Introduction CLPT: More resummation RSD: Gaussian streaming CLPT + RSD Summary Back up
Pairwise Velocity Statistics in ConvolutionLagrangian Perturbation Theory–Application to Redshift Space Distortions–
Lile Wang
Collaborator:Martin White (UC Berkeley/LBNL)
Beth A. Reid (LBNL, Hubble Fellow)
THCA/Dept. of Phys., Tsinghua UniversityLBNL
May 10, 2013
Introduction CLPT: More resummation RSD: Gaussian streaming CLPT + RSD Summary Back up
What’s Next?
1 IntroductionMotivations and BackgroundBefore resummationLagrangian resummation theory (LRT)
2 CLPT: More resummation
3 RSD: Gaussian streaming
4 CLPT + RSD
5 Summary
6 Back up
Introduction CLPT: More resummation RSD: Gaussian streaming CLPT + RSD Summary Back up
Motivations and Background
Why are you giving a report here about theory?
Why perturbation?
We need fitting modelsWeeks of CPU time for simulation?Perturbation theory on a “jailbreaked” iPhoneWe need more insight
Why Lagrangian?
Better bias model (than Eulerian)
So what?
Velocity statistics: More with clusteringRSD: More accurate model with velocity
We already have perturbation with RSD!
Work well in configuration space?“Brute-force”/ ?大力出奇迹
Introduction CLPT: More resummation RSD: Gaussian streaming CLPT + RSD Summary Back up
Before resummation
Lagrangian scheme: Basic introductions I
Eulerian vs. Lagrangian
v , ρ = ρ0(1 + δ) ; vs. x = q + Ψ .
Density: (1 + δm)d3x = d3q
Lagrangian dynamics: Equation of motion of DM
d2Ψ
dt2+ 2H
dΨ
dt= −∇xφ , ∇2
xφ = 4πGρ0a2δm(x) .
Local Lagrangian Bias: 1 + δX = F (δm)
Correlation: ξ(r) = 〈δX(x)δX(x + r)〉x
Introduction CLPT: More resummation RSD: Gaussian streaming CLPT + RSD Summary Back up
Before resummation
Lagrangian scheme: Basic introductions II
Then...
1 + δm =d3q
d3x=
∫d3qδD(x− q−Ψ) ,
δD(x) =
∫d3k
(2π)3eik·x ,
F (δ) =
∫dλ
2πF (λ)eiλδ ,
1 + ξ(r) =
∫d3q
∫d3k
(2π)3eik·(q−r)
∫dλ12π
dλ22π
× F1(λ1)F2(λ2)⟨
ei(λ1δ1+λ2δ2+k·∆)⟩.
Subscript “2” for x + r, “1” for x
Introduction CLPT: More resummation RSD: Gaussian streaming CLPT + RSD Summary Back up
Before resummation
Lagrangian scheme: Basic introductions III
Bias parameters∫dλ
2πF (λ)eλ
2σ2R/2(iλ)n = 〈F (n)〉 ,
F1 ≡ F2 F1 6= F2
(λ1 + λ2) 2〈F ′〉 〈F ′1〉+ 〈F ′2〉(λ21 + λ22) 2〈F ′′〉 〈F ′′1 〉+ 〈F ′′2 〉λ1λ2 〈F ′〉2 〈F ′1〉〈F ′2〉λ21λ
22 〈F ′′〉2 〈F ′′1 〉〈F ′′2 〉
λ1λ2(λ1 + λ2) 2〈F ′〉〈F ′′〉 〈F ′1〉〈F ′′2 〉+ 〈F ′2〉〈F ′′1 〉
Fitting parameters: 〈F ′〉 and 〈F ′′〉Or get 〈F ′1,2〉 and 〈F ′′1,2〉 first, respectively
Introduction CLPT: More resummation RSD: Gaussian streaming CLPT + RSD Summary Back up
Lagrangian resummation theory (LRT)
Matsubara’s scheme I
Theorem: Cumulant expansion
⟨eiX⟩
= exp
[ ∞∑N=1
iN
N !〈XN 〉c
], X = λ1δ1 + λ2δ2 + k ·∆
∆ = Ψ2 −Ψ1
Up to one-loop order, up to O(P 2L)
Matsubara (2008), PRD 77063530 expands everything else,KEEPING −(λ21 + λ22)σR/2 (σR = 〈δ2〉) in the exponent;
If RSD needed: Ψsubstituted by−−−−−−−−−→ Ψ + (z · Ψ)z/H
“LRT”; How does it work?
Introduction CLPT: More resummation RSD: Gaussian streaming CLPT + RSD Summary Back up
Lagrangian resummation theory (LRT)
Matsubara’s scheme II
How does LRT work?
ξ(r): Good enough aroundBAO scale, ∼ 100 Mpc h−1
Fig 8 and 6 in Matsubara (2008)
P (k): not ideal around∼ 10 Mpc h−1
Well, we need something more accurate at lower scale...
How about one more resummation?
Introduction CLPT: More resummation RSD: Gaussian streaming CLPT + RSD Summary Back up
What’s Next?
1 Introduction
2 CLPT: More resummationCLPT: “Convolution” Lagrangian
3 RSD: Gaussian streaming
4 CLPT + RSD
5 Summary
6 Back up
Introduction CLPT: More resummation RSD: Gaussian streaming CLPT + RSD Summary Back up
CLPT: “Convolution” Lagrangian
“Convolution”, or one more resummation
Also keeps −(k ·A · k)/2 in the exponent, A = 〈∆∆〉c
⟨eiX⟩
= e−(1/2)Aijkikje−(1/2)(λ21+λ
22)σ
2R
{1− λ1λ2ξL +
1
2λ21λ
22ξ
2L
− (λ1 + λ2)Uiki +1
2(λ1 + λ2)
2UiUjkikj −i
6Wijkkikjkk
+ λ1λ2(λ1 + λ2)ξLUiki −i
2(λ1 + λ2)A
10ij kikj
+i
2(λ21 + λ22)U
20i ki − iλ1λ2U11
i ki +O(P 3L)
}.
Integrations then conducted: WRT λ1, λ2, k, and q
Integrands (A, U , W etc.) obtained by Lagrangian dynamics
Introduction CLPT: More resummation RSD: Gaussian streaming CLPT + RSD Summary Back up
CLPT: “Convolution” Lagrangian
CLPT for ξ(r) in real-space
Real-space ξ(r) for halos: 12.785 < lg (M/M�) < 13.086
Introduction CLPT: More resummation RSD: Gaussian streaming CLPT + RSD Summary Back up
CLPT: “Convolution” Lagrangian
In redshift-space, looking good...
Redshift-space ξ0(s) for HOD
Carlson et al. (2012) arXiv:1209.0780v1
Introduction CLPT: More resummation RSD: Gaussian streaming CLPT + RSD Summary Back up
CLPT: “Convolution” Lagrangian
... looking good?
Redshift-space ξ2(s) for halos: 12.785 < lg (M/M�) < 13.086
Carlson et al. (2012) arXiv:1209.0780v1
Introduction CLPT: More resummation RSD: Gaussian streaming CLPT + RSD Summary Back up
What’s Next?
1 Introduction
2 CLPT: More resummation
3 RSD: Gaussian streamingLinear and quasi-linear RSD
4 CLPT + RSD
5 Summary
6 Back up
Introduction CLPT: More resummation RSD: Gaussian streaming CLPT + RSD Summary Back up
Linear and quasi-linear RSD
Gaussian?
Reid & White (2011), MNRAS 417, 1913
1 + ξs(rσ, rπ) =
∫dy
[2πσ212(r, µ)]1/2[1 + ξ(r)]
× exp
{− [rπ − y − µv12(r)]2
2σ212(r, µ)
}.
Decouple streaming from PT
Vulnerable: Gaussian assumption
Do not expect Gaussian...But, central limit theorem, at leastStole some images from Reid & White (2011)
Introduction CLPT: More resummation RSD: Gaussian streaming CLPT + RSD Summary Back up
Linear and quasi-linear RSD
... Nope, but not too far away I
Redshift-space ξ2(s) for halos: 12.785 < lg (M/M�) < 13.086
Reid & White (2011) MNRAS 417, 1913
Introduction CLPT: More resummation RSD: Gaussian streaming CLPT + RSD Summary Back up
Linear and quasi-linear RSD
... Nope, but not too far away II
Redshift-space ξ2(s) for halos: 12.785 < lg (M/M�) < 13.086
Reid & White (2011) MNRAS 417, 1913
Introduction CLPT: More resummation RSD: Gaussian streaming CLPT + RSD Summary Back up
What’s Next?
1 Introduction
2 CLPT: More resummation
3 RSD: Gaussian streaming
4 CLPT + RSDVelocity statistics
5 Summary
6 Back up
Introduction CLPT: More resummation RSD: Gaussian streaming CLPT + RSD Summary Back up
Velocity statistics
How to get time derivatives?
Expectation of correlation-weighted v12 and σ212
v12(r)r =〈[1 + δ(x)][1 + δ(x + r)][v(x + r)− v(x)]〉
〈[1 + δ(x)][1 + δ(x + r)]〉
σ212(r, µ) =
⟨[1 + δ(x)][1 + δ(x + r)][v`(x + r)− v`(x)]2
⟩〈[1 + δ(x)][1 + δ(x + r)]〉
But how to get the numerators?
Auxillary function and ∆
λ1δ1 + λ2δ2 + k ·∆ +J · ∆ , ∂/∂J→ ∆
Introduction CLPT: More resummation RSD: Gaussian streaming CLPT + RSD Summary Back up
Velocity statistics
Pairwise infall velocity and dispersion
v12 and σ212 from CLPT
v12(r): ∼ 1 per cent down to∼ 10 Mpc h−1
σ212(r): Not so good below
∼ 20 Mpc h−1
Introduction CLPT: More resummation RSD: Gaussian streaming CLPT + RSD Summary Back up
Velocity statistics
CLPT ξr, v12 and σ212
INSERT−−−−−→ Gaussian streaming
Multipole expansion: ξ(s, µs) =∑
` ξ`(s)L`(µ)
ξ0(r)/ξ0,lin(r): Monopole ξ2(r)/ξ2,lin(r): Quadrupole
Introduction CLPT: More resummation RSD: Gaussian streaming CLPT + RSD Summary Back up
Velocity statistics
Cross-correlation I
v12 and σ212 from CLPT
v12(r): Worse, ∼ 5 per cent σ212(r): Similar to
“auto-correlation”
Difference between different mass bin is NOT ΨBUT bias parameters
Introduction CLPT: More resummation RSD: Gaussian streaming CLPT + RSD Summary Back up
Velocity statistics
Cross-correlation II
Multipole expansion: ξ(s, µs) =∑
` ξ`(s)L`(µ)
ξ0(r)/ξ0,lin(r): Monopole ξ2(r)/ξ2,lin(r): Quadrupole
Introduction CLPT: More resummation RSD: Gaussian streaming CLPT + RSD Summary Back up
What’s Next?
1 Introduction
2 CLPT: More resummation
3 RSD: Gaussian streaming
4 CLPT + RSD
5 SummarySummary and Conclusions
6 Back up
Introduction CLPT: More resummation RSD: Gaussian streaming CLPT + RSD Summary Back up
Summary and Conclusions
Summary
One more resummation: Improvement, only 2(3) parameters
Real-space ξr improvedReal-space v12 improved (σ2
12 a little bit worse)
Scale-independent local Lagrangian bias: Halos applicable (?)
Gaussian streaming gives good output for good input (CLPT)
∼ 1 per cent at <∼ 15 Mpc h−1
∼ 10 per cent at <∼ 10 Mpc h−1
Works well for monopole and quadrupole
Possible applications:
RS correlation → Cosmology + Bias parameters → v12, σ2
Perturbations are still fast enough on an iPhone