Redshift-Space Distortions of the Galaxy Distribution

Alex Szalay, JHU

Redshift Space Distortions Three different distortions   Linear infall (large scales) ◦  Flattening of the correlations (Kaiser 86)

  Thermal motion (small scales) ◦  ‘Fingers of God’ ◦  Cuspy exponential (Davis&Peebles)

  Nonlinear infall (intermediate scales) ◦  Caustics

σ/12

12)( vevP −∝

Finding the BAO in SDSS DR4

 Eisenstein et al (2005) – LRG sample

Correlation function

Redshift Space Correlations

  Azimuthal average wipes out the P2 and P4 terms, only the real space correlation function remains

  β terms contain a second derivative (Hamilton ‘92)   Interesting effect on power spectra with sharp features:

redshift-space distortions make features even sharper!   Baryon Acoustic Oscillations !!!

)(cos)(358)(cos)(

74

34)(

5321),( 44

2

22

2

0

2)( γξ

βγξ

ββξ

ββγξ PrPrrrs +⎟⎟

⎠

⎞⎜⎜⎝

⎛+−⎟⎟

⎠

⎞⎜⎜⎝

⎛++=

[ ] [ ])(''2)(2)(32

020 krjkrjkrj ββ

−=−

ξ(r) from linear theory + BAO

 Mixing of ξ0 , ξ2 and ξ4 ◦ Along the line of sight

)()(21)(

0

22 kPkrjkdkr nn ∫∞

=π

ξ

)(358)(

74

34)(

5321)( 4

2

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2)( rrrrs ξβξββξββξ +⎟⎟

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⎛++=

π

σ

r

r2 ξ(r)

Shift of the BAO Scale with µ? Position of the BAO peak is sensitive to how we measure it (Jeong, Dai, Kamionkowski, Szalay 2015)

  SDSS DR7 MGS, Stripes 9 through 37, Northern Cap only

  0.1

300-750 Mpc/h only, LOS

Wavelet Transform

 Mexican hat wavelet transform ◦ Compensated filter ◦  Enhances localized “bump”

 Zero signal for constant background  Decreases correlations among bins

S/N of the Wavelet Transform

Flat theta weighting

Noise estimated from slices, corrected for correlations

Far Side Infall (140Mpc)

 Center: 140 h-1Mpc, width: 25 h-1Mpc   Still shows some skewness

0

10

20

30

40

50

60

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

ξLOS(140)

Strong Non-Linear Infall (55Mpc)

Distribution of LOS Mexican Hat wavelet coefficients over the 660 slices, ◦ Center at 55 h-1Mpc, width 25 h-1Mpc

0

5

10

15

20

25

30

35

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 -1E-15 0.1

ξLOS(55)

Pairwise Velocity Distribution

R=1 Mpc/h R=15 Mpc/h

0

1

2

3

4

0 5 10 15 20

sigm

a^2/

sigm

a0^2

R[Mpc/h]

L

N1

N2

-2

-1.5

-1

-0.5

0

0 5 10 15 20

Vin

f/V0

r[Mpc/h]

Infall

Pairwise Velocity Moments   Longitudinal 2nd moment 25% more elongated

than transverse   Infall has a maximum at around 4Mpc/h

Using the Indra simulations (see B. Falck’s talk)

Covariance matrix of

Real space Line of sight

Nuala McCullagh (2015) From the Indra simulations

  Redshift space distortions amplify and sharpen BAO features along the line of sight

  Near and far side infall onto the BAO bump   Angular averaging wipes out most of this effect   Evidence for BAO in SDSS DR7 MGS

at ~110 h-1Mpc, potentially constraining the equation of state at low z

  Trough at 55 h-1Mpc indicates effects of strong nonlinear infall on these scales

  Nonlinearities important even on BAO scales!   Fingers of God are quite anisotropic