Taka Matsubara (Nagoya U.) - PowerPoint PPT Presentation

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, Takehara 2011/6/8

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

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

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

Eulerian Local bias is not physical initial ! density ! field linear ! evolution: ! local linear ! density ! field galaxy ! formation: ! local ! or ! nonlocal nonlinear ! evol.: ! nonlocal nonlinear ! evol.: ! nonlocal number ! density ! field mass ! density ! field Nonlocal ! relation ! in ! general Local ! only ! in ! linear ! regime ! & ! local ! galaxy ! formation

Nonlocal Bias • “Functional” instead of function • For a single streaming fluid (quasi- nonlinear) • Taylor expansion of the functional

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.

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?

Eulerian local bias • The Eulerian local bias in nonlinear perturbation theory is dynamically inconsistent

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

When Lagrangian bias is local

Displacement vector Lagrangian (initial) position Eulerian (final) position Lagrangian perturbation theory • Lagrangian perturbation theory • suitable for handling Lagrangian bias • Fundamental variables in Lagrangian picture • Displacement field Buchert (1989) Ψ ( q , t ) = x ( q , t ) − q x ( q , t ) Ψ ( q , t ) q

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 line of sight ˆ z Observer

Lagrangian perturbation theory with Lagrangian (nonlocal) bias • The relation between Eulerian density fluctuations and Lagrangian variables Eulerian Biased field in displacement density field Lagrangian space (& redshift distortions) • Perturbative expansion in Fourier space Kernel of the Lagrangian bias Kernel of the displacement field (& redshift distortions)

Diagrammatics • Introducing diagrammatic rules is useful k 1 k n k b L n ( k 1 , . . . , k n ) k i 1 · · · k i m ⇔ i 1 i m − k k P L ( k ) ⇔ k 1 P ( n ) k 1 L ( k 1 , k 2 , . . ., k n ) ⇔ k n k 2 k 2 i L n,i ( k 1 , k 2 , . . . , k n ) ⇔ k n

Diagrammatics • Shrunk vertices = + = + + + + = + cyc. + cyc. + + + + + cyc. + cyc. + + + • Example: power spectrum + + + + + + · · ·

Multi-point propagator • Multi-point propagator Responses to the nonlinear density field from initial density fluc. • Central role in renormalized perturbation theory • Crocce & Scoccimarro (2006), Bernardeau et al. (2008) Define corresponding quantity in Lagrangian perturbation theory and • Lagrangian bias Γ ( n ) X ( k 1 , . . . , k n ) = k 1 = + + + k n + + + + · · · Renormalization of external vertices •

Multi-point propagator • Example: nonlinear power spectrum in terms of multi-point propagator Bernardeau et al. (2008) k 1 ∞ k � P X ( k ) = n =1 k n • No way of obtaining exact multi-point propagator • In renormalized perturbation theory, large-k limit and one-loop approximation are interpolated by hand

Multi-point propagator • Partial renormalization • Infinite series are partially resummed in Lagrangian bias + Lagrangian perturbation theory j 1 j r k � k � ∞ r 1 ∞ k k � � i 1 i 1 = + = + r = 0 r = 0 i m i m k n k n k 1 k 1 + · · · + + · · · + · · · + + · · ·

Positiveness • Each term in the resummed series is positive and add constructively (common feature with RPT) Crocce & Scoccimarro (2006)

Application: Baryon Acoustic Oscillations N-body This work Linear theory 1-loop SPT Linear theory N-body This work TM ! (2008)

Application: Effects of halo bias on BAO • Apply halo bias (local Lagrangian bias) • redshift-space distortions also included TM ! (2008)

2-loop corrections Okamura, ! Taruya ! & ! TM ! (2011)

2-loop corrections Okamura, ! Taruya ! & ! TM ! (2011)

Halo clustering: Comparison with N- body simulations Sato ! & ! TM ! (2011)

Halo clustering: Comparison with N- body simulations Sato ! & ! TM ! (2011)

Halo clustering: Comparison with N- body simulations Sato ! & ! TM ! (2011)

Application: Scale-dependent bias and prim.nG • Previous methods are not accurate enough real ! space: redshift ! space comparison ! with ! simple ! formula ! ! Y R A N I M I L E R P ! ! is ! accurate ! only ! in ! a ! high-peak ! limit

まとめ 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 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 • 摂動論を観測可能量に直接結びつける • • • • • 摂動論の改善法 •

Jeong et al. (2010)