Perturbation Theory Reloaded Modeling the Power Spectrum in High-redshift Galaxy Surveys Eiichiro Komatsu Department of Astronomy University of Texas at Austin MPE Seminar, July 19, 2007
Large-scale Structure of the Universe (LSS) Millenium Simulation (Springel et al., 2005)
LSS of the universe : What does it tell us? Matter density, Ω m Baryon density, Ω b Amplitude of fluctuations, σ 8 Angular diameter distance, d A ( z ) Expansion history, H ( z ) Growth of structure, D ( z ) Shape of the primordial power spectrum from inflation, n s , α , ... Massive neutrinos, m ν Dark energy, w , dw/da , ... Primordial Non-Gaussianity, f NL , ... Galaxy bias, b 1 , b 2 , ...
How can we extract cosmology from LSS? : Statistics 1 One point statistics Mass function, n ( M ) 2 Two point statistics Power spectrum, P ( k ) 3 Three point statistics Bispectrum, B ( k ) 4 Four point statistics Trispectrum, T ( k ) 5 n -point functions
1 dV 2 2 1 dV dV dV The most popular quantity, ξ ( r ) and P ( k ) Correlation function ξ ( r ) = Strength of clustering at a given separation r = � δ ( x ) δ ( x + r ) � where, δ ( x ) = excess number of galaxies above the mean. r We use P ( k ) , the Fourier transform of ξ ( r ) : � d 3 r ξ ( r ) e − i k · r P ( k ) =
How do we do this? Cosmological parameters Matter density, Ω m Baryon density, Ω b Dark energy density, Ω Λ Dark energy eq. of state, w Hubble constant, H 0 ... We have to be able to predict P ( k ) very accurately, as a function of cosmological models.
iii)Non-lineargrowth instability iv)Galaxyformation ii)Linearperturbation i)Initialcondition comovinghorizon etc. galaxies, bygravitational quantumfluctuation Magnified densityperturbation seedof curvatureperturbation (Gaussianrandomfield) classicalfluctuation Cosmological perturbation theory
Initial Condition from inflation Inflation gives the initial power spectrum that is nearly a power law. “ ” � n s + 1 k 2 α s ln � k 0 k P ( k, η i ) = A k 0 Inflation predicts, and observations have confirmed, that n s ∼ 1 α s ∼ 0
Initial Power Spectrum: Tilting Initial matter power spectrum for various n s : P ( k ) ∝ ( k/k 0 ) n s
Initial Power Spectrum: Running Initial matter power spectrum for various α s
Evolution of linear perturbations Two key equations The Boltzmann equation d f dλ = C [ f ] Perturbed Einstein’s equations δG µν = 8 πGδT µν Dark Neutrinos energy ρ N ρ N Λ metric Dark Photons mater g µν ρ Θ ρ δ m γ m Compton Compton Scattering Scattering Electrons Baryons ρ δ ρ δ Coulomb Coulomb e e b b Scattering Scattering
Basic equations for linear perturbations The equations for linear perturbations Dark matter δ ′ + ikv = − 3Φ ′ : Continuity v ′ + a ′ a v = − ik Ψ : Euler Baryons b + ikv b = − 3Φ ′ : Continuity δ ′ b + a ′ a v b = − ik Ψ+ τ ′ v ′ R ( v b + 3 i Θ 1 ) : Euler with interaction w/ photons Photon temperature, Θ = ∆ T/T Θ ′ + ikµ Θ = − Φ ′ − ikµ Ψ − τ ′ � Θ 0 − Θ + µv b − 1 � 2 P 2 ( µ )Θ 2 Gravity � � k 2 Φ + 3 a ′ Φ ′ − Ψ a ′ = 4 πGa 2 ( ρ m δ m + 4 ρ r Θ r, 0 ) a a k 2 (Φ + Ψ) = − 32 πGa 2 ρ r Θ r, 2 These are well known equations. Observational test?
Prediction: the CMB power spectrum Sound horizon at the photon decoupling epoch = 147 ± 2 Mpc (Spergel et al. 2007)
WMAP 3-year temperature map 3-year ILC Map (Hinshaw et al., 2007)
Triumph of linear perturbation theory 3-year Temperature Power Spectrum (Hinshaw et al. 2007) Experimental Verification of the Linear Perturbation Theory!
How about the matter P ( k ) ?
SDSS Luminous Red Galaxies map ( z < 0 . 474 ) 1000 500 0 -500 -1000 -1000 -500 0 500 1000 SDSS main galaxies and LRGs (Tegmark et al., 2006)
SDSS LRG and main galaxy power spectrum Failure of linear theory is clearly seen. P ( k ) from main (bottom) and LRGs (top) (Tegmark et al., 2006)
BAO from the SDSS power spectrum The BAOs have been measured in P ( k ) successfully (Percival et al. 2006). The planned galaxy surveys (e.g., HETDEX, WFMOS) will measure BAOs with 10x smaller error bars. Is theory ready?
Systematics: Three Non-linearities The SDSS P ( k ) has been used only up to k < 0 . 1 hMpc − 1 . Why? Non-linearities. Non-linear evolution of matter clustering Non-linear bias Non-linear redshift space distortion Can we do better? CMB theory was ready for WMAP’s precision measurement. LSS theory has not reached sufficient accuracy. The planned galaxy surveys = WMAP for LSS. Is theory ready? The goal: LSS theory that is ready for precision measurements of P ( k ) from the future galaxy surveys.
Our approach: Non-linear perturbation theory 3 rd -order expansion in linear density fluctuations, δ 1 . c.f. CMB theory: 1 st -order (linear) theory. Is this approach new? It has been known that non-linear perturbation theory fails at z = 0 ← − too non-linear. HETDEX ( z > 2 ) and CIP ( z > 3 ) are at higher-z, where perturbation theory is expected to perform better.
Upcoming high-z galaxy surveys
Upcoming high-z galaxy surveys
Assumptions and basic equations Assumptions 1 Newtonian matter fluid 2 Matter is the pressureless fluid without vorticity. Good approximation before fluctuations go fully non-linear. It is convenient to use the “velocity divergence”, θ = ∇ · v Equations (Newtonian one component fluid equation) ˙ δ + ∇ · [(1 + δ ) v ] = 0 v + ( v · ∇ ) v = − ˙ a ˙ a v − ∇ φ ∇ 2 φ = 4 πGa 2 ¯ ρδ
Go to Fourier space Equations in Fourier space ˙ δ ( k , τ ) + θ ( k , τ ) d 3 k 1 � � d 3 k 2 δ D ( k 1 + k 2 − k ) k · k 1 = − δ ( k 2 , τ ) θ ( k 1 , τ ) , (2 π ) 3 k 2 1 a 2 θ ( k , τ ) + ˙ a aθ ( k , τ ) + 3˙ ˙ 2 a 2 Ω m ( τ ) δ ( k , τ ) d 3 k 1 d 3 k 2 δ D ( k 1 + k 2 − k ) k 2 ( k 1 · k 2 ) � � = − θ ( k 1 , τ ) θ ( k 2 , τ ) , (2 π ) 3 2 k 2 1 k 2 2 Taylor expanding δ , and θ ∞ n � (2 π ) 3 · · · d 3 q n − 1 d 3 q 1 � � � a n ( τ ) d 3 q n δ D ( δ ( k , τ ) = q i − k ) F n ( q 1 , q 2 , · · · , q n ) δ 1 ( q 1 ) · · · δ 1 ( q n ) , (2 π ) 3 n =1 i =1 ∞ n (2 π ) 3 · · · d 3 q n − 1 d 3 q 1 � � � a ( τ ) a n − 1 ( τ ) � θ ( k , τ ) = − ˙ d 3 q n δ D ( q i − k ) G n ( q 1 , q 2 , · · · , q n ) δ 1 ( q 1 ) · · · δ 1 ( q n ) (2 π ) 3 n =1 i =1
Why 3 rd order? δ = δ 1 + δ 2 + δ 3 where, δ 2 ∝ [ δ 1 ] 2 , δ 3 ∝ [ δ 1 ] 3 The power spectrum from the higher order density field : (2 π ) 3 P ( k ) δ D ( k + k ′ ) ≡ � δ ( k , τ ) δ ( k ′ , τ ) � = � δ 1 ( k , τ ) δ 1 ( k ′ , τ ) � + � δ 2 ( k , τ ) δ 1 ( k ′ , τ ) + δ 1 ( k , τ ) δ 2 ( k ′ , τ ) � + � δ 1 ( k , τ ) δ 3 ( k ′ , τ ) + δ 2 ( k , τ ) δ 2 ( k ′ , τ ) + δ 3 ( k , τ ) δ 1 ( k ′ , τ ) � + O ( δ 6 1 ) Therefore, P ( k ) = P 11 ( k ) + P 22 ( k ) + 2 P 13 ( k )
Why 3 rd order? δ = δ 1 + δ 2 + δ 3 where, δ 2 ∝ [ δ 1 ] 2 , δ 3 ∝ [ δ 1 ] 3 The power spectrum from the higher order density field : (2 π ) 3 P ( k ) δ D ( k + k ′ ) ≡ � δ ( k , τ ) δ ( k ′ , τ ) � = � δ 1 ( k , τ ) δ 1 ( k ′ , τ ) � + � δ 2 ( k , τ ) δ 1 ( k ′ , τ ) + δ 1 ( k , τ ) δ 2 ( k ′ , τ ) � + � δ 1 ( k , τ ) δ 3 ( k ′ , τ ) + δ 2 ( k , τ ) δ 2 ( k ′ , τ ) + δ 3 ( k , τ ) δ 1 ( k ′ , τ ) � + O ( δ 6 1 ) Therefore, P ( k ) = P 11 ( k ) + P 22 ( k ) + 2 P 13 ( k )
Non-linear matter power spectrum: analytic solution (Vishniac 1983; Fry1984; Goroff et al. 1986;Suto & Sasaki 1991; Makino et al. 1992; Jain & Bertschinger 1994; Scoccimarro & Frieman 1996) P δδ ( k, τ ) = D 2 ( τ ) P L ( k ) + D 4 ( τ ) [2 P 13 ( k ) + P 22 ( k )] , where, d 3 q � � 2 � F ( s ) P 22 ( k ) = 2 (2 π ) 3 P L ( q ) P L ( | k − q | ) 2 ( q , k − q ) , � ∞ 2 πk 2 dq 2 P 13 ( k ) = 252 P L ( k ) (2 π ) 3 P L ( q ) 0 � 100 q 2 k 2 − 158 + 12 k 2 q 2 − 42 q 4 × k 4 � k + q � � 3 k 5 q 3 ( q 2 − k 2 ) 3 (2 k 2 + 7 q 2 ) ln + , | k − q | „ « » – q 2 ) 2 − 1 F ( s ) 2 ( q 1 , q 2 ) = 17 21 + 1 q 1 q 2 + q 2 + 2 2 ˆ q 1 · ˆ q 2 ( ˆ q 1 · ˆ q 1 7 3
Prediction: non-linear matter P(k)
Prediction: Baryon Acoustic Oscillations Non-linearity distorts BAOs significantly.
Simulation Set I: Low-resolution (faster) Particle-Mesh (PM) Poisson solver (Ryu et al. 1993) Cosmological parameters Ω m = 0 . 27 , Ω Λ = 0 . 73 , Ω b = 0 . 043 , H 0 = 70 km / s / Mpc , σ 8 = 0 . 8 , n s = 1 . 0 Simulation parameters Box size [Mpc / h] 3 k max [ h Mpc − 1 ] n particle M particle ( M ⊙ ) N realizations 512 3 256 3 2 . 22 × 10 12 60 0 . 24 256 3 256 3 2 . 78 × 10 11 50 0 . 5 128 3 256 3 3 . 47 × 10 10 20 1 . 4 64 3 256 3 4 . 34 × 10 9 15 5
Testing convergence with 4 box sizes
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