Detection of the Baryon Acoustic Peak in the Large-Scale Correlation Function of SDSS Luminous Red Galaxies

0.46 MB·English

by  D. J. Eisenstein

SubmittedtoTheAstrophysical Journal 12/31/2004
PreprinttypesetusingLATEXstyleemulateapjv.11/12/01
DETECTION OF THE BARYON ACOUSTIC PEAK IN THE LARGE-SCALE
CORRELATION FUNCTION OF SDSS LUMINOUS RED GALAXIES
Daniel J. Eisenstein1,2, Idit Zehavi1, David W. Hogg3, Roman Scoccimarro3, Michael R.
Blanton3, Robert C. Nichol4, Ryan Scranton5, Hee-Jong Seo1, Max Tegmark6,7, Zheng
Zheng8, Scott F. Anderson9, Jim Annis10, Neta Bahcall11, Jon Brinkmann12, Scott
Burles7, Francisco J. Castander13, Andrew Connolly5, Istvan Csabai14, Mamoru Doi15,
Masataka Fukugita16, Joshua A. Frieman10,17, Karl Glazebrook18, James E. Gunn11, John
S. Hendry10, Gregory Hennessy19, Zeljko Ivezic´9, Stephen Kent10, Gillian R. Knapp11,
Huan Lin10, Yeong-Shang Loh20, Robert H. Lupton11, Bruce Margon21, Timothy A.
McKay22, Avery Meiksin23, Jeffery A. Munn19, Adrian Pope18, Michael W. Richmond24,
David Schlegel25, Donald P. Schneider26, Kazuhiro Shimasaku27, Christopher
Stoughton10, Michael A. Strauss11, Mark SubbaRao17,28, Alexander S. Szalay18, Istva´n
5 Szapudi29, Douglas L. Tucker10, Brian Yanny10, & Donald G. York17
0 Submitted to The Astrophysical Journal 12/31/2004
0
2 ABSTRACT

n We present the large-scale correlation function measured from a spectroscopic sample of 46,748 lu-
a minous red galaxies from the Sloan Digital Sky Survey. The survey region covers 0.72h−3Gpc3 over
J 3816 square degrees and 0.16 < z < 0.47, making it the best sample yet for the study of large-scale
0 structure. We find a well-detected peak in the correlationfunction at 100h−1Mpc separation that is an
1 excellent match to the predicted shape and location of the imprint of the recombination-epochacoustic
oscillations on the low-redshift clustering of matter. This detection demonstrates the linear growth of

1 structure by gravitational instability between z 1000 and the present and confirms a firm predic-
v ≈
tion of the standard cosmological theory. The acoustic peak provides a standard ruler by which we
1
can measure the ratio of the distances to z = 0.35 and z = 1089 to 4% fractional accuracy and the
7
absolute distance to z = 0.35 to 5% accuracy. From the overall shape of the correlation function, we
1
measurethematterdensityΩ h2 to8%andfindagreementwiththevaluefromcosmicmicrowaveback-
1 m
0 ground (CMB) anisotropies. Independent of the constraints provided by the CMB acoustic scale, we
5 find Ωm =0.273 0.025+0.123(1+w0)+0.137ΩK. Including the CMB acousticscale, we find that the
±
0 spatial curvature is Ω = 0.010 0.009 if the dark energy is a cosmologicalconstant. More generally,
K
− ±
/ our results provide a measurement of cosmological distance, and hence an argument for dark energy,
h
based on a geometric method with the same simple physics as the microwave background anisotropies.
p
- The standard cosmological model convincingly passes these new and robust tests of its fundamental
o properties.
r
t Subject headings: cosmology: observations — large-scale structure of the universe — distance scale —
s
a cosmologicalparameters — cosmic microwave background — galaxies: elliptical and
: lenticular, cD
v
i
X
1 Steward Observatory, University of Arizona, 933 N. CherryAve., Pa´zma´nyP´eter s´eta´ny1,H-1518Budapest, Hungary
r Tucson, AZ85121 15 Institute of Astronomy, School of Science, The University of
a 2 AlfredP.SloanFellow Tokyo,2-21-1Osawa,Mitaka,Tokyo181-0015,Japan
3CenterforCosmologyandParticlePhysics,DepartmentofPhysics, 16 InstituteforCosmicRayResearch,Univ.ofTokyo,Kashiwa277-
NewYorkUniversity,4WashingtonPlace,NewYork,NY10003 8582,Japan
4 InstituteofCosmologyandGravitation,MercantileHouse,Hamp- 17 University of Chicago, Astronomy & Astrophysics Center, 5640
shireTerrace,UniversityofPortsmouth,Portsmouth,P012EG,UK S.EllisAve.,Chicago, IL60637
5 Department of Physics and Astronomy, University of Pittsburgh, 18 Department ofPhysicsandAstronomy,TheJohns HopkinsUni-
Pittsburgh,PA15260 versity,3701SanMartinDrive,Baltimore,MD21218
6 Department of Physics, University of Pennsylvania, Philadelphia, 19 United States Naval Observatory, Flagstaff Station, P.O. Box
PA19104 1149,Flagstaff,AZ86002
7 Department of Physics, Massachusetts Institute of Technology, 20CenterforAstrophysicsandSpaceAstronomy,Univ.ofColorado,
Cambridge,MA02139 Boulder,Colorado,80803
8 School of Natural Sciences, Institute forAdvanced Study, Prince- 21 SpaceTelescopeScience Institute, 3700SanMartinDrive,Balti-
ton,NJ08540 more,MD21218
9 DepartmentofAstronomy,UniversityofWashington,Box351580, 22 Dept.ofPhysics,Univ.ofMichigan,AnnArbor,MI48109-1120
Seattle WA98195-1580 23 Institute for Astronomy, University of Edinburgh, Royal Obser-
10FermilabNationalAcceleratorLaboratory,P.O.Box500,Batavia, vatory,BlackfordHill,Edinburgh,EH93HJ,UK
IL60510 24 Department of Physics, Rochester Institute of Technology, 85
11 Princeton University Observatory, Peyton Hall, Princeton, NJ LombMemorialDrive,Rochester,NY14623-5603
08544 25 Lawrence Berkeley National Lab, 1 Cyclotron Road, MS 50R-
12 ApachePointObservatory,P.O.Box59,Sunspot,NM88349 5032,Berkeley,CA94720-8160
13Institutd’EstudisEspacialsdeCatalunya/CSIC,GranCapita`2-4, 26 Department ofAstronomyandAstrophysics, Pennsylvania State
E-08034Barcelona,Spain University,UniversityPark,PA16802
14 Department of Physics of Complex Systems, Eo¨tvo¨s University, 27 Department of Astronomy, School of Science, The University of
1
2
1. introduction z 1000, since a fully baryonic model produces an effect
∼
muchlargerthanobserved;and3)itwouldprovideachar-
In the last five years, the acoustic peaks in the cosmic
acteristic and reasonably sharp length scale that can be
microwavebackground(CMB)anisotropypowerspectrum
measuredatawiderangeofredshifts,therebydetermining
have emerged as one of the strongest cosmological probes
purelybygeometrytheangular-diameter-distance-redshift
(Miller et al.1999;de Bernardiset al.2000;Hananyet al.
relationandtheevolutionoftheHubbleparameter(Eisen-
2000; Halversonet al. 2001;Lee et al. 2001;Netterfield et
stein,Hu,&Tegmark1998;Eisenstein2003). Thelastap-
al.2002;Benoˆitetal.2003;Pearsonetal.2003;Bennettet
plicationcanprovidepreciseandrobustconstraints(Blake
al.2003). Theymeasurethecontentsandcurvatureofthe
& Glazebrook2003;Hu & Haiman2003;Linder 2003;Seo
universe (Jungman et al. 1996;Knox & Page 2000;Lange
& Eisenstein 2003; Amendola et al. 2004; Dolney, Jain, &
etal.2001;Jaffeetal.2001;Knoxetal.2001;Efstathiouet
Takada 2004; Matsubara 2004) on the acceleration of the
al.2002;Percivaletal.2002;Spergelet al.2003;Tegmark
expansionrateofthe universe(Riess etal.1998;Perlmut-
et al. 2004b), demonstrate that the cosmic perturbations
ter et al. 1999). The nature of the “dark energy” causing
are generated early (z 1000) and are dominantly adia-
≫ this acceleration is a complete mystery at present (for a
batic (Hu & White 1996a,b; Hu et al. 1997; Peiris et al.
review, see Padmanabhan 2004), but sorting between the
2003; Moodley et al. 2004), and by their mere existence
various exotic explanations will require superbly accurate
largely validate the simple theory used to support their
data. The acoustic peak method could provide a geomet-
interpretation (for reviews, see Hu et al. 1997; Hu & Do-
ric complement to the usual luminosity-distance methods
delson 2002).
such as those based on type Ia supernovae (e.g. Riess et
The acoustic peaks occur because the cosmologicalper-
al. 1998; Perlmutter et al. 1999; Knop et al. 2003; Tonry
turbations excitesoundwavesinthe relativisticplasmaof
et al. 2003; Riess et al. 2004).
theearlyuniverse(Peebles&Yu1970;Sunyaev&Zel’dovich
Unfortunately, the acoustic features in the matter cor-
1970; Bond & Efstathiou 1984, 1987; Holtzmann 1989).
relations are weak (10% contrast in the power spectrum)
The recombination to a neutral gas at redshift z 1000
≈ andonlargescales. Thismeansthatonemustsurveyvery
abruptly decreases the sound speed and effectively ends largevolumes,oforder1h−3Gpc3,to detectthe signature
the wave propagation. In the time between the forma-
(Tegmark1997;Goldberg& Strauss1998;Eisenstein,Hu,
tion of the perturbations and the epoch of recombination,
& Tegmark 1998). Previous surveys have not found clean
modesofdifferentwavelengthcancompletedifferentnum-
detections, due to sample size and (in some cases) survey
bers of oscillation periods. This translates the character-
geometry. Earlysurveys(Broadhurstetal.1990;Landyet
istic time into acharacteristiclengthscaleandproducesa
al.1996;Einastoetal.1997)foundanomalouspeaksthat,
harmonic series of maxima and minima in the anisotropy
thoughunlikelytobeacousticsignatures(Eisensteinetal.
power spectrum.
1998), did not reappear in larger surveys. Percival et al.
Becausetheuniversehasasignificantfractionofbaryons,
(2001) favored baryons at 2 σ in a power spectrum anal-
cosmologicaltheory predicts that the acoustic oscillations
ysis of data from the 2dF Galaxy Redshift Survey (here-
in the plasma will also be imprinted onto the late-time
after 2dFGRS; Colless et al. 2001), but Tegmark et al.
power spectrum of the non-relativistic matter (Peebles &
(2002) did not recover the signal with 65% of the same
Yu 1970; Bond & Efstathiou 1984; Holtzmann 1989; Hu
data. Miller et al. (2002) argued that due to smearing
& Sugiyama 1996; Eisenstein & Hu 1998). A simple way
from the window function, the 2dFGRS result could only
tounderstandthisistoconsiderthatfromaninitialpoint
be due to the excess of power on large scales in baryonic
perturbationcommontothedarkmatterandthebaryons,
modelsandnotduetoadetectionoftheoscillationsthem-
the dark matter perturbation grows in place while the
selves. A spherical harmonic analysis of the full 2dFGRS
baryonic perturbation is carried outward in an expanding
(Percivaletal.2004)didnotfindasignificantbaryonfrac-
sphericalwave(Bashinsky&Bertschinger2001,2002). At
tion. Miller et al. (2001) presented a “possible detection”
recombination, this shell is roughly 150 Mpc in radius.
(2.2 σ) from a combination of three smaller surveys. The
Afterwards, the combined dark matter and baryon per-
analysis of the power spectrum of the SDSS Main sample
turbationseedstheformationoflarge-scalestructure. Be-
(Tegmark et al. 2004a,b)did not address the question ex-
cause the central perturbation in the dark matter is dom-
plicitly but would not have been expected to detect the
inant comparedto the baryonicshell, the acoustic feature
oscillations. The analysis of the correlation function of
ismanifestedasasmallsinglespikeinthecorrelationfunc-
the 2dFGRS (Hawkins et al. 2003) and the SDSS Main
tion at 150 Mpc separation.
sample(Zehavietal.2004b)did notconsiderthese scales.
The acoustic signatures in the large-scale clustering of
Large quasar surveys (Outram et al. 2003; Croom et al.
galaxies yield three more opportunities to test the cos-
2004a,b;Yahata et al. 2004) are too limited by shot noise
mological paradigm with the early-universe acoustic phe-
to reachthe requiredprecision,althoughthe highredshift
nomenon: 1) it would provide smoking-gun evidence for
doesgiveleverageondarkenergy(Outrametal.2004)via
our theory of gravitational clustering, notably the idea
the Alcock-Paczynski(1979) test.
that large-scale fluctuations grow by linear perturbation
Inthispaper,wepresentthelarge-scalecorrelationfunc-
theory from z 1000 to the present; 2) it would give
∼ tionofalargespectroscopicsampleofluminous,redgalax-
another confirmation of the existence of dark matter at
ies(LRGs)fromtheSloanDigitalSkySurvey(SDSS;York
Tokyo, 7-3-1Hongo,Bunkyo,Tokyo113-0033, Japan etal.2000). Thissamplecovers3816squaredegreesoutto
28 AdlerPlanetarium,1300S.LakeShoreDrive,Chicago,IL60605 a redshift of z = 0.47 with 46,748 galaxies. While it con-
29 Institute for Astronomy, University of Hawaii, 2680 Woodlawn tainsfewergalaxiesthanthe 2dFGRSorthe Mainsample
Drive,Honolulu,HI,96822 of the SDSS, the LRG sample (Eisenstein et al. 2001)has
BaryonAcoustic Oscillations 3
been optimized for the study of structure on the largest of Zehavi et al. (2004a). The LRG sample is unusual as
scales and as a result it is expected to significantly out- comparedto flux-limited surveysbecause ituses the same
perform those samples. First results on intermediate and type of galaxy (luminous early-types) at all redshifts.
small-scale clustering of the LRG sample were presented We modelthe radialandangularselectionfunctions us-
by Zehavi et al. (2004a) and Eisenstein et al. (2004), and ing the methods in the appendix of Zehavi et al. (2004a).
the sample is also being used for the study of galaxyclus- In brief, we build a model of the redshift distribution of
ters and for the evolution of massive galaxies (Eisenstein the sample by integratingan empiricalmodel of the lumi-
et al.2003). Herewe focus onlargescalesandpresentthe nosity function and color distribution of the LRGs with
first clear detection of the acoustic peak at late times. respectto the luminosity-colorselectionboundariesofthe
The outline of the paper is as follows. We introduce sample. Themodelissmoothonsmallscalesandincludes
the SDSS and the LRG sample in 2. In 3, we present the subtle interplay of the color-redshift relation and the
§ §
the LRG correlation function and its covariance matrix, color selection boundaries. To include slow evolutionary
along with a discussionof tests we have performed. In 4, effects, we force the model to the observed redshift his-
§
we fit the correlation function to theoretical models and togramusinglow-passfiltering. Wewillshowin 3.2that
§
constructmeasurementsonthecosmologicaldistancescale thisfilteringdoesnotaffectthecorrelationfunctiononthe
and cosmological parameters. We conclude in 5 with a scales of interest.
§
generaldiscussionofthe results. Readerswishingto focus The angularselectionfunction is basedonthe spherical
on the results rather than the details of the measurement polygondescriptionoflss sample14(Blantonetal.2004).
may want to skip 3.2, 3.3, and 4.2. We model fiber collisions and unobserved plates by the
§ § §
methods of Zehavi et al. (2004a). Regions with complete-
2. the sdss luminous red galaxy sample ness below 60% (due to unobserved plates) are dropped
entirely; the 3816 square degrees we use are highly com-
The SDSS (York et al. 2000; Stoughton et al. 2002;
plete. The survey mask excludes regions around bright
Abazajian et al. 2003, 2004) is imaging 104 square de-
stars, but does not otherwise model small-scale imperfec-
grees of high Galactic latitude sky in five passbands, u, g,
tions in the data.
r, i, and z (Fukugita et al. 1996; Gunn et al. 1998). Im-
The typical redshift of the sample is z = 0.35. The co-
age processing (Lupton et al. 2001;Stoughton et al. 2002; ordinate distance to this redshift is 962h−1Mpc for Ω =
Pier et al. 2003; Ivezi´c et al. 2004) and calibration (Hogg m
0.3,Ω =0.7. Atthisdistance,100h−1Mpcsubtends6.0◦
et al. 2001; Smith, Tucker et al. 2002) allow one to se- Λ
on the sky and corresponds to 0.04 in redshift. The LRG
lectgalaxies,quasars,andstarsforfollow-upspectroscopy correlationlengthof10h−1Mpc(Zehavietal.2004a)sub-
with twin fiber-fed double-spectrographs. Targets are as-
′
tendsonly40 ontheskyatz =0.35;viewedonthisscale,
signed to plug plates with a tiling algorithm that ensures
the survey is far more bulk than boundary.
nearly complete samples(Blantonet al.2003a);observing
A common way to assess the statistical reach of a sur-
eachplategenerates640spectra covering3800˚Ato 9200˚A
vey, including the effects of shot noise, is by the effective
with a resolution of 1800.
volume(Feldman,Kaiser,&Peacock1994;Tegmark1997)
We select galaxies for spectroscopy by two algorithms.
The primarysample (Strausset al.2002),referredto here n(~r)P(k) 2
V (k)= d3r (1)
as the SDSS Main sample, targets galaxies brighter than eff
Z (cid:18)1+n(~r)P(k)(cid:19)
r =17.77. Thesurfacedensityofsuchgalaxiesisabout90
where n(~r) is the comoving number density of the sample
per square degree, and the median redshift is 0.10 with a
at every location ~r. The effective volume is a function of
tailouttoz 0.25. TheLRGalgorithm(Eisensteinetal.
∼ the wavenumber k via the power amplitude P. For P =
2001) selects 12 additional galaxies per square degree,
∼ 104h−3Mpc3 (k 0.15hMpc−1), we find 0.13h−3Gpc3;
using color-magnitudecuts ing,r, andito selectgalaxies
to a Petrosian magnitude r < 19.5 that are likely to be forP =4 104h−3≈Mpc3(k 0.05hMpc−1),0.38h−3Gpc3;
luminous early-types at redshifts up to 0.5. All fluxes andforP×=105h−3Mpc3(k≈ 0.02hMpc−1),0.55h−3Gpc3.
are corrected for extinction (Schlegel et∼al. 1998) before The actual survey volume is≈0.72h−3Gpc3, so there are
use. The selection is extremely efficient, and the redshift roughly 700 cubes of 100h−1Mpc size in the survey. The
success rate is very high. A few additional galaxies (3 per relativesparsenessoftheLRGsample,n 10−4h3Mpc−3,
∼
square degree at z > 0.16) matching the rest-frame color is well suited to measuring power on large scales (Kaiser
and luminosity properties of the LRGs areextractedfrom 1986).
the SDSS Main sample; we refer to this combined set as In Figure 1, we show a comparison of the effective vol-
the LRG sample. ume of the SDSS LRG survey to other published surveys.
For our clustering analysis, we use 46,748 luminous red While this calculation is necessarily a rough ( 30%) pre-
∼
galaxiesover3816squaredegreesandintheredshiftrange dictorofstatisticalperformanceduetoneglectoftheexact
0.16 to 0.47. The sky coverage of the sample is shown in survey boundary and our detailed assumptions about the
Hoggetal.(2004)andissimilartothatofSDSSDataRe- amplitude of the power spectrum and the number density
lease3(Abazajianetal.2004). Werequirethatthegalax- of objects for each survey, the SDSS LRG is clearly the
ies have rest-frame g-band absolute magnitudes 23.2 < largest survey to date for studying the linear regime by a
M < 21.2 (h = 1, H = 100h km s−1 Mpc−1−) where factor of 4. The LRG sample should therefore outper-
g − 0 ∼
we have applied k corrections and passively evolved the form these surveys by a factor of 2 in fractional errors on
galaxiesto a fiducialredshift of 0.3. The resulting comov- large scales. Note that quasar surveys cover much more
ing number density is close to constant out to z = 0.36 volume than even the LRG survey, but their effective vol-
(i.e., volume limited) and drops thereafter; see Figure 1 umes are worse, even on large scales, due to shot noise.
4
linear regime fluctuations and acoustic effects.
We compute the redshift-space correlation function us-
ing the Landy-Szalay estimator (Landy & Szalay 1993).
Random catalogs containing at least 16 times as many
galaxiesastheLRGsamplewereconstructedaccordingto
theradialandangularselectionfunctionsdescribedabove.
We assume a flat cosmology with Ω =0.3 and Ω =0.7
m Λ
when computing the correlation function. We place each
data point in its comoving coordinate location based on
itsredshiftandcomputethecomovingseparationbetween
twopointsusingthe vectordifference. We usebinsinsep-
aration of 4h−1Mpc from 10 to 30h−1Mpc and bins of
10h−1Mpc thereafter outto 180h−1Mpc, for atotalof20
bins.
Weweightthesampleusingascale-independentweight-
ing that depends on redshift. When computing the corre-
lationfunction, eachgalaxyandrandompointisweighted
Fig. 1.—Theeffectivevolume(eq.[1])asafunctionofwavenum- by 1/(1 + n(z)Pw) (Feldman, Kaiser, & Peacock 1994)
berforvariouslargeredshiftsurveys. Theeffectivevolumeisarough where n(z) is the comoving number density and P =
guidetotheperformanceofasurvey(errorsscalingasVe−ff1/2)but 40,000h−3Mpc3. WedonotallowPw tochangewithswcale
shouldnotbetrustedtobetterthan30%. Tofacilitatecomparison, soastoavoidscale-dependentchangesintheeffectivebias
we have assumed 3816 square degrees for the SDSS Main sample,
causedby differential changes in the sample redshift. Our
thesameareaastheSDSSLRGsamplepresentedinthispaperand
similartotheareainDataRelease3. Thisisabout50%largerthan choice of Pw is close to optimal at k 0.05hMpc−1 and
≈
thesampleanalyzedinTegmarketal.(2004a),whichwouldbesim- within 5% of the optimal errors for all scales relevant to
ilartothecurveforthefull2dFGalaxyRedshiftSurvey(Collesset the acoustic oscillations (k . 0.15hMpc−1). At z < 0.36,
al.2003). Wehaveneglectedthepotentialgainsonverylargescales
nP is about 4, while nP 1 at z = 0.47. Our results
from the 99 outrigger fields of the 2dFGRS. The other surveys are w w
≈
theMXsurveyofclusters(Miller&Batuski2001),thePSCzsurvey do not qualitatively depend upon the value of Pw.
ofgalaxies(Sutherlandetal.1999), andthe2QZsurveyofquasars Redshift distortionscause the redshift-spacecorrelation
(Croom et al.2004a). TheSDSS DR3quasar survey(Schneider et
function to vary according to the angle between the sep-
al.2005)issimilarineffectivevolumetothe2QZ.Fortheamplitude
aration vector and the line of sight. To ease comparison
of P(k), we have used σ8 = 1 for 2QZ and PSCz and 3.6 for the
MXsurvey. Weusedσ8=1.8forSDSSLRG,SDSSMain,andthe to theory, we focus on the spherically averaged correla-
2dFGRS; Forthelatter two,thisvaluerepresents theamplitudeof tion function. Because of the boundary of the survey, the
clustering of the luminous galaxies at the surveys’ edge; at lower
numberofpossibletangentialseparationsissomewhatun-
redshift, the number density is so high that the choice of σ8 is ir-
relevant. Reducing SDSS Main or 2dFGRS to σ8 = 1, the value derrepresentedcomparedtothe numberofpossibleline of
typical ofnormalgalaxies,decreases theirVeff by30%. sightseparations,particularlyatverylargescales. Tocor-
rect for this, we compute the correlationfunctions in four
angular bins. The effects of redshift distortions are ob-
3. the redshift-space correlation function vious: large-separation correlations are smaller along the
line of sight direction than along the tangential direction.
3.1. Correlation function estimation
Wesumthesefourcorrelationfunctionsintheproportions
In this paper, we analyze the large-scale clustering us- correspondingtothefractionofthesphereincludedinthe
ing the two-point correlation function (Peebles 1980). In angular bin, thereby recovering the spherically averaged
recentyears,thepowerspectrumhasbecomethecommon redshift-space correlation function. We have not yet ex-
choice on large scales, as the power in different Fourier plored the cosmological implications of the anisotropy of
modes of the linear density field is statistically indepen- the correlationfunction (Matsubara & Szalay 2003).
dentinstandardcosmologytheories(Bardeenetal.1986). Theresultingredshift-spacecorrelationfunctionisshown
However, this advantage breaks down on small scales due in Figure 2. A more convenientview is in Figure 3, where
to non-linear structure formation, while on large scales, we have multiplied by the square of the separation, so as
elaborate methods are required to recover the statistical to flatten out the result. The errors and overlaid models
independence in the face of survey boundary effects (for will be discussed below. The bump at 100h−1Mpc is the
discussion, see Tegmark et al. 1998). The power spec- acoustic peak, to be described in 4.1.
trum and correlation function contain the same informa- The clustering bias of LRGs is§known to be a strong
tion in principle, as they are Fourier transforms of one function of luminosity (Hogg et al. 2003; Eisenstein et al.
another. The property of the independence of different 2004; Zehavi et al. 2004a) and while the LRG sample is
Fourier modes is notlost inrealspace,but rather it is en- nearly volume-limited out to z 0.36, the flux cut does
codedintothe off-diagonalelementsofthe covariancema- produce a varying luminosity cu∼t at higher redshifts. If
trix via a linear basis transformation. One must therefore larger scale correlations were preferentially drawn from
accurately track the full covariance matrix to use the cor- higher redshift, we would have a differential bias (see dis-
relation function properly, but this is feasible. An advan- cussion in Tegmark et al. 2004a). However, Zehavi et al.
tageofthecorrelationfunctionisthat,unlikeinthepower (2004a) have studied the clustering amplitude in the two
spectrum, small-scaleeffects likeshotnoiseandintra-halo limitingcases,namelytheluminositythresholdatz <0.36
astrophysics stay on small scales, well separated from the and that at z = 0.47. The differential bias between these
BaryonAcoustic Oscillations 5
Fig. 3.—AsFigure2,butplottingthecorrelationfunctiontimes
s2. Thisshowsthevariationofthepeakat20h−1Mpcscalesthatis
controlled by the redshiftof equality (and hence by Ωmh2). Vary-
ingΩmh2 alterstheamountoflarge-to-smallscalecorrelation,but
boosting the large-scale correlations too much causes an inconsis-
tency at 30h−1Mpc. The pure CDM model (magenta) is actually
closetothebest-fitduetothedatapointsonintermediatescales.
Fig. 2.—Thelarge-scaleredshift-spacecorrelationfunctionofthe
SDSS LRGsample. The errorbarsarefromthe diagonal elements
ofthemock-catalogcovariancematrix;however,thepointsarecor- catalogaresimplypickedrandomlyfromtheobservedred-
related. Note that the vertical axis mixes logarithmic and linear
shifts,producesanegligiblechangeinthecorrelationfunc-
scalings. The inset shows an expanded view with a linear vertical
axis. The models are Ωmh2 = 0.12 (top, green), 0.13 (red), and tion. This of course corresponds to complete suppression
0.14(bottom withpeak,blue),allwithΩbh2=0.024andn=0.98 of purely radial modes.
and with a mild non-linear prescription folded in. The magenta TheselectionofLRGsishighlysensitivetoerrorsinthe
line shows a pure CDM model (Ωmh2 = 0.105), which lacks the photometriccalibrationoftheg,r,andibands(Eisenstein
acoustic peak. It is interesting to note that although the data ap-
pears higherthan themodels,thecovariance between the points is et al. 2001). We assess these by making a detailed model
soft as regards overall shifts in ξ(s). Subtracting 0.002 from ξ(s) of the distribution in color and luminosity of the sample,
athteabllessct-afiletsχm2abkyesonthlye1p.3lo.tTlohoekbcuomsmpeattic1a0ll0yh−p1erMfepctc,sbcuatlec,hoanntghees includingphotometricerrors,andthencomputingthevari-
ation of the number of galaxies accepted at each redshift
other hand,isstatisticallysignificant.
with small variations in the LRG sample cuts. A 1% shift
in the r i color makes a 8-10% change in number den-
−
twosamplesonlargescalesismodest,only15%. Wemake sity; a 1% shift in the g r color makes a 5% changes in
−
a simple parameterizationof the bias asa function of red- number density out to z = 0.41, dropping thereafter; and
shift and then compute b2 averaged as a function of scale a 1% change in all magnitudes together changes the num-
overthepaircountsintherandomcatalog. Thebiasvaries ber density by 2% out to z = 0.36, increasing to 3.6% at
bylessthan0.5%asafunctionofscale,andsoweconclude z =0.47. Thesevariationsareconsistentwiththechanges
thatthereisnoeffectofapossiblecorrelationofscalewith in the observed redshift distribution when we move the
redshift. This test also shows that the mean redshift as a selection boundaries to restrict the sample. Such photo-
function of scale changes so little that variations in the metriccalibrationerrorswouldcauseanomaliesinthecor-
clustering amplitude at fixed luminosity as a function of relationfunctionasthesquareofthe numberdensityvari-
redshift are negligible. ations, as this noise source is uncorrelated with the true
sky distribution of LRGs.
3.2. Tests for systematic errors
Assessments of calibration errors based on the color of
We have performed a number of tests searching for po- the stellar locus find only 1% scatter in g, r, and i (Ivezi´c
tentialsystematicerrorsinourcorrelationfunction. First, et al. 2004), which would translate to about 0.02 in the
we have tested that the radialselection function is not in- correlationfunction. However,thesituationismorefavor-
troducingfeaturesintothecorrelationfunction. Ourselec- able, because the coherence scale of the calibration errors
tion function involves smoothing the observed histogram islimitedbythefactthattheSDSSiscalibratedinregions
◦ ◦
with a box-car smoothing of width ∆z = 0.07. This cor- about0.6 wideandupto15 long. Thismeansthatthere
responds to reducing power in the purely radial mode at are 20 independent calibrations being applied to a given
k =0.03hMpc−1 by50%. Purelyradialpoweratk =0.04 6◦ (100h−1Mpc) radius circular region. Moreover, some
(0.02)hMpc−1 isreducedby13%(86%). Theeffectofthis of the calibration errors are even more localized, being
suppressionis negligible, only 5 10−4 (10−4) on the cor- caused by small mischaracterizations of the point spread
relation function at the 30 (100×) h−1Mpc scale. Simply function and errors in the flat field vectors early in the
put, purely radial modes are a small fraction of the total survey (Stoughton et al. 2002). Such errors will average
at these wavelengths. We find that an alternative radial down on larger scales even more quickly.
selection function, in which the redshifts of the random The photometric calibration of the SDSS has evolved
6
slightlyovertime(Abazajianetal.2003),butofcoursethe
LRG selection was based on the calibrations at the time
of targeting. We make our absolute magnitude cut on the
latest(‘uber’)calibrations(Blantonetal.2004;Finkbeiner
et al. 2004), although this is only important at z < 0.36
as the targeting flux cut limits the sample at higher red-
shift. We test whether changes in the calibrations might
alter the correlation function by creating a new random
catalog,inwhichthe differencesineachbandbetweenthe
targetepochphotometryandthe‘uber’-calibrationvalues
are mapped to angularly dependent redshift distributions
usingthenumberdensityderivativespresentedabove. Us-
ingthisrandomcatalogmakesnegligibledifferencesinthe
correlation function on all scales. This is not surprising:
while an rms error of 2% in r i would give rise to a
−
0.04excessinthecorrelationfunction,scaleslargeenough
that this would matter have many independent calibra-
tionscontributingtothem. Ontheotherhand,the‘uber’- Fig. 4.—Thecorrelationfunctionfortwodifferentredshiftslices,
calibration of the survey does not necessarily fix all cali- 0.16 < z < 0.36 (filled squares, black) and 0.36 < z < 0.47 (open
squares,red). Thelatterissomewhatnoisier,butthetwoarequite
bration problems, particularly variations within a single
similarandbothshowevidencefortheacousticpeak. Notethatthe
survey run, and so this test cannot rule out an arbitrary verticalaxismixeslogarithmicandlinearscalings.
calibration problem.
Wecontinueoursearchforcalibrationerrorsbybreaking
the survey into 10 radial pieces and measuring the cross- 3.3. Covariance Matrix
correlations between the non-adjacent slabs. Calibration
Because of the large number of separation bins and the
errors would produce significant correlations on large an-
large scales being studied, it is infeasible to use jackknife
gular scales. Some cross-correlations have amplitudes of
sampling to construct a covariance matrix for ξ(r). In-
2-3%, but many others don’t, suggesting that this is sim-
stead, we use a large set of mock catalogs to construct a
plynoise. Wealsotakethefullmatrixofcross-correlations
covariancematrixandthentestthatmatrixwithasmaller
at a given separation and attempt to model it (minus the
number of jackknife samples.
diagonalandfirstoff-diagonalelements)asanouterprod-
OurmockcatalogsareconstructedusingPTHalos(Scoc-
uctofvectorwithitself, aswouldbeappropriateifitwere
cimarro&Sheth 2002)witha halooccupationmodel(Ma
dominated by a single type of radialperturbation, but we
& Fry 2000; Seljak 2000; Peacock & Smith 2000; Scocci-
do not find plausible or stable vectors, again indicative of
marro et al. 2001; Cooray & Sheth 2002; Berlind et al.
noise. Hence, we conclude that systematic errors in ξ(r)
2003) that matches the observed amplitude of clustering
due to calibration must be below 0.01.
of LRGs (Zehavi et al. 2004a). PTHalos distributes dark
It is important to note that calibration errors in the
matterhalosaccordingtoextendedPress-Schechtertheory
SDSSproducelarge-anglecorrelationsonlyalongthescan
conditioned by the large-scale density field from second-
direction. Eveniferrorswerenoticeablylarge,theywould
notproducenarrowfeaturessuchasthatseenatthe100h−1 orderLagrangianperturbationtheory. Thisgeneratesden-
sity distributions that fully include Gaussian linear the-
Mpc scale for two reasons. First, the projections from the
ory, but also include second-order clustering, redshift dis-
three-dimensional sphere to one-dimension strips on the
tortions, and the small-scale halo structure that domi-
sky necessarily means that a given angular scale maps to
nates the non-Gaussian signal. We use a cosmology of
awiderangeofthree-dimensionalseparations. Second,the
Ω = 0.3, Ω = 0.7, h = 0.7 for the mock catalogs. We
comoving angular diameter distance used to translate an- m Λ
subsample the catalogs so as to match the comoving den-
gles into transverse separations varies by a factor of three
sity of LRGs as a function of redshift exactly, but do not
from z =0.16to 0.47,so that a preferredangle wouldnot
attempt to include the small redshift dependence in the
map to a narrow range of physical scales. We therefore
amplitude of the bias. Our catalogs match the angular
expect that calibration errors would appear as a smooth
geometry of the survey except in some fine details involv-
anomalous correlation, rolling off towards large scales.
ing less than 1% of the area. Two simulation boxes, each
Breaking the sample into two redshift slices above and (1250h−1Mpc)2 2500h−1Mpc, one for the NorthGalac-
belowz =0.36yieldssimilarcorrelationfunctions(Fig.4). ×
ticCapandtheotherfortheSouth,arepastedtogetherin
Errors in calibration or in the radial selection function
eachcatalog. Thesetworegionsofthesurveyareverywell
would likely enter the two redshift slices in different man-
separated in space, so this division is harmless. We gen-
ners, but we see no sign of this. In particular, the bump
erate 1278 mock catalogs with independent initial condi-
in the correlation function appears in both slices. While
tions, compute the correlation function in each, and com-
this could in principle give additional leverage on the cos-
pute the covariance matrix from the variations between
mological distance scale, we have not pursued this in this
them.
paper.
Theresultingmatrixshowsconsiderablecorrelationsbe-
tween neighboring bins, but with an enhanced diagonal
due to shot noise. The power on scales below 10h−1Mpc
creates significant correlation between neighboring scales.
BaryonAcoustic Oscillations 7
Acuriousaspectisthatonlargescales,whereourbinsare during the radiation-dominated period of the universe.
10h−1Mpc wide, the reduced inverse covariance matrix is This introduces the characteristic turnover in the CDM
quiteclosetotri-diagonal;thefirstoff-diagonalistypically power spectrum at the scale of the horizon at matter-
around 0.4, but the second and subsequent are typically radiationequality. Thislengthscalesas(Ω h2)−1(assum-
m
a few percent. Such matrices correspond to exponential ing the standard cosmic neutrino background). Second,
decays of the off-diagonal correlations (Rybicki & Press the acoustic oscillations have a characteristic scale known
1994). asthe soundhorizon,whichis the comovingdistance that
We test the covariance matrix in several ways. First, a sound wavecantravelbetween the big bang and recom-
as described in 4.3, the best-fit cosmological model has bination. This depends both on the expansion history of
χ2 = 16.1 on 17§degrees of freedom (p = 0.52), indicat- the early universe and on the sound speed in the plasma;
ing that the covariances are of the correct scale. Next, we formodelsclosetotheconcordancecosmology,thislength
subdivide the survey into ten large, compact subregions scalesas(Ω h2)−0.25(Ω h2)−0.08 (Hu2004). Thespectral
m b
and look at the variations between jack-knifed samples tiltandmassiveneutrinostiltthelow-redshiftpowerspec-
(i.e., computing ξ while excluding one region at a time). trum (Bond & Szalay 1983), but don’t otherwise change
The variations of ξ(r) between the ten jackknife samples these two scales. Importantly, given Ω h2 and Ω h2, the
m b
matches the diagonal elements of the covariance matrix two physical scales are known in absolute units, with no
to within 10%. We then test the off-diagonal terms by factors of H−1.
0
askingwhetherthe jack-kniferesidualsvaryappropriately We use CMBfast (Seljak & Zaldarriaga 1996; Zaldar-
relative to the covariance matrix. If one defines the resid- riaga et al. 1998; Zaldarriaga & Seljak 2000) to compute
ual ∆ξ between the jth jackknife sample and the mean, the linear power spectra, which we convert to correlation
j
thentheconstruction ∆ξ (r )C−1∆ξ (r ),whereC is functionswithaFouriertransform. Itisimportanttonote
ab j a ab j b
the covariancematrixPandthe sums are overthe 20 radial that the harmonic series of acoustic peaks found in the
bins, should havea mean (averagedover the ten jackknife power spectrum transformto a single peak in the correla-
sample) of about 20/(10 1) 2.2. The mean value is tion function (Matsubara 2004). The decreasing envelope
2.7 0.5, in reasonable ag−reem≈ent. of the higher harmonics, due to Silk damping (Silk 1968)
W±egetsimilarresultsforcosmologicalparameterswhen ornon-lineargravity(Meiksinet al.1999),correspondsto
using a covariance matrix that is based on constructing abroadeningofthesinglepeak. Thisbroadeningdecreases
the Gaussian approximation (Feldman, Kaiser, & Pea- the accuracy with which one can measure the centroid of
cock1994;Tegmark1997)ofindependentmodesinFourier the peak, equivalent to the degradation of acoustic scale
space. We use the effective volume at each wavenumber, measurements caused by the disappearanceof higher har-
plus an extra shot noise term to represent non-Gaussian monics in the power spectrum (Seo & Eisenstein 2003).
halos,andthenrotatethismatrixfromthediagonalFourier Examples of the model correlation functions are shown
basis into the real-space basis. This matrix gives reason- in Figures 2 and 3, each with Ωbh2 = 0.024 and h = 0.7
able χ2 values, satisfies the jackknife tests, and gives sim- but with three values of Ωmh2. Higher values of Ωmh2
ilar values and errors on cosmologicalparameters. correspond to earlier epochs of matter-radiation equality,
Finally, we test our results for cosmological parameters which increase the amount of small-scale power compared
by using the following hybrid scheme. We use the mock- to large, which in turn decreases the correlations on large
catalogcovariancematrix to find the best-fit cosmological scales when holding the small-scale amplitude fixed. The
model for each of the ten jackknife samples, and then use acousticscale,onthe other hand,depends only weaklyon
the rms of the ten best-fit parameter sets to determine Ωmh2.
the errors. This means that we are using the mock cata-
logs to weight the ξ(r) measurements, but relying on the 4.2. Non-linear corrections
jackknifed samples to actually determine the variance on
The precision of the LRG correlations is such that we
the cosmologicalparameters. We willquotethe numerical cannotrelyentirelyonlineartheoryevenatr>10h−1Mpc
results in 4.3, but here we note that the resulting con- (wavenumbersk <0.1hMpc−1). Non-lineargravity(Meiksin
§
straints on the acoustic scale and matter density match
et al. 1999), redshift distortions (Kaiser 1987; Hamilton
the errors inferred from the fitting with the mock-catalog
1998;Scoccimarro 2004), and scale-dependent bias all en-
covariancematrix. Hence,weconcludethatourcovariance
teratasubtle level. We addressthese byapplyingcorrec-
matrix is generating correct results for our model fitting
tions derived from N-body simulations and the Smith et
and cosmologicalparameter estimates.
al. (2003) halofit method.
Seo&Eisenstein(2005)presentasetof51N-body sim-
4. constraints on cosmological models
ulations of the WMAP best-fit cosmology (Spergel et al.
4.1. Linear theory 2003). Eachsimulationhas2563particlesandis512h−1Mpc
GiventhevalueofthematterdensityΩ h2,thebaryon insize. The outputsatz =0.3wereanalyzedtofindtheir
m
density Ω h2, the spectral tilt n, and a possible neutrino power spectra and correlation functions, in both real and
b
redshift-space, including simple halo bias. This provides
mass, adiabatic cold dark matter (CDM) models predict
anaccuratedescriptionofthenon-lineargravitationaland
the linear matter power spectrum (and correlation func-
redshift distortion effects on large scales.
tion) up to an amplitude factor. There are two primary
For a general cosmology, we begin from the CMBfast
physicalscalesatwork(fordiscussions,seeHuetal.1997;
linear power spectrum. We next correct for the erasureof
Eisenstein & Hu 1998; Hu & Dodelson 2002). First, the
thehigheracousticpeakthatoccursduetomode-coupling
clustering of CDM is suppressed on scales small enough
(Meiksin et al. 1999). We do this by generating the “no-
to have been traversed by the neutrinos and/or photons
8
large scales that we simply include into the large-scale
clustering bias. After removingthis asymptotic value, the
remainingpieceoftheratioisonly10%atr=10h−1Mpc.
We fit this to a simple smoothfunction andthen multiply
themodelcorrelationfunctionsbythisfit. Figure5shows
theN-body resultsfortheratioofredshift-spaceξ toreal-
space ξ for the halos and the ratio of the real-space halo
correlation function to that of the matter. The solid line
shows our fit to the product. Tripling the mass thresh-
oldofthe biasmodelincreasesthe correctionbyonly30%
(i.e., 1.12 to 1.16 on small scales).
All of the correctionsin Figure 5 are clearly small, only
10% at r > 10h−1Mpc. This is because 10h−1Mpc sep-
arations are larger than any virialized halo and are only
affected by the extremes of the finger of God redshift dis-
tortions. While our methods are not perfect, they are
plausibly matched to the allowed cosmology and to the
Fig. 5.— Scale-dependent corrections derived from 51 N-body biasofthe LRGs. Assuch,webelievethatthecorrections
simulations,each512h−1Mpccomovingwith2563 particles(Seo& shouldbe accurateto a few percent,whichis sufficientfor
Eisenstein2005). Thecrossesshowtheratiobetweenthenon-linear
our purposes. Similarly, while we havederivedthe correc-
matter correlationfunction andthelinearcorrelationfunction; the
dashed line is the model we use from Smith et al. (2003). The tions for a single cosmological model, the data constrain
solid points are the ratio between the biased correlation function the allowedcosmologiesenoughthat variationsinthe cor-
(usingasimplehalomasscut)tothenon-linearmatter correlation rections will be smaller than our tolerances. For example,
function. Theopensquaresaretheratioofthebiasedredshift-space
increasingσ to 1.0 for the halofit calculationchangesthe
correlation function to the biased real-space correlation function, 8
afterremovingthelarge-scaleasymptoticvalue(Kaiser1987),which corrections at r >10h−1Mpc by less than 2%.
wesimplyfoldintothecorrelationamplitudeparameter. Theopen We stress that while galaxy clustering bias does rou-
triangles show the product of these two effects, and the solid line tinelyaffectlarge-scaleclustering(obviouslysointheLRG
is our fit to this product. These corrections are of order 10% at
10h−1Mpc separations and decrease quickly on larger scales. In sample, with bias b 2), it is very implausible that it
≈
additiontothesecorrections,wemimictheerasureofthesmall-scale would mimic the acoustic signature, as this would require
acoustic oscillations in the power spectrum by using a smoothed galaxy formation physics to have a strong preferred scale
cross-over at k = 0.14hMpc−1 between the CMBfast linear power at 100h−1 Mpc. Galaxy formation prescriptions that in-
spectrum andtheno-wiggleformfromEisenstein&Hu(1998).
volveonlysmall-scalephysics,suchas thatinvolvingdark
matter halos or even moderate scale radiation transport,
necessarily produce smooth effects on large scales (Coles
wiggle”approximationfromEisenstein&Hu(1998),which
1993;Fry & Gaztanaga1993;Scherrer& Weinberg 1998).
matches the overall shape of the linear power spectrum
Evenlong-rangeeffects that mightbe invokedwouldneed
but with the acoustic oscillations edited out, and then
toaffect100h−1Mpcscalesdifferentlyfrom80or130. Our
smoothly interpolating between the linear spectrum and
detection of the acoustic peak cannot reasonably be ex-
the approximate one. We use a Gaussian exp[ (ka)2],
withascalea=7h−1Mpcchosentoapproximatel−ymatch plained as an illusion of galaxy formation physics.
the suppression of the oscillations seen in the power spec-
4.3. Measurements of the acoustic and equality scales
trum of the N-body simulations.
WenextusetheSmithetal.(2003)packagetocompute TheobservedLRGcorrelationfunctioncoulddifferfrom
the alterations in power from non-linear gravitationalcol- that of the correct cosmological model in amplitude, be-
lapse. We use a model with Γ = 0.162 and σ = 0.85. cause of clustering bias and uncertain growth functions,
8
This model has the same shape for the power spectrum and in scale, because we may have used an incorrect cos-
on small scales as the WMAP best-fit cosmology and is mologyinconvertingfromredshiftintodistance. Ourgoal
therefore a reasonable zero-baryon model with which to is to use the comparisonbetween observations and theory
compute. We find the quotient of the non-linear and lin- to infer the correct distance scale.
ear power spectra for this model and multiply that ratio Notethatinprincipleachangeinthecosmologicalmodel
ontoourbaryonicpowerspectrum. WethenFouriertrans- would change the distances differently for different red-
formthe powerspectrumtogeneratethe real-spacecorre- shifts, requiring us to recompute the correlation function
lation function. The accuracy of this correction is shown for each model choice. In practice, the changes are small
in the bottom of Figure 5: the crosses are the ratio in the enough and the redshifts close enough that we treat the
N-bodysimulationsbetweenthez =0.3andz =49corre- variation as a single dilation in scale (similar to Blake &
lation functions, while the dashed line is the Smith et al. Glazebrook2003). This wouldbe asuperbapproximation
(2003) derived correction. at low redshift, where all distances behave inversely with
Next we correct for redshift distortions and halo bias. theHubbleconstant. Byz =0.35,theeffectsofcosmolog-
We use the N-body simulations to find the ratio of the icalaccelerationarebeginningtoenter. However,wehave
redshift-spacebiasedcorrelationfunctiontothereal-space checked explicitly that our single-scale approximation is
matter correlationfunction for ahalo massthresholdthat good enough for Ω between 0.2 and 0.4. Relative to our
m
approximately matches the observed LRG clustering am- fiducialscaleatz =0.35,thechangeindistanceacrossthe
plitude. This ratio approaches an asymptotic value on redshift range 0.16 < z < 0.47 is only 3% peak to peak
BaryonAcoustic Oscillations 9
Fig. 7.— The likelihood contours of CDM models as a func-
tion of Ωmh2 and DV(0.35). The likelihood has been taken to be
proportional to exp(−χ2/2), and contours corresponding to 1 σ
through 5 σ for a 2-d Gaussian have been plotted. The one-
dimensional marginalized values are Ωmh2 = 0.130±0.010 and
DV(0.35) = 1370±64 Mpc. We overplot lines depicting the two
major degeneracy directions. The solid (red) line is a line of con-
Fig. 6.—Theχ2valuesofthemodelsasafunctionofthedilation stantΩmh2DV(0.35), whichwouldbethedegeneracy directionfor
a pure CDM model. The dashed (magenta) line is a line of con-
DofVth(0e.3sc5a)lereolfattihveectoorrtehleatbioansefluinnectcioosnm. oTlohgisycoofrrΩesp=on0.d3s,tΩoΛal=ter0i.n7g, sdteavnitatseofurnodmhtohreizpounr,ehColDdMingdΩegbehn2er=acy0,.0im24p.lyTinhgetchoanttothuerspcelaekaralyt
hva=lue0.o7f.ΩEmachh2,li0n.e11(,sa0v.1e3t,haendma0g.1e5ntfarolminele)ftintothreigphlto.tΩisbha2d=iff0e.r0e2n4t 100h−1Mpcisconstrainingthefits.
and n = 0.98 are used in all cases. The amplitude of the model
has been marginalized over. The best-fit χ2 is 16.1 on 17 degrees
of freedom, consistent with expectations. The magenta line (open w(z) are subsumed within D (0.35). We assume h = 0.7
sbyemstbχol2s)osfh2o7w.8s,thwehipcuhreisCrDejMectemdodaetl3w.4ithσ.ΩmNoht2e=th0a.t10t;hiist chuarsvae whencomputingthescaleatwVhichtoapplythenon-linear
corrections;havingsetthosecorrections,wethendilatethe
is also much broader, indicating that the lack of an acoustic peak
makes thescalelessconstrainable. scale of the final correlationfunction.
The WMAP data (Bennett et al. 2003),as well as com-
binations of WMAP with large-scalestructure (Spergelet
for Ωm = 0.2 compared to 0.3, and even these variations al. 2003; Tegmark et al. 2004b), the Lyman-alpha forest
largelycancelaroundthez =0.35midpointwherewewill (McDonald et al. 2004; Seljak et al. 2004), and big bang
quote our cosmologicalconstraints. nucleosynthesis (e.g., Burles et al. 2001; Coc et al. 2004),
The other error in our one-scale-parameter approxima- constrainΩ h2 andn ratherwellandso to begin, we hold
b
tion is to treat the line-of-sight dilation equivalently to these parameters fixed (at 0.024 and 0.98, respectively),
the transverse dilation. In truth, the Hubble parame- and consider only variations in Ω h2. In practice, the
m
ter changesdifferentlyfromthe angulardiameterdistance sound horizon varies only as (Ω h2)−0.08, which means
b
(the Alcock-Paczynski(1979)effect). Forsmalldeviations that the tight constraints from WMAP (Spergel et al.
from Ωm = 0.3, ΩΛ = 0.7, the change in the Hubble pa- 2003) and big bang nucleosynthesis (Burles et al. 2001)
rameter at z = 0.35 is about half of that of the angular make the uncertainties in the baryon density negligible.
diameter distance. We model this by treating the dilation Figure 6showsχ2 as afunction ofthe dilationfor three
scale asthe cube rootofthe productofthe radialdilation different values of Ω h2, 0.11, 0.13, and 0.15. Scanning
m
timesthesquareofthetransversedilation. Inotherwords, acrossallΩ h2,thebest-fitχ2is16.1on17degreesoffree-
m
we define dom (20 data points and 3 parameters: Ω h2, D (0.35),
m V
cz 1/3 and the amplitude). Figure 7 shows the contours of equal
D (z)= D (z)2 (2)
V (cid:20) M H(z)(cid:21) χ2 inΩmh2 andDV(0.35),correspondingto 1σ upto5 σ
for a 2-dimensional Gaussian likelihood function. Adopt-
where H(z) is the Hubble parameter and DM(z) is the ing a likelihood proportional to exp( χ2/2), we project
comoving angular diameter distance. As the typical red- the axes to find Ω h2 = 0.130 0.01−0 and D (0.35) =
m V
shift of the sample is z =0.35,we quote our resultfor the ±
1370 64Mpc (4.7%), where these are 1 σ errors.
dilation scale as D (0.35). For our fiducial cosmology of ±
V Figure7alsocontainstwolinesthatdepictthetwophys-
Ωm =0.3, ΩΛ =0.7, h=0.7, DV(0.35)=1334 Mpc. ical scales. The solid line is that of constant Ω h2D ,
We compute parameter constraints by computing χ2 m V
whichwouldplacethe(matter-radiation)equalityscaleat
(usingthefullcovariancematrix)foragridofcosmological
a constant apparent location. This would be the degener-
models. In addition to cosmological parameters of Ω h2,
m acy direction for a pure cold dark matter cosmology and
Ω h2,andn,weincludethedistancescaleD (0.35)ofthe
b V would be a line of constant Γ=Ω h were the LRG sam-
m
LRG sample and marginalize over the amplitude of the
ple at lower redshift. The dashed line holds constant the
correlation function. Parameters such as h, Ω , Ω , and
m K soundhorizondividedbythedistance,whichistheappar-
10
Table 1
Summaryof ParameterConstraints fromLRGs
Ωmh2 0.130(n/0.98)1.2±0.011
DV(0.35) 1370±64Mpc(4.7%)
R0.35≡DV(0.35)/DM(1089) 0.0979±0.0036(3.7%)
A≡DV(0.35) ΩmH02/0.35c 0.469(n/0.98)−0.35±0.017(3.6%)
q
NOTES.—WeassumeΩbh2=0.024throughout,butvariationsper-
mitted byWMAPcreate negligiblechanges here. Weuse n=0.98,
butwherevariationsby0.1wouldcreate1σchanges,weincludean
approximate dependence. The quantity A is discussed in§4.5. All
constraints are1σ.
ent location of the acoustic scale. One sees that the long
axisofthecontoursfallsinbetweenthesetwo,andthefact Fig. 8.—AsFigure7,butnow withscalesbelow18h−1Mpcex-
that neither direction is degenerate means that both the cludedfromtheχ2computation. Thisleaves18separationbinsand
equality scale and the acoustic scale have been detected. 15 degrees of freedom. The contours are now obviously aligned to
NotethatnoinformationfromtheCMBonΩmh2hasbeen thelineofconstantsoundhorizon,andtheconstraintsintheΩmh2
used in computing χ2, and so our constraint on Ωmh2 is adtirescctailoensabreelowwea1k8ehn−ed1bMyp4c0%he.lpAstoFicgounrset3rawinouΩldmshu2g,getwsti,stthinegdtahtae
separate from that from the CMB. contours towards the pureCDMdegeneracy. Droppingthesmaller
The best-fit pure CDM model has χ2 = 27.8, which scalesdoesn’taffecttheconstraintonR0.35;wefind0.0973±0.0038
means that it is disfavored by ∆χ2 = 11.7 compared to ascomparedto0.0979±0.0036before.
the model with Ω h2 = 0.024. Note that we are not
b
marginalizing over the baryon density, so these two pa-
rameterspaces havethe same number ofparameters. The
baryonsignatureisthereforedetectedat3.4σ. As amore
stringent version of this, we find that the baryon model
is preferred by ∆χ2 = 8.8 (3.0 σ) even if we only include
data points between 60 and 180h−1Mpc. Figure 6 also
shows χ2 for the pure-CDM model as a function of di-
lation scale; one sees that the scale constraint on such a
model is a factor of two worse than the baryonic models.
This demonstrates the importance of the acoustic scale in
our distance inferences.
As most of our distance leverage is coming from the
acoustic scale, the most robust distance measurement we
can quote is the ratio of the distance to z = 0.35 to the
distance to z = 1089 (the redshift of decoupling, Bennett
et al. 2003). This marginalizes over the uncertainties in
Ω h2andwouldcanceloutmoreexoticerrorsinthesound
m
horizon, suchas fromextra relativistic species (Eisenstein Fig. 9.—AsFigure7,butnowwithaspectraltiltofn=0.90. The
& White 2004). We denote this ratio as bestfithasχ2=17.8. TheprimaryeffectisashifttolargerΩmh2,
0.143±0.011. However, this shift occurs at essentially constant
DV(0.35) R0.35; we find 0.0986±0.0041. Again, the acoustic scale robustly
R0.35 . (3) determines the distance, even though the spectral tilt biases the
≡ D (1089)
M measurementoftheequalityscale.
NotethattheCMBmeasuresapurelytransversedistance,
while the LRG sample measures the hybrid in Equation
(2). D (1089)=13700 Mpc for the Ω =0.3, Ω =0.7, 0.136 0.014),butthecontoursremainwellconfinedalong
M m Λ ±
h = 0.7 cosmology (Tegmark et al. 2004b), with uncer- the direction of constant acoustic scale (the dashed line).
tainties due to imperfect measurement of the CMB an- We find a distance ratio of 0.0973 0.0038, essentially
gular acoustic scale being negligible at < 1%. We find identical to what we found above, w±ith a best χ2 of 13.7
R = 0.0979 0.0036, which is a 4% measurement of on 15 degrees of freedom.
0.35
the relative dist±ance to z = 0.35 and z = 1089. Table 1 Varying the spectral tilt, which has a similar effect to
summarizes our numericalresults on these basic measure- including massive neutrinos, is partially degenerate with
ments. Ωmh2, but the ratio of the distances is very stable across
To stress that the acoustic scale is responsible for the the plausible range. Repeating the fitting with n = 0.90
distanceconstraint,werepeatourfittinghavingdiscarded changes the distance ratio R0.35 to 0.0986 0.0041,a less
the two smallest separation bins (10 < s < 18h−1Mpc) than1%change. Thedistanceitselfchange±sbyonly2%to
from the correlation function. This is shown in Figure 8. DV(0.35)= 1344 70. The change in the likelihood con-
One sees that the constraints on Ωmh2 have degraded (to toursisshowninF±igure9. Thebest-fitΩmh2 forn=0.90