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宇宙の大規模構造における観測可能量と摂動論
Taka Matsubara (Nagoya U.)
@Daikoen, Takehara2011/6/8
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Observables in LSS
• Redshift survey is a useful probe of density fields in cosmology
• number density of galaxies ~ mass density
• Issues to resolve:
• Redshift space distortions
• galaxy biasing
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Bias between mass and objects
• Mass density number density (in general)
• Densities of both mass and astronomical objects are determined by initial density field
• There should be a relation
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Eulerian Local Bias model• Local bias: A simple model usually adopted in
the nonlinear perturbation theory
• The number density is assumed to be locally determined by (smoothed) mass density
• Apply a Taylor expansion
• Phenomenological model, just for simplicity, but divergences in loop corrections
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Eulerian Local bias is not physical
initial density field
linear evolution: local
mass density fieldnumber density field
linear density field
nonlinear evol.: nonlocal
galaxy formation: local or nonlocal nonlinear evol.: nonlocal
Nonlocal relation in generalLocal only in linear regime & local galaxy formation
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Nonlocal Bias• “Functional” instead of function
• For a single streaming fluid (quasi-nonlinear)
• Taylor expansion of the functional
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Perturbation theory with nonlocal bias
• Perturbative expansions in Fourier space:
• nonlocal bias:
• nonlinear dynamics
• It is straightforward to calculate observables, such as power spectrum, bispectrum, etc.
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The problem with Eulerian bias
• No physical model of Eulerian nonlocal bias !!
• Physical models of bias known so far is provided in Lagrangian space
• e.g., Halo bias model, Peak bias model,...
• In those models, conditions of galaxy formation are imposed on initial (Lagrangian) density field
• What is the relation between Eulerian bias and Lagrangian bias?
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Eulerian local bias• The Eulerian local bias in nonlinear
perturbation theory is dynamically inconsistent
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Eulerian and Lagrangian bias• Equivalence of Eulerian and Lagrangian
nonlocal bias
• Nonlocal Eulerian and Lagrangian biases are equivalent. Only representations are different
• The relations can be explicitly derived in perturbation theory:
local biases are incompatible! At least one must be nonlocal
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When Lagrangian bias is local
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Lagrangian perturbation theory
• Lagrangian perturbation theory
• suitable for handling Lagrangian bias
• Fundamental variables in Lagrangian picture
• Displacement field
Ψ(q, t) = x(q, t)− q
q
x(q, t)
Ψ(q, t)
Lagrangian (initial) position
Displacement vector
Eulerian (final) position
Buchert (1989)
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Redshift-space distortions• Redshift-space distortions are easily incorporated to
Lagrangian perturbstion theory
• Mapping from real space to redshift space is exactly linear in Lagrangian variables c.f.) nonlinear in Eulerian
• Mapping of the displacement field
Observer
line of sight
z
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Lagrangian perturbation theory with Lagrangian (nonlocal) bias
• The relation between Eulerian density fluctuations and Lagrangian variables
• Perturbative expansion in Fourier space
Eulerian density field
Biased field in Lagrangian space
displacement(& redshift distortions)
Kernel of the displacement field (& redshift distortions)
Kernel of the Lagrangian bias
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Diagrammatics• Introducing diagrammatic rules is useful
⇔ PL(k)k −k
k1
k2
kn⇔ P (n)
L (k1,k2, . . . ,kn)
k
k1
⇔ bLn(k1, . . . ,kn)ki1 · · · kim
k1k2
kn
⇔ Ln,i(k1,k2, . . . ,kn)
kn
im
i1
i
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Diagrammatics• Shrunk vertices
• Example: power spectrum
= +
= + + +
= + + +cyc.+
+ + cyc. ++ + cyc.
+ + cyc.
+
+ +
+ + +
+ · · ·
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Multi-point propagator• Multi-point propagator
• Responses to the nonlinear density field from initial density fluc.
• Central role in renormalized perturbation theory
• Define corresponding quantity in Lagrangian perturbation theory and Lagrangian bias
• Renormalization of external vertices
=
Γ (n)X (k1, . . . ,kn) =
k1
kn
+ + +
+ · · ·+ + +
Crocce & Scoccimarro (2006), Bernardeau et al. (2008)
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Multi-point propagator• Example: nonlinear power spectrum in terms of
multi-point propagator
• No way of obtaining exact multi-point propagator
• In renormalized perturbation theory, large-k limit and one-loop approximation are interpolated by hand
Bernardeau et al. (2008)
PX(k) =∞∑
n=1
k1
kn
k
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Multi-point propagator• Partial renormalization
• Infinite series are partially resummed in Lagrangian bias + Lagrangian perturbation theory
= +
∞∑
r=0
j1 jr
knk1
k i1
im
+ · · · + + · · ·
= +
∞∑
r=0
k′1 k′r
knk1
k i1
im
+ · · · + + · · ·
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Positiveness• Each term in the resummed series is positive and
add constructively (common feature with RPT)
Crocce & Scoccimarro (2006)
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Application: Baryon Acoustic Oscillations
Linear theory1-loop SPT
N-body This work
This work
N-body
Linear theory
TM (2008)
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Application:Effects of halo bias on BAO
• Apply halo bias (local Lagrangian bias)
• redshift-space distortions also included
TM (2008)
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2-loop corrections
Okamura, Taruya & TM (2011)
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2-loop corrections
Okamura, Taruya & TM (2011)
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Halo clustering: Comparison with N-body simulations
Sato & TM (2011)
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Halo clustering: Comparison with N-body simulations
Sato & TM (2011)
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Halo clustering: Comparison with N-body simulations
Sato & TM (2011)
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Application:Scale-dependent bias and prim.nG
real space:comparison with simple formula
redshift space
• Previous methods are not accurate enough
!! PRELIMINARY
!!
is accurate only in a high-peak limit
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まとめ• 摂動論を観測可能量に直接結びつける
• これまでの現象論的な局所的なオイラー・バイアスは非線形領域で非整合的
• オイラー・バイアスとラグランジュ・バイアスの関係を導出
• ラグランジュ空間の局所バイアスはハローモデルという成功例があり、整合的に摂動論と結びつけることが可能
• 赤方偏移変形はラグランジュ摂動論での取り扱いが便利
• 摂動論の改善法
• 部分的な無限和を取ることが可能で、数値シミュレーションと比較すると、標準的摂動論を実際に改善している
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Jeong et al. (2010)