Analyzing BAO in Galaxy Surveys

Taka Matsubara (Nagoya Univ.)

“Probing the Dark Universe with Subaru and Gemini”

(Waikoloa, Hawaii)　11/6/2005

Historical Remark (1)

• Alcock & Paczynski (1979)

– AP test: assume spherical objects, use only ratio:

　　 comoving space　 redshift space (z-space)

observer

)(zH∝

)(zDz A∝

Q0

Q3QM

2M

3

0 13exp)1()1()1()1(

)( Ω˜ˆ

ÁÁËÊ

+++Ω−Ω−++Ω+= ∫

z

z

dzwzzz

H

zH

˜ˆ

ÁÁËÊ

′′

Ω−Ω−Ω−Ω−

= ∫z

A zH

zdH

HzD

0QM0

QM0 )(1sinh

1

1)(

)(:)( zDzzH A

Histrical Remark (2)• Ryden (1995)

– Voids as spherical objects in AP test: need too many voids

• TM & Suto (1996), Ballinger et al. (1996)– z-space distortions of correlation function (or power spectrum)

can be useful in determining lambda

– Velocity distortion + cosmological distortion

– Extension of the Kaiser’s formula (1987) in low-z universe tothe formula in high-z universe

TM & Suto (1996)

Spatial Correlation Function

• Totsuji & Kihara (1969)– Spatial correlation function of galaxies

Clustering pattern of galaxies, as a function of scales

dV1

r

dV2

( )[ ]rdVdVndP ξ+= 1212

Probability of havinggalaxies in both cells

Spatial correlationfunction

Power spectrum and Correlation function

• Power spectrum and Correlation function– Fourier transforms to each other

• Power spectrum– Direct prediction from theory– Need homogeneous, contiguous survey volume– Theory-friendly statistic

• Correlation function– Directly measured from galaxy distribution– Can handle patchy survey volume– Observation-friendly statistic

•• PS and CF are complementary statisticsPS and CF are complementary statistics

( )∫ ‡−= || )( 3 rrk ξierdkP

)(kP )(rξ

Power spectrum and Correlation function

• Baryon Acoustic Oscillation in PS and CF

Power spectrum:

Many baryon wiggles

Correlation function:

Single baryon peak

TM (2004)

2D Power spectrum, 2D correlation function

• 2D PS– Theoretical formula is simple (under distant-observer approx.)

Non-trivial to incorporate wide-angle effect

• 2D CF– Theoretical formula is tedious (but straightforward)

Wide-angle effects and selection effects are incorporated

2D power spectrum and 2D correlation function

• BAO features in 2D PS and 2D CF

Hu & Haiman (2003) TM (2004)

2D Power spectrum:

“Baryonic Rings”2D Correlation function:

“Baryonic Ridge”

Error forecasts from CF analysis

• Fisher Matrix, assuming WFMOS-like survey

Analysis of SDSS LRG survey

• BAO in 2DCF of SDSS LRG: in progress (Okumura etal.)– Z ~ 0.3, ~ 1Gpc3

very p

relim

inary

very p

relim

inary

Karhunen-Loeve Eigenmode analysis

• KL modes– Fourier modes are not

statistically independent infinite survey geometries

– Statistically independentmodes in a given surveygeometry: KL modes

Vogeley & Szalay 1996

– Optimal analysis w.r.t. S/N

– KL analysis with BAO

Formulated by TM, Szalay &Pope (2004)

Summary

• BAO in PS and CF– Power spectrum

theoretically simple,

baryonic features spread into many rings

– Correlation function

Theoretical formula is tedious (but already implemented)

Baryonic features accumulated as a single ridge

• SDSS LRG constraining dark energy by BAO– in progress (same analysis applicable to WFMOS)

• KL Eigenmode analysis : an optimal method– BAO analysis by Karhunen-Loeve method will also be useful

Cosmological Distortion Effect

• Cosmological distortion effect and dark energy

M0Ω 0w 1wK0Ω

1.0=z

3.0=z

1=z

3=z

( )zwwzw 10)( +=

Line of sight