Modeling and Measuring Redshift Space
Distortions and the Alcock-Paczynski
Effect in the Baryon Oscillation
Spectroscopic Survey

Beth Reid
Hubble Fellow
Lawrence Berkeley National Lab
in collaboration with Martin White, Lado Samushia,
Will Percival, BOSS galaxy clustering working group
Image Courtesy Chris Blake and Sam Moorfield
Tuesday, January 31, 2012Outline• Motivation• Basic redshift space distortions (RSD) in configuration
space• Reid and White 2011 configuration space RSD model(+ connections to other recent RSD work)• From halos to galaxies...
• BOSS DR9 first results: BAO, RSD and AP constraints• Future prospects
Beth Reid
Tuesday, January 31, 2012

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RSD motivation: Testing
General Relativity• Once we know the expansion history H(a),we know how perturbations grow in GR:
δ(k, a) = aG(H(a))δi(k)• We want to test both scale (k) and time (a)dependence

Beth Reid
Tuesday, January 31, 2012

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RSD in 3d Galaxy Mapsθ, φ, redshiftdepends on thegeometry of
the universe

χ(z) = χtrue + vp/aH(a)χ(z) =0∫z c dz’/H(z’)
comoving coordinates: x, y, z
Beth Reid
Tuesday, January 31, 2012

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RSD in configuration spacereal to redshift space separations∇ ⋅ vp = -aHf δmzxx|vp| ~ d σ8/d ln a = σ8 * fisotropic

squashed along line of sightf = d ln σ8 /d ln a

Beth Reid
Tuesday, January 31, 2012

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RSD: Anisotropy in ξ(rσ, rπ)

BOSS DR9: Reid et al., Samushia et al. (in prep)
Beth Reid
Tuesday, January 31, 2012

6White et al. 2011 mock catalogs
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Linear RSD (Kaiser 1987)

Beth Reid
Tuesday, January 31, 2012

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Linear RSD: Legendre Polynomial moments
General Expansion

Linear theory prediction

Beth Reid
Tuesday, January 31, 2012

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Legendre Polynomial moments
General Expansion

Relation to Pℓ(k)Beth Reid
Tuesday, January 31, 2012

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Linear theory Legendre polynomial
moments: scale dependence

Beth Reid
Tuesday, January 31, 201210

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RSD in configuration space: newquantities of interest

2v̅r, σ rrπrvz
yv̅t=0, σ2trπrσ
Beth Reid
Tuesday, January 31, 201211

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Linear theory pairwise velocities (δg = bδm)

Beth Reid
Tuesday, January 31, 2012

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Fisher 1995: the Kaiser formula inconfiguration space• δ(x), v(x’) correlated Gaussian fields• Expand around y = rπ

• Equivalent to Kaiser formula
Beth Reid
Tuesday, January 31, 201213

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Pairwise velocity statistics in linear theory

Beth Reid
Tuesday, January 31, 201214

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Outline• Motivation• Basic redshift space distortions (RSD) in configurationspace• Reid and White 2011 configuration space RSD model(+ connections to other recent RSD work)• From halos to galaxies...
• BOSS DR9 first results: BAO, RSD and AP constraints• Future prospects
Beth Reid
Tuesday, January 31, 201215

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Recent work: Matter Density Field

Blake et al., arXiv:1105.2862; see also Scoccimarro 2004

Beth Reid
Tuesday, January 31, 201216

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Why halos?

• Galaxies live there!
• Halos occupy “special”
places in the density
field; θ is a volumeaveraged statistic• Dependence on halobias is complex; studies
of matter correlations
not easily generalizedBeth Reid
Tuesday, January 31, 201217

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Recent Work: Halos• Tinker, Weinberg, Zheng 2006; Tinker
2007 (+ galaxies in halo model)• Matsubara 2008ab [LPT with biasing]• Tang, Kayo, Takada arXiv:1103.3614• Nishimichi, Taruya arXiv:1106.4562• ....
Beth Reid
Tuesday, January 31, 201218

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N-body Simulations• White et al. 2011; arXiv:1010.4915• 67.5 (Gpc/h) total volume3(for BOSS galaxies V ~ 5 (Gpc/h)3)

Beth Reid
Tuesday, January 31, 201219

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N-body simulations vs Linear andLagrangian Perturbation Theories• LPT works onBAO scales• See Matsubara

PRD 78, 083519;
arXiv:1105.5007

Beth Reid
Tuesday, January 31, 201220

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N-body simulations vs Linear andLagrangian Perturbation Theories•Beth Reid
Tuesday, January 31, 2012

ξ2 suppressed by 2.5-7.5% at 50 h-1 Mpc,depends strongly on bias
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Our approach• Two distinct sources of nonlinearity:• Nonlinear growth of structure/biasing --affects both halo clustering and velocities(study in N-body sims/perturbation theory)• Nonlinear mapping from real to redshift
space coordinates (non-perturbative)• Recall: to get Kaiser: dv /dz small (P) orz

expand around y = rπ (ξ)

Beth Reid
Tuesday, January 31, 201222

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The scale-dependent Gaussian streaming
model ansatz

2v̅r, σ rrπrvz
yv̅t=0, σ2trπrσ
Beth Reid
Tuesday, January 31, 201223

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The scale-dependent Gaussian streaming
model ansatz• Non-perturbative!
• Approximate pairwise

velocity PDF P(vz, r) witha Gaussian; match 1stand 2nd moments• Agrees at linear orderwith Kaiser/exact

Beth Reid
Tuesday, January 31, 201224

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The scale-dependent Gaussian streaming
model ansatz: “linear” theory predictions
b=2.7

b=0.5
Kaiser limit
Gaussian streaming model
Beth Reid
Tuesday, January 31, 201225

3b correction !!
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The scale-dependent Gaussian streaming
model ansatz vs N-body simulations• Start with ξ(r), v(r), σ² (r) measured from N-body⊥,∥halos in real space• Compare with N-body halo clustering in redshift
space

Beth Reid
Tuesday, January 31, 2012

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The scale-dependent Gaussian streaming
model ansatz vs N-body simulations: ξ0Beth Reid
Tuesday, January 31, 201227

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The scale-dependent Gaussian streaming
model ansatz vs N-body simulations: ξ0Beth Reid
Tuesday, January 31, 201228

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The scale-dependent Gaussian streaming
model ansatz vs N-body simulations: ξ2

Beth Reid
Tuesday, January 31, 201229

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The scale-dependent Gaussian streaming
model ansatz vs N-body simulations: ξ2

Beth Reid
Tuesday, January 31, 201230

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Can we predict real space halo clustering/velocity PDFs using perturbation theory?

• LPT (including nonlinearbias) predicts halo ξ(r)down to 25 Mpc/h

Beth Reid
Tuesday, January 31, 201231

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Velocity statistics in standard
perturbation theory: new results• Pair-weighted, not volume weighted!

Beth Reid
Tuesday, January 31, 201232

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Velocity statistics in standard
perturbation theory: new results* assumes linear bias
Beth Reid
Tuesday, January 31, 201233

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Velocity statistics in standard
perturbation theory: new results
Pair-weighting correction

Linear theory

PT correction to PδθBeth Reid
Tuesday, January 31, 2012

Bispectrum terms: Bδδθ34

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Velocity statistics in standard
perturbation theory: new results
Bispectrum terms: Bδδθ
Bδδθ , Pδθ terms

appear in Tang et al.,Nishimishi et al.
PT correction to Pδθtotal PT correction
Pair-weighting correction
Beth Reid
Tuesday, January 31, 201235

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Putting it all together: fully analytic model• Error dominated byerror in v12(r) slope•

Works where b2L = 0(i.e., for BOSSgalaxies)• New LPT calculationin prep: Carlson etal., 2012

Beth Reid
Tuesday, January 31, 201236

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Summary: Two distinct effects• Non-linear gravitational

evolution: MUST be
accounted for given
current statistical errors:ξ2 suppressed by 2.5-7.5%
at 50 h-1 Mpc!

• Non-linear real-to-redshift
space mapping: b3 term

Beth Reid
Tuesday, January 31, 201237

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Outline• Motivation• Basic redshift space distortions (RSD) in configuration
space• Reid and White 2011 configuration space RSD model(+ connections to other recent RSD work)• From halos to galaxies...
• Interlude: SDSS DR9 first results• Future prospects
Beth Reid
Tuesday, January 31, 201238

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Dominant impact of galaxies: Fingers-of-God
Central galaxies
REAL SPACE: r ~ 1 Mpc/hx

Satellite galaxies
x

REDSHIFT SPACE: r ~ 15 Mpc/hFinger-of-God features mix small and large scale powerxx

Beth Reid
Tuesday, January 31, 2012x

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Fingers-of-God in ξ(rσ, rπ)Beth Reid
Tuesday, January 31, 201240

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From halos to galaxies• In principle, straightforward to model in ξ(r , r )σπ-- just another convolution with intrahalo velocityPDF• In practice 3 (broad) distinct PDFs: cs, ss (1h), ss(2h)• Inaccarucy of halo ξ(r , r ) on small scales inhibits
σπthis approach

Beth Reid
Tuesday, January 31, 201241

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Safe on quasilinear scales...

• One-halo (classicalFOG) unimportant• ~ 10% effect at 25
Mpc/h for BOSSgalaxies

Beth Reid
Tuesday, January 31, 201242Nagoya Feb 1

From halos to galaxies• Marginalizing over additional Gaussian dispersionworks!

RW11LPTFOGs

RSD model vs 70 (Gpc/h)3 of sims

Beth Reid
Tuesday, January 31, 2012

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Outline• Motivation• Basic redshift space distortions (RSD) in configuration
space• Reid and White 2011 configuration space RSD model(+ connections to other recent RSD work)• From halos to galaxies...
• BOSS DR9 first results: BAO, RSD and AP constraints• Future prospects
Beth Reid
Tuesday, January 31, 2012

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Galaxy clustering lightningtheory review• Theory 1: underlying matter power spectrum(determined at z >~ zCMB, neglecting ν)• Theory II: Expansion history H(0 < z < zTuesday, January 31, 2012GAL)

Matter Power Spectrum• Entire P(k) (not just BAO) acts as standard ruler
determined by CMB• We marginalize over the (negligible) uncertaintyP(k)2r ξ(r)Mpc-1
Tuesday, January 31, 2012

Mpc

Theory II: geometry• We measure θ, φ, and z for each galaxy, and use acosmological model to convert to comoving
coordinatesz1

1/H(z)θ
χ(z) (or DA(z))

Tuesday, January 31, 2012z2

Theory II: Alcock-Paczynski• ξ(r , π) appears anisotropic if you assume thepwrong cosmological model (constrain ηAP = DA * H)

χ(z) =0∫z c dz’/H(z’)

180ξ(rp, π)160

BAO in ξ0(s) determines“geometric mean”
DV ∝ (DA2 H-1)1/3

c Δz / H(z)

140

1120

ʌ

100

080-1
6040-22020406080100rp

χ(z)*Δθ
Tuesday, January 31, 2012

2120

140

160

180-3

Image from Tian et al. arXiv:1011.2481

3Fitting to 2d clustering•Use full model of ξ0,2(s ≥ 25 h-1 Mpc) to constrain:• growth of structure (fσ )• D ∝ (D /H)• Alcock-Paczynski (η ∝ D (z ) * H(z ))• marginalizing over shape of underlying linear
8V2
A

1/3

APP(k), bσ8, σ2FOG

Tuesday, January 31, 2012

A

eff

eff

Alcock-Paczynski in multipoles

Tuesday, January 31, 2012

DR9 spectroscopic results:
preliminary!

• DR9 data final (public July 2012), clustering/covariances ~final, cosmological constraints
preliminary• Current uncertainties reported, not centralvalues

Tuesday, January 31, 2012

BOSS “CMASS” (zeff = 0.57)
galaxy sample in perspective

Eisenstein et al. arXiv:1101.1529
Tuesday, January 31, 2012

BAO fits in P(k)/ξ(r) consistent
X. Xu et al. (in prep; DR7)
BOSS Galaxy Clustering (in prep.)• 2-3% uncertainty
plot of BAO feature hereon BAO positionin angle-averaged
P(k)/ξ(r)• Constrains
D ∝ (D /H)
VTuesday, January 31, 2012

2A1/3

The CMASS measurements• 26 log bins in s for ξ and ξ = 52 DOF
02plot of ξ0 and ξ2 here

Tuesday, January 31, 2012

Model Fits

• We test the LCDM hypothesis in 4 models,always marginalizing over P(k) shape and σ2FOG:• LCDM (bσ )• LCDM + fσ : (bσ , fσ )• LCDM + geometry: (bσ , D , D *H)• LCDM++: (bσ , fσ , D , D *H)
8888

8

8

Tuesday, January 31, 2012

8VVA

A

Current status• D /DVV,fid = x ± 0.019 (i.e., minimal information

gain on DV compared to BAO only!)• Geometry LCDM: fσ = xx ± 0.03 (7%)
8[WMAP7 LCDM: 0.45 ± 0.025]• fσ LCDM: η = xx ± 0.04 (4%)
8

[WMAP7 LCDM: 1.00 ± 0.012]• Fit both: fσ = xx ± 0.07, η = xx ± 0.07
8

Tuesday, January 31, 2012

Testing alternative models with amplitude ofpeculiar velocities

BOSS DR9

Tuesday, January 31, 2012

Expansion rate at z=0.57

Tuesday, January 31, 2012Outline• Motivation• Basic redshift space distortions (RSD) in configuration
space• Reid and White 2011 configuration space RSD model(+ connections to other recent RSD work)• From halos to galaxies...
• Interlude: SDSS DR9 first results• Future prospects
Beth Reid
Tuesday, January 31, 2012

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Future Prospects: “convolutional” LPT(Jordan Carlson, et al., in prep)• Fourier transform formal LPT P(k, μ) expression; use
cumulant expansion thm + Gaussian integrals• Recovers Zel’dovich approximation exactly ->b3 nonlinear mapping term (?)

Beth Reid
Tuesday, January 31, 2012

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Future Prospects: “convolutional” LPT(Jordan Carlson, et al., in prep)
CLPTLinear theory
Matsubara LPT

halo real space ξ0 (preliminary!)
Beth Reid
Tuesday, January 31, 2012

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Future Prospects: “convolutional” LPT(Jordan Carlson, et al., in prep)
CLPT
Linear theory
Matsubara LPT

halo redshift space ξ0 (preliminary!)
Beth Reid
Tuesday, January 31, 2012

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Future Prospects: “convolutional” LPT(Jordan Carlson, et al., in prep)
CLPT
Linear theory
Matsubara LPT
matter ξ2 (preliminary!)

Beth Reid
Tuesday, January 31, 2012

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Future Prospects: “convolutional” LPT(Jordan Carlson, et al., in prep)
CLPT
Linear theory
Matsubara LPT
halo ξ2 (preliminary!)

Beth ReidTuesday, January 31, 201264

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Future Prospects: “convolutional” LPT(Jordan Carlson, et al., in prep)• New real space ξ(r) fits to ~ 10 Mpc/h !!
• Repeat v (r), σ (r) calculations in LPT (including122⊥,∥Lb2 ) may extend analytic Gaussian streaming model tosmaller scales

Beth Reid
Tuesday, January 31, 2012

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Future Prospects: using small-scale
2clustering to infer σ FOG

IF we can determine σ fromsmall-scale clustering (e.g.,HOD), gain factor of 2 on RSDXXsmin, WiggleZ
Beth Reid
Tuesday, January 31, 2012

smin, BOSS
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Conclusions• Configuration space simplifies many conceptual issues
in modeling RSD• Worked example of developing/modelling target(BOSS) galaxies: 2% accurate to 25 h-1 Mpc• 7% measurement of fσ in DR9 CMASS galaxies, ~4%
8

final (barring further modeling improvements)• Further development underway (CLPT, small scale/
HOD modeling, ...)

Beth Reid
Tuesday, January 31, 2012

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