Modeling and Measuring Redshift Space Distortions and the Alcock ...
Modeling and Measuring Redshift Space Distortions and the Alcock ...
Modeling and Measuring Redshift Space Distortions and the Alcock ...
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<strong>Modeling</strong> <strong>and</strong> <strong>Measuring</strong> <strong>Redshift</strong> <strong>Space</strong><br />
<strong>Distortions</strong> <strong>and</strong> <strong>the</strong> <strong>Alcock</strong>-Paczynski<br />
Effect in <strong>the</strong> Baryon Oscillation<br />
Spectroscopic Survey<br />
Beth Reid<br />
Hubble Fellow<br />
Lawrence Berkeley National Lab<br />
in collaboration with Martin White, Lado Samushia,<br />
Will Percival, BOSS galaxy clustering working group<br />
Image Courtesy Chris Blake <strong>and</strong> Sam Moorfield<br />
Tuesday, January 31, 2012
Outline<br />
• Motivation<br />
• Basic redshift space distortions (RSD) in configuration<br />
space<br />
• Reid <strong>and</strong> White 2011 configuration space RSD model<br />
(+ connections to o<strong>the</strong>r recent RSD work)<br />
• From halos to galaxies...<br />
• BOSS DR9 first results: BAO, RSD <strong>and</strong> AP constraints<br />
• Future prospects<br />
Beth Reid Nagoya Feb 1<br />
2<br />
Tuesday, January 31, 2012
RSD motivation: Testing<br />
General Relativity<br />
• Once we know <strong>the</strong> expansion history H(a),<br />
we know how perturbations grow in GR:<br />
δ(k, a) = aG(H(a))δi(k)<br />
• We want to test both scale (k) <strong>and</strong> time (a)<br />
dependence<br />
Beth Reid Nagoya Feb 1<br />
3<br />
Tuesday, January 31, 2012
RSD in 3d Galaxy Maps<br />
θ, φ, redshift<br />
depends on <strong>the</strong><br />
geometry of<br />
<strong>the</strong> universe<br />
χ(z) = χtrue + vp/aH(a)<br />
χ(z) =0∫ z c dz’/H(z’)<br />
comoving coordinates: x, y, z<br />
Beth Reid<br />
Tuesday, January 31, 2012<br />
4<br />
Nagoya Feb 1
RSD in configuration space<br />
real to redshift space separations<br />
∇ ⋅ vp = -aHf δm<br />
z<br />
x<br />
x<br />
|vp| ~ d σ8/d ln a = σ8 * f<br />
isotropic<br />
squashed along line of sight<br />
f = d ln σ8 /d ln a<br />
Beth Reid<br />
5<br />
Nagoya Feb 1<br />
Tuesday, January 31, 2012
RSD: Anisotropy in ξ(rσ, rπ)<br />
BOSS DR9: Reid et al., Samushia et al. (in prep)<br />
Beth Reid<br />
6<br />
White et al. 2011 mock catalogs<br />
Nagoya Feb 1<br />
Tuesday, January 31, 2012
Linear RSD (Kaiser 1987)<br />
Beth Reid Nagoya Feb 1<br />
7<br />
Tuesday, January 31, 2012
Linear RSD: Legendre Polynomial moments<br />
General Expansion<br />
Linear <strong>the</strong>ory prediction<br />
Beth Reid<br />
Tuesday, January 31, 2012<br />
8<br />
Nagoya Feb 1
Legendre Polynomial moments<br />
General Expansion<br />
Relation to P(k)<br />
Beth Reid<br />
9<br />
Nagoya Feb 1<br />
Tuesday, January 31, 2012
Linear <strong>the</strong>ory Legendre polynomial<br />
moments: scale dependence<br />
Beth Reid<br />
10 Nagoya Feb 1<br />
Tuesday, January 31, 2012
RSD in configuration space: new<br />
quantities of interest<br />
v̅r, σ 2 r<br />
rπ<br />
r<br />
vz<br />
y<br />
rπ<br />
v̅t=0, σ 2 t<br />
rσ<br />
Beth Reid<br />
11<br />
Nagoya Feb 1<br />
Tuesday, January 31, 2012
Linear <strong>the</strong>ory pairwise velocities (δg = bδm)<br />
Beth Reid<br />
12 Nagoya Feb 1<br />
Tuesday, January 31, 2012
Fisher 1995: <strong>the</strong> Kaiser formula in<br />
configuration space<br />
• δ(x), v(x’) correlated Gaussian fields<br />
• Exp<strong>and</strong> around y = rπ<br />
• Equivalent to Kaiser formula<br />
Beth Reid<br />
13<br />
Nagoya Feb 1<br />
Tuesday, January 31, 2012
Pairwise velocity statistics in linear <strong>the</strong>ory<br />
Beth Reid<br />
14 Nagoya Feb 1<br />
Tuesday, January 31, 2012
Outline<br />
• Motivation<br />
• Basic redshift space distortions (RSD) in configuration<br />
space<br />
• Reid <strong>and</strong> White 2011 configuration space RSD model<br />
(+ connections to o<strong>the</strong>r recent RSD work)<br />
• From halos to galaxies...<br />
• BOSS DR9 first results: BAO, RSD <strong>and</strong> AP constraints<br />
• Future prospects<br />
Beth Reid Nagoya Feb 1<br />
15<br />
Tuesday, January 31, 2012
Recent work: Matter Density Field<br />
Blake et al., arXiv:1105.2862; see also Scoccimarro 2004<br />
Beth Reid<br />
16<br />
Nagoya Feb 1<br />
Tuesday, January 31, 2012
• Galaxies live <strong>the</strong>re!<br />
•<br />
Halos occupy “special”<br />
places in <strong>the</strong> density<br />
field; θ is a volumeaveraged<br />
statistic<br />
• Dependence on halo<br />
bias is complex; studies<br />
of matter correlations<br />
not easily generalized<br />
Why halos<br />
Beth Reid<br />
17 Nagoya Feb 1<br />
Tuesday, January 31, 2012
Recent Work: Halos<br />
• Tinker, Weinberg, Zheng 2006; Tinker<br />
2007 (+ galaxies in halo model)<br />
• Matsubara 2008ab [LPT with biasing]<br />
• Tang, Kayo, Takada arXiv:1103.3614<br />
• Nishimichi, Taruya arXiv:1106.4562<br />
• ....<br />
Beth Reid<br />
18 Nagoya Feb 1<br />
Tuesday, January 31, 2012
N-body Simulations<br />
• White et al. 2011; arXiv:1010.4915<br />
• 67.5 (Gpc/h) 3 total volume<br />
(for BOSS galaxies V ~ 5 (Gpc/h) 3 )<br />
Beth Reid<br />
19 Nagoya Feb 1<br />
Tuesday, January 31, 2012
N-body simulations vs Linear <strong>and</strong><br />
Lagrangian Perturbation Theories<br />
• LPT works on<br />
BAO scales<br />
• See Matsubara<br />
PRD 78, 083519;<br />
arXiv:1105.5007<br />
Beth Reid<br />
20 Nagoya Feb 1<br />
Tuesday, January 31, 2012
N-body simulations vs Linear <strong>and</strong><br />
Lagrangian Perturbation Theories<br />
• ξ2 suppressed by 2.5-7.5% at 50 h -1 Mpc,<br />
depends strongly on bias<br />
Beth Reid<br />
21 Nagoya Feb 1<br />
Tuesday, January 31, 2012
Our approach<br />
• Two distinct sources of nonlinearity:<br />
• Nonlinear growth of structure/biasing --<br />
affects both halo clustering <strong>and</strong> velocities<br />
(study in N-body sims/perturbation <strong>the</strong>ory)<br />
• Nonlinear mapping from real to redshift<br />
space coordinates (non-perturbative)<br />
• Recall: to get Kaiser: dvz/dz small (P) or<br />
exp<strong>and</strong> around y = rπ (ξ)<br />
Beth Reid<br />
22 Nagoya Feb 1<br />
Tuesday, January 31, 2012
The scale-dependent Gaussian streaming<br />
model ansatz<br />
v̅r, σ 2 r<br />
rπ<br />
r<br />
vz<br />
y<br />
rπ<br />
v̅t=0, σ 2 t<br />
rσ<br />
Beth Reid<br />
23<br />
Nagoya Feb 1<br />
Tuesday, January 31, 2012
The scale-dependent Gaussian streaming<br />
model ansatz<br />
• Non-perturbative!<br />
• Approximate pairwise<br />
velocity PDF P(vz, r) with<br />
a Gaussian; match 1 st<br />
<strong>and</strong> 2 nd moments<br />
• Agrees at linear order<br />
with Kaiser/exact<br />
Beth Reid<br />
24 Nagoya Feb 1<br />
Tuesday, January 31, 2012
The scale-dependent Gaussian streaming<br />
model ansatz: “linear” <strong>the</strong>ory predictions<br />
b=2.7<br />
b=0.5<br />
Kaiser limit<br />
Gaussian streaming model b 3 correction !!<br />
Beth Reid<br />
25<br />
Nagoya Feb 1<br />
Tuesday, January 31, 2012
The scale-dependent Gaussian streaming<br />
model ansatz vs N-body simulations<br />
• Start with ξ(r), v(r), σ²⊥,∥(r) measured from N-body<br />
halos in real space<br />
• Compare with N-body halo clustering in redshift<br />
space<br />
Beth Reid<br />
26 Nagoya Feb 1<br />
Tuesday, January 31, 2012
The scale-dependent Gaussian streaming<br />
model ansatz vs N-body simulations: ξ0<br />
Beth Reid<br />
27 Nagoya Feb 1<br />
Tuesday, January 31, 2012
The scale-dependent Gaussian streaming<br />
model ansatz vs N-body simulations: ξ0<br />
Beth Reid<br />
28 Nagoya Feb 1<br />
Tuesday, January 31, 2012
The scale-dependent Gaussian streaming<br />
model ansatz vs N-body simulations: ξ2<br />
Beth Reid<br />
29 Nagoya Feb 1<br />
Tuesday, January 31, 2012
The scale-dependent Gaussian streaming<br />
model ansatz vs N-body simulations: ξ2<br />
Beth Reid<br />
30 Nagoya Feb 1<br />
Tuesday, January 31, 2012
Can we predict real space halo clustering/<br />
velocity PDFs using perturbation <strong>the</strong>ory<br />
• LPT (including nonlinear<br />
bias) predicts halo ξ(r)<br />
down to 25 Mpc/h<br />
Beth Reid<br />
31<br />
Nagoya Feb 1<br />
Tuesday, January 31, 2012
Velocity statistics in st<strong>and</strong>ard<br />
perturbation <strong>the</strong>ory: new results<br />
• Pair-weighted, not volume weighted! Nagoya Feb 1<br />
Beth Reid<br />
32<br />
Tuesday, January 31, 2012
Velocity statistics in st<strong>and</strong>ard<br />
perturbation <strong>the</strong>ory: new results<br />
* assumes linear bias<br />
Beth Reid<br />
33<br />
Nagoya Feb 1<br />
Tuesday, January 31, 2012
Velocity statistics in st<strong>and</strong>ard<br />
perturbation <strong>the</strong>ory: new results<br />
Pair-weighting correction<br />
Linear <strong>the</strong>ory<br />
PT correction to Pδθ<br />
Bispectrum terms: Bδδθ<br />
Beth Reid<br />
34<br />
Nagoya Feb 1<br />
Tuesday, January 31, 2012
Velocity statistics in st<strong>and</strong>ard<br />
perturbation <strong>the</strong>ory: new results<br />
Bispectrum terms: Bδδθ<br />
Bδδθ , Pδθ terms<br />
appear in Tang et al.,<br />
Nishimishi et al.<br />
PT correction to Pδθ<br />
total PT correction<br />
Pair-weighting correction<br />
Beth Reid<br />
35<br />
Nagoya Feb 1<br />
Tuesday, January 31, 2012
Putting it all toge<strong>the</strong>r: fully analytic model<br />
• Error dominated by<br />
error in v12(r) slope<br />
• Works where b 2 L = 0<br />
(i.e., for BOSS<br />
galaxies)<br />
• New LPT calculation<br />
in prep: Carlson et<br />
al., 2012<br />
Beth Reid<br />
36<br />
Nagoya Feb 1<br />
Tuesday, January 31, 2012
Summary: Two distinct effects<br />
• Non-linear gravitational<br />
evolution: MUST be<br />
accounted for given<br />
current statistical errors:<br />
ξ2 suppressed by 2.5-7.5%<br />
at 50 h -1 Mpc!<br />
• Non-linear real-to-redshift<br />
space mapping: b 3 term<br />
Beth Reid<br />
37 Nagoya Feb 1<br />
Tuesday, January 31, 2012
Outline<br />
• Motivation<br />
• Basic redshift space distortions (RSD) in configuration<br />
space<br />
• Reid <strong>and</strong> White 2011 configuration space RSD model<br />
(+ connections to o<strong>the</strong>r recent RSD work)<br />
• From halos to galaxies...<br />
• Interlude: SDSS DR9 first results<br />
• Future prospects<br />
Beth Reid Nagoya Feb 1<br />
38<br />
Tuesday, January 31, 2012
Dominant impact of galaxies: Fingers-of-God<br />
REAL SPACE: r ~ 1 Mpc/h<br />
x<br />
Central galaxies<br />
Satellite galaxies<br />
x<br />
REDSHIFT SPACE: r ~ 15 Mpc/h<br />
Finger-of-God features mix small <strong>and</strong> large scale power<br />
x<br />
x<br />
x<br />
Beth Reid<br />
39<br />
Nagoya Feb 1<br />
Tuesday, January 31, 2012
Fingers-of-God in ξ(rσ, rπ)<br />
Beth Reid<br />
40 Nagoya Feb 1<br />
Tuesday, January 31, 2012
From halos to galaxies<br />
• In principle, straightforward to model in ξ(rσ, rπ)<br />
-- just ano<strong>the</strong>r convolution with intrahalo velocity<br />
PDF<br />
• In practice 3 (broad) distinct PDFs: cs, ss (1h), ss<br />
(2h)<br />
• Inaccarucy of halo ξ(rσ, rπ) on small scales inhibits<br />
this approach<br />
Beth Reid Nagoya Feb 1<br />
41<br />
Tuesday, January 31, 2012
Safe on quasilinear scales...<br />
• One-halo (classical<br />
FOG) unimportant<br />
• ~ 10% effect at 25<br />
Mpc/h for BOSS<br />
galaxies<br />
Beth Reid Nagoya Feb 1<br />
42<br />
Tuesday, January 31, 2012
From halos to galaxies<br />
• Marginalizing over additional Gaussian dispersion<br />
works!<br />
RW11<br />
LPT<br />
FOGs<br />
RSD model vs 70 (Gpc/h) 3 of sims<br />
Beth Reid Nagoya Feb 1<br />
43<br />
Tuesday, January 31, 2012
Outline<br />
• Motivation<br />
• Basic redshift space distortions (RSD) in configuration<br />
space<br />
• Reid <strong>and</strong> White 2011 configuration space RSD model<br />
(+ connections to o<strong>the</strong>r recent RSD work)<br />
• From halos to galaxies...<br />
• BOSS DR9 first results: BAO, RSD <strong>and</strong> AP constraints<br />
• Future prospects<br />
Beth Reid Nagoya Feb 1<br />
44<br />
Tuesday, January 31, 2012
Galaxy clustering lightning<br />
<strong>the</strong>ory review<br />
• Theory 1: underlying matter power spectrum<br />
(determined at z >~ zCMB, neglecting ν)<br />
• Theory II: Expansion history H(0 < z < zGAL)<br />
Tuesday, January 31, 2012
Matter Power Spectrum<br />
• Entire P(k) (not just BAO) acts as st<strong>and</strong>ard ruler<br />
determined by CMB<br />
• We marginalize over <strong>the</strong> (negligible) uncertainty<br />
P(k)<br />
r 2 ξ(r)<br />
Mpc -1<br />
Mpc<br />
Tuesday, January 31, 2012
Theory II: geometry<br />
• We measure θ, φ, <strong>and</strong> z for each galaxy, <strong>and</strong> use a<br />
cosmological model to convert to comoving<br />
coordinates<br />
z1<br />
z2<br />
1/H(z)<br />
θ<br />
χ(z) (or DA(z))<br />
Tuesday, January 31, 2012
Theory II: <strong>Alcock</strong>-Paczynski<br />
• ξ(rp, π) appears anisotropic if you assume <strong>the</strong><br />
wrong cosmological model (constrain ηAP = DA * H)<br />
3<br />
χ(z) =0∫ z c dz’/H(z’)<br />
BAO in ξ0(s) determines<br />
“geometric mean”<br />
DV ∝ (DA 2 H -1 ) 1/3<br />
c Δz / H(z)<br />
π<br />
180<br />
160<br />
140<br />
120<br />
100<br />
80<br />
60<br />
40<br />
20<br />
ξ(rp, π)<br />
20 40 60 80 100 120 140 160 180<br />
2<br />
1<br />
0<br />
-1<br />
-2<br />
-3<br />
Image from Tian et al. arXiv:1011.2481<br />
r p<br />
χ(z)*Δθ<br />
Tuesday, January 31, 2012
Fitting to 2d clustering<br />
• Use full model of ξ0,2(s ≥ 25 h -1 Mpc) to constrain:<br />
• growth of structure (fσ8)<br />
• D V ∝ (DA 2 /H) 1/3<br />
• <strong>Alcock</strong>-Paczynski (ηAP ∝ DA(zeff) * H(zeff))<br />
• marginalizing over shape of underlying linear<br />
P(k), bσ8, σ 2 FOG<br />
Tuesday, January 31, 2012
Tuesday, January 31, 2012<br />
<strong>Alcock</strong>-Paczynski in multipoles
DR9 spectroscopic results:<br />
preliminary!<br />
• DR9 data final (public July 2012), clustering/<br />
covariances ~final, cosmological constraints<br />
preliminary<br />
• Current uncertainties reported, not central<br />
values<br />
Tuesday, January 31, 2012
BOSS “CMASS” (zeff = 0.57)<br />
galaxy sample in perspective<br />
Eisenstein et al. arXiv:1101.1529<br />
Tuesday, January 31, 2012
BAO fits in P(k)/ξ(r) consistent<br />
X. Xu et al. (in prep; DR7)<br />
BOSS Galaxy Clustering (in prep.)<br />
• 2-3% uncertainty<br />
on BAO position<br />
plot of BAO feature here<br />
in angle-averaged<br />
P(k)/ξ(r)<br />
• Constrains<br />
DV ∝ (DA 2 /H) 1/3<br />
Tuesday, January 31, 2012
The CMASS measurements<br />
• 26 log bins in s for ξ0 <strong>and</strong> ξ2 = 52 DOF<br />
plot of ξ0 <strong>and</strong> ξ2 here<br />
Tuesday, January 31, 2012
Model Fits<br />
• We test <strong>the</strong> LCDM hypo<strong>the</strong>sis in 4 models,<br />
always marginalizing over P(k) shape <strong>and</strong> σ 2 FOG:<br />
• LCDM (bσ8)<br />
• LCDM + fσ8: (bσ8, fσ8)<br />
• LCDM + geometry: (bσ8, DV, DA*H)<br />
• LCDM++: (bσ8, fσ8, DV, DA*H)<br />
Tuesday, January 31, 2012
Current status<br />
• D V/DV,fid = x ± 0.019 (i.e., minimal information<br />
gain on DV compared to BAO only!)<br />
• Geometry LCDM: fσ8 = xx ± 0.03 (7%)<br />
[WMAP7 LCDM: 0.45 ± 0.025]<br />
• fσ8 LCDM: η = xx ± 0.04 (4%)<br />
[WMAP7 LCDM: 1.00 ± 0.012]<br />
• Fit both: fσ8 = xx ± 0.07, η = xx ± 0.07<br />
Tuesday, January 31, 2012
Testing alternative models with amplitude of<br />
peculiar velocities<br />
BOSS DR9<br />
Tuesday, January 31, 2012
Tuesday, January 31, 2012<br />
Expansion rate at z=0.57
Outline<br />
• Motivation<br />
• Basic redshift space distortions (RSD) in configuration<br />
space<br />
• Reid <strong>and</strong> White 2011 configuration space RSD model<br />
(+ connections to o<strong>the</strong>r recent RSD work)<br />
• From halos to galaxies...<br />
• Interlude: SDSS DR9 first results<br />
• Future prospects<br />
Beth Reid Nagoya Feb 1<br />
59<br />
Tuesday, January 31, 2012
Future Prospects: “convolutional” LPT<br />
(Jordan Carlson, et al., in prep)<br />
• Fourier transform formal LPT P(k, μ) expression; use<br />
cumulant expansion thm + Gaussian integrals<br />
• Recovers Zel’dovich approximation exactly -><br />
b 3 nonlinear mapping term ()<br />
Beth Reid Nagoya Feb 1<br />
60<br />
Tuesday, January 31, 2012
Future Prospects: “convolutional” LPT<br />
(Jordan Carlson, et al., in prep)<br />
CLPT<br />
Linear <strong>the</strong>ory<br />
Matsubara LPT<br />
halo real space ξ0 (preliminary!)<br />
Beth Reid<br />
61 Nagoya Feb 1<br />
Tuesday, January 31, 2012
Future Prospects: “convolutional” LPT<br />
(Jordan Carlson, et al., in prep)<br />
CLPT<br />
Linear <strong>the</strong>ory<br />
Matsubara LPT<br />
halo redshift space ξ0 (preliminary!)<br />
Beth Reid<br />
62 Nagoya Feb 1<br />
Tuesday, January 31, 2012
Future Prospects: “convolutional” LPT<br />
(Jordan Carlson, et al., in prep)<br />
CLPT<br />
Linear <strong>the</strong>ory<br />
Matsubara LPT<br />
matter ξ2 (preliminary!)<br />
Beth Reid<br />
63 Nagoya Feb 1<br />
Tuesday, January 31, 2012
Future Prospects: “convolutional” LPT<br />
(Jordan Carlson, et al., in prep)<br />
CLPT<br />
Linear <strong>the</strong>ory<br />
Matsubara LPT<br />
halo ξ2 (preliminary!)<br />
Beth Reid<br />
64 Nagoya Feb 1<br />
Tuesday, January 31, 2012
Future Prospects: “convolutional” LPT<br />
(Jordan Carlson, et al., in prep)<br />
• New real space ξ(r) fits to ~ 10 Mpc/h !!<br />
• Repeat v12(r), σ 2 ⊥,∥(r) calculations in LPT (including<br />
b2 L ) may extend analytic Gaussian streaming model to<br />
smaller scales<br />
Beth Reid Nagoya Feb 1<br />
65<br />
Tuesday, January 31, 2012
Future Prospects: using small-scale<br />
clustering to infer σ 2 FOG<br />
IF we can determine σ from<br />
small-scale clustering (e.g.,<br />
HOD), gain factor of 2 on RSD<br />
X<br />
X<br />
smin, WiggleZ<br />
smin, BOSS<br />
Beth Reid<br />
66 Nagoya Feb 1<br />
Tuesday, January 31, 2012
Conclusions<br />
• Configuration space simplifies many conceptual issues<br />
in modeling RSD<br />
• Worked example of developing/modelling target<br />
(BOSS) galaxies: 2% accurate to 25 h -1 Mpc<br />
• 7% measurement of fσ8 in DR9 CMASS galaxies, ~4%<br />
final (barring fur<strong>the</strong>r modeling improvements)<br />
• Fur<strong>the</strong>r development underway (CLPT, small scale/<br />
HOD modeling, ...)<br />
Beth Reid Nagoya Feb 1<br />
67<br />
Tuesday, January 31, 2012