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Modeling and Measuring Redshift Space Distortions and the Alcock-Paczynski
Effect in the Baryon Oscillation Spectroscopic Survey
Beth ReidHubble Fellow
Lawrence Berkeley National Labin collaboration with Martin White, Lado Samushia, Will Percival, BOSS galaxy clustering working group
Image Courtesy Chris Blake and Sam Moorfield
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 Nagoya Feb 12
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 Nagoya Feb 13
Tuesday, January 31, 2012
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RSD in 3d Galaxy Maps
depends on the geometry of the universe
θ, φ, redshift
comoving coordinates: x, y, z
χ(z) = χtrue + vp/aH(a)
Beth Reid
χ(z) =0∫z c dz’/H(z’)
Nagoya Feb 14
Tuesday, January 31, 2012
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RSD in configuration space
Beth Reid 5
x x
real to redshift space separations
|vp| ~ d σ8/d ln a = σ8 * f
isotropic squashed along line of sight
z∇ ⋅ vp = -aHf δm
Nagoya Feb 1
f = d ln σ8 /d ln a
Tuesday, January 31, 2012
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RSD: Anisotropy in ξ(rσ, rπ)
Beth Reid 6
White et al. 2011 mock catalogs
Nagoya Feb 1
BOSS DR9: Reid et al., Samushia et al. (in prep)
Tuesday, January 31, 2012
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Linear RSD (Kaiser 1987)
Beth Reid Nagoya Feb 17
Tuesday, January 31, 2012
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Linear RSD: Legendre Polynomial moments
Beth Reid
General Expansion
Linear theory prediction
Nagoya Feb 18
Tuesday, January 31, 2012
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Legendre Polynomial moments
Beth Reid 9
General Expansion
Relation to Pℓ(k)
Nagoya Feb 1
Tuesday, January 31, 2012
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Linear theory Legendre polynomial moments: scale dependence
Beth Reid 10 Nagoya Feb 1
Tuesday, January 31, 2012
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RSD in configuration space: new quantities of interest
Beth Reid 11
rσ
rπr vz
v̅r, σ2r
v̅t=0, σ2ty rπ
Nagoya Feb 1
Tuesday, January 31, 2012
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Linear theory pairwise velocities (δg = bδm)
Beth Reid 12 Nagoya Feb 1
Tuesday, January 31, 2012
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Fisher 1995: the Kaiser formula in configuration space
Beth Reid 13
• δ(x), v(x’) correlated Gaussian fields
• Expand around y = rπ
• Equivalent to Kaiser formula
Nagoya Feb 1
Tuesday, January 31, 2012
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Pairwise velocity statistics in linear theory
Beth Reid 14 Nagoya Feb 1
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 Nagoya Feb 115
Tuesday, January 31, 2012
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Recent work: Matter Density Field
Beth Reid 16
Blake et al., arXiv:1105.2862; see also Scoccimarro 2004
Nagoya Feb 1
Tuesday, January 31, 2012
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Why halos?
• Galaxies live there!
• Halos occupy “special” places in the density field; θ is a volume-averaged statistic
• Dependence on halo bias is complex; studies of matter correlations not easily generalized
Beth Reid 17 Nagoya Feb 1
Tuesday, January 31, 2012
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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 18 Nagoya Feb 1
Tuesday, January 31, 2012
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N-body Simulations
• White et al. 2011; arXiv:1010.4915
• 67.5 (Gpc/h)3 total volume (for BOSS galaxies V ~ 5 (Gpc/h)3)
Beth Reid 19 Nagoya Feb 1
Tuesday, January 31, 2012
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N-body simulations vs Linear and Lagrangian Perturbation Theories
• LPT works on BAO scales
• See Matsubara PRD 78, 083519; arXiv:1105.5007
Beth Reid 20 Nagoya Feb 1
Tuesday, January 31, 2012
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N-body simulations vs Linear and Lagrangian Perturbation Theories
• ξ2 suppressed by 2.5-7.5% at 50 h-1 Mpc, depends strongly on bias
Beth Reid 21 Nagoya Feb 1
Tuesday, January 31, 2012
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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: dvz/dz small (P) or expand around y = rπ (ξ)
Beth Reid 22 Nagoya Feb 1
Tuesday, January 31, 2012
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The scale-dependent Gaussian streaming model ansatz
Beth Reid 23
rσ
rπr vz
v̅r, σ2r
v̅t=0, σ2ty rπ
Nagoya Feb 1
Tuesday, January 31, 2012
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The scale-dependent Gaussian streaming model ansatz
• Non-perturbative!
• Approximate pairwise velocity PDF P(vz, r) with a Gaussian; match 1st and 2nd moments
• Agrees at linear order with Kaiser/exact
Beth Reid 24 Nagoya Feb 1
Tuesday, January 31, 2012
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The scale-dependent Gaussian streaming model ansatz: “linear” theory predictions
Beth Reid 25
Kaiser limitGaussian streaming model b3 correction !!
b=2.7
b=0.5
Nagoya Feb 1
Tuesday, January 31, 2012
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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 26 Nagoya Feb 1
Tuesday, January 31, 2012
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The scale-dependent Gaussian streaming model ansatz vs N-body simulations: ξ0
Beth Reid 27 Nagoya Feb 1
Tuesday, January 31, 2012
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The scale-dependent Gaussian streaming model ansatz vs N-body simulations: ξ0
Beth Reid 28 Nagoya Feb 1
Tuesday, January 31, 2012
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The scale-dependent Gaussian streaming model ansatz vs N-body simulations: ξ2
Beth Reid 29 Nagoya Feb 1
Tuesday, January 31, 2012
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The scale-dependent Gaussian streaming model ansatz vs N-body simulations: ξ2
Beth Reid 30 Nagoya Feb 1
Tuesday, January 31, 2012
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Can we predict real space halo clustering/velocity PDFs using perturbation theory?
Beth Reid 31
• LPT (including nonlinear bias) predicts halo ξ(r) down to 25 Mpc/h
Nagoya Feb 1
Tuesday, January 31, 2012
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Velocity statistics in standard perturbation theory: new results
Beth Reid 32
• Pair-weighted, not volume weighted!
Nagoya Feb 1
Tuesday, January 31, 2012
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Velocity statistics in standard perturbation theory: new results
Beth Reid 33
* assumes linear bias
Nagoya Feb 1
Tuesday, January 31, 2012
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Velocity statistics in standard perturbation theory: new results
Beth Reid 34
Pair-weighting correction Linear theory
PT correction to Pδθ
Linear theory
Bispectrum terms: Bδδθ
Nagoya Feb 1
Tuesday, January 31, 2012
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Velocity statistics in standard perturbation theory: new results
Beth Reid 35
Pair-weighting correction
PT correction to Pδθ
Linear theory
Bispectrum terms: Bδδθ
total PT correction
Nagoya Feb 1
Bδδθ , Pδθ terms appear in Tang et al.,
Nishimishi et al.
Tuesday, January 31, 2012
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Putting it all together: fully analytic model
Beth Reid 36
• Error dominated by error in v12(r) slope
• Works where b2L = 0 (i.e., for BOSS galaxies)
• New LPT calculation in prep: Carlson et al., 2012
Nagoya Feb 1
Tuesday, January 31, 2012
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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 37 Nagoya Feb 1
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...
• Interlude: SDSS DR9 first results
• Future prospects
Beth Reid Nagoya Feb 138
Tuesday, January 31, 2012
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Dominant impact of galaxies: Fingers-of-God
Beth Reid 39
x
x
x x
x
REAL SPACE: r ~ 1 Mpc/h
REDSHIFT SPACE: r ~ 15 Mpc/hFinger-of-God features mix small and large scale power
Central galaxies
Satellite galaxies
Nagoya Feb 1
Tuesday, January 31, 2012
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Fingers-of-God in ξ(rσ, rπ)
Beth Reid 40 Nagoya Feb 1
Tuesday, January 31, 2012
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From halos to galaxies
• In principle, straightforward to model in ξ(rσ, rπ) -- just another convolution with intrahalo velocity PDF
• In practice 3 (broad) distinct PDFs: cs, ss (1h), ss (2h)
• Inaccarucy of halo ξ(rσ, rπ) on small scales inhibits this approach
Beth Reid Nagoya Feb 141
Tuesday, January 31, 2012
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Safe on quasilinear scales...
• One-halo (classical FOG) unimportant
• ~ 10% effect at 25 Mpc/h for BOSS galaxies
Beth Reid Nagoya Feb 142
Tuesday, January 31, 2012
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From halos to galaxies• Marginalizing over additional Gaussian dispersion
works!
Beth Reid Nagoya Feb 143
FOGs
RSD model vs 70 (Gpc/h)3 of sims
LPTRW11
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 Nagoya Feb 144
Tuesday, January 31, 2012
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Galaxy clustering lightning theory review
• Theory 1: underlying matter power spectrum (determined at z >~ zCMB, neglecting ν)
• Theory II: Expansion history H(0 < z < zGAL)
Tuesday, January 31, 2012
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P(k)
r2 ξ(r)
Matter Power Spectrum• Entire P(k) (not just BAO) acts as standard ruler
determined by CMB
• We marginalize over the (negligible) uncertainty
MpcMpc-1
Tuesday, January 31, 2012
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Theory II: geometry
• We measure θ, φ, and z for each galaxy, and use a cosmological model to convert to comoving coordinates
θ
z1 z2
χ(z) (or DA(z))
1/H(z)
Tuesday, January 31, 2012
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Theory II: Alcock-Paczynski
χ(z) =0∫z c dz’/H(z’)
20 40 60 80 100 120 140 160 180
20
40
60
80
100
120
140
160
180
-3
-2
-1
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2
3
rp
ξ(rp, π)
Imag
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om T
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:101
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c Δz
/ H
(z)
χ(z)*Δθ
• ξ(rp, π) appears anisotropic if you assume the wrong cosmological model (constrain ηAP = DA * H)
BAO in ξ0(s) determines “geometric mean” DV ∝ (DA2 H-1)1/3
Tuesday, January 31, 2012
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Fitting to 2d clustering
• Use full model of ξ0,2(s ≥ 25 h-1 Mpc) to constrain:
• growth of structure (fσ8)
• DV ∝ (DA2/H)1/3
• Alcock-Paczynski (ηAP ∝ DA(zeff) * H(zeff))
• marginalizing over shape of underlying linear P(k), bσ8, σ2FOG
Tuesday, January 31, 2012
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Alcock-Paczynski in multipoles
Tuesday, January 31, 2012
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DR9 spectroscopic results: preliminary!
• DR9 data final (public July 2012), clustering/covariances ~final, cosmological constraints preliminary
• Current uncertainties reported, not central values
Tuesday, January 31, 2012
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BOSS “CMASS” (zeff = 0.57) galaxy sample in perspective
Eisenstein et al. arXiv:1101.1529Tuesday, January 31, 2012
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BAO fits in P(k)/ξ(r) consistent
• 2-3% uncertainty on BAO position in angle-averaged P(k)/ξ(r)
• Constrains DV ∝ (DA2/H)1/3
X. Xu et al. (in prep; DR7)BOSS Galaxy Clustering (in prep.)
plot of BAO feature here
Tuesday, January 31, 2012
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The CMASS measurements
• 26 log bins in s for ξ0 and ξ2 = 52 DOF
plot of ξ0 and ξ2 here
Tuesday, January 31, 2012
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Model Fits
• We test the LCDM hypothesis in 4 models, always marginalizing over P(k) shape and σ2FOG:
• LCDM (bσ8)
• LCDM + fσ8: (bσ8, fσ8)
• LCDM + geometry: (bσ8, DV, DA*H)
• LCDM++: (bσ8, fσ8, DV, DA*H)
Tuesday, January 31, 2012
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Current status
• DV/DV,fid = x ± 0.019 (i.e., minimal information gain on DV compared to BAO only!)
• Geometry LCDM: fσ8 = xx ± 0.03 (7%) [WMAP7 LCDM: 0.45 ± 0.025]
• fσ8 LCDM: η = xx ± 0.04 (4%) [WMAP7 LCDM: 1.00 ± 0.012]
• Fit both: fσ8 = xx ± 0.07, η = xx ± 0.07
Tuesday, January 31, 2012
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Testing alternative models with amplitude of peculiar velocities
BOSS DR9
Tuesday, January 31, 2012
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Expansion rate at z=0.57
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...
• Interlude: SDSS DR9 first results
• Future prospects
Beth Reid Nagoya Feb 159
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 Nagoya Feb 160
Tuesday, January 31, 2012
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Future Prospects: “convolutional” LPT (Jordan Carlson, et al., in prep)
Beth Reid 61 Nagoya Feb 1
CLPT
Matsubara LPT
Linear theory
halo real space ξ0 (preliminary!)
Tuesday, January 31, 2012
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Future Prospects: “convolutional” LPT (Jordan Carlson, et al., in prep)
Beth Reid 62 Nagoya Feb 1
CLPT
Matsubara LPT
Linear theory
halo redshift space ξ0 (preliminary!)
Tuesday, January 31, 2012
![Page 63: Modeling and Measuring Redshift Space Distortions and the](https://reader033.vdocument.in/reader033/viewer/2022042702/6265315455cf532fbd09bc5c/html5/thumbnails/63.jpg)
Future Prospects: “convolutional” LPT (Jordan Carlson, et al., in prep)
Beth Reid 63 Nagoya Feb 1
CLPT
Matsubara LPTLinear theory
matter ξ2 (preliminary!)
Tuesday, January 31, 2012
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Future Prospects: “convolutional” LPT (Jordan Carlson, et al., in prep)
Beth Reid 64 Nagoya Feb 1
CLPT
Matsubara LPTLinear theory
halo ξ2 (preliminary!)
Tuesday, January 31, 2012
![Page 65: Modeling and Measuring Redshift Space Distortions and the](https://reader033.vdocument.in/reader033/viewer/2022042702/6265315455cf532fbd09bc5c/html5/thumbnails/65.jpg)
Future Prospects: “convolutional” LPT (Jordan Carlson, et al., in prep)
• New real space ξ(r) fits to ~ 10 Mpc/h !!
• Repeat v12(r), σ2⊥,∥(r) calculations in LPT (including
b2L) may extend analytic Gaussian streaming model to smaller scales
Beth Reid Nagoya Feb 165
Tuesday, January 31, 2012
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Future Prospects: using small-scale clustering to infer σ2FOG
Beth Reid 66 Nagoya Feb 1
IF we can determine σ from small-scale clustering (e.g.,
HOD), gain factor of 2 on RSD
smin, WiggleZ smin, BOSS
X X
Tuesday, January 31, 2012
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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 fσ8 in DR9 CMASS galaxies, ~4% final (barring further modeling improvements)
• Further development underway (CLPT, small scale/HOD modeling, ...)
Beth Reid Nagoya Feb 167
Tuesday, January 31, 2012