the best fit, and all data are consistent with the vanilla case w = 1, with 1σ uncer- Although it has been argued that closed inflation mod- tainties in w in the 10% - 30% range,− depending on dark els require particularly ugly fine-tuning [133], a number energy clustering assumptions. of recent papers have considered nearly-flat models ei- As described above, the physical basis of these con- ther to explain the low CMB quadrupole [134], in string straints is similar to those for curvature, since (aside from theory landscape-inspired short inflation models, or for low-ℓ corrections from the late ISW effect and dark en- anthropic reasons [110, 135, 136], so it is clearly interest- ergy clustering), the only readily observable effect of the ing and worthwhile to continue sharpening observational dark energy density history X(z) is to alter dA(zrec) and tests of the flatness assumption. In the same spirit, mea- dA(0.35), and hence the CMB and LRG acoustic angular suring the Hubble parameter h independently of theo- scales. (The dark energy history also affects fluctuation retical assumptions about curvature and measurements growth and hence the power spectrum amplitude, but of galaxy distances at low redshift provides a powerful we do not measure this because our analysis marginal- consistency check on our whole framework. izes over the galaxy bias parameter b.) Figures 15, 16 and 17 illustrate the well-known CMB It has been argued (see, e.g., [138]) that it is inappro- degeneracies between the curvature Ω 1 Ω and priate to assume Ω = 0 when constraining w, since there k ≡ − tot k 23 FIG. 18: 95% constraints in the (Ωm,w) plane. The shaded FIG. 19: 95% constraints in the (ns,r.002) plane for 7-parameter red/grey region is ruled out by WMAP1 alone when the dark tensor models (the vanilla parameters plus r). The large shaded energy equation of state w is added to the 6 vanilla parameters. regions are ruled out by WMAP1 (red/dark grey) and WMAP3 (or- The shaded orange/grey region is ruled out by WMAP3. The yel- ange/grey). The yellow/light grey region is ruled out when adding low/light grey region is ruled out when adding SDSS LRG infor- SDSS LRG information, pushing the upper limit on r.002 down by mation. The region not between the two black curves is ruled out a factor of two to r.002 < 0.33 (95%). The solid black curve with- by WMAP3 when dark energy is assumed to cluster. out shading shows the 68% limit. The two dotted lines delimit the three classes of inflation models known as small-field, large-field and hybrid models. Some single-field inflation models make highly specific predictions in this plane as indicated. From top to bottom, is currently no experimental evidence for spatial flatness the figure shows the predictions for V (φ) ∝ φ6 (line segment; ruled unless w = 1 is assumed. We agree with this critique, ∝ 4 − out by CMB alone), V (φ) φ (star; a textbook inflation model; and merely note that no interesting joint constraints can on verge of exclusion) and V (φ) ∝ φ2 (line segment; the eternal currently be placed on as many as four spacetime geome- stochastic inflation model; still allowed). These predictions assume try parameters (Ω , Ω ,w,h) from WMAP and our LRG that the number of e-foldings between horizon exit of the observed m k fluctuations and the end of inflation is 64 for the φ4 model and measurements alone, since they accurately constrain only between 50 and 60 for the others as per [150]. the two combinations dA(zrec) and dV (0.35). Other data such as SN Ia need to be included for this; [7] do this and obtain w = 1.06+0.13. − −0.08 sian and adiabatic. For the ekpyrotic universe alternative One can also argue, in the spirit of Occam’s razor, that [145], controversy remains about whether it can survive a the fact that vanilla works so well can be taken as evi- “bounce” and whether it predicts ns 1 [146] or ns 3 dence against both Ωk = 0 and w = 1, since it would [147]. ≈ ≈ require a fluke coincidence6 for them6 to− both have signifi- In the quest to measure the five parameters (Q,ns cantly non-vanilla values that conspire to lie on the same − 1, α, r, nt) characterizing inflationary seed fluctuations, dV (0.35) and dA(zrec) degeneracy tracks as the vanilla the first breakthrough was the 1992 COBE discovery that model. Q 10−5 and that the other four quantities were con- sistent∼ with zero [149]. The second breakthrough is cur- rently in progress, with WMAP3 suggesting 1 ns > 0 D. Inflation +0.019 − at almost the 3σ level (1 ns = 0.049−0.015) [7]. This central value is in good agreement− with classic (single Inflation [139–143] remains the leading paradigm for slow-rolling scalar field) inflation models, which generi- what happened in the early universe because it can solve cally predict non-scale invariance in the ballpark 1 ns the flatness, horizon and monopole problems (see, e.g., 2/N 0.04, assuming that the number of e-foldings− be-∼ [144]) and has, modulo minor caveats, successfully pre- tween∼ the time horizon the observed fluctuations exit the dicted that Ω 1, n 1, α 1 and r < 1 as well horizon and the end of inflation is 50 Harrison-Zeldovch (ns =1, r = 0) really is [39, 127, 154– FIG. 20: 95% constraints in the (ωd,fν ) plane. The large shaded 160]. For example, the WMAP team marginalized over regions are ruled out by WMAP1 (red/dark grey) and WMAP3 (orange/grey) when neutrino mass is added to the 6 vanilla param- the SZ-amplitude on small scales, which lowered the ns- estimate by about 0.01, but did not model the CMB lens- eters. The yellow/light grey region is ruled out when adding SDSS LRG information. The five curves correspond to Mν , the sum of ing effect, which would raise the ns-estimate by a com- the neutrino masses, equaling 1, 2, 3, 4 and 5 eV, respectively — parable amount [127]. It has also been argued that im- barring sterile neutrinos, no neutrino can have a mass exceeding proved modeling of point source contamination increases ∼ Mν /3 ≈ 0.3 eV (95%). the ns-estimate [158]. Inclusion of smaller-scale CMB data and LyαF information clearly affects the signifi- cance as well. The bottom line is therefore that even abundance, we obtain a 95% upper limit Mν < 0.9 eV, modest improvements in measurement accuracy over the so combining this with the atmospheric and solar neu- next few years can significantly improve our confidence in trino oscillation results [168, 169], which indicate small distinguishing between competing early-universe models mass differences between the neutrino types, implies that — even without detecting r> 0. none of the three masses can exceed Mν/3 0.3 eV. In other words, the heaviest neutrino (presumably≈ in a hi- erarchical model mostly a linear combination of νµ and E. Neutrinos ντ ) would have a mass in the range 0.04 0.3 eV. − If one is willing to make stronger assumptions about It has long been known [161] that galaxy surveys are the ability to model smaller-scale physics, notably involv- sensitive probes of neutrino mass, since they can detect ing the LyαF, one can obtain the substantially sharper the suppression of small-scale power caused by neutrinos upper bound Mν < 0.17 eV [39]. However, it should be streaming out of dark matter overdensities. For detailed noted that [39] also find that these same assumptions discussion of post-WMAP3 astrophysical neutrino con- rule out the standard model with three active neutrino straints, see [7, 39, 162–167]. species at 2.5σ, preferring more than three species. Our neutrino mass constraints are shown in Figure 20 and in the Mν-panel of Figure 12, where we allow our 6 standard 6 vanilla parameters and fν to be free . As- F. Robustness to data details suming three active neutrinos with standard freezeout Above, we explored in detail how our cosmologi- cal parameter constraints depend on assumptions about 6 physics in the form of parameter priors (Ω = 0, w = 1, It has been claimed that the true limits on neutrino masses from k − the WMAP1 (but not WMAP3) CMB maps are tighter than etc.). Let us now discuss how sensitive they are to details represented in these figures [37, 163, 164]. related to data modeling. 25 1. CMB modeling issues With any data set, it is prudent to be extra cautious regarding the most recent additions and the parts with the lowest signal-to-noise ratio. In the WMAP case, this suggests focusing on the T power spectrum around the third peak and the large-scale E-polarization data, which as discussed in Section IV B 1 were responsible for tight- ening and lowering the constraints on ωm and τ, respec- tively. The large-scale E-polarization data appear to be the most important area for further investigation, because they are single-handedly responsible for most of the dra- matic WMAP3 error bar reductions, yet constitute only a 3σ detection after foregrounds an order of magnitude larger have been subtracted from the observed polar- ized CMB maps [2]. As discussed in [127] and Sec- tion IV B 1, all the WMAP3 polarization information is effectively compressed into the probability distribution for τ, since using the prior τ = 0.09 0.03 instead of the polarized data leaves the parameter± constraints es- sentially unchanged. This error bar ∆τ = 0.03 found in [7] and Table 2 reflects only noise and sample vari- ance and does not include foreground uncertainties. If future foreground modeling increases this error bar sub- FIG. 21: The key information that our LRG measurements add stantially, it will reopen the vanilla banana degeneracy to WMAP comes from the power spectrum shape. Parametrizing described in [33]: Increasing τ and A in such a way this shape by Ωm and the baryon fraction Ωb/Ωm for vanilla mod- s els with n = 1, h = 0.72, the 95% constraints above are seen to that A A e−2τ stays constant, the peak heights re- s peak ≡ s be nicely consistent between the various radial subsamples. More- main unchanged and the only effect is to increase power over, the WMAP+LRG joint constraints from our full 6-parameter on the largest scales. The large-scale power relative to analysis are seen to be essentially the intersection of the WMAP the first peak can then be brought back down to the and “ALL LRG” allowed regions, indicating that these two shape parameters carry the bulk of the cosmologically useful LRG infor- observed value by increasing ns, after which the second mation. peak can be brought back down by increasing ωb. Since quasar observations of the Gunn-Peterson effect allow τ to drop by no more than about 1σ (0.03) [170, 171], the 2. LRG modeling issues main change possible from revised foreground modeling is therefore that (τ, ΩΛ,ωd,ωb, As,ns,h) all increase to- gether [33]. For a more detailed treatment of these issues, Since we marginalize over the overall amplitude of see [172]. LRG clustering via the bias parameter b, the LRG power A separate issue is that, as discussed in Section IV D, spectrum adds cosmological information only through reasonable changes in the CMB data modeling can easily its shape. Let us now explore how sensitive this shape increase ns by of order 0.01 [39, 127, 154–156, 158], weak- is to details of the data treatment. A popular way to ening the significance with which the Harrison-Zeldovich parametrize the power spectrum shape in the literature model (ns =1, r = 0) can be ruled out. has been in terms of the two parameters (Ωm,fb) shown in Figure 21, where f Ω /Ω is the baryon fraction. With the above-mentioned exceptions, parameter mea- b ≡ b m surements now appear rather robust to WMAP modeling Since we wish to use (Ωm,fb) merely to characterize this details. We computed parameter constraints using the shape here, not for constraining cosmology, we will ig- WMAP team chains available on the LAMBDA archive. nore all CMB data and restrict ourselves to vanilla mod- We created our own chains using the CosmoMC pack- els with ns = 1, h = 0.72 and As = 1, varying only the age [102] for the vanilla case (of length 310,817) as a four parameters (Ωm,fb,b,Qnl). Figure 21 suggest that cross-check and for the case with curvature (of length for vanilla models, the two parameters (Ωm,fb) do in 226,456) since this was unavailable on LAMBDA. The pa- fact capture the bulk of this shape information, since the rameter constraints were in excellent agreement between WMAP+LRG joint constraints from our full 6-parameter these two vanilla chains. For a fair comparison between analysis are seen to be essentially the intersection of the WMAP team and CosmoMC-based chains, the best-fit WMAP and “ALL LRG” allowed regions in the (Ωm,fb)- χ2 values listed in Table 3 have been offset-calibrated so plane. that they all give the same value for our best fit vanilla a. Sensitivity to defogging Figure 21 shows good model. consistency between the power spectrum shapes recov- 26 FIG. 23: 1σ constraints on Ωm as a function of the largest k-band included in the analysis. The yellow band shows the result when marginalizing over the baryon density ωb, the thinner cyan/grey band shows the result when fixing ωb at the best-fit WMAP3 value. our above-mentioned 4-parameter fits give highly sta- ble best-fit values Ωm = (0.244, 0.242, 0.243) and fb = FIG. 22: Effect of finger-of-god (FOG) compression. Raising the (0.168, 0.169, 0.168) together with the strongly varying FOG compression threshold δc means that fewer FOGs are iden- best-fit values Q = (27.0, 30.9, 34.2). If we fix the tified and compressed, which suppresses small-scale power while nl leaving the large scale power essentially unchanged. baryon density at the best fit WMAP3 value and vary only the three parameters (Ωm,b,Qnl), the correspond- ing results are Ωm = (0.246, 0.243, 0.244) and Qnl = (27.1, 31.0, 34.3). Note that the cosmological parameter ered from the three radial subsamples. Let us now ex- values do not show a rising or falling trend with δc. For plore in more detail issues related to our nonlinear mod- comparison, the 1σ uncertainty on Ωm from Table 2 is eling. Our results were based on the measurement using ∆Ω 0.02, an order of magnitude larger than these m ≈ FOG compression with threshold δc = 200 defined in [28]. variations. In other words, the Qnl-parameter closely Applied to the LRG sample alone, the FOG compres- emulates the effect of changing δc, so that marginalizing sion algorithm (described in detail in [28]) finds about over Qnl is tantamount to marginalizing over δc, making 20% of the LRGs in FOGs using this threshold; 77% our treatment rather robust to the modeling of nonlinear of these FOGs contain two LRGs, 16% contain three, redshift distortions. and 7% contain more than three. Thus not all LRGs b. Sensitivity to k-cutoff This is all very reassur- are brightest cluster galaxies that each reside in a sep- ing, showing that our cosmological constraints are al- arate dark matter halo. Figure 22 shows a substantial most completely unaffected by major changes in the dependence of P (k) on this δc identification threshold k > 0.1h/Mpc power spectrum. (The reason that we for k > 0.1h/Mpc. This is because FOGs smear out nonetheless∼ perform the Qnl-marginalization is if course galaxy∼ clusters along the line of sight, thereby strongly that we wish to immunize our results against any small reducing the number of very close pairs, suppressing the nonlinear corrections that extend to k < 0.1h/Mpc.) To small-scale power. Figure 22 shows that on small scales, further explore this insensitivity to nonlinearities,∼ we re- the approximate scaling P (k) k−1.3 seen for our de- peat the above analysis for the default δ = 200 case, ∝ c fault FOG compression matches∼ the well-known corre- including measurements for 0.01h/Mpc k kmax, and lation function scaling ξ(r) r−1.7, which also agrees vary the upper limit k . We apply a prior≤ 0≤ Q 50 ∝ max ≤ nl ≤ with the binding energy considerations∼ of [173]. Fit- to prevent unphysical Qnl-values for small kmax-values ting linear power spectra to these P (k) curves would (where Qnl becomes essentially unconstrained). If no clearly give parameter constraints strongly dependent on nonlinear modeling is performed, then as emphasized δc, with less aggressive FOG-removal (a higher thresh- in [43], the recovered value of Ωm should increase with old δc) masquerading as lower Ωm. Using our nonlin- kmax as nonlinear effects become important. In con- ear modeling, however, we find that δc has almost no trast, Figure 23 shows that with our nonlinear model- effect on the cosmological parameters, with the change ing, the recovered Ωm-value is strikingly insensitive to seen in Figure 22 being absorbed by a change in the k . For k 0.07h/Mpc, the constraints are weak max max ≪ Qnl-parameter. For the three cases δc = (100, 200, 337), and fluctuate noticeably as each new band power is in- 27 cluded, but for kmax beyond the first baryon bump at respectively (the new prior picks up a factor from the k 0.07h/Mpc, both the central value and the measure- Jacobian of the parameter transformation). Such prior ment∼ uncertainty remain essentially constant all the way differences could lead to substantial ( 1σ) discrepancies ∼ out to kmax =0.2h/Mpc. on parameter constraints a few years ago, when some pa- The above results tells us that, to a decent approxima- rameters were still only known to a factor of order unity. tion, our k > 0.1/Mpc data are not contributing informa- In contrast, Table 2 shows that most parameters are now tion about∼ cosmological parameters, merely information measured with relative errors in the range 1% 10%. about Q . Indeed, the error bar ∆Ω is larger when As long as these relative measurement errors are− 1, nl m ≪ using k < 0.2h/Mpc data and marginalizing over Qnl such priors become unimportant: Since the popular re- then when using merely k < 0.09h/Mpc data and fixing parametrizations in the literature and in Table 2 involve Qnl. In other words, our cosmological constraints come smooth functions that do not blow up except perhaps almost entirely from the LRG power spectrum shape at where parameters vanish or take unphysical values, the k < 0.1h/Mpc. relative variation of their Jacobian across the allowed pa- ∼c. Comparison with other galaxy P (k)-measurements rameter range will be of the same order as the relative Let us conclude this section by briefly comparing with variation of the parameters ( 1), i.e., approximately ≪ Ωm-values obtained from other recent galaxy clustering constant. Chosing a uniform prior across the allowed re- analyses. gion in one parameter space is thus essentially equivalent Our WMAP3+LRG measurement Ωm = 0.24 0.02 to choosing a uniform prior across the allowed region of has the same central as that from WMAP3 alone± [7], anybody else’s favorite parameter space. merely with a smaller error bar, and the most recent 2dFGRS team analysis also prefers Ω 0.24 [37]. This m ≈ central value is 1.5σ below the result Ωm = 0.30 0.04 V. CONCLUSIONS reported from WMAP1 + SDSS main sample galaxies± in [33]; part of the shift comes from the lower third peak We have measured the large-scale real-space power in WMAP3 as discussed in Section IV B. Post-WMAP3 spectrum P (k) using luminous red galaxies in the Sloan results are also consistent with ours. Analysis of an in- Digital Sky Survey (SDSS) with narrow well-behaved dependent SDSS LRG sample with photometric redshifts window functions and uncorrelated minimum-variance gave best-fit Ωm-values between 0.26 and 0.29 depending errors. The results are publicly available in an easy- on binning [41], while an independent analysis includ- to-use form at http://space.mit.edu/home/tegmark/ ing acoustic oscillations in SDSS LRGs and main sample sdss.html. galaxies preferred Ω 0.256 [148]. m ≈ This is an ideal sample for measuring the large-scale The galaxy power spectra measured from the above- power spectrum, since its effective volume exceeds that of mentioned data sets are likely to be reanalysed as nonlin- the SDSS main galaxy sample by a factor of six and that ear modeling methods improve. This makes it interesting of the 2dFGRS by an order of magnitude. Our results to compare their statistical constraining power. [41] do are robust to omitting purely angular and purely radial so by comparing the error bar ∆Ωm from fitting two pa- density fluctuations and are consistent between differ- rameter (Ω ,b)-models to all k 0.2h/Mpc data, with m ≤ ent parts of the sky. They provide a striking model- all other parameters, including Qnl or other nonlinear independent confirmation of the predicted large-scale modeling parameters, fixed at canonical best fit values. ΛCDM power spectrum. The baryon signature is clearly This gives ∆Ω 0.020 for 2dFGRS and ∆Ω 0.012 m ≈ m ≈ detected (at 3σ), and the acoustic oscillation scale pro- for for the SDSS LRG sample with photometric redshifts vides a robust measurement of the distance to z = 0.35 [41]. Applying the same procedure to our LRGs yields independent of curvature and dark energy assumptions. ∆Ωm = 0.007. This demonstrates both the statistical Although our measured power spectrum provides inde- power of our sample, and that our cosmological analysis pendent cross-checks on Ωm and the baryon fraction, in has been quite conservative in the sense of marginalizing good agreement with WMAP, its main utility for cosmo- away much of the power spectrum information (marginal- logical parameter estimation lies in complementing CMB izing over Qnl doubles the error bar to ∆Ωm =0.014). measurements by breaking their degeneracies; for exam- ple, Table 3 shows that it cuts error bars on Ωm, ωm and h by about a factor of two for vanilla models (ones 3. Other issues with a cosmological constant and negligible curvature, tensor modes, neutrinos and running spectral index) and A fortunate side effect of improved cosmological preci- by up to almost an order of magnitude when curvature, sion is that priors now matter less. Monte Carlo Markov tensors, neutrinos or w are allowed. We find that all Chain generators usually assume a uniform Bayesian these constraints are essentially independent of scales prior in the space of its “work parameters”. For ex- k> 0.1h/Mpc and associated nonlinear complications. ample, if two different papers parametrize the fluctua- Since the profusion of tables and figures in Section IV tion amplitude with As and ln As, respectively, they im- can be daunting to digest, let us briefly summarize them plicitly assign A -priors that are constant and 1/A , and discuss both where we currently stand regarding cos- s ∝ s 28 mological parameters and some outstanding issues. them by factors 2 - 10. The prior assumptions of the vanilla model (Ωk = r = fν = α = 0, w = 1) matter a lot with WMAP alone, and when one of them− is dropped, A. The success of vanilla the best fit values of Ωm and h are typically very different, with much larger errors. These assumptions no longer The first obvious conclusion is that “vanilla rules OK”. matter much when SDSS is included, greatly simplify- We have seen several surprising claims about cosmologi- ing the caveat list that the cautious cosmologist needs to cal parameters come and go recently, such as a running keep in mind. This is quite different from the recent past, spectral index, very early reionization and cosmologically when the joint constraints from older WMAP and SDSS detected neutrino mass — yet the last two rows of Table 3 data were sensitive to prior assumptions such as spatial show that there is no strong evidence in the data for any flatness [33]; a major reason for this change is clearly non-vanilla behavior: none of the non-vanilla parameters the SDSS measurement of the baryon acoustic scale. In- reduces χ2 significantly relative to the vanilla case. The deed, one of the most interesting results of our analysis is the strengthened evidence for a flat universe, with the WMAP team made the same comparison for the CMB- +0.064 only case and came to the same conclusion [7]. Adding a constraint on Ωtot tightening from 1.054−0.046 (WMAP3 2 +0.010 generic new parameter would be expected to reduce χ by only) to 1.003−0.009 (WMAP3+SDSS). about unity by fitting random scatter. Although WMAP In other words, large-scale cosmic clustering data now alone very slightly favor spatial curvature, this preference robustly constrain all the vanilla parameters, even when disappears when SDSS is included. The only non-vanilla any one of (fν , Ωk,r,fν , w) are included as in Table 3. behavior that is marginally favored is running spectral If w is varied jointly with Ωk (as it arguably should index α< 0, although only at 1.6σ. This persistent suc- be [138]), one expects dramatically weakened constraints cess of the vanilla model may evoke disturbing parallels on the two (since two standard rulers cannot determine with the enduring success of the standard model of par- the three parameters (w, Ωk, Ωm)), but rather unaffected ticle physics, which has frustrated widespread hopes for degradation for the rest. surprises. However, the recent evidence for ns < 1 rep- resents a departure from the ns = 1 “vanilla lite” model that had been an excellent fit ever since COBE [149], and as we discuss below, there are good reasons to expect fur- ther qualitative progress soon. C. Other data Our cosmological parameter analysis has been very B. Which assumptions matter? conservative, using the bare minimum number of data sets (two) needed to break all major degeneracies, and When quoting parameter constraints, it is important to using measurements which mainly probe the large-scale know how sensitive they are to assumptions about both linear regime. It is therefore interesting to compare our data sets and priors. The most important data assump- results with the complementary approach of [39] of push- tions discussed in Section IV F are probably those about ing the envelope by using essentially all available data polarized CMB foreground modeling for constraining τ (including LyαF, supernovae Ia and smaller-scale CMB and those about nonlinear galaxy clustering modeling for experiments), which gives tighter constraints at the cost constraining the power spectrum shape. The effect of pri- of more caveats. Comparing with the error bars in Ta- ors on other parameters is seen by comparing the seven ble 1 of [39] shows that the additional data give merely columns of Table 3, and the effect of including SDSS is modest improvements for (ωb,ωd,ns,r,h), a halving of seen by comparing odd and even rows. the error bars on Ωtot (still consistent with flatness), WMAP alone has robustly nailed certain parameters and great gains for α and Mν. These last two param- so well that that neither adding SDSS information nor eters are strongly constrained by the small-scale LyαF changing priors have any significant effect. Clearly in this information, with [39] reporting α = 0.015 0.012 and − ± camp are the baryon density ωb (constrained by WMAP Mν /3 < 0.06 eV (95%), a factor of six below our con- even-odd peak ratios) and the reionization optical depth straint and bumping right up against the atmospheric τ (constrained by WMAP low-ℓ E-polarization); indeed, lower bound 0.04 eV. On the other hand, the same Table 1 in [39] shows that adding LyαF and other CMB analysis also rules∼ out the standard model with three ac- and LSS data does not help here either. The spectral in- tive neutrino species at 2.5σ [39]; one can always worry dex ns is also in this nailed-by-WMAP category as long about pushing the envelope too far by underestimating as we assume that α is negligible; generic slow-roll infla- modeling uncertainties and systematics. [39] also high- tion models predict α < 10−3, well below the limits of light interesting tension at the 2σ-level between the LyαF detectability with current| | ∼ data sets. and WMAP3 data regarding the fluctuation amplitude For many other parameters, e.g., Ωm, h and tnow, the σ8, and weak gravitational lensing may emerge as the de- WMAP-only constraints are extremely sensitive to pri- cisive arbiter here, by directly pinning down the matter ors, with the inclusion of SDSS information tightening fluctuation amplitude independently of bias [174, 175]. 29 clustering constraints are only now beginning to bump up against theory and other measurements, so that fur- ther sensitivity gains give great discovery potential. We have (ns, r, α, Ωk) to test inflation, Mν to cosmologically detect neutrino mass, w and more generally X(z) to con- strain dark energy, and σ8 to resolve tension between different cosmological probes. Cosmology has now evolved from Alan Sandage’s “search for two numbers” (h, Ωm) to Alan Alexander Milne’s “Now we are six” (h, Ωb, Ωc, σ8,ns, τ). Each time a non-trivial value was measured for a new parameter, nature gave up a valuable clue. For example, Ωc > 0 revealed the existence of dark matter, ΩΛ > 0 revealed the existence of dark energy and the recent evidence for ns < 1 may sharpen into a powerful constraint on infla- tion. Milestones clearly within reach during the next few years include a measurement of ns < 1 at high signifi- FIG. 24: 95% constraints in the (Ωm,Qnl) plane for vanilla mod- cance and Mν > 0 from cosmology to help uncover the els. The shaded regions are ruled out by WMAP1 (red/dark grey), neutrino mass hierarchy. If we are lucky and r 0.1 (as WMAP3 (orange/grey) and when adding SDSS LRG information. suggested by classic inflation and models such as∼ [152]), an r> 0 detection will push the frontier of our ignorance back to 10−35s and the GUT scale. Then there is always D. Future challenges the possibility of a wild surprise such as Ωtot = 1, large α , X(z) = 1, demonstrable non-Gaussianity,6 isocurva- | | 6 The impressive improvement of cosmological measure- ture contributions, or something totally unexpected. Our ments is likely to continue in coming years. For example, results have helped demonstrate that challenges related the SDSS should allow substantially better cosmologi- to survey geometry, bias and potential systematic errors cal constraints from LRGs for several reasons. When can be overcome, giving galaxy clustering a valuable role the SDSS-II legacy survey is complete, the sky area cov- to play in this ongoing quest for greater precision mea- ered should be about 50% larger than the DR4 sample surements of the properties of our universe. we have analyzed here, providing not only smaller er- ror bars, but also narrower window functions as the gaps Acknowledgments: in Figure 3 are filled in. Global photometric calibration We thank Angelica de Oliveira-Costa, Kirsten Hub- will be improved [176]. Various approaches may allow di- bard, Oliver Zahn and Matias Zaldarriaga for helpful rect measurements of the bias parameter b, e.g., galaxy comments, and Dulce Gon¸calves de Oliveira-Costa for lensing [178], higher-order correlations [177], halo lumi- ground support. We thank the WMAP team for mak- nosity modeling [179] and reionization physics [180]. A ing data and Monte Carlo Markov Chains public via the bias measurement substantially more accurate than our Legacy Archive for Microwave Background Data Analy- 11% constraint from redshift space distortions would be sis (LAMBDA) at http://lambda.gsfc.nasa.gov, and a powerful degeneracy breaker. Figure 24 shows that our Anthony Lewis & Sarah Bridle for making their Cos- other galaxy nuisance parameter, Qnl, is somewhat de- moMC software [102] available at http://cosmologist. generate with Ωm, so improved nonlinear modeling that info/cosmomc. Support for LAMBDA is provided by the reliably predicts the slight departure from linear theory NASA Office of Space Science. in the quasilinear regime from smaller scale data would MT was supported by NASA grants NAG5-11099 and substantially tighten our cosmological parameter con- NNG06GC55G, NSF grants AST-0134999 and 0607597, straints. More generally, any improved modeling that the Kavli Foundation, and fellowships from the David allows inclusion of higher k will help. and Lucile Packard Foundation and the Research Corpo- As a result of such data progress in many areas, pa- ration. DJE was supported by NSF grant AST-0407200 rameter constraints will clearly keep improving. How and by an Alfred P. Sloan Foundation fellowship. good is good enough? The baryon density ωb is a pa- Funding for the SDSS has been provided by the Al- rameter over which it is tempting to declare victory and fred P. Sloan Foundation, the Participating Institutions, move on: The constraints on it from cosmic clustering the National Science Foundation, the U.S. Department are in good agreement, and are now substantially tighter of Energy, the National Aeronautics and Space Adminis- than those from the most accurate competing technique tration, the Japanese Monbukagakusho, the Max Planck against which it can be cross-checked (namely Big Bang Society, and the Higher Education Funding Council for nucleosynthesis), and further error bar reduction appears England. The SDSS Web Site is http://www.sdss.org. unlikely to lead to qualitatively new insights. In con- The SDSS is managed by the Astrophysical Research trast, there are a number of parameters where cosmic Consortium for the Participating Institutions. The Par- 30 ticipating Institutions are the American Museum of Nat- formalism that also incorporates the PKL method, see ural History, Astrophysical Institute Potsdam, Univer- [181].) As long as one uses Nr Nd random points, sity of Basel, Cambridge University, Case Western Re- they will contribute negligible Poisson≫ noise; their role serve University, University of Chicago, Drexel Univer- is in effect to evaluate certain cumbersome integrals by sity, Fermilab, the Institute for Advanced Study, the Monte Carlo integration. Japan Participation Group, Johns Hopkins University, Let us define the quantity the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the ξ[f] w(r)δ(r)w(r′)δ(r′)f( r r′ )d3rd3r′. (A4) ≡ | − | Korean Scientist Group, the Chinese Academy of Sci- Z Z ences (LAMOST), Los Alamos National Laboratory, the Hereb w(r) and f(d)b are the above-mentionedb weight func- Max-Planck-Institute for Astronomy (MPIA), the Max- Planck-Institute for Astrophysics (MPA), New Mexico tions that depend on position and distance, respectively. As we will see, the “DD-2DR+RR”, FKP and FFT meth- State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton Univer- ods simply correspond to different choices of w and f. Substituting equations (A1)-(A3) into equation (A4), we sity, the United States Naval Observatory, and the Uni- find that versity of Washington. ξ[f]= ξ [f] 2ξ [f]+ ξ [f], (A5) dd − dr rr APPENDIX A: POWER SPECTRUM where we have defined ESTIMATION DETAILS b b b b N N d d w(r )w(r ) ξ [f] i j f( r r ), (A6) 1. Relation between methods for measuring the dd ≡ n¯(r )¯n(r ) | i − j | i=1 j=1 i j power spectrum and correlation function X X b Nd Nr w(r )w(s ) ξ [f] α i j f( r s ), (A7) In this section, we clarify the relationship between dif- dr ≡ n¯(r )¯n(s ) | i − j | i=1 j=1 i j ferent popular techniques for quantifying galaxy cluster- X X b Nr Nr ing with pair-based statistics, including correlation func- w(s )w(s ) tion estimation with the “DD-2DR+RR” method [52, 53] ξ [f] α2 i j f( s s ), (A8) rr ≡ n¯(s )¯n(s ) | i − j | and power spectrum estimation with the FKP [19], FFT i=1 j=1 i j X X [26, 29, 30, 38, 43] and PKL [23, 24, 27, 28, 34, 58] meth- b ods. As a first example, let us consider the FKP method [19]. This corresponds to [181] Suppose we have Nd data points giving the comoving redshift space position vectors ri of galaxies numbered f(d) = j0(kd), (A9) i =1,Nd, and Nr random points si from a mock catalog n¯(r) which has the same selection functionn ¯(r) as the real w(r) , (A10) data. The number densities of data points and random ∝ 1+¯n(r)P (k) points are then sums of Dirac δ-functions: and turns ξ into the FKP estimator of the window- N d convolved power spectrum P (k). Here j0(x) sin(x)/x, nd(r) = δ(r ri), (A1) r 2 3 ≡ a pri- − w is normalizedb so that w( ) d r = 1 and P is an i=1 ori guess as to what the galaxy power spectrum is. For X R Nr details, see [181] around equations (25) and (56). The n (r) = δ(r s ). (A2) r − i main point is that Fourier transforming δ and averaging i=1 X δ(k) 2 over a spherical shell in k-space gives the factor | −i|k·|r−r′| ′ By definition of the selection functionn ¯(r), the quantity e dΩk/4π = j0(k r r ) = f.b We apply this methodb to our LRG data and| − compare| the results with n (r) αn (r) Rthose of [43] in Figure 25, finding good agreement. δ(r) d − r , (A3) ≡ n¯(r) The FFT method [26, 29, 30, 38, 43] is identical to the FKP method except for two simplifications: P in b where α Nd/Nr, is then an unbiased estimator of the equation (A10) is taken to be a k-independent constant underlying≡ density fluctuation field δ(r) in the sense that and the density field is binned onto a three-dimensional δ = δ, where the averaging is over Poisson fluctua- grid to replace the time-consuming double sums above htionsi as customary. Except for the PKL method, all with a fast Fourier transform. techniquesb we will discuss take the same general form, The “DD-2DR+RR” method [52, 53] estimates the weighting galaxy pairs in a form that depends only on correlation function ξ(r) by the Landy-Szalay estimator the position of each galaxy and on the distance between the two, so we will now describe them all with a uni- ξdd 2ξdr + ξrr ξLS = − , (A11) fied notation. (For an even more general pair-weighting ξrr b b b b b 31 timators is clearly the same, since dividing by ξrr in equation (A11) is a reversible operation. Moreover, it is straightforward to show that the FKP estimator ofbP (k) is simply the 3D Fourier transform of the convolved cor- relation function estimator as long as the same weight- ing function w(r) is used for both. [182] also comment on this. (Note that this is a quite different statement from the well-known fact that P (k) is the 3D Fourier transform of the correlation function ξ(r).) This im- plies that the measured FKP power spectrum and the measured correlation function contain exactly the same information. In particular, it means that cosmological constraints from one are no better than cosmological con- straints from the other, since they should be identical as long as window functions, covariance matrices, etc., are handled correctly. (An analogous correspondence for purely angular data is discussed in [183].) In contrast, the information content in the PKL measurement of the power spectrum is not identical; it uses a more general pair weighting than equation (A4) and by construction contains more cosmological information; a more detailed discussion of this point is given in Appendix A.3 in [184]. Third, this Fourier equivalence between the convolved correlation function estimator and the FKP power spec- FIG. 25: Comparison of power spectrum estimation techniques. trum estimator sheds light on the fact that the decon- Our FKP measurement without defogging is seen to agree quite volved correlation function estimator ξLS is unbiased well with the measurement of [43] considering that the latter in- ( ξ (d)= ξ(d), the true correlation function), whereas cludes also main sample galaxies with different β and small-scale h iLS clustering properties. These curves cannot be directly compared the expectation value of the FKP estimatorb is merely the with the PKL measurements or theoretical models, because they trueb power spectrum convolved with a so-called window are not corrected for the effects of redshift distortions, window func- function. This difference stems from the division by ξ tions and the integral constraint; the qualitative agreement that is rr nonetheless seen is as good as one could expect given these caveats. in equation (A11): Multiplication by ξrr in real space cor- responds to convolution with the Fourier transform of ξbrr (the window function) in Fourier space.b The reason that which is often written informally as (DD 2DR + one cannot deconvolve this windowing in Fourier spaceb − RR)/RR. Here two common weighting choices in the lit- is that one cannot Fourier transform ξLS, as it is com- erature are w(r)=¯n(r) [52] and w(r)=¯n(r)/[1+¯n(r)J] pletely unknown for large d-values that exceed all pair r [53], where J ξ(r′)d3r′ tends to be of the same order separations in the survey. b ≡ 0 of magnitude as P (k). To measure the binned correla- Fourth, this equivalence implies that gridding errors in R tion function using equations (A6)-(A8), one thus sets the 3D FFT method (which become important at large k f(d) = 1 when d is inside the bin and f(d) = 0 other- [30]) can be completely eliminated by simply computing wise. the correlation function with w(r)=n ¯(r)/[1+n ¯(r)P ] These close relationships between the FKP, FFT and by summation over pairs and then transforming the con- “DD-2DR+RR” methods lead to interesting conclusions volved correlation function with the kernel j0(kr). regarding all three methods. Figure 25 compares the LRG power spectra measured First, it can be interesting for some applications to re- with the different techniques discussed above. A direct place J by P when measuring the correlation function, comparison between our PKL P (k)-measurement and using w(r)=¯n(r)/[1+¯n(r)P ], as was done for the analy- that of [43] is complicated both by window function ef- sis of the QDOT survey in [19] and for the LRG analysis fects and by the fact that the latter was performed in red- in [36]. For instance, one could use a constant P evalu- shift space without FOG compression, with SDSS MAIN ated at the baryon wiggle scale if the goal is to measure galaxies mixed in with the LRG sample. To facilitate the baryon bump in the correlation function. comparison, we performed our own FKP analysis using Second, there is an interesting equivalence between the the direct summation method as described above, with methods. For reasons that will become clear below, let us constant P = 30000(h−1Mpc)3 and α 0.06. 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