Teppei OTeppei OKUMURAKUMURA (Nagoya University, Japan)(Nagoya University, Japan)Takahiko MatsubaraTakahiko Matsubara11, Daniel Eisenstein, Daniel Eisenstein22,,

Issha KayoIssha Kayo11, Chiaki Hikage, Chiaki Hikage11, & Alex Szalay, & Alex Szalay33,,SDSS CollaborationSDSS Collaboration

11Nagoya, Nagoya, 22Arizona, Arizona, 33Johns HopkinsJohns Hopkins

Anisotropic Correlation Anisotropic Correlation Function of Large-Scale Function of Large-Scale

Galaxy Distribution from the Galaxy Distribution from the SDSS LRG SampleSDSS LRG Sample

OQSCM @ Imperial College LondonMar. 29, 2007

What we didWhat we did We calculated an Anisotropic Correlation Function, We calculated an Anisotropic Correlation Function, ξ(sξ(s

⊥⊥,s,s////)), from SDSS LRG sample, focusing on anisotropy , from SDSS LRG sample, focusing on anisotropy of baryon acoustic oscillations.of baryon acoustic oscillations.

We then constrained cosmological parameters, We then constrained cosmological parameters, ΩΩmm, Ω, Ωbb, h, h, and , and ww, by comparing it with a corresponding theor, by comparing it with a corresponding theoretical model.etical model.

Cosmological parameters are Cosmological parameters are constrained with high precision. constrained with high precision. However, to understand the However, to understand the

properties of dark energy properties of dark energy (cosmological constant? time (cosmological constant? time evolution? spatial clustering??), we evolution? spatial clustering??), we need both more accurate need both more accurate observations and analyses.observations and analyses.

MotivationMotivation

Baryon Acoustic Baryon Acoustic Oscillations in Oscillations in

LSSLSS Correlation FunctionCorrelation Function Eisenstein et al.(2005)Eisenstein et al.(2005)

Power SpectrumPower Spectrum Cole et al. (2005) Cole et al. (2005) Tegmark et al.(2006) Tegmark et al.(2006) Percival et al.(2007a,b)Percival et al.(2007a,b) Padmanabhan et al. (2007)Padmanabhan et al. (2007)

Eisenstein et al.(2005)

Tegmark et al.(2006)

s2ξ(s)

kP(k)Our analysis can be complementary to the previous analyses above.

Sound Horizon scale at decoupling

Cosmological Information in the Redshift-SCosmological Information in the Redshift-Space Correlation Functionpace Correlation Function

1. Mass Power Spectrum in 1. Mass Power Spectrum in Comoving SpaceComoving Space ΩΩmmh, h, ΩΩbb//ΩΩmm, (h), (h) ΩΩmmhh

ffbb==ΩΩbb//ΩΩmm

real space

redshift spacenon-linearlinear

∝ H(z)

∝ z/DA(z)

2. Dynamical Redshift Disto2. Dynamical Redshift Distortionrtion β= Ωβ= Ωmm

0.60.6/b/b(for Kaiser’s effect)(for Kaiser’s effect)

3. Geometrical Distortion3. Geometrical Distortion ΩΩmm, Ω, ΩΛΛ, w, w

Cosmological Information in the Redshift-SCosmological Information in the Redshift-Space Correlation Functionpace Correlation Function

1. Mass Power Spectrum 1. Mass Power Spectrum in Comoving Spacein Comoving Space ΩΩmmh, h, ΩΩbb//ΩΩmm, (h), (h) ΩΩmmhh

ffbb==ΩΩbb//ΩΩmm

real space

redshift spacenon-linearlinear

∝ H(z)

∝ z/DA(z)

2. Dynamical Redshift Disto2. Dynamical Redshift Distortionrtion β= Ωβ= Ωmm

0.60.6/b/b(for Kaiser’s effect)(for Kaiser’s effect)

3. Geometrical Distortion3. Geometrical Distortion ΩΩmm, Ω, ΩΛΛ, w, w

To include all of these information, we calculate a correlation function as two variables, ξ(s⊥,s//), from the SDSS LRGs.

ξ ＜ 0 ξ≧0

Anisotropic Correlation Function of LRGsAnisotropic Correlation Function of LRGs

(left)Analytical Formulae (Matsubara 2004)

Baryon Ridges Correspond to the 1D Baryon Peak scale detected by Eisenstein et al.

Angle average!

(right)SDSS LRG Correlation Function

Anisotropic Correlation Function of LRGsAnisotropic Correlation Function of LRGs

(left)Analytical Formulae (Matsubara 2004)

(right)SDSS LRG Correlation Function

Baryon Ridges Correspond to the 1D Baryon Peak scale detected by Eisenstein et al.

Dynamical distortionDynamical distortion is due to the peculiar is due to the peculiar velocity of galaxiesvelocity of galaxies

Anisotropic Correlation Function of LRGsAnisotropic Correlation Function of LRGs

(left)Analytical Formulae (Matsubara 2004)

(right)SDSS LRG Correlation Function

Baryon Ridges Correspond to the 1D Baryon Peak scale detected by Eisenstein et al.

Geometrical distortionGeometrical distortion can be also measured can be also measured when deviation of ridgewhen deviation of ridges from the ideal sphere s from the ideal sphere in comoving space is in comoving space is detected. (Alcocdetected. (Alcock-Paczynski)k-Paczynski)

The Covariance Matrix for the The Covariance Matrix for the Measured Correlation Measured Correlation

FunctionFunction Much more realizations than the degrees of freedom of Much more realizations than the degrees of freedom of

the binned data points are needed, ~2000 realizations.the binned data points are needed, ~2000 realizations. Possible Methods for Mock Catalogs and CovariancePossible Methods for Mock Catalogs and Covariance

Jackknife resamplingJackknife resampling The easiest way, but it is unsure whether this can provide The easiest way, but it is unsure whether this can provide

a reliable of estimator of the cosmic variance. a reliable of estimator of the cosmic variance. N-body SimulationsN-body Simulations

A robust and reliable way, but it is too expensive. A robust and reliable way, but it is too expensive. 2LPT code (Crocce, Pueblas & Scoccimarro 2006) 2LPT code (Crocce, Pueblas & Scoccimarro 2006)

　 　 + Biased selection of galaxies with weighting + Biased selection of galaxies with weighting of of ∝∝ eebbδδ

We use in this workWe use in this work

mm

Correlation Functions Measured Correlation Functions Measured from Our Mocksfrom Our Mocks

The averaged correlatioThe averaged correlation function measured fron function measured from our mocks match the m our mocks match the one of LRGs well as for one of LRGs well as for ξξ(s)(s)

LRG

2LPT + biased selection

s2ξ(s)

Correlation Functions Measured Correlation Functions Measured from Our Mocksfrom Our Mocks

The averaged correlatioThe averaged correlation function measured fron function measured from our mocks match the m our mocks match the one of LRGs well as for one of LRGs well as for ξξ(s)(s)and and ξ(sξ(s⊥⊥,s,s////))

LRG2LPT +

bias

We generate 2,500 mock catalogs to construct the covariance matrix.

We consider 5D Parameter We consider 5D Parameter SpaceSpace

Results(1): Results(1): Fundamental Fundamental ParametersParameters

1 ,0

)b , , , ,(

0

8

−==ΩΩΩ=

wh

k

bm σθr

40<s<200Mpc/h

60<s<150Mpc/h

(Extended) Alcock-Paczynski(Extended) Alcock-Paczynski

Results(2) : Dark Energy Results(2) : Dark Energy ParametersParameters

WMAP3

0 ,024.0

)b , , , ,(2

8

=Ω=Ω

Ω= Λ

kbh

hw σθrWMAP3

+SN IaOur results

Future Works: Future Works: Toward Precision Toward Precision

CosmologyCosmology Covariance Matrix Covariance Matrix For more accurate covariance, we should run a huFor more accurate covariance, we should run a hu

ge number of N-body simulations with independenge number of N-body simulations with independent initial conditions.t initial conditions.

Nonlinear regionsNonlinear regions 　　 (≦ 40~60 Mpc/h)(≦ 40~60 Mpc/h) Also contain abundant cosmological information. Also contain abundant cosmological information.

However we have discarded all of them in this anHowever we have discarded all of them in this analysis. In addition, the baryonic signature is affectalysis. In addition, the baryonic signature is affected by nonlinearity. We should estimate non-linear ed by nonlinearity. We should estimate non-linear corrections somehow. (e.g. using N-body simulaticorrections somehow. (e.g. using N-body simulation or higher-order perturbations)on or higher-order perturbations)

SummarySummary We have calculated the correlation function of We have calculated the correlation function of

SDSS LRGs as a function of 2-variables, SDSS LRGs as a function of 2-variables, ξξ(s(s⊥⊥,,ss////)), ,

and have estimated cosmological parameters and have estimated cosmological parameters using only the data of linear-scale regions. using only the data of linear-scale regions.

We have obtained the consistent results with We have obtained the consistent results with the previous LRG works.the previous LRG works. This method can be useful in probing Dark Energy This method can be useful in probing Dark Energy

(like Seo & Eisenstein 2003, Hu & Haiman 2003, (like Seo & Eisenstein 2003, Hu & Haiman 2003, and Glazebrook & Blake 2005), when a future deand Glazebrook & Blake 2005), when a future deep redshift survey such as WFMOS (Wide-Field Mep redshift survey such as WFMOS (Wide-Field Multi-Object Spectrograph) gets available.ulti-Object Spectrograph) gets available.

AppendixAppendixCorrelation Function in Redshift SpaceCorrelation Function in Redshift Space

General Formulae of Correlation Function in Redshift Space derived from a General Formulae of Correlation Function in Redshift Space derived from a Linear Perturbation Theory Linear Perturbation Theory （（ Matsubara 2000; 2004Matsubara 2000; 2004 ））